   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 69



        CELESTIAL NAVIGATION

             The close relationship between navigation and astronomy as 
        well as the development of accurate time keeping is no accident, 
        as related elsewhere in this text. In this age of constellations 
        of artificial navigation satellites (NAVSTAR Global Positioning 
        System) and precision inertial guidance or navigation systems 
        (INS), it is easy to forget how difficult navigation was when the 
        only "instrument" available may have been keen eyesight or a 
        simple compass. The captain of a modern ship or aircraft only 
        needs to glance at a digital readout to know his position within 
        a few meters. It has not always been so, and indeed the 
        occasional human or instrument failures which result in disaster 
        remind us that navigational skills (and common sense) still need 
        to be kept handy when traveling long distances.
             Navigation, which might be defined as the skills required to 
        determine how to move from Point A to Point B, may be divided 
        into two reasonably distinct classes: visual and calculated. 
        Visual navigation is a skill we each practice every time we move 
        about; it involves those actions and reactions necessary to move 
        from our present location to a second location, whether across 
        the room or across town, which is always in view or via 
        intermediate way points always in view. Regardless of the 
        conditions or obstacles we encounter, we automatically make any 
        adjustments in our course required to keep us heading toward our 
        objective. The outcome is usually certain and we seldom think 
        much about the processes involved. Even longer distance travel by 
        automobile is primarily visual navigation, with occasional 
        reference to a map to remind us of the landmarks to watch for; 
        although some practice at map reading may be helpful, few 
        calculations are required.
             True long distance travel, whether by land, sea, or more 
        recently in the air, requires navigation. The goal is to attain a 
        known destination which is not in view through conditions which 
        may be unknown or which may constantly be changing. Once again, 
        those of us who are merely passengers think little of the 
        processes involved. Unfortunately in a few cases, those charged 
        with our safety sometimes assume that Nature will unfailingly 
        cooperate and that they have correctly supplied all required 
        information to instruments which are (and will continue to be) 
        working perfectly. Airline pilots, ship captains, and weekend 
        sailors may occasionally fallen victim to these dangerous 
        assumptions with deadly results.
             To be successful, any scheme of navigation must include 
        certain essential ingredients: the correct (UT) time, where you 
        are now, where you want to go, and how to measure or calculate 
        your progress towards that destination. The text which follows 
        describes two ingredients which are in common use today (if only 
        as backup skills in the event of electronic failure), "Navigation 
        by Dead Reckoning" and "Calculation of Position by Sight 
        Reduction".
             ASTROCLK uses both of these methods to provide navigational 
        information and calculations. The equations required for these 
        calculations are given in the Nautical Almanac 1989 (see 
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 70


        BIBLIOGRAPHY). Beginning in 1989, the Nautical Almanac has 
        included a new section which describes the "Formulae and method 
        ... for use with an electronic calculator or microcomputer for 
        the determination of position at sea". I have adapted this 
        material for use in ASTROCLK.
             One final comment regarding ASTROCLK's navigation functions 
        is in order. So as to minimize the code and memory required to 
        perform these tasks, ASTROCLK utilizes common subroutines to 
        perform many of the calculations and display functions. These are 
        used for navigation and otherwise. In particular, navigational 
        positions may be shown to a precision of 0.01 seconds of arc and 
        such accuracy is far beyond even the most sophisticated satellite 
        navigation equipment today, never mind ASTROCLK. There are many 
        possible sources of error, both human and electronic, in the data 
        used for dead reckoning and celestial navigation, any one of 
        which could contribute a position difference of some minutes of 
        arc or more. I have made every effort to achieve reasonable 
        accuracy, but the user should keep possible error factors in mind 
        when using the navigation functions.


        Setting UT TIME ZONE OFFSET

             Prior to the inclusion of the Navigation Mode (Version 8943 
        and higher), ASTROCLK always assumed that the computer's internal 
        clock was set correctly to the local time and based all other 
        time calculations upon that assumption. Navigation, however, 
        presumes that the computer may be moving from place to place and 
        that the longitude (and therefore the local time and time zone) 
        may be changing. Under these circumstances, what ASTROCLK really 
        needs to know is Universal Time, UT1, and for our purposes UT = 
        UT1 = UTC to sufficient accuracy in most cases except extremely 
        precise navigation and astronomical measurements.
             One way to accomplish that end is to simply set the 
        computer's clock to UT and be done with it; most users, myself 
        included, would object to that inconvenience especially when 
        using the computer outside of ASTROCLK. The alternative is to 
        introduce a constant which tells ASTROCLK how to calculate UT 
        from the setting of the computer's clock. I have chosen to use 
        the second method and to call the constant "UT TIME ZONE OFFSET". 
        When operated in this mode, the computer clock remains set to 
        "home" local time; UT time is always available by applying the UT 
        TIME ZONE OFFSET, and the correct local time is obtained directly 
        from the current longitude (either calculated or manually 
        entered). However, since local time is always calculated, no ZONE 
        CORRECTION is permitted in the Navigation Mode and any ZONE 
        CORRECTION in effect will be cleared when the UT TIME ZONE OFFSET 
        is set. For users accustomed to the "old" versions of ASTROCLK 
        (Versions 8935 and earlier) and who are not concerned with the 
        navigation features, program operation is essentially unchanged 
        and the ZONE CORRECTION is permitted if the UT TIME ZONE OFFSET 
        is left disabled (the default condition).
             Several advantages result from the use of the UT TIME ZONE 
        OFFSET. Most importantly, ASTROCLK can always calculate UT time, 
        and therefore all of the celestial time and position information 
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 71


        regardless of the actual location of the user. Once properly set, 
        the user may "move" his computer from place to place and the time 
        will remain correct. Unlike operation in the normal real time 
        mode, the user may select another location using F6 and the 
        correct local time (as determined by the longitude) will be 
        displayed. By selecting a starting point (a navigational "fix") 
        and entering the true course and speed, the user may place 
        himself upon a moving vessel, calculate the current position by 
        dead reckoning (see following section), and maintain real time 
        coordinates for planetary or celestial bodies based upon the 
        current estimated position. Finally, the user may accurately 
        calculate his current geographic position using two or three star 
        sights, ASTROCLK's version of classical celestial navigation.
             At the same time, several minor penalties must be paid for 
        these additional capabilities. First, as noted above, the ZONE 
        CORRECTION is not permitted. This may represent an inconvenience 
        for users in local time zones different from that calculated by 
        ASTROCLK. Second, for users with slower computers not equipped 
        with a math coprocessor, additional time is required for 
        calculations in all modes and performance for those computers is 
        slightly degraded. Performance degradation of AT and 386 
        computers, whether or not equipped with a math coprocessor, is 
        not significant. Last of all, additional RAM memory is required 
        for ASTROCLK to accomodate these features. [See the section 
        PROGRAM OPERATION, Required ASTROCLK Files, for additional 
        discussion.]
             When ASTROCLK is first started, the UT TIME ZONE OFFSET is 
        disabled and program operation is essentially unchanged from 
        prior versions. The UT TIME ZONE OFFSET may be enabled or 
        disabled at any time by pressing Function Key F10 (NAVIGATION) 
        and then F10 again. If currently disabled, the main ASTROCLK 
        NAVIGATION menu will appear in the main window the first time F10 
        is selected:


                  ASTROCLK NAVIGATION INFORMATION

                  Navigation functions available are:

                  F1 = Show current NAVIGATION DATA       
                  F5 = Select USNO Navigation Stars

                  Before using other NAVIGATION functions,
                  you must use F10 to set the time offset 
                  between your computer clock and UT Time.

                  F10 = Set Computer UT Time Zone Offset   

                  Select function or press RETURN to cancel:   


        Note that except for displaying current NAVIGATION DATA and 
        selecting USNO Navigational Stars, no other navigation functions 
        are available until the UT TIME ZONE OFFSET has been set. If the 
        UT TIME ZONE OFFSET is enabled, other functions will be available 
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 72


        for selection (see below). Pressing F10 the second time, to set 
        the UT TIME ZONE OFFSET, will display the following:


                  ASTROCLK NAVIGATION INFORMATION

                  In order to calculate positions and times
                  correctly, ASTROCLK must know the time zone
                  offset from UT Time to which your computer
                  is now set. If LOCAL and UT times are both
                  correctly displayed press '*'; otherwise enter
                  the time offset in hours. Press RETURN to skip
                  or F10 to disable UT OFFSET and NAVIGATION.

                  The current UT OFFSET is: (disabled)

                  Enter UT TIME ZONE OFFSET (hours):


        The display reproduced above shows that the UT TIME ZONE OFFSET 
        is now disabled; if the UT TIME ZONE OFFSET were active, the 
        actual offset in hours would be displayed instead of 
        "(disabled)". If the local and UT times displayed in the small 
        windows on the right side of the screen are correct and no ZONE 
        CORRECTION is in effect, simply enter '*' (without the 
        apostophes) and ASTROCLK will calculate the offset. Otherwise, 
        enter the correct offset in hours followed by RETURN. Decimal 
        fractions of an hour are permitted. If the UT TIME ZONE OFFSET is 
        now active (a number such as "-7.00" is displayed instead of 
        "(disabled)") and you wish to disable the function, press F10 
        again. If you wish to retain the present value, press RETURN.
             The required value for UT TIME ZONE OFFSET will be positive 
        for East longitudes and negative for West longitudes. For 
        example, the correct value is -8.00 for Pacific Standard Time or 
        -7.00 for Pacific Daylight Time and -5.00 for Eastern Standard 
        Time or -4.00 for Eastern Daylight Time. CAUTION: If your time 
        zone is non-standard (that is, if you must normally use a ZONE 
        CORRECTION to obtain the correct local and UT time displays), you 
        must enter the value that corresponds to your time zone as 
        calculated based upon your longitude and subtract an hour if 
        daylight time is in effect. Any ZONE CORRECTION in effect will be 
        cleared.
             Verify that local time and UT time are both correct when 
        ASTROCLK resumes normal operation and repeat the process if 
        necessary. For locations with "standard" time zones, there will 
        be no apparent difference so long as the current longitude 
        remains in the original time zone. All standard time zones extend 
        7-1/2 degrees on either side of the 15 degree meridians. Once 
        set, the UT TIME ZONE OFFSET is saved in file ASTROCLK.INI and 
        will continue in effect until disabled.
             You may verify the operation of the UT TIME ZONE OFFSET by 
        using Function Key F6. First, press "1" to select Local Time in 
        the main display window, then press F6. If you live in the 
        Western United States, enter "USNO" as the location and Eastern 
        Standard or Daylight Time will be shown, as determined by the 
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 73


        current setting of the DAYLIGHT FLAG. If you live in the Eastern 
        United States, enter "RPV" as the location and Pacific Standard 
        or Daylight Time will be shown. Press F6 again and restore your 
        correct local name and coordinates and observe that the display 
        returns to your correct local time. UTC Time will not change as 
        you change location.
             Even if you do not plan to use the other navigation features 
        of ASTROCLK, you may find it helpful to set the UT TIME ZONE 
        OFFSET. Once properly set, you may change ASTROCLK's local 
        coordinates using Function Key F6 to any desired location, 
        display the correct local time for that location, and view 
        planetary or celestial coordinates as they appear at that 
        location for the current time. For example, you may determine 
        where a particular object would appear in the sky (or if it is 
        below the visible horizon) at a given instant for Los Angeles, 
        Chicago, New York, London, and so forth. This may be done in real 
        time, or the clocks may be stopped and set to any desired time 
        and/or date.


        Navigation by Dead Reckoning

             The oldest and perhaps the most basic method of navigation 
        is called Dead Reckoning. The name derives from the fact that you 
        assume you are proceeding along the course you have "reckoned", 
        come what may. In theory it is quite simple: if you know where 
        you started and your course and speed, you may calculate your 
        present position; similarly, if you know where you are and where 
        you want to go, you may calculate the course, speed and time 
        necessary to get there. To improve accuracy, you may also take 
        into account the effects of wind, currents and other factors as 
        they occur. Provided all these things are known to sufficient 
        accuracy and are correctly included in your calculations, easy to 
        say but more difficult in practice, you will know your present 
        position and will likely reach your destination.
             It is a considerable credit to the navigators of old that, 
        long before the development of the nautical chronometer, they 
        were able to sail for days and sometimes weeks relying entirely 
        upon dead reckoning and still come reasonably close to their 
        intended destination. Captain William Bligh, of "Mutiny on the 
        Bounty" fame (or notoriety, if you prefer), may never qualify as 
        Mr. Nice Guy but he nevertheless performed what must rank as one 
        of the most amazing feats of navigation ever recorded. This in 
        1789 by sailing a small boat on open seas nearly four thousand 
        miles from the point where he and 18 others were set adrift from 
        the Bounty all the way across the South Pacific to Timor in the 
        East Indies, arriving some six and a half weeks later. Even with 
        ASTROCLK along, I'm not sure I'd like to try to duplicate that 
        trick! Less spectacular but equally impressive feats were almost 
        a matter of routine for the master navigators of that age and 
        earlier.
             ASTROCLK takes a somewhat simple minded approach to 
        navigation by dead reckoning. Four items of information are 
        required to specify the last "fix" or position from which future 
        movements are calculated: longitude, latutude, time, and date. 
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 74


        The longitude and latitude may either be obtained from star 
        sights (as described in the following section) or be taken from 
        charts or other sources. CAUTION: West longitude and South 
        latitude are negative; not all sources use the same sign 
        conventions, particularly with respect to longitude. To avoid 
        confusion with respect to time zones, all navigational times and 
        dates are UT (Universal Time), still referred to by most 
        navigators as GMT or Greenwich Mean Time; even NASA retains the 
        older designation.
             Once these data are entered, the current position is then 
        calculated by taking the true course and speed (in knots, 
        nautical miles per hour) and calculating the direction and 
        distance traveled in the time elapsed since the last position. 
        If the last position is accurate and if, as is less likely, the 
        course and speed correctly take into account all those effects 
        such as wind and current, the calculated position will be 
        accurate. Even when the current course and speed are less well 
        known, dead reckoning can provide a useful confirmation for other 
        methods of position determination.
             ASTROCLK uses true bearings rather than magnetic bearings 
        since the magnetic declination, the difference between true North 
        and magnetic North as shown by a compass, varies considerably and 
        changes very slowly with time. The direction of the declination 
        is given as "East" or "West", meaning that true North is in the 
        specified direction from magnetic North. Magnetic declination 
        should not be confused with astronomical declination. In the 
        United States, the magnetic declination ranges from about 20 
        degrees West in the extreme Northeast to 22 degrees East in the 
        extreme Northwest. The line of zero magnetic declination, where 
        the true and magnetic bearings are the same, passes near Chicago, 
        Illinois and Tallahasse, Florida. Local magnetic anomalies can 
        also cause significant changes in the magnetic declination. Most 
        large ships and aircraft use satellite or inertial navigation 
        systems which provide true bearings but smaller craft (and the 
        air traffic control system) use magnetic bearings. Knowing the 
        local magnetic declination is therefore important in navigation 
        and can also be helpful for the alignment of telescopes when the 
        North star is not visible (i.e. during daylight hours).
             When traveling long distances, life is not quite so simple 
        if the navigator wishes to minimize the distance covered. The 
        bearing or "true course" to a distant destination, that course 
        plotted directly on a conventional map or chart, does not 
        represent the ideal course. Depending upon the projection used in 
        the preparation of the chart, the minimum distance and best 
        course are not necessarily represented by a straight line. For 
        distances under several hundred miles, the difference is usually 
        trivial and can be ignored. However, for distances of hundreds of 
        miles or more which involve significant differences in longitude, 
        the navigator should plot his "great circle" course. A few 
        minutes spent with a globe and a piece of string stretched taut 
        between two locations will suffice to demonstrate that a great 
        circle route can be considerably shorter than what appears to be 
        the most direct route on a flat map. The polar route used by 
        aircraft from the Western United States to Europe is an example 
        of a frequently used great circle route. It is important to note 
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 75


        that, unlike lines of equal latitude, the meridians (lines of 
        equal longitude) are already great circle routes; therefore, 
        voyages which are primarily North-South gain little or no benefit 
        from plotting a great circle route.
             Exactly following a typical great circle route involves 
        constant changes in course, since the route follows an arc rather 
        than a straight line when plotted on a standard projection chart. 
        In practice, therefore, navigators usually select a series of way 
        points along the desired route and follow a set course between 
        each point. The more way points selected, the better the 
        approximation to the great circle route -- and the greater the 
        chance for human error. It has been suggested that Korean Air 
        Flight 007 may have met disaster because of an entry error on a 
        way point, most likely a digit transposition in the longitude 
        coordinate, thereby crossing restricted Russian airspace rather 
        than being well out over the Pacific ocean and thus setting the 
        stage for what followed.
             Although ASTROCLK calculates the distance already traveled 
        from the last navigation fix and the current position using dead 
        reckoning (current speed times elapsed time in the direction 
        specified by the current course), the distance to a selected 
        destination (or way point) is computed using the great circle arc 
        from the current position to that destination. To a first order 
        approximation, each degree of that arc is sixty nautical miles. 
        (The conversion is exact by definition along the equator but 
        becomes slightly less accurate as the latitude increases due to 
        the flattening of the Earth at the poles. ASTROCLK takes this 
        factor into account in its distance-to-destination calculations.) 
        The displayed distance to the destination is therefore always the 
        current great circle distance from the current position; whether 
        or not the destination lies along the present course is of no 
        consequence to the calculations, and that fact must be kept in 
        mind when using the data for navigation.
             If the current speed is entered as zero, ASTROCLK may be 
        used to calculate the great circle distance from the current 
        navigation point to the selected destination. The distance is 
        shown in nautical miles and kilometers. Also shown is the "chart 
        course" from the navigation point to the destination.
             By deliberately picking an off-course destination, you may 
        take advantage of this method and watch for the point of closest 
        approach as you pass by. By setting the speed to zero, which 
        forces the current position to remain at the navigation fix or 
        geographic location, ASTROCLK may also be used to calculate the 
        great circle distance between any two points on the globe.
             Point-to-point navigation by either true or magnetic 
        bearings, as opposed to great circle routes, is most accurate in 
        the mid latitudes and over moderate distances. As the route 
        approaches polar regions and as the distances become longer, 
        inaccuracies become more and more significant; these inaccuracies 
        are almost entirely due to the coordinate system used to project 
        the surface of a sphere onto a flat surface. Since ASTROCLK and 
        most navigators use that same coordinate system, some care must 
        be used in these cases. The problem is easily illustrated by an 
        example: plot a straight line course of 45 degrees (Northeast) on 
        a typical Mercator or cylindrical projection of the world. Sooner 
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 76


        or later you will arrive at the top edge of the map, and that 
        entire upper edge represents the North Pole. Therefore, ANY 
        northerly course will eventually wind up at the North Pole; the 
        same applies with respect to the South Pole for southerly 
        courses. Transferring the plot to a globe will trace a gradually 
        curving course toward the pole. If the course were 80 degrees 
        rather than 45 degrees, it would trace a spiral route toward the 
        pole through successive revolutions around the globe.
             Of course, no navigator would ever steer 80 degrees in hope 
        of eventually reaching the North Pole, but ASTROCLK must know how 
        to handle such a case in the event that a course is entered and 
        left alone for some days or even months. Having reached the Pole, 
        regardless of the circuitous route, the program must select a 
        reasonable and consistent method of processing continuing travel. 
        The most obvious choice is to assume that, having reached the 
        Pole, the voyage should continue on the opposite side of the 
        globe with a course 180 degrees different from the initial 
        course. Using this method, an initial course of 0 degrees (North) 
        will result in polar circumnavigation of the globe, just as 
        expected; reaching the North Pole, the course becomes 180 degrees 
        and continues to the South Pole where the process is reversed. 
        This is the algorithm which ASTROCLK uses over long distances but 
        it can yield results which appear rather peculiar taken out of 
        context.

             When first started, the navigation functions of ASTROCLK 
        are disabled. Before attempting to enable these functions, the UT 
        TIME ZONE OFFSET must be set as described above. If the 
        navigation functions are enabled, they may be disabled at any 
        time by using Function Key F6 to enter new local coordinates. 
        This disables navigation without clearing the data; the 
        navigation data may be re-enabled with Function Key F10 followed 
        by F2 and then pressing RETURN to select the old data.
             Once the UT TIME ZONE OFFSET has been set, pressing Function 
        Key F10 displays the full Navigation Menu:

                  ASTROCLK NAVIGATION INFORMATION

                  Navigation functions available are:
                       F1 = Show current NAVIGATION DATA
                       F2 = Set current NAVIGATION DATA
                       F3 = Set current DESTINATION DATA
                       F4 = Set current STAR SIGHT DATA
                       F5 = Select USNO Navigation Stars

                      F10 = Set Computer UT Time Zone Offset

                  Select function or press RETURN to cancel:

             Pressing Function Key F1 will display the Navigation Data 
        now stored, whether or not navigation is active. A typical 
        display contains the following data:

                  ASTROCLK NAVIGATION INFORMATION

   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 77


                  The current NAVIGATIONAL DATA are:
                    Nav LONGITUDE:        -15.000000
                    Nav LATITUDE:          32.000000
                    Nav POSITION TIME:     20.0 UT
                    Nav POSITION DATE:     16-06-1989
                    Nav COURSE (true):    325.000000
                    Nav SPEED (knots):     20.00
                    DISTANCE Traveled:     10.00 nm =   18.53 km
                    ELAPSED TIME:          0.5000 hrs

                  Press RETURN to resume ASTROCLK:

        When you have finished reviewing the data, press RETURN to resume 
        normal operation. You may use Function Key F7 to select the 
        preferred format of displaying angles and time.
             Pressing Function Key F2 will display the current 
        navigational data as above except that the prompt at the bottom 
        of the window is changed to:

                  Press SPACE to enter NEW Navigation Data, or
                  press RETURN to ACCEPT, or F10 to CANCEL:

        Press RETURN to accept the data as shown, press Function Key F10 
        to cancel data entry and return to normal operation, or press the 
        SPACE BAR to enter new or changed data. If you press RETURN, you 
        will be prompted for each of the six required items and the 
        current value of that item will be shown.

                    Nav LONGITUDE:        -15.000000
                    Nav LATITUDE:          32.000000
                    Nav POSITION TIME:     20.0 UT
                    Nav POSITION DATE:     16-06-1989
                    Nav COURSE (true):    325.000000
                    Nav SPEED (knots):     20.00

        If the current value of an item is correct, press RETURN for that 
        item. If you wish to change the item, enter the new value 
        followed by RETURN. The input format is very flexible, and the 
        longitude, latitude and course may be entered in degrees, degrees 
        and minutes, or degrees and minutes and seconds. Any item may 
        have a fractional decimal part. Use the comma as a separator. If 
        you wish to use the local coordinates and the current time as the 
        navigation fix, enter "*" (without the quotation marks) followed 
        by RETURN in response to the prompt for LONGITUDE. In this case, 
        only the COURSE and SPEED will remain to be entered.
             When all items have been processed, the original display 
        will be repeated with any new or changed values shown and the 
        same prompt:

                  Press SPACE to enter NEW Navigation Data, or
                  press RETURN to ACCEPT, or F10 to CANCEL:

        Press RETURN to accept the values shown and enable navigation or 
        press SPACE BAR if some values must be corrected. This process 
        may be repeated as many times as necessary and at any time. Once 
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 78


        the navigation functions have been enabled, the local coordinates 
        window will display the current calculated position based upon 
        the data just entered above using dead reckoning. If a non-zero 
        speed has been entered, the local coordinates window will display the 
        title "Calculated Position" instead of a location name, and that 
        position will be calculated in real time when the clocks are on. 
        The local time will be adjusted according to the current 
        longitude and all celestial and planetary positions and other 
        data will be calculated dynamically.
             Once the navigation data has been entered, the main display 
        window may be set to the Navigation Mode by pressing the "N" key. 
        A typical navigation data display contains the following data:

                  ASTROCLK NAVIGATION INFORMATION

                  Data relative to: NAVIGATION DATA
                       LONGITUDE:        -15 00'00.00"
                       LATITUDE:          32 00'00.00"
                       DISTANCE:         280.00 nm =  519.49 km
                       ELAPSED TIME:      14:00:00
                       Current COURSE:   325 00'00.00"

                  (No DESTINATION DATA entered)

        In this example, Function Key F7 has been used to set the angle 
        and time formats as shown. Note that the distance traveled is 
        based solely upon the elapsed time multiplied by the current 
        speed and does not necessarily bear any relationship to the 
        distance between the navigational position or fix and the current 
        position. Note also that if the current speed has been set to 
        zero, the DISTANCE and COURSE data will not be displayed.
             Even when actual navigation is not intended, ASTROCLK may be 
        used to measure great circle distances between the current 
        navigation point (or local coordinates) and any other geographic 
        location by setting the speed equal to zero. In this case, 
        certain items which do not apply, such as distance traveled, are 
        eliminated from the displays.
             One final step is required to fully set up a navigation 
        or distance measuring situation: entering a "destination". The 
        destination may be the intended destination, a way point along 
        the projected course, or simply a point of interest. Two methods 
        are available for entering the destination data: Function Keys 
        F10 and SHIFT-F6; both methods accomplish the same purpose but by 
        slightly different techniques. To manually enter the destination 
        data, press F10 and then F3. The current destination information, 
        if any, will be displayed:

                  ASTROCLK NAVIGATION INFORMATION

                  Current DESTINATION DATA:
                    NAME:        Way Point A
                    LONGITUDE:   -15 14'57.84"
                    LATITUDE:     32 20'57.47"

        You will be prompted in turn to enter new or changed information:
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 79



                  Enter NAME (SPACE to cancel):

                  Enter LONGITUDE (W = negative):

                  Enter LATITUDE  (S = negative):

        To clear all destination information, enter SPACE followed by 
        RETURN instead of a name or designation. Press RETURN to leave an 
        item unchanged. Note that West longitudes and South latitudes are 
        entered as negative numbers. The input format is very flexible, 
        and the longitude and latitude may be entered in degrees, degrees 
        and minutes, or degrees and minutes and seconds. Any item may 
        have a fractional decimal part. Use the comma as a separator.
             Function Key SHIFT-F6 may also be used to enter destination 
        data, especially when that data is available in a "city file" on 
        disk. For example, file USWEST.VOR is available which includes 
        complete data for the 287 VOR's (VHF Omni-Directional Range, a 
        radio navigation aid for aircraft) in the 11 western states. A 
        navigator might wish to prepare a special file of navigation 
        points for use in an upcoming trip. Operation of SHIFT-F6 is 
        identical to that used to set the local coordinates with Function 
        Key F6.
             Once destination data has been entered, pressing the "N" key 
        to enable the Navigation Mode will automatically include the 
        calculation of your present position compared to that 
        destination:

                  ASTROCLK NAVIGATION INFORMATION

                  Data relative to: NAVIGATION DATA
                       LONGITUDE:        -15 00'00.00"
                       LATITUDE:          32 00'00.00"
                       DISTANCE:         280.00 nm =  519.49 km
                       ELAPSED TIME:      14:00:00
                       Current COURSE:   325 00'00.00"
                  Data relative to: WAY POINT A
                       LONGITUDE:        -15 14'57.84"
                       LATITUDE:          32 20'57.47"
                       DISTANCE:         253.54 nm =  470.40 km
                       TIME TO DEST:      12:40:37
                       Chart COURSE:     140 04'26.57"

        In this example, we have obviously sailed well past Way Point A 
        by some 254 nautical miles (great circle distance), and the 
        course back to that point as plotted on a conventional chart is 
        approximately 140 degrees. At the present speed, it would require 
        about 12 hours and 40 minutes to return to Way Point A IF we 
        follow the great circle route. For longer distances, the great 
        circle route and the chart course will NOT be the same, as 
        discussed above. For short and moderate distances, the two 
        courses will be approximately the same.
             When the speed has been set to zero (using Function Key F2), 
        information which does not apply in that case is deleted from the 
        navigation mode display:
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 80



                  ASTROCLK NAVIGATION INFORMATION

                  Data relative to: NAVIGATION DATA           
                      LONGITUDE:       -120 34'00.00"         
                      LATITUDE:          38 09'00.00"         
                      ELAPSED TIME:       0:34:15             

                  Data relative to:  Crazy Woman, WY    CZI   
                      LONGITUDE:       -106 26'06.00"         
                      LATITUDE:          43 59'54.00"         
                      DISTANCE:         727.77 nm = 1350.24 km
                                                  =  838.03 mi
                      Chart COURSE:      67 31'04.73"         

        In this example, the destination has been set to the aircraft VOR 
        at Crazy Woman, Wyoming (VOR code "CZI"), and the navigation fix
        is for Calaveras County, California. Using SHIFT-F6, you may 
        select different destinations from the current city file and 
        obtain a display of the coordinates, distance and chart course 
        relative to the navigation fix. Note the addition of the distance 
        in statute miles ("mi") in this version of the display.


        Celestial Navigation with Star Sights

             To be effective, any method of navigation requires that the 
        initial position be known as precisely as possible. Departing a 
        location whose coordinates are known provides that initial data 
        but within a relatively short time, depending upon the speed of 
        travel, a navigator needs to determine a new position both to 
        check the accuracy of his dead reckoning calculations as well as 
        to serve as a new basis for position calculations. Failure to do 
        so can have unfortunate results.
             One of the most accurate methods of establishing a position, 
        or "fix", has been to take sights of the Sun, Moon, planets or
        selected bright stars, and use that information to compute a 
        position. This technique is known as celestial navigation. To do 
        this, a triangle known as the "celestial triangle" or 
        "navigational triangle" is formed between the observer, the North 
        or South Celestial Pole, and the selected star or other celestial 
        object. These three points are projected onto a sphere and the 
        solution of the angles of the resulting celestial triangle using 
        spherical trigonometry provides the position information the 
        navigator seeks.
             A number of different methods have been used over past 
        centuries to obtain the solution to the celestial triangle. Early 
        methods were very cumbersome and difficult to solve accurately. 
        In the nineteenth century a technique called the Altitude-
        Intercept Method was developed by the Frenchman Marc St. Hilaire 
        using two trigonometric equations (known as the Cosine-Haversine 
        formulas) to solve the problem. Although this new method was a 
        considerable improvement over earlier methods, it was still quite 
        a chore to manually calculate a position. About 1930 a Japanese, 
        Ogura, developed a simplified solution based upon sight reduction 
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 81


        tables. These tables gave the position of the Sun and selected 
        stars and planets at regular intervals throughout the year. By 
        recording the altitude of two or preferably three celestial 
        objects whose positions were tablulated, along with the time of 
        each measurement and the vessel's course and speed, the navigator 
        could determine his position at a specific time and calculate his 
        present estimated position.
             The Nautical Almanac, jointly published every year by the U. 
        S. Naval Observatory and H. M. Nautical Almanac Office, gives 
        similar, improved tables today that form the basis for manual 
        calculation of a position by sight reduction. Data are given for 
        the Sun, Moon, Venus, Mars, and Saturn for each hour of each day, 
        and the positions of the 57 USNO navigational stars for each 
        three day period (since the rate of change of stellar positions 
        is relatively slow). The method involves little more than noting 
        the date and time, looking up numbers in the tables, and then 
        performing various interpolations, additions, and subtractions. 
        Simple as that may sound, the calculations must be performed 
        correctly and with sufficient precision in order to obtain a 
        reliable position.
             With the advent of electronic calculators and, more 
        recently, portable computers, attention has again been focused 
        on St. Hilaire's original Cosine-Haversine formulas developed in 
        1875. Using the formulas directly instead of tables derived from 
        them makes electronic calculation relatively straightforward once 
        the formulas themselves have been properly entered. ASTROCLK uses 
        this method with observations of any of the 57 USNO Standard 
        Navigational Stars, as described in the Nautical Almanac 1989. 
        (However "straightforward" the data entry process may be, a brief 
        look at ASTROCLK's inner workings will reveal that setting up all 
        the information needed to use the formulas is a non-trivial 
        task!)
             Regardless of which of these methods is employed, sight 
        reduction tables or formulas, everything depends upon taking 
        accurate star sights and knowing the correct time. Taking a 
        sextant sight on a moving vessel requires considerable skill and 
        practice as well as an accurate instrument. ASTROCLK and a good 
        short wave radio can provide the time to sufficient accuracy 
        almost anywhere in the world. The resulting position calculations 
        are more accurate than the typical star sights by an average 
        navigator.

             Star sights are typically made using a marine sextant or a 
        bubble sextant. One of the important differences between these 
        two instruments is the method by which the horizon is determined. 
        The marine sextant uses the apparent horizon (which must 
        therefore be visible at the time of measurement) and the 
        resulting star altitudes must be corrected for "horizon dip", the 
        lowering of the apparent horizon as the elevation of the observer 
        increases. The bubble sextant, on the other hand, uses an 
        artificial (true) horizon formed by a bubble in a liquid, much 
        like the common carpenter's level, and needs no horizon 
        correction.
             Depending upon the type of instrument being used, the 
        elevation must be set to the actual elevation of the observer's 
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 82


        eye above mean sea level (marine sextant) or to zero (bubble 
        sextant) using ALT-F6. ASTROCLK then makes the appropriate 
        correction for horizon dip using a standard formula. Failure to 
        set the elevation to the correct value can cause appreciable
        position errors.
             Some care should be used in the selection of the stars to be 
        used for celestial navigation. Since objects near the zenith 
        (directly overhead) are difficult to observe with a marine 
        sextant, they should be avoided; similarly, errors due to 
        refraction increase near the horizon. It is therefore recommended 
        that the selected stars be at observed altitudes of from about 15 
        degrees to 80 degrees.
             Before the actual navigation calculations can be made, an 
        estimated position or navigational fix must be entered and the 
        celestial star sights must be taken. Using Function Key F7, set 
        the display format to your preference (i.e. the same format as 
        your sextant or navigational instrument uses). Then press 
        Function Key F10 followed by F2 (a combination referred to as 
        "Navigation Function Key F2") to enter the navigational fix data. 
        The longitude and latitude of the navigational fix need only be 
        entered to an accuracy of several degrees; a less accurate 
        estimate simply means a few more calculations for ASTROCLK to 
        achieve the desired accuracy. UT Time, UT Date, course and speed 
        complete the required items. If you are in a fixed position, 
        enter zero for course and speed. (See the Dead Reckoning section 
        above for a more detailed description of setting the navigational 
        fix.)
             Taking an accurate star sight typically requires from five 
        to fifteen minutes. Record the UT Time when the sight is taken 
        along with the observed altitude. While you may wish to check the 
        azimuth of the star, ASTROCLK does not require that information 
        for its calculations. Star sights may be made before or after the 
        time of the estimated position.
             HINT: If you set the estimated position as the current 
        coordinates using Function Key F6, you may then use ASTROCLK to 
        help select suitable stars for your location and time; select a 
        USNO Standard Navigational Star using Function Key F5 followed by 
        F1 and check the Target Tracking Display to see that it is 
        observable.
             ASTROCLK internally "plots" each of your star sights to 
        determine a Line of Position (LOP) starting with the given 
        altitudes and times, and processes the internal star database 
        along with the course and speed to determine the various required 
        functions. The initial estimated position and calculated position 
        for each star sight should lie approximately along the Line of 
        Position. ASTROCLK then generates a calculated position from 
        these data and compares this calculated position with the initial 
        estimated position. If these differ appreciably, it substitutes 
        the new calculated position for the estimated position and 
        repeats the process until the difference in positions reaches a 
        minimum. The result is the final calculated position.

             To begin ASTROCLK's celestial navigation calculations, press 
        Function Keys F10 and then F4. The program reminds you that you 
        must take either two or three star sights and have previously 
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 83


        entered your estimated position:

                  ASTROCLK NAVIGATION INFORMATION

                  Celestial Navigation requires observed data
                  for two or three USNO Navigational Stars.

                  For this data to be valid, you must first
                  have entered your last Navigation Fix using
                  Navigation Function Key F2.

                  Press RETURN to begin data entry or press
                  any other key to cancel:

        Press RETURN if you are ready to enter the star sight data or 
        press any other key to cancel and resume normal program 
        operation.
             After pressing RETURN, ASTROCLK requests that you enter the 
        instrument INDEX ERROR to be used in correcting the altitude 
        measurements:

                  Altitude measurements made with a sextant or
                  other instrument often have an associated
                  INDEX ERROR which must be removed from each
                  measurement prior to performing calculations.
                  Enter the INDEX ERROR (minutes) for your
                  instrument or press RETURN to enter zero.

                  The Index Error will be SUBTRACTED.

                  Enter Index Error:

        Enter the index error in minutes of arc or press RETURN to enter 
        an index error of zero. Once you have entered an index error 
        value, ASTROCLK retains that value until the program is halted. 
        Note that the index error entered will be subtracted from your 
        altitude measurements.
             ASTROCLK now requests that you select the USNO Standard 
        Navigational Star for the first star sight:

                  Select USNO Standard Navigational Star

                  Enter STAR NAME or STAR NUMBER:

        You may enter either the star name, using upper or lower case and 
        sufficient letters to unambiguously identify the star, or the 
        star number, 1 to 57. Use "DENEB ", with a trailing space, to 
        select Deneb rather than Denebola. If you select star #49, for 
        example, the program will look up the star, display its full 
        name, and prompt you for the UT TIME of the star sight and the 
        observed altitude:

                  USNO Star #49 - a Lyrae - Vega           

                  Enter UT TIME for Star Sight #1:
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 84


                  Enter Observed Altitude [Ho]:

        ASTROCLK interprets any time entered as UT TIME, without adding a 
        trailing "U". The time may be before or after the time entered 
        for the navigation fix, but in practice the star sights should be 
        made at approximately the same time as the estimated fix in order 
        to minimize the dead reckoning errors if you are on a moving 
        vessel. The observed altitude is the reading directly from the 
        instrument; ASTROCLK will apply the horizon dip, index error and 
        refraction corrections automatically.
             Repeat the last steps for the second (and third) star sight, 
        as prompted. If you are entering only two star sights, press 
        RETURN when requested for the USNO star number for the third 
        sight. ASTROCLK uses the least squares method of calculating the 
        position described in the Nautical Almanac 1989. However, 
        ASTROCLK uses its own internal algorithms to calculate altutides 
        and azimuths rather than those given in the NA; the results are 
        essentially the same using either method. While two "perfect" 
        sights are sufficient to do the calculations, three sights are 
        preferred to minimize potential errors. After a brief delay, the 
        results of the calculations are displayed:

                  The sextant altitudes have been corrected to:
                    Ho ALTITUDE 1:      20 02'31.94"
                    Ho ALTITUDE 2:      29 28'28.19"
                    Ho ALTITUDE 3:      43 55'22.80"

                  The Celestial Navigation calculations have
                  estimated the Navigational Fix Position as:
                    Nav LONGITUDE:     -15 14'58.27"
                    Nav LATITUDE:       32 20'59.27"

                  Press RETURN to ACCEPT the calculated posi-
                  tion or any other key to discard:

        The first section of data are the corrected values for the 
        observed altitudes. If data for only two sights have been 
        entered, no data will be shown for a third sight. The second 
        section of data are the results of the sight reduction 
        calculations: the calculated longitude and latitude.
             If you wish to accept the new position, press RETURN; the 
        new position will then appear in the local coordinates window and 
        ASTROCLK will resume normal operation. Use Navigation Function 
        Key F2 to set the new position as the current navigation fix.


        Selecting USNO Navigational Stars

             Before star sights can be used with ASTROCLK's celestial 
        navigation functions, the two or three USNO Navigational Stars 
        must be selected. While the skilled star gazer or navigator will 
        immediately recognize the USNO stars, the casual observer may 
        have more difficulty. Navigation Function F5 scans all 57 USNO 
        stars, calculates the horizon coordinates (Altitude and Azimuth), 
        then displays the first 20 which may be found above 15 degrees 
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 85


        and below 80 degrees referred to the actual horizon. (Be sure 
        that the ELEVATION is correctly set!) All calculations are based 
        upon the current local coordinates and time.
             In order to display this list, select Navigation Function 
        F5. A brief delay (longer for computers not equipped with a math 
        coprocessor!) will follow and then ASTROCLK will display the 
        selected stars. The first 20 stars which are suitable will be 
        displayed. Since the USNO stars are well distributed around the 
        celestial sphere, from 15 to 20 stars are usually acceptable at a 
        given time and place.

                  USNO STARS: 15 < ALTITUDE < 80

                   #   ALT     AZ   MAG     #   ALT     AZ   MAG
                   3  67.1    5.1   2.2  | 47  21.7  318.4   2.2
                   4  38.0  174.7   2.0  | 49  22.3  302.7   0.0
                   6  65.6  108.5   2.0  | 51  22.5  265.6   0.8
                   8  43.1  120.6   2.5  | 53  46.2  302.5   1.3
                   9  53.8   50.0   1.8  | 54  45.9  246.7   2.4
                  10  31.8   90.7   0.9  | 56  23.2  200.8   1.2
                  12  34.9   54.0   0.1  | 57  63.9  229.5   2.5
                  13  16.2   93.2   1.6
                  14  27.1   72.2   1.6
                  40  20.7  350.1   2.1

                  Press RETURN to continue ...

        The example above indicates that 17 USNO stars were considered 
        suitable for navigation purposes using the current local 
        coordinates and the current time. The following information is 
        displayed for each star: USNO number, Apparent Altitude (ALT, 
        degrees), Azimuth (AZ, degrees in the sense NESW), and standard 
        visual magnitude (MAG). The Altitude has been corrected for 
        refraction and horizon dip and therefore corresponds to the 
        apparent position in horizon coordinates where the star may be 
        found. Note that a star is brightest when its magnitude is 
        smallest; negative magnitudes are brightest of all.
             Since the calculations are based upon the current location 
        and time, the navigator may use the current calculated position 
        or set an anticipated location and time (using F6 and F3) before 
        taking star sights and select "suitable" stars in advance. If the 
        current position is reasonably close to the expected position, 
        only the time need be set; this avoids disabling and then re-
        enabling navigation mode when F6 is used.
             The non-navigator may also find the display useful: by 
        setting the SPEED to zero (as discussed above), you may see an 
        immediate display of the current positions of the visible USNO 
        navigational stars (which also, by no coincidence, are the 
        brightest stars) visible at the current position and time. Star 
        gazers not yet accustomed to using horizon coordinates, altitude 
        and azimuth, may find the information helpful in orienting their 
        view of the night sky and in locating these stars.


   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 86


        Celestial Navigation Example

             It is often helpful to examine a worked out problem to see 
        how entries are made and calculations performed. The following 
        example illustrates how ASTROCLK can compute a position using 
        celestial navigation and is based upon the example on pages 282 
        and 283 of the Nautical Almanac 1989. The original objective, of 
        course, was to verify ASTROCLK's accuracy using known data.

        1.   Using Function Key F3, set the time and date to 20:00:00 UTC 
             ("20U") on 16 June 1989 ("16,6,1989"). Note the "U" to 
             signify Universal Time.

        2.   Using Function Key F7, set the display format for degrees to 
             "ddd.dddddd" in order to agree with the format displayed in 
             the Nautical Almanac. (The display format makes no 
             difference to ASTROCLK.)

        3.   Using Function Key ALT-F6, set the Elevation to 0. Leave all 
             other local conditions at their default values.

             In practice, the elevation should be be set to zero if the 
             instrument provides an accurate artificial horizon; 
             otherwise, set  the elevation (height of the observer's eye  
             above mean sea level) so as to compensate for the dip of the 
             apparent horizon. The pressure and temperature should be set 
             to the current conditions, if known.

        4.   Using Function Key F10 followed by F2, set the navigation 
             fix to the coordinates, time, date, course, and speed 
             required. The following display should appear if all 
             information has been entered correctly:

                  ASTROCLK NAVIGATION INFORMATION

                  The current NAVIGATIONAL DATA are:
                    Nav LONGITUDE:        -15.000000
                    Nav LATITUDE:          32.000000
                    Nav POSITION TIME:     20.0 UT
                    Nav POSITION DATE:     16-06-1989
                    Nav COURSE (true):    325.000000
                    Nav SPEED (knots):     20.00
                    DISTANCE Traveled:      0.00 nm =    0.00 km
                    ELAPSED TIME:          0.0000 hrs

             Note that because the navigational fix time and date are the 
             same as the time and date set in Step 1, the calculated 
             distance traveled and the elapsed time are both zero.

        5.   Using Function Key F10 followed by F3, set the destination 
             name and coordinates. Use the following values:

                  Name:           NA-1989
                  Longitude:     -15.2494
                  Latitude:       32.3493
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 87



             These values are the final position estimate calculated in 
             the Nautical Almanac and will be used to compare ASTROCLK's 
             position calculation. The destination information is not 
             required for ASTROCLK to perform the celestial navigation 
             calculations and it is included here only for purposes of 
             comparison.

        6.   Using Function Key F10 followed by F4, enter the star sight 
             information as follows:

                  Index Error:        0

                  Star #1:            49 (or "Vega")
                  UT Time:            20:00
                  Altitude:           20.08481

                  Star #2:            21 (or "Pollux")
                  UT Time:            19:50
                  Altitude:           29.50204

                  Star #3:            33 (or "Spica")
                  UT Time:            19:40
                  Altitude:           43.93917

             Either the USNO Star Number or its proper name (sufficient 
             letters to unambiguously identify it, upper or lower case) 
             may be used without the quotation marks. The time entry does 
             not require the "U" to signify UTC. The altitude is shown 
             entered in degrees and decimal fraction, but may be entered 
             in any of the usual formats.

             Note that if you were using an actual sextant, an index 
             error would normally be entered and automatically subtracted 
             from the measured altitudes. Once entered, the index error 
             is retained by ASTROCLK until the program is next restarted, 
             on the assumption that all altitude measurements will be 
             performed with the same instrument. The Nautical Almanac 
             example does not include any index error, hence no error is 
             entered here.

        7.   When the data have all been entered, the following display 
             will appear to enable you to check your data:

                  ASTROCLK NAVIGATION INFORMATION

                  The sextant altitudes have been corrected to:
                    Ho ALTITUDE 1:      20.042198
                    Ho ALTITUDE 2:      29.474503
                    Ho ALTITUDE 3:      43.923001

                  The Celestial Navigation calculations have
                  estimated the Navigational Fix Position as:
                    Nav LONGITUDE:     -15.249526
                    Nav LATITUDE:       32.349772
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 88



                  Press RETURN to ACCEPT the calculated posi-
                  tion or any other key to discard:

             The data input in Step 6 have been "rigged" to yield the 
             observed altitudes (Ho) in the display above. Comparison of 
             these corrected data with that published in the Nautical 
             Almanac will show a difference of no more than 0.000003 
             degrees, a trivial amount. The reason for rigging the data 
             is that the Nautical Almanac uses fully corrected data while 
             ASTROCLK automatically corrects the sextant altitude for 
             refraction. The input data have been adjusted so that the 
             observed altitudes agree after that refraction correction.

             The second set of data are the position coordinates which 
             ASTROCLK has calculated from the input data. Press RETURN to 
             accept this position, or press any other key to discard the 
             calculation; either choice will return to ASTROCLK. 
             Accepting the data will change the local coordinates window 
             to the new longitude and latitude.

        8.   Now press "N" to change to Navigation Mode. The following 
             display will appear:

                  ASTROCLK NAVIGATION INFORMATION

                  Data relative to: NAVIGATION DATA
                       LONGITUDE:        -15.249526
                       LATITUDE:          32.349772
                       DISTANCE:           0.00 nm =    0.00 km
                       ELAPSED TIME:       0.0000 hrs
                       Current COURSE:   325.000000
                  Data relative to: NA-1989
                       LONGITUDE:        -15.249400
                       LATITUDE:          32.349300
                       DISTANCE:           0.03 nm =    0.05 km
                       TIME TO DEST:       0.0014 hrs
                       Chart COURSE:     165.108506

             The first portion of the display shows the data relative to 
             the last navigation fix (which is the data ASTROCLK has just 
             calculated in Step 7) and is obvious. The distance and time 
             are both zero because ASTROCLK is set to the time of the 
             navigation fix. The course is as set in Step 4.

             The second portion of the display shows the data relative to 
             the "destination", set to the results of the calculation in 
             the Nautical Almanac; note that the longitude and latitude 
             are exact. The distance is therefore the great circle 
             distance bewteen the fix ASTROCLK has just calculated and 
             the Nautical Almanac result, shown in nautical miles (nm) 
             and kilometers (km). In this case, ASTROCLK produced a 
             result within less than 200 feet of the Nautical Almanac.

             Note that the initial position estimate entered in Step 4 is 
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 89


        quite close to the end result, following the example in the 
        Nautical Almanac. In fact, the estimated longitude and latitude 
        may be off by five or ten degrees in either direction with little 
        effect on the final result except to increase the computation 
        time on computers not equipped with a math coprocessor.
             When using these celestial navigation functions, it is 
        important to note that the accuracy of ASTROCLK's calculations is 
        actually far better than can likely be achieved in practice. Not 
        only is it all but impossible to read a sextant or similar 
        instrument to the accuracy and precision used in the example, but 
        changing atmospheric conditions especially near the horizon 
        (which are difficult to measure from the Earth's surface without 
        a fully equipped observatory) can cause the refraction to vary by 
        as much as several arc seconds from the calculated value. The 
        purpose here is to provide a method which introduces little or no 
        additional error in the celestial navigation calculations. This 
        example demonstrates that ASTROCLK's apparent geocentric 
        equatorial star positions are typically within one arc second of 
        the values published in the Astronomical Almanac and the Nautical 
        Almanac as well as those generated by USNO's Interactive Computer 
        Ephemeris and Floppy Almanac, and that the resulting navigational 
        calculations are essentially exact.
             For comparison with current state of the art navigation and 
        position determination equipment, manufacturers are claiming an 
        accuracy of better than 50 feet with military versions of the 
        NavStar Global Positioning System (GPS) receivers. Commercial 
        versions of the GPS receiver, which cannot decode some of the 
        special signals on NavStar (which are required for maximum 
        accuracy), are expected to achieve accuracies on the order of 300 
        feet or less.
             In practice, ASTROCLK's navigation calculations can all be 
        made with the clocks running; the current calculated position is 
        displayed in real time and all celestial and planetary data are 
        similarly calculated. However in the case of this example from 
        the Nautical Almanac, because the date of June of 1989 is now 
        long past, the resulting calculated position after many months at 
        20 knots is not correct. By setting the computer clock and date 
        to some time shortly after the time of the last star sight (use 
        Function Key ALT-F4 to enable the SIMULATION mode, or use 
        Function Key F9 to return to DOS and then use the TIME and DATE 
        commands), the "real" situation can be simulated and the actual 
        running position will be displayed in the local coordinates 
        window, labeled "Calculated Position". At that point, you may 
        select a star or planet in the usual manner, display its 
        coordinates, and they will be referenced to the current 
        calculated position in real time.
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 90



        SIDEREAL TIME AND EQUATORIAL COORDINATES

             The concept of sidereal time is perhaps a bit difficult for 
        the layman to grasp. Even the idea that time is not absolute may 
        be a little unsettling to some and confusing to others.  However, 
        visualizing a "celestial sphere" with the Sun (heliocentric) or 
        the Earth (geocentric) at the center and with the stars, planets, 
        and other astronomical objects on its surface is relatively 
        straightforward. Using this approach, the stars remain in more or 
        less fixed positions on the sphere (although the planets 
        continuously change their positions) and the sphere appears to 
        rotate around us. Thus, stars appear to rotate about the 
        celestial pole in a counter-clockwise direction in the Northern 
        Hemisphere. Given this constantly changing scenario, astronomers 
        had to develop a coordinate system which would allow them to 
        unambiguously locate each celestial object. Although there are 
        several coordinate systems in use depending upon the application, 
        the most common is called Equatorial Coordinates and uses Right 
        Ascension and Declination, roughly analogous to geographical 
        longitude and latitude, respectively, to locate an object. This 
        is the coordinate system used in catalogs of star positions.
             The problem, and the reason for sidereal time, is that the 
        Earth is rotating about its axis as it orbits the Sun. As a 
        result of this, when viewed at the same time each night the stars 
        appear to change their position by a small amount. After a full 
        year, they are back in their original positions. If we divide the 
        360 degrees around the celestial sphere into 24 hours (much the 
        same as our earthly time zones) and call the resulting coordinate 
        Right Ascension, we have described what is sometimes called "star 
        time" but is more properly termed Mean Sidereal Time. 
        (Declination, the second coordinate, specifies the number of 
        degrees above or below the celestial equator.)
             Because of the Earth's rotation, sidereal time runs just a 
        bit faster than regular (mean solar) time; the difference is 
        about 4 minutes per day. If you are sufficiently patient, you can 
        watch one of ASTROCLK's sidereal clocks and see it skip a second 
        about every six minutes. Further, variations in the orbit and 
        rotation of the Earth and other considerations cause true 
        sidereal time not to be constant and astronomers therefore 
        usually use mean (or average) sidereal time.
             The difference between solar and sidereal time is best 
        illustrated by an example. Remembering that the Earth makes one 
        complete orbit around the Sun in about 365 days, it follows that 
        the Earth moves through approximately one degree each day 
        (360/365). Since solar time is measured from noon to noon, the 
        Earth must therefore rotate through approximately 361 degrees 
        each day in order for a given point on the Earth's surface to 
        again be directly facing the Sun. But the sidereal day is the 
        time elapsed for the Earth to make exactly one revolution of 360 
        degrees. That one degree difference distinguishes the two methods 
        of time measurement and means that the solar day is about 4 
        minutes longer than the sidereal day (3 minutes 56.6 seconds mean 
        solar time, actually).
             Both solar and sidereal time use the same units: days, 
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 91


        hours, minutes, and seconds; care must be taken that the type of 
        time being used is specified in order to avoid errors. The mean 
        sidereal times in ASTROCLK are calculated to a precision of 
        0.0001 seconds and have been checked against the Astronomical 
        Almanac for accuracy and are exact. All times displayed in the 
        small windows on the right of the screen have been rounded to the 
        nearest second. Near the vernal equinox each year (March 20th in 
        1988), sidereal time is exactly 12 hours different from mean 
        solar time. Similarly, sidereal time equals mean solar time near 
        the autumnal equinox in September.
             The current sidereal time corresponds to the right ascension 
        that is on your meridian (the "line" running from the North 
        celestial pole to the South celestial pole and passing directly 
        overhead) at that instant. Therefore, if you know a star's right 
        ascension, you know that the star may be found somewhere on the
        line from the North Pole through a point directly above you when 
        that right ascension equals the sidereal time. Where the star 
        will appear on that line is determined by its declination; +90 
        degrees corresponds to the North Pole, zero to the celestial 
        equator, and -90 degrees to the South Pole. If you hold your fist 
        out at arms' length with the thumb folded out of sight, its width 
        corresponds to about 10 degrees of arc (declination), or 40 
        minutes of time (right ascension) near the celestial equator. As 
        you move toward the poles, the lines of right ascension come 
        closer together, just as a section of orange is narrower at each 
        end. Another useful guide is that the stars most easily visible 
        at a given time will have right ascensions within a couple of 
        hours of the current sidereal time. Some stars, called 
        circumpolar stars, will always be visible if their declination is 
        greater than your latitude. If you stand at one of the poles, 
        naturally, all the stars you can see are circumpolar.
             When you are far away from clocks, books, and program 
        ASTROCLK, you can estimate sidereal time or right ascension using 
        the two pointer stars of the Big Dipper; the right ascension of 
        both stars is very close to 11 hours. Using the meridian 
        connecting those two stars and the North celestial pole as a 
        starting point, you can imagine a "clock" in the heavens to tell 
        you the sidereal time and to estimate the right ascension of a 
        star. That's the good news; the bad news is that this simple 
        sounding analogy is complicated by the fact that the celestial 
        clock must be divided into 24 hours instead of 12 hours, and that 
        the hour numbers go around in the opposite direction from a 
        "normal" clock, or counter-clockwise. Even so, it's worth giving 
        it a try just to familiarize yourself with the concept and to 
        practice locating a few well known stars. See the following 
        section for the Equatorial Coordinates of a number of bright 
        stars selected by USNO as Standard Navigational Stars.
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 92



        USNO COMPUTER EPHEMERIS PROGRAMS

             Beginning in the mid-1980's, the U. S. Naval Observatory 
        (USNO) supplemented its printed almanacs and ephemerides with a 
        disk-based program called the Floppy Almanac and designed to 
        execute on IBM-compatible personal computers (among others). 
        The Floppy Almanac was produced for years through 1999. While 
        not all Floppy Almanac data were equal in accuracy to that 
        contained in the Astonomical Almanac and other similar 
        publications, the Floppy Almanac provided more than sufficient 
        accuracy for most purposes and made reliable astronomical data 
        available to the vast majority of would-be users. USNO produced a 
        custom Floppy Almanac for each calendar year (400 days, actually, 
        with an small overlap from year to year).
             Starting with 1988, all Floppy Almanac versions used a 
        common set of data files and by adding the custom Floppy Almanac 
        program for each year the user could produce astronomical data 
        for the years 1988 through 1999. One of the more useful features 
        of ASTROCLK (from my perspective, at least) is to automatically 
        execute the Floppy Almanac. When Function Key ALT-F9 is pressed, 
        ASTROCLK examines the current date, writes a default data file of 
        initial values, and then executes the proper version of the 
        Floppy Almanac.
             The only significant problem with the Floppy Almanac has 
        been that each user must acquire a different version of the 
        program for each calendar year, plus or minus a few days. To 
        address this problem, USNO in early 1989 released a new program, 
        the Interactive Computer Ephemeris or ICE. ICE uses a common 
        program to process data for a 250 year period, from December 21, 
        1800, through June 7, 2049. A set of highly compressed ephemeris 
        data files (EPH01.DAT through EPH24.DAT), each covering 
        approximately 4000 days, allows the program to cover this 
        extended time span. For the approximate period 1980 through 1999, 
        only the data files EPH18.DAT and EPH19.DAT are required.
             This added capability and convenience has its price, 
        however. Each data file (except the first and the last) requires 
        approximately 37K bytes of disk storage and the complete package 
        requires approximately 1.1M bytes of disk storage. The Floppy 
        Almanac for a given year, by comparison, easily fits on a single 
        360K byte floppy disk. Each time it is executed, ICE must select 
        and then decompress the appropriate ephemeris data file. 
        Particularly when executed on a computer without a math 
        coprocessor, ICE therefore runs more slowly than FA. ICE and FA 
        appear to have essentially the same accuracy.
             In view of these factors, some users may may decide to 
        continue using the Floppy Almanac in preference to the 
        Interactive Computer Ephemeris. I have no information as to 
        whether or not USNO will continue to make the Floppy Almanac 
        available; I presently have FA versions for 1988 through 1992 and 
        these are available via my bulletin board system.
             As of Version 8915, ASTROCLK allows the user to select which 
        USNO program will be executed via ALT-F9, or ALT-F9 may be 
        disabled if neither program is available. This selection is made 
        using ALT-F10. See the section SETTING PROGRAM OPTIONS for 
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 93


        additional information on using ALT-F10 to select the desired 
        USNO program and to set the proper drive and path names.
             Both USNO programs operate in essentially the same manner. 
        Users familiar with the Floppy Almanac will have no difficulty 
        using ICE. The program name has changed, of course, and the 
        compressed ephemeris data files are a new fetaure. The star 
        catalog file names are unchanged and appear to be identical to 
        those supplied with the Floppy Almanac although the file dates 
        and times are different. The default parameter files, FA.DFT and 
        ICE.DFT, are slightly different; because of the time span 
        covered, the date parameter in the first line now requires the 
        full year for ICE. ASTROCLK correctly formats a default parameter 
        file for either program.
             Each time ALT-F9 is invoked, the default parameter file, 
        ICE.DFT or FA.DFT, is written with the current ASTROCLK date and 
        time, the current local geographical coordinates, and the local 
        time zone referred to UTC; the parameters "Time Step" and "Num of 
        Positions" are each set to +1.00. The USNO program is therefore 
        ready to use immediately upon entry.
             The following is a typical ICE.DFT file as written by 
        ASTROCLK (FA.DFT is the same except the Starting Date would read 
        "890328" rather than "19890328"):

                       Starting Date =   19890328.005806
                       Latitude      =         38.150000
                       Longitude     =       -120.566667
                       Time Step     =          1.0000
                       Num of Pos'ns =          1.0
                       Time Zone     =         -8.0

             Use F1 after starting the program to adjust these parameters 
        if desired. See the User's Guide for each program for more 
        information on operation and features. Upon exit from ICE or FA 
        (using F10), ASTROCLK automatically resumes normal operation. 
        Operation of ASTROCLK with the USNO programs has been tested with 
        ICE Beta (test) version 0.50 and with FA versions 2.11.88 and 
        2.11.89.
             If ICE has been selected (using ALT-F10), pressing ALT-F9 
        will automatically execute the ephemeris provided the current 
        date falls within ICE's time span and the proper ICE data files 
        are available. ICE may be used for any date from December 21, 
        1800 through June 7, 2049 inclusive. An error message is 
        displayed if the date falls outside these limits and ICE will not 
        be executed. The ICE ephemeris data files, EPH01.DAT through 
        EPH24.DAT, cover approximately 4000 days each; EPH18.DAT and 
        EPH19.DAT are sufficient for dates from about 1980 through 2000.
             If FA has been selected (using ALT-F10), pressing ALT-F9 
        will automatically execute the Floppy Almanac if the current 
        ASTROCLK date falls within the years 1988 through 1999. An error 
        message is displayed if the date falls outside these limits and 
        FA will not be executed. (NOTE: ASTROCLK allows the use of FA88 
        for the last 15 days of December, 1987 and of FA99 for the first 
        15 days of January, 2000.) The proper Floppy Almanac program 
        (FA88.EXE through FA99.EXE) must be present in the ASTROCLK 
        directory or the Floppy Almanac drive and path must have been set 
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 94


        using ALT-F10, SETTING PROGRAM OPTIONS. 
             ASTROCLK assumes that neither ICE nor FA is present when it 
        is first started. Use ALT-F10 to first select the USNO ephemeris 
        program you desire, then to set the drive and/or path where the 
        program and its data files may be found. If the drive and/or path 
        for the selected ephemeris program is not set or is set 
        incorrectly, the ephemeris may fail to execute or it may warn the 
        user that it has used its internal default files. The default 
        selection for ASTROCLK is that both USNO programs are disabled 
        and ALT-F9 will have no effect.
             NOTE: ASTROCLK remains in memory while ICE or FA is 
        executing; systems with less than 640K of main memory or which 
        have large Terminate and Stay Resident (TSR) programs active may 
        have insufficient memory for this feature. Also for this reason, 
        ICE and FA cannot be executed from ASTROCLK when using the 
        QuickBASIC interpreter rather than the complied program.
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 95



        USNO STANDARD NAVIGATIONAL STARS

             The U. S. Naval Observatory (USNO) has designated 57 stars 
        as Standard Navigational Stars and publishes their coordinates 
        (along with those of other important stars) in a number of 
        their publications including the Almanac for Computers. For 
        convenience, I have added Polaris to the USNO list as number 
        zero. Throughout this text, the phrase "Standard Navigational 
        Stars" will mean the 57 USNO stars plus Polaris. The stars are 
        listed by Standard Navigational Star Number, Bayer Designation, 
        Proper or Common Name, Right Ascension (RA, hours), and 
        Declination (DEC, degrees). The Bayer designation consists of two 
        parts: a Greek letter, such as Alpha, to designate the particular 
        star in a constellation and usually in descending order of 
        brightness; and the name of the constellation in the Latin 
        genitive (possessive) case, such as Ursae Minoris and meaning "of 
        Ursa Minor". The names of the 88 constellations are always given 
        in Latin regardless of the origin of the name. Most of the common 
        names for stars are inherited from Arabic (the scientists and 
        mathematicians in North Africa being the conduit for much of our 
        knowledge of ancient astronomy and astronomers), with a few from 
        Greek and other languages. For an explanation and a listing of 
        constellation names, see the following section CONSTELLATIONS AND 
        NAMES.
             The actual star data has been extracted from the USNO Floppy 
        Almanac 1988, Version 2.11.88, file STAR1.CAT, and is for Epoch 
        J2000.0. Not shown in the table below but included within the 
        program are constants for adjusting the data for proper motion. 
        The data represent the "mean place" of the star, described by 
        USNO in the Almanac for Computers 1988 as "a fundamental 
        reference point with no simple geometric or observational 
        significance. The apparent place of a star is the geocentric 
        position, referred to the true equinox and equator of date, at 
        which the star is observed. Thus, the apparent place is the 
        position needed for navigation, calibration of telescope setting 
        circles, computation of transit time, etc." Star catalogs with 
        earlier epochs, such as B1950.0, use "mean catalog place" which 
        has a slightly different meaning.

        #   Bayer Designation and Name              RA           DEC
        ----------------------------------------------------------------
        0   Alpha Ursae Minoris, Polaris       2.530195556  89.264088889
                                                                        
        1   Alpha Andromedae, Alpheratz         .139795833  29.090438889
        2   Alpha Phoenicis, Ankaa              .438063889 -42.306058333
        3   Alpha Cassiopeiae, Schedar          .675125000  56.537350000
        4   Beta Ceti, Diphda/Deneb Kaitos      .726492222 -17.986616667
        5   Alpha Eridani, Achernar            1.628570000 -57.236716667
        6   Alpha Arietis, Hamal               2.119556389  23.462405556
        7   Theta1 Eridani, Acamar             2.971026667 -40.304713889
        8   Alpha Ceti, Menkar                 3.037992500   4.089702778
        9   Alpha Persei,Mirfak                3.405379167  49.861205556
        10  Alpha Tauri, Aldebaran             4.598676944  16.509275000
        11  Beta Orionis, Rigel                5.242296667  -8.201661111
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 96


        #   Bayer Designation and Name              RA           DEC
        ----------------------------------------------------------------
        12  Alpha Aurigae, Capella             5.278153611  45.998027778
        13  Gamma Orionis, Bellatrix           5.418849167   6.349650000
        14  Beta Tauri, Elnath                 5.438197500  28.607408333
        15  Epsilon Orionis, Alnilam           5.603558056  -1.201950000
        16  Alpha Orionis, Betelgeuse          5.919529722   7.407041667
        17  Alpha Carinae, Canopus             6.399199722 -52.695694444
        18  Alpha Canis Majoris, Sirius        6.752464167 -16.716108333
        19  Epsilon Canis Majoris, Adhara      6.977096667 -28.972083333
        20  Alpha Canis Minoris, Procyon       7.655031389   5.225016667
        21  Beta Geminorum, Pollux             7.755262778  28.026183333
        22  Epsilon Carinae, Avior             8.375231389 -59.509586111
        23  Lambda Velae, Suhail               9.133271111 -43.432605556
        24  Beta Carinae, Miaplacidus          9.219988056 -69.717208333
        25  Alpha Hydrae, Alphard              9.459790833  -8.658652778
        26  Alpha Leonis, Regulus             10.139531944  11.967191667
        27  Alpha Ursae Majoris, Dubhe        11.062129444  61.750894444
        28  Beta Leonis, Denebola             11.817661111  14.572041667
        29  Gamma Corvi, Gienah               12.263435000 -17.541936111
        30  Alpha1 Crucis, Acrux              12.443297500 -63.099050000
        31  Gamma Crucis, Gacrux              12.519424722 -57.113194444
        32  Epsilon Ursae Majoris, Alioth     12.900485556  55.959852778
        33  Alpha Virginis, Spica             13.419885278 -11.161308333
        34  Eta Ursae Majoris, Alkaid         13.792342778  49.313319444
        35  Beta Centauri, Hadar              14.063724444 -60.372997222
        36  Theta Centauri, Menkent           14.111375278 -36.370008333
        37  Alpha Bootis, Arcturus            14.261021389  19.182419444
        38  Alpha Centauri A, Rigil Kentaurus 14.659968056 -60.835400000
        39  Alpha2 Librae, Zubenelgenubi      14.847975833 -16.041783333
        40  Beta Ursae Minoris, Kochab *      14.845096111  74.155494444
        41  Alpha Coronae Borealis, Alphecca  15.578132222  26.714705556
        42  Alpha Scorpii A, Antares          16.490121944 -26.431986111
        43  Alpha Triangulii, Atria           16.811074722 -69.027727778
        44  Eta Ophiuchi, Sabik               17.172966944 -15.724919440
        45  Lambda Scorpii, Shaula            17.560148333 -37.103811111
        46  Alpha Ophiuchi, Rasalhague        17.582243333  12.560038889
        47  Gamma Draconis, Eltanin           17.943435278  51.488947222
        48  Epsilon Sagittarii, Kaus Aust.    18.402868611 -34.384647222
        49  Alpha Lyrae, Vega                 18.615647778  38.783658333
        50  Sigma Saggittarii, Nunki          18.921090000 -26.296730556
        51  Alpha Aquilae, Altair             19.846389444   8.868341667
        52  Alpha Pavonis, Peacock            20.427458889 -56.735105556
        53  Alpha Cygni, Deneb                20.690532500  45.280363889
        54  Epsilon Pegasi, Enif              21.736434444   9.874977778
        55  Alpha Gruis, Al Na'ir             22.137222222 -46.960997222
        56  Alpha Piscis Austrini, Formalhaut 22.960848611 -29.622250000
        57  Alpha Pegasi, Markab              23.079349444  15.205250000
                                               
        *  The Right Ascension for Beta Ursae Minoris, #40, appears in 
        error but is correct. The USNO J1988.5 list was in strict 
        descending order of SHA (Sidereal Hour Angle, directly related to 
        RA) but proper motion and precession changes to J2000.0 have 
        changed the RA. To avoid possible confusion, I have retained the 
        original USNO order and numbering (Almanac for Computers, 1988).
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 97



        CONSTELLATIONS AND NAMES

             One of my early "novice" problems when trying to identify a 
        star or constellation was to learn the names of the various 
        constellations and their standard 3-letter IAU abbreviations. 
        Some were easy to guess but others were less obvious and it was 
        some time before I discovered a reference with the proper 
        information. There are still a few that I have not yet memorized.
             I have divided the list which follows into three sections, 
        Northern, Zodiacal, and Southern. The Northern and Southern 
        designations correspond roughly to the location of the 
        constellations with respect to the celestial equator. The twelve 
        constellations of the zodiac, of course, are closely linked with 
        astrology, a "science" once considered a part of astronomy, and 
        span a band of approximately eight degrees on either side of the 
        ecliptic following the course of the Sun through the heavens. The 
        names marked with an asterisk are those known to the Egyptian 
        astronomer Ptolemy and, for the most part, the ancient Greeks; 
        many of these names have survived essentially unchanged for two 
        thousand years and more although all 88 constellations are now 
        known by the Latin version of their names, whatever the origin.
             To the ancients, and continuing almost to modern times, the 
        constellations were more or less casual groups of stars usually 
        clustered around one of the brighter stars easily visible to the 
        naked eye. Descriptions of the ancient Greek constellations are 
        found in the poetry of Homer (9th century B.C.) and Aratus (3rd 
        century B.C.). Ptolemy (2nd century A.D.) cataloged about 1022 
        stars, divided into 48 different constellations, that could be 
        seen from Alexandria. His chief work, the Almagest, remained the 
        definitive authority until the European voyages of discovery in 
        the sixteenth century brought navigators into Southern latitudes. 
        The first star atlas, published by Johann Bayer in 1603, employed 
        a method of identification still in use today and added 12 new 
        Southern constellations.
             During the three hundred plus years which have followed 
        Bayer, more constellations have been added to the list, old 
        constellations have been split into several new groupings, and 
        new names have been adopted or proposed. Some of these changes 
        stuck, some did not. Since about 1750, no changes to the 
        constellation names have been accepted except that since about 
        the mid-1800's Ptolemy's constellation Argo Navis (Argo the Ship) 
        has usually been divided into three parts representing the keel 
        (Carina), the stern (Puppis), and the sails (Vela). The compass 
        (Pyxis) is also sometimes considered part of the original Argo 
        Navis.
             With the advent of the telescope, many more stars were 
        visible and the practice of naming and cataloging stars according 
        to the constellation in which they appeared continued. 
        Unfortunately, the boundaries of the constellations were not well 
        defined and there was occasional confusion. The boundary problems 
        were codified in 1930 when the International Astronomical Union 
        (IAU) agreed upon precise definitions. The new divisions were 
        drawn along lines of right ascension and declination for Epoch 
        1875.0 and were made to zigzag in order to retain the ancient 
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 98


        figures. One result of this new precision, however, was that a 
        few stars previously known as part of one constellation became 
        part of another. For example, one of the four stars of the Great 
        Square of Pegasus became part of the constellation Andromeda and 
        is now known as Alpha Andromedae. Because of precession since 
        1875, the boundary lines are no longer nicely aligned with the 
        coordinate scales.
             Only a few stars have a common or proper name such as 
        Polaris or Arcturus. The remaining stars, those few out of the 
        uncounted billions that have names or numbers at all, were named 
        by the individuals who cataloged them. Since there are many 
        different catalogs, stars often have multiple names, another 
        source of possible confusion and errors. One catalog often 
        includes a star or other objects with coordinates very slightly 
        different from a comparable object in another catalog, probably 
        the same object but not always. More confusion!
             Many different methods have been used to name or number 
        stars, but one of the most common is still the Bayer designation. 
        Each star in a constellation was assigned a Greek letter, usually 
        starting with the brightest (alpha), and the name of the 
        constellation was appended. The Greek letter may be followed by a 
        superscript to distinguish multiple stars. The constellation name 
        is used in the Latin genitive (possessive) case, meaning "of" or 
        "belonging to". Thus the first and brightest star of the 
        constellation Andromeda is Alpha Andromedae, and in Ursa Minor we 
        have Alpha Ursae Minoris (Polaris), and so forth. In most 
        references, however, both the Greek letter and the constellation 
        name are abbreviated.
             The first three lists show the standard IAU abbreviation, 
        Latin constellation name, Latin genitive name, and common English 
        translation for the three groups of constellations. The final 
        list gives the standard abbreviations for the letters of the 
        Greek alphabet. Using these lists, the abbreviated Bayer 
        designation of a star can easily be "decoded"; for example, OMI 
        CVN is Omicron Canum Venaticorum.

        NORTHERN CONSTELLATIONS (28)

        AND  *Andromeda            Andromedae          Andromeda
        AQL  *Aquila               Aquilae             Eagle
        AUR  *Auriga               Aurigae             Charioteer
        BOO  *Bootes               Bootis              Herdsman
        CAM   Camelopardis         Cameloparids        Giraffe
        CVN   Canes Venatici       Canum Venaticorum   Hunting Dogs
        CAS  *Cassiopeia           Cassiopeia          Cassiopeia
        CEP  *Cephus               Cephi               Cephus
        COM   Coma Berenices       Comae Berenices     Berenice's Hair
        CRB  *Corona Borealis      Coronae Borealis    Northern Crown
        CYG  *Cygnus               Cygni               Swan
        DEL  *Delphinus            Delphini            Dolphin
        DRA  *Draco                Draconis            Dragon
        EQU  *Equuleus             Equulei             Little Horse/Colt
        HER  *Hercules             Herculis            Hercules
        LAC   Lacerta              Lacertae            Lizard
        LMI   Leo Minor            Leonis Minoris      Little Lion
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 99


        LYN   Lynx                 Lyncis              Lynx
        LYR  *Lyra                 Lyrae               Harp
        OPH  *Ophiuchus            Ophiuchi            Ophiuchus
        PEG  *Pegasus              Pegasi              Pegasus
        PER  *Perseus              Persei              Perseus
        SGE  *Sagitta              Sagittae            Arrow
        SER  *Serpens              Serpentis           Serpent
        TRI  *Triangulum           Trianguli           Triangle
        UMA  *Ursa Major           Ursae Majoris       Big Bear
        UMI  *Ursa Minor           Ursae Minoris       Little Bear
        VUL  *Vulpecula            Vulpeculae          Little Fox

        CONSTELLATIONS OF THE ZODIAC (12)

        AQR  *Aquarius             Aquarii             Water Bearer
        ARI  *Aries                Arietis             Ram
        CNC  *Cancer               Cancri              Crab
        CAP  *Capricornus          Capricorni          Goat
        GEM  *Gemini               Geminorum           Twins
        LEO  *Leo                  Leonis              Lion
        LIB  *Libra                Librae              Scales
        PSC  *Pisces               Piscium             Fish
        SGR  *Sagittarius          Sagittarii          Archer
        SCO  *Scorpius             Scorpii             Scorpion
        TAU  *Taurus               Tauri               Bull
        VIR  *Virgo                Virginis            Virgin

        SOUTHERN CONSTELLATIONS (48)

        ANT   Antlia               Antilae             Pump
        APS   Apus                 Apodis              Bird of Paradise
        ARA  *Ara                  Arae                Altar
        CAE   Caelum               Caeli               Chisel
        CMA  *Canis Major          Canis Majoris       Big Dog
        CMI  *Canis Minor          Canis Minoris       Little Dog
        CAR   Carina               Carinae             Ship's Keel
        CEN  *Centaurus            Centauri            Centaur
        CET  *Cetus                Ceti                Whale
        CHA   Chamaeleon           Chamaeleonis        Chameleon
        CIR   Circinus             Circini             Compass
        COL   Columba              Columbae            Dove
        CRA  *Corona Australis     Coronae Australis   Southern Crown
        CRV  *Corvus               Corvi               Crow
        CRT  *Crater               Crateris            Cup
        CRU   Crux                 Crucis              Southern Cross
        DOR   Dorado               Doradus             Swordfish
        ERI  *Eridanus             Eridani             River Eridanus
        FOR   Fornax               Fornacis            Furnace
        GRU   Grus                 Gruis               Crane
        HOR   Horologium           Horologii           Clock
        HYA  *Hydra                Hydrae              Water Snake
        HYI   Hydrus               Hydri               Water Snake
        IND   Indus                Indi                Indian
        LEP  *Lepus                Leporis             Hare
        LUP  *Lupus                Lupi                Wolf
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 100


        MEN  *Mensa                Mensae              Table
        MIC   Microscopium         Microscopii         Microscope
        MON   Monoceros            Monocerotis         Unicorn
        MUS   Musca                Muscae              Fly
        NOR   Norma                Normae              Level
        OCT   Octans               Octantis            Octant
        ORI  *Orion                Onionis             Orion
        PAV   Pavo                 Pavonis             Peacock
        PHE   Phoenix              Phoenicis           Phoenix
        PIC   Pictor               Pictoris            Easel
        PSA   Piscis Austrinus     Picis Austrini      Southern Fish
        PUP   Puppis               Puppis              Ship's Stern
        PYX   Pyxis                Pyxidis             Ship's Compass
        RET   Reticulum            Reticulii           Net
        SCL   Sculptor             Sculptoris          Sculptor
        SCT   Scutum               Scuti               Shield
        SEX   Sextans              Sextantis           Sextant
        TEL   Telescopium          Telescopii          Telescope
        TRA   Triangulum Australe  Trianguli Australis Southern Triangle
        TUC   Tucana               Tucanae             Toucan
        VEL   Vela                 Velorum             Ship's Sails
        VOL   Volans               Volantis            Flying Fish

        GREEK LETTER ABBREVIATIONS

        ALP   Alpha              NU    Nu
        BET   Beta               XI    Xi
        GAM   Gamma              OMI   Omicron
        DEL   Delta              PI    Pi
        EPS   Epsilon            RHO   Rho
        ZET   Zeta               SIG   Sigma
        ETA   Eta                TAU   Tau
        THE   Theta              UPS   Upsilon
        IOT   Iota               PHI   Phi
        KAP   Kappa              CHI   Chi
        LAM   Lambda             PSI   Psi
        MU    Mu                 OME   Omega
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 101



        USING EXTERNAL STAR CATALOGS

             ASTROCLK stores all the data for the 57 USNO Standard 
        Navigational Stars plus Polaris internally. USNO has prepared a 
        catalog, STAR1.CAT, of 1536 bright stars (the first 57 of which 
        are the Standard Navigational Stars) in conjunction with their 
        Floppy Almanacs. This catalog is from the Fifth Fundamental 
        Catalog, FK5, with one star (Eta Ophiuchi) added. A second USNO 
        catalog, MESSIER.CAT, contains data for the 109 standard Messier 
        objects; M40 has always been "missing" but an entry with null 
        data occupies its place in the catalog. All catalog data is for 
        Epoch J2000.0.
             These two USNO catalogs have been converted to an ASCII 
        format (using the USNO program CATALOG) and combined to form 
        ASTROCLK.CAT with a total of 1645 stars and objects included. The 
        catalog is fairly large, requiring approximately 160K of disk 
        space. For those users short of space and who might wish to omit 
        the catalog from their disk, ASTROCLK will issue a warning 
        message if a search is requested and ASTROCLK.CAT cannot be 
        found; press RETURN to resume normal operation. The Messier 
        catalog is also available separately in ASCII format as 
        MESSIER.CAT.
             ASTROCLK can perform two types of catalog searches: search 
        for USNO Name or Number or, search for star closest to specified 
        RA/DEC or ALT/AZ position as selected by Function Key F5 followed 
        by F3, F4, and F5 respectively. 
             Each entry in the catalog is assigned a "catalog number" 
        corresponding to its position in the file. If you wish to examine 
        the whole file, you may print it with your favorite print utility 
        (adding sequential line numbers, if desired) or look at it with 
        your favorite editor. The names assigned by USNO follow standard 
        IAU conventions but may take a bit of getting used to for the 
        novice user.
             USNO allows up to three different 8-character names for each 
        star. In the following explanation each type of name is followed 
        by an example in parenthesis. The first name is either the Bayer 
        Designation (BET AND or ALP2 LIB) if one has been assigned to 
        that star, or the Messier Number (M 23). The second name, if any, 
        is the common name usually associated with the star (Polaris) or 
        the NGC number for the Messier object (NGC 1976). The third name 
        is the DM Number (Bonner Durchmusterang Catalogue) for the star 
        (-15 3996) or the common name for the Messier object (Orion). 
        Note that a SPACE is required between two part names. Many stars, 
        particularly those toward the end of the STAR1.CAT catalog, have 
        only the DM Number as a name and a printout of the catalog is 
        almost essential if these stars are to be used with ASTROCLK. Any 
        name field may be left blank and names have been truncated to 8 
        characters if necessary.
             Press F3 to search by name or number. After the requested 
        name or number has been entered, ASTROCLK will capitalize the 
        name and adjust the spacing if necessary to that required by the 
        catalog. ASTROCLK then locates the catalog file (ASTROCLK.CAT 
        unless another catalog has been designated using ALT-F10). If a 
        catalog number has been entered, ASTROCLK reads the corresponding 
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 102


        data immediately; when a name is entered, a search of the catalog 
        is required. Floppy disk based computer systems may notice a 
        considerable delay for stars located near the end of the catalog 
        and for searches which require testing the whole catalog. For 
        floppy disk systems and slower hard disk systems, a considerable 
        improvement in search time can be obtained if you have sufficient 
        memory and use a "RAM DISK" to store the catalog and specify the 
        new drive and name using Function Key ALT-F10.
             Searches by ALT/AZ or RA/DEC also search the entire catalog; 
        F4 is used for RA/DEC (Right Ascension and Declination), and F5 
        for ALT/AZ (Altitude and Azimuth). Pressing F4 gives the follwing 
        prompt (F5 is the same except ALTITUDE and AZIMUTH will be 
        requested):

                  SET TARGET COORDINATES

                  Search external STAR CATALOG for nearest 
                  star using Right Ascension & Declination:


                  Enter RIGHT ASCENSION (hours):
                  Enter DECLINATION (degrees):
                  Show nearby star list [Y/n]:

        Enter the coordinates as requested. Searches can be made in two 
        modes: find the 10 stars nearest to the coordinates given, or 
        find the single star nearest to the coordinates given. The search 
        mode is determined by the third prompt: "Y" (or RETURN) will find 
        10 stars and display a list of those stars; "N" will find the 
        nearest star and immediately switch to the Target Tracking 
        Display. Searches for a single star are somewhat quicker than 
        searches for 10 stars, due to the additional sorting required. 
        The following is a typical list of 10 stars (the degree symbol 
        has been omitted):

                  CAT #  Diff  RtAscension    Declination   Mag
                     49  0.04  18:36:56.33   38 47'01.16"   0.0
                    536  4.33  18:19:51.70   36 03'52.43"   4.3
                   1050  5.30  18:15:38.79   42 09'33.61"   5.6
                    208  6.00  18:50:04.80   33 21'45.65"   3.5
                    390  6.23  18:55:20.11   43 56'46.00"   4.0
                    894  6.57  19:07:18.12   36 06'00.61"   5.3
                   1593  6.63  18:53:36.00   33 02'00.00"   0.0
                   1475  7.49  18:33:47.66   46 13'09.02"   6.7
                    193  7.52  18:58:56.61   32 41'22.42"   3.2
                    585  7.73  19:16:22.10   38 08'01.46"   4.4
                  Press RETURN for #49 or enter CAT #:

             The first column gives the catalog number for each star. The 
        stars on the list are displayed in order of increasing angular 
        separation (in degrees) from the requested coordinates, given in 
        the second column. Only stars with a declination within 10 
        degrees of that given will be displayed. The remaining columns 
        are the Right Ascension, Declination, and Magnitude. This display 
        was prepared using the standard catalog, ASTROCLK.CAT, which 
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 103


        includes the 109 Messier objects at the end of the file and have 
        magnitudes of 0.0. Note star #1593 in the sample above; this is a 
        Messier object rather than a true star. Note also that all 
        searches are made using the "raw" catalog data, in this case 
        J2000.0 Mean Place. The first 57 stars in ASTROCLK.CAT are the 
        standard USNO Navigational Stars, identical to the ASTROCLK 
        internal star database.
             Press RETURN to select the first star in the list, #49 
        (Vega) in the sample, or enter the catalog number of another 
        star (which does not necessarily have to appear on the list). The 
        data for the selected star will be displayed in the Target 
        Tracking Display.
             The message "SEARCHING ..." is displayed at the upper right 
        and the on-screen clocks are stopped during searchs. Once 
        started, a search may be cancelled by pressing SPACE BAR. When 
        the requested star has been selected, its catalog number 
        (prefixed by the letter "C" to indicate "Catalog") and all valid 
        names are displayed in the Tracking Display title, the star data 
        is read from the file, and the coordinates are displayed as with 
        internal star data. If a requested star cannot be found, ASTROCLK 
        displays a warning message; press RETURN to resume normal 
        operation.
             For those interested in the technical details, ASTROCLK 
        expects the standard USNO ASCII catalog format of 96 characters 
        plus CR/LF per record as described in The Floppy Almanac User's 
        Guide, 2nd Edition, Appendix A. Provided the exact format is 
        maintained, the user may edit the catalog file or prepare a new 
        one. The following field definitions are extracted from that 
        appendix:

        Field     Field
        Position  Format    Contents                      Units
        ----------------------------------------------------------------
         1- 8     A8        Name1, left justified         -----
         9-16     A8        Name2, left justified         -----
        17-24     A8        Name3, left justified         -----
        25-38     F14.10    J2000.0 Right Ascension       hours
        39-52     F14.10    J2000.0 Declination           degrees
        53-62     F10.4     J2000.0 Proper Motion in RA   sec/J Cent*
        63-72     F10.4     J2000.0 Proper Motion in DEC  arcsec/J Cent*
        73-80     F8.4      Parallax                      arcsec
        81-88     F8.4      Radial Velocity               km/sec
        89-96     F8.4      Visual Magnitude (or flux)    mag, Jy
        97-98     CR/LF     Carriage Return + Line Feed

        * Proper motion is given in seconds (RA) or arcseconds (DEC) per 
        Julian Century of 36525 days.
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 104



        PRECESSION AND STELLAR MOTION

             The Earth's pole of rotation is tilted approximately 23 
        degrees 27 minutes from the plane of the ecliptic, that plane 
        which describes the Earth's orbit about the Sun. Rather than 
        constantly pointing to some fixed point in the heavens, however, 
        the gravitational influence of the Moon, Sun, and to a far lesser 
        extent the planets, cause the Earth to "wobble" slightly and the 
        pole describes a small circle with a period of about 25,770 
        years. This phenomena is known as lunisolar precession. A number 
        of other phenomena, such as nutation, also contribute to a lesser 
        extent to changes in the orientation of the Earth relative to the 
        plane of the ecliptic.
             One of the by-products of precession is that Polaris, whose 
        proper name is Alpha Ursae Minoris, has not always been the pole 
        star. In ancient times Beta Ursae Minoris, (whose Arabic name 
        Kochab derives from the words "pole star", about 1,200 B.C.), 
        Alpha Draconis (about 3,000 B.C.) and Vega (about 13,000 B.C. and 
        again in about 13,000 A.D.) have been nearer to the true pole 
        than Polaris. Polaris will actually be closest to the true pole 
        in about the year 2,102 A.D. Some 25,000 years from now, Polaris 
        will again be the pole star as the cycle continues.
             Another by-product of precession is that the standard 
        celestial coordinate system, using units of right ascension and 
        declination, changes gradually. The origin (0,0) of these 
        coordinates is the point on the ecliptic of the vernal equinox, 
        the intersection of the equator and the plane of the ecliptic. 
        This is commonly known as "The First Point of Aries", but over 
        the centuries since it acquired its name precession has caused it 
        to move out of that constellation and into the constellation 
        Pisces.
             Time standards and terms of reference have also changed 
        considerably over the last fifty years adding to the possible 
        confusion. Better technology and demands for greater precision by 
        science and industry have been the driving causes. Over the past 
        decade or so new standards of time measurement and reference have 
        been adopted by the International Astronomical Union, the 
        governing body for all astronomical measurements.
             Because of these changes and in order to provide a 
        consistent standard frame of reference, astronomers select an 
        "epoch", usually every 50 years, and base all of their 
        measurements against that standard point in time. Until recently, 
        the standard reference epoch has been 1950, now usually written 
        as B1950.0 (for Besselian epoch, another story related to the 
        time standard changes). Most references and publications have now 
        switched to the new standard epoch, J2000.0 (Julian epoch). 
        References requiring very high precision (such as the USNO 
        Almanacs) or calculated positions of the planets often use the 
        "equator and equinox of date", meaning the present epoch; in mid-
        1988, for example, that is J1988.5.
             When looking up the coordinates for a star or other object, 
        an astronomer must also note the epoch as well as the coordinates 
        themselves. If the epoch is different from that used for aligning 
        his instruments and/or is different from other objects to be 
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 105


        viewed, the data should be "precessed" or adjusted to account for 
        precession. The vernal equinox moves westward approximately 50 
        seconds of arc per year. The calculation of precession is 
        relatively complex and many writers choose to use an 
        approximation method which is sufficiently accurate only for 
        casual astronomical viewing or over very short time periods.
             Unfortunately, a computer program such as ASTROCLK can be 
        used to cycle back and forth between epochs almost at will. The 
        "quick and dirty" approximations of the simpler methods can yield 
        cumulative errors that soon become unacceptable. A more rigorous 
        calculation for precession, the Improved IAU System, was adopted 
        in 1984; it is this method that is used in ASTROCLK. An earlier 
        method, developed in 1897 and published in 1906 by the American 
        astronomer Simon Newcomb, was used in earlier versions of 
        ASTROCLK and yielded comparable results. (Similar expressions 
        were published in Germany in 1830 by F. W. Bessel and 
        subsequently by others.) Although these calculations take 
        considerably more computer processing time, they produce errors 
        that are about two orders of magnitude less than typical 
        approximations. ASTROCLK also always resets the internal star 
        data to Epoch J2000.0 prior to precession calculations so as to 
        avoid cumulative errors. Since manually entered data cannot be 
        "reset"in this way, repetitive cycling from one epoch to another 
        will yield modest cumulative errors. The formulas employed are 
        described in the main text and the supplement of the 1984 
        Astronomical Almanac. When the internal data is precessed to 
        J1988.5, the results are in good agreement with USNO data for 
        that epoch given in Almanac for Computers 1988, pages E2 through 
        E10.
             Further complicating the picture is the fact that the Earth 
        and the stars themselves are not stationary. The Earth's orbit 
        about the Sun causes parallax for nearby stars but the effect is 
        periodic and relatively small; it has been ignored for this 
        version of ASTROCLK. The changing position of the stars is known 
        as "proper motion". While stellar motion is extremely difficult 
        to measure for distant stars, proper motion data has been 
        collected on a large number of stars (including those used in 
        this program). ASTROCLK calculates the proper motion of stars 
        prior to calculating the effects of precession. The effects of 
        nutation and annual aberration are also included.
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 106



        DATES AND THE GREGORIAN CALENDAR

             For convenience and standardization, many astronomical 
        calculations reference a unique point in time known as the 
        "Fundamental Epoch". This is defined as 12:00:00 at the Prime
        Meridian (Greenwich) on 1 January, -4713 (often written as -4713 
        JAN 1.5). Note that the day starts at noon in conformance with 
        astronomical convention and corresponds to the time at which 
        accurate sun sights could be made. The time elapsed since then is 
        measured in units of days and the current date and time may thus 
        be expressed as a single number, UTC JULIAN DATE (usually known 
        simply as the Julian Date or JD). The number of days appears to 
        the left of the decimal point, and the time is represented by a 
        decimal fraction of a day. Years "before Christ" or "B.C." (but 
        not prior to 1 January 4713 B.C. for this program) are given as 
        negative numbers with no zero year. The Julian Date should not be 
        confused with the Day-of-the-Year, the number of days elapsed 
        during the current year, which is popularly and incorrectly also 
        sometimes referred to as the Julian Date.
             However, astronomers delight (it would seem) in changing 
        their units of measure at depressingly frequent intervals; 
        multiple systems are sometimes in use simultaneously. Readers are 
        cautioned that some authors, especially in older works, include a 
        zero year in their calendars; using that scheme, 4713 B.C. 
        becomes year -4712. In the references I have used, for example, 
        Meeus prefers the zero year method while Duffet-Smith uses the 
        same method as ASTROCLK with no zero year; see BIBLIOGRAPHY for 
        references. I find the no zero year method far more convenient 
        and less confusing: years B.C have the same number and are simply 
        prefixed by a negative sign. Not all astronomers would agree.
             To add to the potential confusion, prior to 1925 astronomers 
        considered that each calendar day commenced at NOON, agreeing 
        with the standard astronomical day numbering convention but in 
        conflict with civil practice. Modern astronomical convention, 
        however, begins the calendar day at MIDNIGHT, the same as the 
        civil calendar, and the practice is to apply the convention to 
        all dates -- even those prior to 1925. Care must therefore be 
        taken when interpreting older dates and times to ensure that the 
        date conventions employed are understood and converted if 
        necessary. This in addition to the various calendars in use! All 
        in all, a good argument for the use of Julian Dates which are 
        completely unambiguous -- if you ignore Julian Ephemeris Dates!
             On an historical note, the Julian Date has been in use for 
        centuries by astronomers, geophysicists, chronologers, and others 
        who needed to have an unambiguous dating system based upon 
        continuing day counts. In fact, the the "Julian" part of Julian 
        Date has nothing to do with the Julian Calendar introduced by 
        Julius Caesar in 46 B.C. The Julian Date was so named by the 
        mathematician Scaliger when he introduced this method of day 
        counting in 1582, allegedly after his father, Julius. True or 
        not, the name has stayed with us regardless of its origins.
             The starting date of January 1, -4713, for the Julian Date 
        was based upon the time it takes from one coincidence to the next 
        of a solar cycle (28 years), a lunar cycle (19 years), and the 
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 107


        Roman Indiction (a Roman tax cycle of 15 years imposed by the 
        Emperor Diocletion during the period 284-305 A.D. and whose 
        connection to astronomy completely escapes me). However, the 
        product of those three cycle periods yields 7980 years, the 
        Julian Period. That period is of interest only with respect to 
        the selection of the starting time and date for the day counting 
        method, at which time all of the cycles, counted backwards, were 
        in coincidence. The real purpose of selecting such a date, of 
        course, was that it be distant enough in time that the resulting 
        day numbers would always be positive for events of interst.
             Not too surprisingly, most historical dates were not 
        recorded using their Julian Date; for ancient dates, of course, 
        the Julian Calendar hadn't been invented yet, and for more recent 
        dates it was not (and still is not) in popular use. Enter the 
        calendar in all its varieties. Calendars have long been an 
        important part of almost every known civilization, especially 
        those dependent upon agriculture. Being able to predict the time 
        for planting and harvest was essential if the community was to 
        continue to have an adequate food supply. Stonehenge in England, 
        for example, is generally acknowledged to have been an 
        astronomical observatory of sorts, used to predict the equinoxes 
        and probably was also used for various religious and social 
        events as well. Except for the stones themselves and their 
        careful alignment, little else is known of the society they 
        represent. But, given the massive effort that was involved in its 
        construction, the importance of the calendar and the prediction 
        of the seasons to its builders is clear. The ancient Egyptians 
        watched Sirius (known to them as Sothis) for its appearance close 
        to the Sun in the morning sky, the First Heliacal Rising. This 
        marked the start of their 365 day calendar and coincided with the 
        rising of the Nile and the fertilizing of the Egyptian plain by 
        her waters. Almost without exception, every civilization of note 
        used a calendar, although their accuracy varied considerably.
             The calendar having the most direct bearing on our present 
        system is the Roman Republican Calendar of ancient Rome and her 
        empire. Although the year started on the first of what is now 
        March (after Mars, the planet and also the God of War), the basic 
        structure of the calendar is quite similar to that in use today. 
        Its immediate successor, the Julian Calendar, came about as a 
        result of centuries of "adjustments" (more properly called 
        intercalation, the addition of extra days in the calendar) to 
        accommodate social, political, religious or other goals. Rulers 
        and court astronomers would insert or delete days seemingly 
        almost at random.
             By the time of Julius Caesar, the Roman Republican Calendar 
        was more than two months out of synchronization with the seasons 
        and nothing was happening when it was supposed to. Spring was 
        occurring in winter months, winter in the fall, and so forth. 
        Caesar's Greek astronomer, Sosigenes, (inherited from Cleopatra 
        of Ptolemaic Egypt) figured out what should be done: a "final 
        adjustment" of 67 days would be made and the (then) last month of 
        the year, February, would be given an extra day every four years. 
        As a consequence, the year 46 B.C. became known as "The Year of 
        Confusion" and is the longest year on record, some 432 days. 
        Although the Julian Calendar was not consistently used for civil 
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 108


        purposes until 8 A.D., the need for a "standard" dating method 
        has led chronologers to extrapolate the Julian Calendar back in 
        time, calling it the Julian Proleptic Calendar to distinguish it 
        from other calendars in use.
             Under the Julian Calendar, therefore, each year contained 
        365 days unless the year was divisible by four, in which case the 
        year contained 366 days. The additional day was inserted at the 
        end of February. The average length of the Julian year was thus 
        365.25 days. Given the relatively simple instruments and 
        mathematics of the time, the calendar that was devised then was 
        remarkably accurate and it continued in force until 1582.
             Unfortunately, however, the tropical year (the time required 
        for the Earth to make one complete orbit around the sun and the 
        fundamental unit of our calendar) is actually 365.242199 days 
        rather than the 365.25 days used for the Julian Calendar. By 1582 
        that relatively small annual error, 0.007801 days or about 11 
        minutes 14 seconds, had accumulated and the calendar was again 
        out of step with the seasons, this time by some ten days. 
        Following a number of false starts by prior pontiffs, Pope 
        Gregory XIII ordered the use of an improved calendar, now known 
        as the Gregorian Calendar and in general civil use throughout 
        most of the world (sometimes in conjunction with an older, 
        religious calendar).
             The new calendar directed that the dates 5 October through 
        14 October 1582 inclusive were to be abolished and that 
        henceforth all centennial years, years ending in "00", be leap 
        years only if divisible by 400. Therefore, 1700, 1800 and 1900 
        would NOT be Leap Years under the new calendar; 1600 and 2000 
        would still be Leap Years as before. Using the new Gregorian 
        method, 400 civil years contained 400 * 365 + 100 - 3 or 146097 
        days and the average length of the civil year was 365.2425 days 
        for a remaining error of approximately 0.0003 days. After all 
        that fuss and bother, the calendar is still some 26 seconds per 
        year too long, but it will take almost 3,000 more years, or until 
        about 4882 AD, for us to accumulate a one day error.
             Some references (Encyclopaedia Britannica, for one) assert 
        that a further adjustment has been proposed to the Gregorian 
        Calendar: eliminate the Leap Year in years evenly divisible by 
        4000. This would reduce the error even further and it would be 
        some 20,000 years before a one day error would be accumulated! 
        Perhaps because the year 4000 A.D. is yet some time distant and 
        much may happen between then and now, most authors do not mention 
        or calculate the 4000 year adjustment. Given the lack of 
        unanimity in my sources, ASTROCLK also does not use the 4000 year 
        cycle in its calculation of future dates and the adjustment does 
        not apply to past dates.
             The Gregorian Calendar, or the "New Style" as it was then 
        called, was of course immediately adopted by the catholic 
        countries: France, Portugal, Spain, and Italy as well as by 
        Denmark and the Netherlands. Catholic Scotland adopted it in 1600 
        but since England did not, this caused considerable confusion 
        between the two countries. The German Protestants waited 120 
        years or so, and it was not until 1752 that England and her 
        colonies finally adopted the new calendar. By then the error had 
        risen to 11 days (1700 was a Leap Year under the Julian Calendar 
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 109


        and was not under the Gregorian Calendar), and 3 September 
        through 13 September 1752 inclusive were abolished, accompanied 
        by much confusion and widespread disturbances. Even after formal 
        adoption of the new calendar, many English communities still 
        clung to the "Old Style" and the legend "O.S." may still be seen 
        on old tombstones. Following the French Revolution, France 
        abandoned the Gregorian Calendar for a new calendar beginning on 
        September 22, 1792; its use was short lived, however, and France 
        returned to the fold on January 1, 1806 and resumed use of the 
        Gregorian Calendar.
             Good news travels slowly, it seems. Japan adopted the "new" 
        calendar in 1873 and China followed in 1911. But it wasn't until 
        the Bolsheviks came to power in 1917 and Pope Gregory had been 
        dead for more than 300 years that the Russians changed their 
        calendar. By then the error had further increased, to 13 days, 
        still the difference in 1988. (Halloween, October 31, 1988 is 
        October 18, 1988 using the Julian Calendar.) Not to be outdone by 
        the West, however, the Russians adopted the Greek Orthodox 
        calendar rule for a centennial year such that it is a leap year 
        only if, after dividing the year by 900, the remainder is either 
        200 or 600. The Soviet calendar is about five times more accurate 
        than the original Gregorian Calendar.
             Because of all of this change and confusion, ASTROCLK simply 
        follows the original Gregorian Calendar as adopted in October of 
        1582 as the default calendar method. Dates prior to October of 
        1582 (and prior to 46 B.C. as well) are based upon the Julian 
        Calendar. However, as an option, ASTROCLK can use the British 
        date for the adoption of the Gregorian calendar in 1752, or it 
        can use the strict Julian calendar for all dates. Local dates in 
        other countries which did not immediately adopt the Gregorian 
        calendar must be adjusted for the period from October, 1582 (or 
        September, 1752 if that calendar is selected) through the date of 
        adoption. Dates for countries which use or used other calendars 
        are left as an exercise for the reader.
             By setting ASTROCLK's internal CALENDAR FLAG (see SETTING 
        PROGRAM OPTIONS for details), dates may easily be converted 
        between the three calendar conventions. For example, select the 
        Perpetual Calendar (Display Mode 6), set the desired date (using 
        F3), then observe the date and calendar differences as you change 
        from one calendar convention to another (using ALT-F10). Because 
        ASTROCLK monitors the computer's internal clock (which includes 
        the current date), real time operation using the Julian Calendar 
        is not allowed; the situation is confusing enough without 
        ASTROCLK having to ignore part of the computer's time data. 
        Naturally, all three calendar conventions show the same date 
        prior to October of 1582; after September of 1752, both Gregorian 
        calendars are in synchronization and may be operated in real 
        time.
             Quite oblivious to religion, politics and computers, the 
        Julian days have been spinning right along since 4713 B.C. They 
        have served their purpose well for astronomers and other 
        scientists. However, the true Julian Date (JD) is a rather large 
        number (4 February 1988 = 2447195.5) and the precision of some 
        calculators and micro-computer software is inadequate to the 
        task. Fortunately for those calculators and computers, the 
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 110


        International Astronomical Union (IAU) at their Dublin meeting in 
        1955 adopted a special Dublin Julian Date (DJD) starting at noon 
        on January 0, 1900 or 1900 January 0.5 and which may be defined 
        as DJD=JD-2415020. The date can be confusing, however, since 
        there obviously is no 0th of January; the selected date is a 
        matter of astronomical convenience and actually is the same as 
        1899 December 31.5. The resulting number has only five digits to 
        the left of the decimal point (3 February 1988 = 32175.5). Both 
        methods, JD and DJD, are used internally by various ASTROCLK 
        routines. Note that the Julian Date cycles at noon rather than at 
        midnight as is the more usual practice for civil time; this can 
        easily cause confusion in calculations.
             The Modified Julian Date (MJD) is a third method of 
        recording the Julian Date which also only requires five digits 
        (3 February 1988 = 47195.0) and is sufficient for most modern 
        purposes. Introduced in the late 1950's by space scientists, it 
        is defined as MJD=JD-2400000.5. An interesting side effect of 
        this purely mathematical definition is the rather unlikely 
        starting point of midnight (00:00:00 UT) on 17 November, 1858. 
        Like DJD above, this method reduces the precision required for 
        calculations but it also subtracts a half day so that the day 
        starts at midnight in conformance with civil time reckoning. 
        Although still mathematically accurate, MJD loses its advantage 
        of lower precision requirements if used prior to about 1600 A.D. 
        It is frequently used as a substitute for the true Julian Day by 
        many scientific organizations and publications. The MJD has been 
        sanctioned by various international commissions such as the 
        International Astronomical Union (IAU), the Consultative 
        Committee for Radio (CCIR), the advisory committee to the 
        International Telecommunications Union (ITU), and others who 
        recommend it as a decimal day count which is independent of the 
        civil calendar in use.
             In addition to MJD, NASA also sometimes uses what they call 
        the Truncated Modified Julian Date or TJD; it is simply MJD less 
        the first digit, or TJD=JD-2440000.5. Like MJD, the day starts at 
        midnight rather than at noon (3 February 1988 = 7195.0). The 
        range of usefulness for TJD, based upon its having fewer digits, 
        is generally restricted to the current century. Mathematically, 
        of course, it is as accurate as any of the other methods.
             Summarizing, the four standard methods of Julian day 
        counting in common use are:

                                        00:00:00 UT
                 Name   Starting Date   04 FEB 1988    Related to JD
                 ----   -------------   -----------    -------------
                   JD  -4713 JAN  1.5   2,447,195.5
                  MJD   1858 NOV 17.0      47,195.0    JD-2400000.5
                  DJD   1900 JAN  0.5      32,175.5    JD-2415020.0
                  TJD   1968 MAY 24.0       7,195.0    JD-2440000.5

             The Julian Ephemeris Date (JED) is a slightly different 
        method of day counting based upon Ephemeris Time (ET, used pre-
        1984) and Terrestrial Dynamical Time (TDT, used post-1983); JED
        differs from the conventional Julian Date (JD) by a matter of 
        some seconds in this century (extrapolated to be 56.3 seconds in 
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 111


        1989, according to the Astronomical Almanac 1989). The actual 
        difference, called Delta T = ET/TDT-UT, is calculated well after 
        the fact using astronomical observations. For most astronomical 
        calculations, JED and JD may be used more or less interchangeably 
        unless high precision is required. However, for solar, lunar, and 
        planetary calculations, JED is usually required as an invariant 
        time system independent of the Earth's motion. Readers should use 
        care because many authors are somewhat casual on the subject and 
        may use the abbreviation "JD" to refer to either or both JD and 
        JED, and the correct usage may not be obvious.
             The Julian Epoch (JE) and Besselian Epoch (BE) are two 
        additional astronomical dating methods, generally used when lower 
        precision is required or when the phenomenae of interest change 
        slowly with time; star catalogs and planetary tables are common 
        examples. The epoch dating methods are based upon the Julian 
        Century (36525 days) and the Tropical Century (36524.2199 days) 
        respectively. Texts written prior to about 1984 will write the 
        epoch without a prefix letter and the Besselian Epoch is assumed 
        (as in B1950.0). Again, however, recent authors often neglect to 
        add the prefix even when different epoch dating methods are 
        assumed; B1950.0 and J2000.0 are frequent examples. Most recent 
        star catalogs and publications reference astronomical data to the 
        current standard epoch, J2000.0. However, NASA and many planetary 
        tables and formulae still reference the prior standard epoch, 
        B1950.0, and some current data is referenced to the equinox of 
        date (or mid-year), such as J1988.5. Conversion is often required 
        in order that all data use the same reference epoch.
             Last of all, Greenwich Sidereal Date (GSD) represents the 
        date using the sidereal day rather than the mean solar day. The 
        starting point for GSD is about 0.6 days earlier than JD but, due 
        to the shorter sidereal day, the date increases more rapidly than 
        JD; GSD is presently some 6700 days ahead of JD. I have not seen 
        it used in calculations, but the Astronomical Almanac includes 
        GSD in some of its tables.
             The following list shows the value of these eight dating 
        methods at 00:00:00 UT on 04 FEB 1988:

                   JD         2447195.500000
                  MJD           47195.000000
                  DJD           32175.500000
                  TJD            7195.000000
                  JED         2447195.500649
                   JE           J1988.091718
                   BE           B1988.092741
                  GSD         2453896.370521

             All of these dating methods are calculated by ASTROCLK and 
        used as required in its calculations. Display Mode 7, Julian Date 
        Information, displays this information except for GSD, which is 
        found using Display Mode 8, Precision Time Display #1.
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 112



        WHAT TIME IS IT?

             This is a crucial question for astronomers and navigators 
        alike and is one of the reasons the two disciplines have been so 
        closely linked from time immemorial. Of course, both are 
        interested in the stars themselves, the first for scientific 
        reasons and the second for more practical purposes. From the 
        earliest recorded history, references are found to Polaris and 
        Kochab, each the pole star at different times, and the nearby Big 
        Dipper, two of whose stars serve as a "pointer" to Polaris. Their 
        principal use was as aids to navigation, both on land and on sea. 
        Not only does Polaris indicate true North with a fair degree of 
        accuracy, but its height above the horizon represents the 
        approximate latitude of the observer, the angle down from the 
        pole or up from the Equator.
             So long as navigation was restricted to relatively confined 
        areas, such as the Mediterranean Sea, voyages stood a reasonably 
        good chance of reaching their intended destinations if the 
        navigator knew his direction and approximate speed. Polaris (and 
        later the magnetic compass, first described by an Englishman in 
        1180 but probably in use much earlier) could establish the 
        direction being traveled and the observation of speed, winds, and 
        tides could be combined with that direction to determine a ship's 
        probable course and position, a procedure known as "dead 
        reckoning". Elaborate charts covered with rhumb lines (lines 
        corresponding to various wind directions) were produced in the 
        13th century to aid the navigator in setting and plotting his 
        true course.
             But as ships ventured further and further from known 
        landmarks, it became clear that this was not enough. Knowing only 
        their latitude (North-South position) and the direction of the 
        pole star, sailors found that they were often nowhere near their 
        destination. When sailing down the West coast of Africa, the 
        Portuguese, for example, adopted the practice of sailing South to 
        the desired latitude, then sailing East for however long it took 
        until they reached their destination. Columbus used this same 
        technique on his return trips to America. To further complicate 
        matters, the carefully drawn rhumb line charts assumed a flat 
        surface; the greater the distance traveled the greater the error 
        due to the fact that the Earth is a sphere and not a plane.
             In an interesting footnote to history, the ancient Greeks 
        had concluded that the Earth was a sphere and described a more or 
        less circular orbit about the Sun -- or vice versa. Starting some 
        time around 450 B.C. give or take a few years and continuing for 
        more than 700 years, Greek astronomers proposed astronomical 
        theories and counter-theories culminating in Ptolemy's Almagest 
        in the middle of the second century AD. Erathosthenes (276-196 
        B.C.) made the first fairly accurate determination of the Earth's 
        diameter. He noticed that at Syene, Egypt (near present Aswan), 
        sunlight struck the bottom of a vertical well at noon. At the 
        same time and date in Alexandria, 5000 stadia north of Syene, he 
        noticed that the Sun's rays made an angle with the vertical of 
        about 1/50 of a circle (about 7 degrees). He therefore calculated 
        that the Earth's circumference must be 50 * 5000 or 250,000 
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 113


        stadia. Unfortunately, there were several stadia (the Greek unit 
        of length) in use and, depending upon which one you assume 
        Erasthones was using, his calculation could have been accurate to 
        within 1 percent or 20 percent too large.
             Somewhere along the way this important bit of information, 
        the spherical Earth, was lost, misplaced or simply not believed 
        and by the middle ages many people in Europe were certain that 
        the Earth was flat. I'm not convinced that any of the great 
        navigators of the time were quite so naive and ill informed, but 
        maps drawn with that assumption in mind became less and less 
        accurate as voyages covered greater distances.
             But the Earth, of course, really is a sphere (an oblate 
        spheroid, actually) and what was needed were maps based upon 
        latitude and longitude, not simply bearings. In 1569 Gerardus 
        Mercator published his world map based on a "true projection 
        suitable for navigation" and within a few decades navigators had 
        maps and tables which would permit the approximate determination 
        of position. The Mercator Projection is still used today for many 
        types of maps. Unfortunately, the maps of the day were not always 
        accurate, especially for unexplored areas of the globe, and even 
        when they were accurate everything depended upon being able to 
        estimate longitude as well as latitude.
             The fifteenth and sixteenth centuries saw notable advances, 
        particularly in England, in the determination of longitude using 
        techniques such as lunar distances or the eclipses of the 
        satellites of Jupiter. The first astronomical ephemeris by 
        Regiomontanus was published in Nurnberg in 1474 and other 
        increasingly accurate ephemerides (tables of astronomical data) 
        useful to navigators were produced over the next two hundred 
        years. In 1675 the Royal Observatory was founded in Greenwich 
        with the specific object of providing the sailor with 
        astronomical data of the precision required for reliable 
        navigation.
             Medieval astronomers knew that the time of a lunar eclipse 
        could be used to determine the local longitude, but that wasn't 
        very handy on a day to day basis. By the sixteenth century it was 
        also recognized that longitude could be determined by noting the 
        precise time and the position of the stars. Away from a stable 
        land platform and good instruments, however, knowing the time 
        accurately was all but impossible and time was a critical factor 
        in the longitude calculations. In 1714, following a series of 
        naval disasters caused by bad navigation, the English Parliament 
        established the Board of Longitude to address the problem. The 
        Board offered a prize of 20,000 pounds sterling, a princely sum 
        in those days, to anyone who could determine longitude to an 
        accuracy of thirty miles after a voyage of six weeks. An 
        Englishman by the name of John Harrison ultimately won the prize 
        on his fourth attempt using a marine chronometer fashioned in the 
        shape of a watch. And so began the practice of determining 
        position at sea by taking timed observations of the stars and 
        planets. The Royal Greenwich Observatory, situated on the Thames 
        River downstream from London, provided essential time services to 
        the Royal Navy and merchant seamen alike, and each captain would 
        carefully set his chronometer upon departure. Small wonder that 
        chronometer was among the most carefully guarded objects on board 
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 114


        ship, for their very lives might well depend upon its continuing 
        accuracy.
             With little need for precision evident ashore, however, 
        local time was often a rather casual affair and based upon 
        apparent solar time, the time indicated by a sundial. Each town 
        or village would establish its own local time independent of its 
        neighbors. But apparent solar time is subject to considerable 
        variation as a result of the Earth's elliptical orbit and the 
        changes in the speed of rotation of the Earth. The difference 
        from day to day is relatively small, but the cumulative 
        difference can add up to about fifteen minutes over the course of 
        several months, a phenomena known as the Equation of Time. The 
        gradual improvement of clocks and watches during the seventeenth 
        century made these variations more obvious and forced the use of 
        mean solar time, apparent solar time averaged over a year, and 
        eventually caused the establishment of uniform time zones. The 
        railroads became prime movers in the push to standardize 
        timekeeping; schedules would be impossible to understand if every 
        stop used a different time convention. Most countries in Europe 
        therefore established single time zones using the time determined 
        at a single point such as Greenwich or Paris, but the United 
        States was forced by its size to adopt multiple time zones in 
        order to keep local times reasonable compared to the Sun. As 
        transportation and communication speeds continued to improve, the 
        various time zones were ultimately standardized in 1884 with 
        Greenwich selected as the Prime Meridian, and thus GMT or 
        Greenwich Mean Time became a worldwide standard. [However, until 
        1925, 0 hours GMT occured at noon rather than at midnight, 
        another source of possible confusion. The use of the designation 
        GMT has now been discontinued for the most part and replaced by 
        UTC, Coordinated Universal Time.]
             The globe was marked with 24 standard meridians spaced at 15 
        degree (one hour) intervals and the meridian at 180 degrees was 
        designated the International Date Line. Most time zones are now 
        an integral number of hours different from Greenwich, 
        corresponding to the nearest standard meridian, and a few are at 
        a half hour multiples for local convenience (India, for example). 
        However, there still remain some odd zones here and there.
             The accuracy and precision of our time measurements have 
        continued to improve as technology has advanced and in response 
        to the demands of the scientific and industrial community. 
        Traditionally, the fundamental unit of time measurement, the 
        second, was defined as 1/86,400 of a mean solar day. With the 
        improved accuracy of timekeeping came the need for a more 
        absolute standard and at the Dublin conference in 1955 the second 
        was redefined as 1/31,556,925.9747 of the tropical year as 
        measured on 1900 January 0.5, the same point selected for the 
        start of the Dublin Julian Date (DJD). This didn't last too long, 
        however, and in 1964 the International Committee on Weights and 
        Measures officially adopted the transition between two specific 
        energy levels of cesium-133 as the definition of the second with 
        the introduction of the atomic clock.
             Timekeeping has now become internationally standardized and 
        the official custodian of the world's clocks is the Bureau 
        International de l'Heure (BIH) in Paris. Here in the United 
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 115


        States time standards and observation are the responsibility of 
        the National Bureau of Standards (NBS) and the U. S. Naval 
        Observatory (USNO). In 1965, after almost three hundred years as 
        the de facto time standard in the world, the Royal Greenwich 
        Observatory was restructured into more of a pure research 
        organization and has subsequently lost interest in, and ceased 
        most support for, time and time standards.
             With the standardization and improved accuracy of our 
        timekeeping has come increased complexity. The old phrase 
        Greenwich Mean Time or GMT has now been officially discontinued 
        by most of the world, Great Britain and to a lesser extent the 
        United States (because of our close cooperation with Great 
        Britain on the Astronomical and Nautical Almanacs and related 
        works) being almost alone in continuing to use it, and then 
        primarily for navigators. Old habits die slowly, however, and 
        many people continue to use the old phrase, often unaware of the 
        change in name. GMT has generally been replaced by Coordinated 
        Universal Time, UTC, which is the time broadcast by the National 
        Bureau of Standards via WWV in Boulder, Colorado, and WWVH in 
        Honolulu, Hawaii, as well as other national radio time services. 
        For most purposes, those requiring accuracy to about one second, 
        GMT and UTC may be considered interchangeable. Individuals with a 
        military or aviation background will recognize ZULU Time, also 
        equivalent to Universal Coordinated Time.
             For scientific work requiring high precision, however, 
        things are not nearly so simple. There are now four "standard" 
        Universal Times which take into account in varying degrees the 
        various phenomena that cause changes in time measurements over 
        long periods. In addition, a number of other time systems are 
        used including International Atomic Time (TAI) and Terrestrial 
        Dynamical Time (TDT). In 1984 TDT replaced Ephemeris Time (ET) as 
        the astronomical standard of time, the time system actually used 
        by most astronomers and computed well after the fact. UTC, tied 
        to the (irregular) rotation of the Earth, is currently "slow" 
        relative to TDT by slightly less than one minute; extrapolated 
        values given in the Astronomical Almanac 1989, Page K9, are 55.8 
        seconds for 1988 and 56.3 seconds for 1989. For the present, all 
        calculations within ASTROCLK assume UT1 and ignore differences 
        with other UT time standards.
             The following simplified definitions describe the various 
        time standards in general use at the present time, or which have 
        been in common use during this century.

             A.1  U.S. Naval Observatory Atomic Time, used from January 
                  1, 1958 through December 31, 1971. ET = A.1 + 32.15 
                  seconds. Replaced by TAI (qv) on January 1, 1972.

             ET   Ephemeris Time, replaced in 1984 by Terrestrial 
                  Dynamical Time (qv).

             GAST Greenwich Apparent Sidereal Time. Greenwich Hour Angle 
                  of the true equinox of date.

             GMST Greenwich Mean Sidereal Time. Greenwich Hour Angle of 
                  the mean equinox of date.
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 116



             GMT  Greenwich Mean Time, a term now used almost exclusively 
                  in the United Kingdom and for navigation. Most modern 
                  references now use UT1 (qv) instead. Prior to 1925, 0 
                  hours GMT occured at noon rather than at midnight; care 
                  must be used when referencing older documents to take 
                  this change into account.

             TAI  International Atomic Time. The unit of TAI time is the 
                  SI (Systeme International) second. This time standard 
                  is based upon the analysis of the atomic time standards 
                  of many countries and is related to the radiation of 
                  Cesium 133. "Atomic time, in the general relativistic 
                  sense, probably keeps the proper time of a moving 
                  observer in a gravitational field." [Taff, p 102, see 
                  BIBLIOGRAPHY.] TAI was adopted as a standard on January 
                  1, 1972, replacing A.1 (USNO Atomic Time) which was 
                  used from January 1, 1958.

             TDT  Terrestrial Dynamical Time, used for astronomical 
                  ephemerides for observations from the surface of the 
                  Earth. TDT/ET = TAI + 32.184 seconds. For most 
                  purposes, ET (up to 1983 December 31) and TDT (from 
                  1984 January 1) can be regarded as a continuous time 
                  scale. In 1989, TDT is ahead of UT by approximately 
                  56.3 seconds; the difference is 56.7 seconds for 1990.

             TDB  Barycentric Dynamical Time, used for high precision 
                  astronomical ephemerides referred to the barycenter 
                  (center of mass) of the solar system. TDB never varies 
                  from TDT by more than 1.7 milliseconds and is not used 
                  by ASTROCLK. TDB was previously known as Coordinate 
                  Time.

             UT0  Classical universal time, based upon the mathematical 
                  relationship between mean solar time and mean sidereal 
                  time. Not directly used or calculated in ASTROCLK.

             UT1  UT0 corrected for precession, the polar motion of the 
                  Earth. This slow wobbling motion describes a circle 
                  about 30 feet in radius over a period of approximately 
                  25,800 years. The combined gravitational fields of the 
                  Sun and Moon acting upon the non-spherical Earth cause 
                  the direction of the Earth's rotation axis to gyrate 
                  slowly. UT1 is now the official designation for, and is 
                  the same as, Greenwich Mean Time, GMT. In program 
                  ASTROCLK, the abbreviation UT is used to mean UT1 and 
                  is used for all calculations and displays unless 
                  specifically noted otherwise. Except in the Precision 
                  Time Displays, ASTROCLK ignores the difference between 
                  UT1 and UTC, considering them identical.

             UT2  UT1 corrected for a slight (maximum seasonal difference
                  of approximately 0.035 second) periodic variation in 
                  the speed of rotation of the Earth caused by the 
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 117


                  varying distances and relative directions of the Sun 
                  and Moon which in turn continuously alter the strength 
                  and direction of the gravitational field. Not used or  
                  calculated in ASTROCLK.

             UTC  Coordinated Universal Time. UTC was originally a 
                  smoothed version of UT2 (pre-1972) and is now based 
                  directly upon TAI. On January 1, 1972 the difference 
                  between TAI and UTC was exactly 10 seconds. Since that 
                  date, adjustments of exactly one second are made as 
                  required on June 30th or December 31st in order to keep 
                  UTC and UT1 within 0.9 seconds of each other. When a 
                  change is required, the last minute of those months 
                  will have 59 or 61 seconds. UTC is the basis for most 
                  radio time services (including WWV/WWVH) and our civil 
                  and legal time systems. It is also, of course, the time 
                  signal most of us use to synchronize time-dependent 
                  equipment and (directly or indirectly) to set our 
                  clocks. As noted above, ASTROCLK generally assumes 
                  UT1=UTC unless noted otherwise; the difference is less 
                  than the setting/running errors of the average micro-
                  computer system clock.

             ZULU A distinctive phonetic acronym having no particular 
                  meaning. ZULU time is equivalent to UTC and is used in 
                  commercial avaition and by the U. S. military services 
                  in order to avoid confusion over local time zones.
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 118



        PRECISION AND ACCURACY TESTS

             A number of tests have been performed to examine the 
        precision and/or accuracy of various calculations made by 
        ASTROCLK. The principal data used for testing and comparison are 
        derived from: Astronomical Almanac 1988 and 1989 (both usually 
        refered to as AA, unless a more specific reference is required), 
        USNO Almanac for Computers, 1988 (AFC88); USNO Floppy Almanac 
        1988 and 1989 (FA generally, or FA88 and FA89 if required); 
        USNO Interactive Computer Ephemeris (ICE); Astronomical Formulae 
        for Calculators (AFC); and, Astronomy with Your Personal Computer 
        (AYPC). See BIBLIOGRAPHY for the full references. Unless noted 
        otherwise, all tests and comparisons were made using a Zenith Z-
        248 computer (IBM PC/AT compatible with 80286 processor) equipped 
        with an 80287 math coprocessor. Representative tests were 
        repeated on a Zenith Z-183 laptop (IBM PC/XT compatible with 
        80C88 processor) with or without a math coprocessor with no 
        differences observed other than execution speed.
             Strict mathematicians and scientists may complain about the 
        precision to which data is typically displayed by ASTROCLK. The 
        reader is reminded at various points in this text that the 
        displayed precision may exceed the accuracy of the data, a 
        practice which is definitely frowned upon in scientific circles, 
        but I plead special circumstances for ASTROCLK.
             First and foremost, ASTROCLK has been developed over a 
        considerable period of time, and the process still continues. The 
        accuracy of all data have been consistently improved over that 
        time, and many items have gradually been improved to the point 
        where the accuracy and the displayed precision are roughly the 
        same -- the desired objective. In some cases, stellar Apparent 
        Geocentric Equatorial Coordinates for example, the improvement 
        has reached the limits of the QuickBASIC compiler and the 
        accuracy is essentially equal to the best available sources.
             Second, many different items are displayed using the same 
        units and in multiple formats but having different or unknown 
        accuracy; it is convenient from a programming standpoint to use 
        common subroutines for display purposes. Attempting to tailor 
        the display each of the dozens of quantities calculated by 
        ASTROCLK to the probably accuracy is impractical.
             Finally, even in cases where the accuracy is known to be 
        lower than the displayed precision, trends and relative 
        magnitudes of change can be observed and are reasonably accurate; 
        these second order effects are of some interest to me (and 
        perhaps others), and would be lost if the data were truncated to 
        the known accuracy.

        COMPILER

             Microsoft QuickBASIC, Version 4.50, is the language used for 
        ASTROCLK. Code may be executed in a quasi-interpretive mode or it 
        may be compiled to an executable file. Two different Microsoft 
        programs, QB and BC, are used for the two methods respectively. 
        The distribution version of ASTROCLK is the compiled version of 
        the code. When required for precision, the double-precision 
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 119


        floating point format has been used for numeric data; this eight 
        byte format has a precision of 15 or 16 digits and an approximate 
        magnitude range of from 4.9E-324 to 1.8E+308. [As of Version 
        8903, the program could still be compiled with QuickBASIC Version 
        4.00b, but that compatibility may not be maintained and will not 
        be tested for future ASTROCLK versions.]
             Unfortunately, testing (and confidence) is complicated by 
        the fact that the interpreted version appears to be very 
        sensitive to the order of evaluation and/or to mixing variable 
        types within an expression. For example, using Version 4.00 
        (since updated to Version 4.50), typical calculated results for 
        mean sidereal time varied by plus or minus 0.000011 hours simply 
        by changing the type of variables. Compiled results were the same 
        for all calculations tested, regardless of type or order, and 
        have been used for all comparisons with other data. In spite of 
        the interpreter situation, however, I have concluded that the 
        flexibility and ease of use of QuickBASIC outweighs concern over 
        its problems. In any event, the accuracy and precision seem 
        sufficient for the intended use in ASTROCLK.

        CALENDAR DATES

             The calendar algorithms used are either modeled upon those 
        given in AFC and AYPC or have been developed specifically for 
        ASTROCLK. The calendar displays for October, 1582 and September, 
        1752 use special algorithms to allow for the 10 or 11 missing 
        days. The default ASTROCLK calendar strictly follows the Julian 
        Calendar from its adoption in 46 B.C. through the Gregorian 
        Calendar at its adoption in 1582. Alternatively, the user may 
        select the strict Julian Calendar for ALL dates, or select the 
        British date of adpotion of the Gregorian Calendar in 1752. See 
        the section SETTING PROGRAM OPTIONS for additional details. Dates 
        prior to 46 B.C. are merely an extension of the Julian Calendar 
        back into time, known as the Julian Proleptic Calendar, and 
        bear no particular relationship to calendar(s) in actual use. For 
        times subsequent to 46 B.C., extensive tests have disclosed no 
        errors. Dates for countries adopting the Gregorian Calendar 
        subsequent to October, 1582 or September, 1752 must be adjusted 
        manually. The intercalation proposed and/or adopted for 4000 A.D. 
        and thereafter on a 4000 year cycle has not been included.
             As a matter of personal preference and in company with some 
        (but not all!) of my references, I have adopted a year numbering 
        scheme which includes no zero year. Readers should note that 
        other authors prefer year numbering WITH a zero year, and 
        feelings seem to run high on the subject. Mathematically, of 
        course, any continuous set of numbers must include zero. However, 
        common usage does not include a zero year. The confusion and 
        errors which may result from converting common years such as 4713 
        B.C. into year number -4712 seem too high a price to pay to 
        maintain conformance with the mathematical niceties. Since 
        opinion and practice in the astronomical community is divided 
        anyway, the reader must always check negative dates to determine 
        the year numbering system being used by a given author.

   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 120


        JULIAN DATES

             Julian Dates have been compared with various Astronomical 
        Almanacs and other sources and are exact. The algorithm used is 
        modeled upon that given in AFC. The Julian Date calculations 
        should be accurate from -4713 onward. Note that ASTROCLK uses a 
        year numbering scheme with no year zero (see above); other 
        authors prefer a scheme with a zero year. The day count is also
        presented in three other formats: MJD, DJD, and TJD. See the 
        section JULIAN DATES AND THE GREGORIAN CALENDAR for additional 
        discussion.

        UNIVERSAL TIMES

             Coordinated Universal Time (UTC), the time broadcast by 
        WWV/WWVH and others, is not the same as Universal Time (UT=UT1) 
        but the difference is maintained at less than 0.9 seconds and for 
        most purposes this difference can be ignored. ASTROCLK assumes UT 
        for all time and date calculations and displays with one 
        exception: the Precision Time Display. In this case, the correct 
        UTC time is calculated and displayed to full accuracy for the 
        period 1972 through 1989 when data from AA, Pages K8 and K9, may 
        be applied. Outside this time period, I have made more or less 
        arbitrary assumptions to supply missing data. AA does not include 
        data for Delta UT = UT-UTC; the following tabulation was made 
        using ASTROCLK data for 00:00:00.00 UT and the UT date shown.

                     1988 DELTA UT = UT - UTC (seconds)
                     ----------------------------------
                  
                  JAN 1     +0.18          JUL 1     -0.06
                  FEB 1     +0.14          AUG 1     -0.11
                  MAR 1     +0.10          SEP 1     -0.15
                  APR 1     +0.06          OCT 1     -0.19
                  MAY 1     +0.02          NOV 1     -0.23
                  JUN 1     -0.02          DEC 1     -0.27

        TERRESTRIAL DYNAMICAL TIME

             Delta T, defined as TDT/ET - UT, is determined retro-
        spectively approximately one year after the fact. Since most 
        planetary phenomena require the use of TDT/ET but ASTROCLK is 
        based upon UT, Delta T is required to relate the two time scales.
             Data for reduction of UT versus TDT (Terrestrial Dynamical 
        Time) times are given in AA, Pages K8 and K9, annually for the 
        period 1620 through 1987 with extrapolated data for 1988 through 
        1990. ASTROCLK uses the published values for Delta T as of 
        0h UT January 1 each year for the available interval. For 
        simplicity, I have assumed that Delta T varies linearly from 
        datum to datum; interpolation would probably yield more accurate 
        results, but the difference would not be significant for most of 
        ASTROCLK's calculations. Prior to 1984, the designation changes 
        to Ephemeris Time (ET). The two time scales are considered 
        continuous by ASTROCLK.
             Data for the future behaviour of the rotation of the Earth 
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 121


        is, of course, mostly well-informed speculation. However, it 
        determines how Universal Time will change with respect to 
        Terrestrial Dynamical Time and, in the context of ASTROCLK, is 
        required for planetary positions especially. Similarly, while the 
        data for the last several hundred years can at least be inferred 
        from historical records with some degree of confidence, little or 
        no accurate information exists for ancient times. A number of 
        formulae have been published which allow the estimation of Delta 
        T over extended periods.
             Versions of ASTROCLK prior to 8848 calculated Delta T using 
        a formula by Meeus (AFC); Version 8848 changes to a formula by 
        Morrison and Stephenson (1982) and used by Bretagnon and Simon 
        (1986). [See BIBLIOGRAPHY for reference.] The two methods produce 
        values of Delta T that differ by about three hours at 4000 BC, 
        out of approximately thirty hours. I have no particular reason to 
        believe one formula more accurate than the other, but I switched 
        to Bretagnon and Simon because their planetary position formulae 
        are widely recognized as some of the more accurate which are 
        suitable for micro-computers. Their planetary data, therefore, 
        form a useful basis for comparison with ASTROCLK's planetary 
        position calculations at any instant in time; using the same time 
        scales makes this comparison far simpler. However, TDT or ET 
        should be used with caution outside the period 1620 through 1990.

        INTERNATIONAL ATOMIC TIME (TAI)

             Data for reduction of TAI versus UTC times (Delta AT) is 
        given in AA, Page K9, annually for the period January, 1972 
        through July, 1985. I do not recall any subsequent Leap Seconds 
        until December 31, 1987 and have therefore increased Delta AT to 
        +24 on January 1, 1988. Prior to its adoption as a standard in 
        1972, TAI is replaced by USNO A.1 (see below). Subsequent to 
        1988, I have arbitrarily adjusted TAI by inserting one or more 
        Leap Seconds so that the difference between UT and UTC is always 
        less than one second. The difference between TAI and TDT/ET is 
        32.184 seconds. TAI should be used with caution outside the 
        period January 1972 through December 1988.

        USNO ATOMIC TIME (A.1)

             Prior to the adoption of International Atomic Time, the U.S. 
        Naval Observatory maintained its own atomic time standard, known 
        as A.1. On January 1, 1958, the difference between A.1 and UTC 
        was exactly zero seconds. By January 1, 1972 (when TAI was 
        adopted), the difference was ten seconds. In calculating Delta AT 
        for A.1, I have assumed a linear rate of change and that 
        adjustments are made on June 30th or December 31st as appropriate 
        to maintain the proper relationship with UT. The difference 
        between A.1 and ET is 32.15 seconds.

        SIDEREAL TIMES

             Greenwich mean and apparent sidereal times at 00:00:00 UT 
        for each day of the year are given in AA, pages B8 through B15; 
        selected dates are also given in AFC88, page A3, or they may be 
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 122


        computed for any time using FA. ASTROCLK computed Greenwich Mean 
        Sidereal Times are exact compared to AA and FA88 using the 
        Precision Time Display #1, Display Mode 8. The displayed values 
        for Greenwich Apparent Sidereal Times have a lower accuracy (due 
        to the complex calculations required to compute nutation and the 
        Equation of the Equinoxes); the accuracy is substantially better 
        than 0.01 seconds.
             A comparison using FA88 for 1 January at 0h UT and 12h UT at 
        each of the decades 1950 through 1990 showed GMST to be exact at 
        the displayed precision of 0.0001 seconds for all samples, and 
        GAST to have an average error of -0.0007 seconds and maximum 
        errors of +0.0013 and -0.0025 seconds. The GAST average error 
        works out to about 1/100,000,000 (10E-8). LMST is GMST adjusted 
        for the local longitude and is therefore as accurate as the 
        longitude data. LAST also depends upon longitude; using the same 
        longitude for ASTROCLK and FA88, comparison of LAST showed 
        results comparable to GAST.
             The algorithms for time calculations in general and for 
        the sidereal time calculations in particular were revised and 
        refined at Version 8826 and again at Version 8831, with an 
        improvement in accuracy of at least an order of magnitude. The 
        Precision Time Displays were also added at Version 8826. [Thanks 
        to Ward Harman for detecting an error at other than 0h UT.] If 
        you wish to calculate the data shown in AA, switch to the 
        Precision Time Display #1. Display Mode 8, and enter the time and 
        date in UT using Function Key F3 as follows (April 1988 is used 
        as an example):

                  0U             (time: 00:00:00 UT)
                  1,4,1988       (date: APR 1, 1988)

        Use Function Key F7 to select the desired data format.
        
        PRECESSION

             Precessing the preset internal star database, derived from 
        USNO FA88 data, from J2000.0 to J1988.5 yields coordinates in 
        good agreement with USNO Almanac for Computers 1988 to the 
        precision given there, although the accuracy decreases slightly 
        for declinations nearer the poles. Beginning with Version 8905, 
        the precession method was changed from Newcomb (B1900.0) to 
        Improved IAU System (J2000.0) as described in the main text and 
        the supplement to the 1984 Astronomical Almanac. The resulting 
        precessed data are little changed.
             Representative test results are shown below. Prior to 
        precessing any star in the internal star database, ASTROCLK 
        automatically restores all data to J2000.0 in order to eliminate 
        cumulative errors. Proper motion for objects entered manually may 
        also be entered, or set to zero if not known; tracking data which 
        is precessed over long periods of time when proper motion 
        parameters are set to zero should be used with caution. Solar 
        system objects should always be entered with proper motion 
        parameters set to zero.
             Care should be taken when manually entering data to ensure
        that the data epoch is the same as that of the internal database. 
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 123


        In order to maintain consistent data within ASTROCLK, the 
        internal star database should first be precessed to a data epoch, 
        then manual data referenced to that epoch should be entered. 
        After that, all data may be precessed to the final epoch; using 
        this procedure, both the manually entered data as well as the 
        internal data will all refer to the same epoch.


                       SAMPLE PRECESSION DATA FOR J1988.5

                                         AFC88         ASTROCLK
                  #    Star Name       SHA/DEC          SHA/DEC
                  ----------------------------------------------
                  0    Polaris        325.0618       325.064979
                                       89.2126        89.212613

                 10    Aldebaran      291.1851       291.185085
                                       16.4868        16.486829

                 20    Procyon        245.3249       245.324922
                                        5.2551         5.255091

                 30    ACrux          173.5116       173.512040
                                      -63.0354       -63.035405

                 40    Kochab         137.3174       137.317359
                                       74.2025        74.202525

                 50    Nunki           76.3618        76.361780
                                      -26.3118       -26.311751

             The data from AFC88 (Almanac for Computers 1988) is given 
             there for Mean Place (J1988.5) as shown. The data from 
             ASTROCLK has been precessed from J2000.0 to J1988.5 using 
             Function Key F8. Note slightly degraded accuracy near the 
             North and South poles.

             SHA: Sidereal Hour Angle in degrees, first line. SHA is 
             related to Right Ascension (in hours) by the formula 
             SHA=360-RA*15. The data format shown for ASTROCLK is 
             obtained using Function Key ALT-F7 (for SHA) and Function 
             Key F7 (for degrees and decimal fractions of a degree).

             DEC: Declination in degrees, second line. The data format 
             shown for ASTROCLK is obtained using Function Key F7 (for 
             degrees and decimal fractions of a degree).

             A similar comparison with the Astronomical Almanac 1989, 
        Appendix H ("Bright Stars, J1989.5"), yields accuracies of 0.1 
        seconds in Right Ascension and 1 second of arc in Declination 
        when the ASTROCLK data are rounded to the same precision as that 
        given in the Astronomical Almanac.
             In its discussion of rigorous precession, the Astronomical 
        Almanac 1989 includes an example of the reduction of celestial 
        coordinates for a fictitious star on page B40. The time is given 
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 124


        as 0h TDT 1989 JAN 1. Entering the relevant example data 
        (including proper motion, but not parallax or velocity) into 
        ASTROCLK yields the following data:

             EQUATORIAL COORDINATES [J2000.0]:
               RIGHT ASCENSION (hours):        14:39:36.09 
               DECLINATION (degrees):         -60 50'07.13"

             APPARENT COORDINATES [J1989.0]:
               RIGHT ASCENSION (hours):        14:38:49.34 
               DECLINATION (degrees):         -60 47'17.56"

        The J2000.0 Equatorial Coordinates shown above are the mean data 
        at the standard epoch, essentially identical to those entered 
        from the Astronomical Almanac. The Right Ascension is correct 
        when the data is rounded to the precision shown; the Declination 
        is low by 0.01 arcseconds and results from internal rounding 
        and/or precision errors. The computed J1989.0 Apparent Geocentric 
        Equatorial Coordinates given in the Astronomical Almanac are:

               RIGHT ASCENSION (hours):        14:38:49.394
               DECLINATION (degrees):         -60 47'17.49"

        Even without the inclusion of velocity factors, the results from 
        ASTROCLK agree with the Astronomical Almanac to -0.05 seconds in 
        Right Ascension and +0.07 seconds in Declination. These errors 
        approach the limits imposed by the double precision floating 
        point representation of numbers within QuickBASIC and probably 
        represent the best accuracy attainable in this context.
             Beginning with Version 8903, the internal or external 
        catalog value for the visual magnitude of the selected star or 
        object is displayed at the lower right of the window border in 
        the Tracking Display, Display Mode 0.

        SOLAR POSITION CALCULATIONS

             The computation of the position of the Sun is crucial to 
        many of ASTROCLK's other calculations. I have selected the 
        Apparent Geocentric Coordinates as representative of the accuracy 
        of the calculated solar position; these values are more or less 
        "at the end of the line" in the series of solar calculations and 
        therefore should provide a good basis for comparision with other 
        sources as well as implying the accuracy of prior calculations.
             In the table which follows, the data source is noted in the 
        right hand column: AA is the Astronomical Almanac, 1988, Pages C4 
        through C18; FA is the USNO Floppy Almanac, 1988, Version 2.11.88 
        with time and date set automatically from ASTROCLK using ALT-F9; 
        and, AC is ASTROCLK, Version 8903, Precision Data Display #2. All 
        data are for 0 hours TDT.
             Use Function Key F3 and enter "0T" to set TDT time in order 
        to obtain the same ASTROCLK results for a given date. Note that 
        the date displayed by ASTROCLK is UTC DATE, which differs from 
        the TDT DATE by some +56 seconds in 1988; the UTC DATE will 
        therefore show as the prior day for all months and the prior year 
        for January 1.
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 125


             Use Function Key F3 and enter "0U" to set UT time in order 
        to obtain the same Floppy Almanac results for a given date. Note 
        that ASTROCLK writes the UT time to the file FA.DFT but the 
        Floppy Almanac assumes the time as TDT for Apparent Geocentric 
        Positions calculations. [Other Floppy Almanac calculations 
        correctly interpret the time from FA.DFT as UT.]
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 126


                 1988 APPARENT GEOCENTRIC COORDINATES OF THE SUN
                                    @ 0h TDT

                  Right Ascen          Declination         Distance
                  HH MM SS.SS          DD MM SS.SS         (AU)
        -----------------------------------------------------------------
        JAN 1     18 42 32.35         -23 04 58.0         0.9832806    AA
                        32.351               57.98        0.9832806    FA
                        32.09                59.13        0.98328271   AC

        FEB 1     20 55 10.26         -17 22 51.2         0.9852225    AA
                        10.263               51.19        0.9852225    FA
                        10.10                52.74        0.98522551   AC

        MAR 1     22 48 28.17          -7 35 04.3         0.9908696    AA
                        28.166               04.27        0.9908696    FA
                        28.10                05.36        0.99087354   AC

        APR 1      0 42 13.66           4 32 33.1         0.9993011    AA
                        13.657               33.09        0.9993011    FA
                        13.57                32.50        0.99930318   AC

        MAY 1      2 33 39.46          15 04 46.1         1.0076058    AA
                        39.462               46.07        1.0076058    FA
                        39.27                45.59        1.00760326   AC

        JUN 1      4 36 31.28          22 03 24.4         1.0140599    AA
                        31.285               24.39        1.0140599    FA
                        30.97                24.52        1.01405250   AC

        JUL 1      6 40 49.08          23 06 40.5         1.0166665    AA
                        49.076               40.53        1.0166665    FA
                        48.79                41.54        1.01665800   AC

        AUG 1      8 45 34.69          18 01 10.9         1.0149312    AA
                        34.695               10.93        1.0149312    FA
                        34.61                11.55        1.01492350   AC

        SEP 1     10 41 32.50           8 16 56.4         1.0091422    AA
                        32.500               56.39        1.0091422    FA
                        32.68                54.99        1.00913318   AC

        OCT 1     12 29 29.89          -3 11 08.2         1.0010858    AA
                        29.888               08.23        1.0010858    FA
                        30.28                11.19        1.00107716   AC

        NOV 1     14 25 35.77         -14 25 48.7         0.9924284    AA
                        35.772               48.75        0.9924284    FA
                        36.18                51.13        0.99242345   AC

        DEC 1     16 29 14.33         -21 48 17.1         0.9860075    AA
                        14.332               17.11        0.9860075    FA
                        14.59                17.81        0.98600708   AC
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 127


        MAJOR PLANET POSITION CALCULATIONS

             Care must be taken when comparing ASTROCLK's planetary data 
        with other sources to ensure that the data are calculated for the 
        same time, date, and epoch. No extensive accuracy comparisons 
        have yet been performed for ASTROCLK's planetary position 
        calculations, but spot checks against the Astronomical Almanac, 
        USNO Floppy Almanac, Bretagnon and Simon, Sky & Telescope 
        Magazine, and Astronomy Magazine indicate good agreement.
             As compared against the monthly magazine positions, ASTROCLK 
        provides essentially the same data, and can generate the data for 
        any date rather than for selected dates within a month. In 
        general, predicted errors for the algorithms used by ASTROCLK are 
        on the order of 10" for the calculated positions, and typical 
        errors for a small number of samples have been of that order of 
        magnitude as compared against the USNO Floppy Almanac. The 
        position of Pluto is calculated using osculating elements as of 
        1988 JAN 1, and the errors will increase as the time difference 
        from that date becomes greater.
             The Astronomical Almanac 1988 includes Geocentric Distance 
        and Coordinates for the planets. The coordinates for Venus are 
        given on pages E18 through E21. The data are given at 0h TDT for 
        each day of 1988. Entering 0h TDT 1988 DEC 25 into ASTROCLK and 
        selecting Venus yields the following data:

             Heliocentric Longitude:         214 52'30.43"
             Heliocentric Latitude:            2 15'35.90"
             Heliocentric Radius (AU):         0.722754   
             Appar Geocentric Longitude:     249 03'00.77"
             Appar Geocentric Latitude:        1 05'37.50"
             Geocentric Distance (AU):         1.4936045    <===
             Apparent Right Ascen [J1988.9]:  16:30:05.90   <===
             Apparent Declination [J1988.9]: -20 43'44.27"  <===
             Apparent Right Ascen [J2000.0]:  16:30:44.93
             Apparent Declination [J2000.0]: -20 45'08.39"
             Angular Size (arcsec):           11.33       

        The True Geocentric Distance and Apparent Equatorial Coordinates 
        given in the Astronomical Almanac for that date are:

             GEOCENTRIC DISTANCE (AU):         1.4935568
             RIGHT ASCENSION (hours):         16:30:05.982
             DECLINATION (degrees):          -20 43'44.04"

        The data compare extremely well. The ASTROCLK errors are 
        +0.0000477 AU in Geocentric Distance, -0.082 seconds in Right 
        Ascension, and +0.24 arcseconds in Declination.
             Beginning with Version 8903, the approximate visual 
        magnitude of the selected planet is also displayed on the 
        Tracking Display, Display Mode 0, at the lower right of the 
        window border.

   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 128


        MINOR PLANET POSITION CALCULATIONS

             It is difficult to directly compare minor planet data from 
        the available sources. ASTROCLK computes all minor planet data in 
        the same way as for major planets: apparent position as of the 
        ecliptic and equinox of date. The Astronomical Almanac gives 
        geocentric positions as Astrometric J2000.0 Right Ascension and 
        Declination, and other sources use B1950.0. However, data for the 
        major planets are available as of the ecliptic and equinox of 
        date; using the osculating elements given in the Astronomical 
        Almanac for the major planets and processing these data through 
        ASTROCLK's minor planet software yields position data generally 
        accurate to a second or arcsecond at or very near the date of 
        osculation. This is better accuracy than ASTROCLK's internal 
        major planet data and algorithms usually provide. I have 
        interpreted these results to mean that my methodology is 
        essentially accurate and correct.
             For example, using the minor planet catalog PLANETS.MPC 
        (which contains osculating elements @ 1989 MAR 15.0 for the eight 
        major planets from the Astronomical Almanac 1989), the following 
        heliocentric and geocentric results were obtained:


             HELIOCENTRIC POSITION FOR MERCURY @ 1989 MAR 15.0

             Longitude      Latitude      Radius Vec   Source
             --------------------------------------------------
             243 44 08.0    -6 22 52.2    0.4440226    AA 1989
             243 44 07.53   -6 22 52.25   0.4440219    ASTROCLK


             GEOCENTRIC POSITION FOR MERCURY @ 1989 MAR 15.0

             Rt. Ascension  Declination   Delta        Source
             --------------------------------------------------
             22 37 19.211   -11 05 56.36  1.2713558    AA 1989
             22 37 19.52    -11 05 55.27  1.2713511    ASTROCLK

        CELESTIAL NAVIGATION CALCULATIONS

             ASTROCLK's celestial navigation calculations are adapted 
        from the material presented in the Nautical Almanac 1989, pages 
        277 and following. ASTROCLK was tested by using the Nautical 
        Almanac data adjusted to offset ASTROCLK's automatic internal 
        refraction calculations, the same practice used by the Nautical 
        Almanac. These data therefore represent the result when extremely 
        precise altitude measurements have been taken and when the 
        atmospheric refraction, horizon dip, course, and speed are 
        precisely known. In practice, altitude measurements to this 
        precision are all but impossible outside an observatory, and 
        atmospheric refraction can seldom be predicted to an accuracy of 
        much better than approximately 0.5 minutes of arc.
             Under these circumstances and using the example data on page 
        282 of the Nautical Almanac 1989, ASTROCLK calculates the 
        position of a moving ship to an accuracy of 0.03 nautical miles 
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 129


        (0.05 kilometers or less than 200 feet) as compared to the 
        results calculated in the Nautical Almanac. This level of 
        accuracy is unlikely to be achieved in actual use. In addition to 
        the potential errors mentioned in the previous paragraph, note in 
        particular that ASTROCLK assumes UTC = UT (or that the computer 
        is set to UT rather than UTC); if UTC is used, the resulting time 
        difference (0.9 seconds maximum) can introduce an error in 
        longitude as much as plus or minus 0.2'.
             The example in the text only shows that ASTROCLK will 
        produce essentially the same result as the Nautical Almanac. The 
        Nautical Almanac does not give the "correct" position for the 
        example data nor does it characterize the errors to be expected 
        using its method. By testing ASTROCLK against itself, we can 
        measure the inherent accuracy of the calculations in another way. 
        Setting the time to 05:00UT on 11 November 1989, the local 
        coordinates to the preset location "CAL", and using the Target 
        Tracking Display to make our three "star sights", the following 
        data are obtained:

                 Star          Ho            Hc            Hc'
             -----------------------------------------------------
             12  Capella   36 22 03.11   36 20 46.88   36 20 46.93
             49  Vega      25 44 32.72   25 42 36.22   25 42 36.38
             51  Altair    23 23 04.13   23 20 54.20   23 20 54.40

        Ho is the Apparent Altitude displayed by ASTROCLK and used as the 
        Altitude input for star sights, Hc is the calculated Altitude 
        displayed by ASTROCLK, and Hc' is the calculated Altitude derived 
        from the Apparent Altitude and displayed with the navigation 
        results. This incidentally shows that the internal refraction 
        calculation is reversible. The results obtained are:

                         Actual          Calculated
                  ---------------------------------
                  -120 34 00.00       -120 34 15.96
                    38 09 00.00         38 09 00.63

        The calculated position is 0.27 nm (0.49 km) from the actual 
        position, a very respectable result but somewhat different from 
        the comparison with the Nautical Almanac example. It is probably 
        more representative of the accuracy of the celestial navigation 
        calculations.
             The determination of position by dead reckoning is dependent 
        only upon the accuracy of the initial position and the course and 
        speed parameters. No attempt has been made to compensate for 
        other factors such as wind and/or current; ASTROCLK assumes that 
        the course and speed data have already been corrected for those 
        factors as required.
             Determining the current position using ASTROCLK's celestial 
        and dead reckoning navigation functions requires that the 
        procedures given in the text be followed carefully and that 
        accurate position fixes or star sights be used. Users should take 
        note that the celestial navigation portion of ASTROCLK can be 
        very sensitive to input data errors and should therefore use 
        these functions with care.
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 130



        J2000.0 INTERNAL STAR DATABASE

             Versions of ASTROCLK prior to 8811 used star data manually 
        entered from SKY CATALOGUE 2000.0 (Sky Publishing, 1982). The 
        current star data was extracted from FA88 Version 2.11.88 (star 
        catalog file STAR1.CAT dated 03-02-87) and was substituted in 
        Version 8811 and following. The visual magnitudes for each star 
        were manually added at Version 8903. This substitution was more a 
        matter of personal preference and judgment than the result of any 
        explicit information regarding the inherent accuracy of one 
        source over the other. Be that as it may, my reasons were: a more 
        recent publication date; FA88 data are used for scientific and 
        navigational purposes and I have therefore assumed higher 
        accuracy for the stars selected by USNO; FA88 data are given to 
        higher precision; and, finally, the data were transferred to 
        ASTROCLK directly. A secondary reason for the use of the FA88 
        data is that the AFC88 data for J1988.5 presumably uses the same 
        USNO master data base as FA88 and therefore provides a useful 
        basis for the comparison of ASTROCLK's internal precession 
        calculations (see PRECESSION above for representative results).
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 131



        ASTROCLK MESSAGES AND ERRORS

             Program ASTROCLK generally tries to detect or work around 
        anticipated error conditions which might interfere with program 
        operation. Most non-critical error conditions or warning messages 
        are displayed in the ASTROCLK error window at the lower left of 
        the screen; these conditions usually do not prevent normal 
        program operation (although the operation causing the error may 
        not be performed). Warning messages are displayed with a BROWN 
        (or YELLOW, depending upon your color monitor) background and 
        the word "CAUTION" appearing in the window title; error messages 
        are displayed with a RED background and the word "ERROR" 
        appearing in the window title. Monochrome monitors, of course, 
        won't display in color! The ASTROCLK error number appears in the 
        lower right of the window border. After you understand the 
        message, press RETURN to resume program execution. Other 
        corrective action may be indicated in the message description 
        below.
             However, certain error conditions may not be detected or 
        processed within ASTROCLK, and may cause QuickBASIC or DOS error 
        messages to be displayed or may cause the program to fail to 
        operate as expected; typical such messages or conditions (shown 
        in parentheses) are described at the end of this section. When 
        such an error is detected, an error message is displayed giving 
        the QuickBASIC error number and error message (if available, see 
        Page 410 of the QuickBASIC 4.50 Reference Manual for a list of 
        the normal error messages). Press RETURN and ASTROCLK is aborted 
        and the user is returned to DOS.

        ASTROCLK Numbered Errors and Cautions:
        --------------------------------------

        [01] CAUTION: ASTROCLK is not
             accurate before -4713!

                  The date has been set prior to the year -4713 using 
             Function Key F3. Many of ASTROCLK's date and time algorithms 
             either fail or are inaccurate prior to -4713. You should use 
             Function Key F3 to set a legal date. If the one of the 
             Julian Date or Epoch formats was used for date input, the 
             date is set to JD 0.000000 rather than the date entered; 
             otherwise, the date is left as entered. Execution is allowed 
             to continue after pressing RETURN.

        [02] Illegal Longitude!
             -180 <= Longitude <= 180

        [03] Illegal Latitude!
             -90 <= Latitude <= 90

                  An illegal value was entered for the Longitude or 
             Latitude when setting new local coordinates with F6. The 
             Longitude must be between -180 degrees (west) and 180 
             degrees (east); the Latitude must be greater than or equal 
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 132


             to -90 degrees (south) and less than or equal to 90 degrees 
             (north). Re-enter the correct value.

        [04] Illegal Rt. Ascension!
             0 <= RtAscen < 24

        [05] Illegal Declination!
             -90 <= Decln <= 90

                  An illegal value was entered for the Right Ascension or 
             Declination when setting target coordinates with F5. The 
             Right Ascension must be greater than or equal to zero hours 
             and less than 24 hours; the Declination must be greater than 
             or equal to -90 degrees (south) and less than or equal to 90 
             (north) degrees.  Re-enter the correct value.

        [06] Illegal Altitude!
             -90 <= Altitude <= 90

        [07] Illegal Azimuth!
             0 <= Azimuth <360

                  An illegal value was entered for the Visual Altitude or 
             Visual Azimuth when setting the visual coordinates with F5. 
             The Visual Altitude must be greater than or equal to -90 
             degrees and less than or equal to 90 degrees; the Visual 
             Azimuth must be greater than or equal to zero degrees and 
             less than 360 degrees. Re-enter the correct value.

        [10] Catalog file not found!
             (Check with ALT-F10)

                  A search of the external star catalog was requested 
             with F5 and the external catalog could not be found. Use 
             ALT-F10 to set the correct file name and/or path.

        [11] External Catalog Search
             cancelled by operator!

                  While searching the external star catalog, the operator 
             pressed the ESC key and cancelled the search. The current 
             data are left unchanged.

        [12] Requested star Name/Number
             not found. Try again!

                  While searching the external star catalog for a 
             specified star name or star number, the requested item could 
             not be found in the catalog. Verify the name or number and 
             try again.

        [13] City file not found!
             (Check with ALT-F10)

                  A search of the external file of city names was 
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 133


             requested with F6 and the file could not be found. Use ALT-
             F10 to set the correct file name and/or path.

        [14] Requested city not found
             in current city file!

                  A search of the external file of city names was 
             requested with F6 and the specified city or abbreviation 
             could not be found in the file. Verify that the correct city 
             file is being used. Check the city name or verify the city 
             file with a text editor to see if the city is included.

        [22] Check PATH; should include
             the backslash char (\)!

                  ASTROCLK has detected that the default path or the path 
             just entered does not include the backslash character. The 
             backslash character should normally be the first character 
             of any path so that the path may be properly found. Repeat 
             the path selection process from the start to correct the 
             incorrect path(s). See the section SETTING PROGRAM OPTIONS 
             for additional information. This is a CAUTION message; press 
             RETURN to resume ASTROCLK operation.

        [23] Check ASTROCLK path; add a
             drive specification!

                  ASTROCLK detected a drive specification (such as "D:") 
             in the path for the Floppy Almanac but not in the path for 
             ASTROCLK. If the path for the Floppy Almanac includes a 
             drive then the path for ASTROCLK must also include a drive. 
             For example, if the Floppy Almanac path is "D:\FA", then the 
             ASTROCLK path should have the form "C:\ASTROCLK". If the 
             drive is the same for both paths, do not include the drive 
             in either path, e.g. "\FA" and "\ASTROCLK". If you do not 
             intend to use the Floppy Almanac, enter SPACE to clear the 
             Floppy Almanac path. Repeat the path selection process from 
             the start to correct the incorrect path(s). See the section 
             SETTING PROGRAM OPTIONS for additional information. This is 
             a CAUTION message; press RETURN to resume ASTROCLK 
             operation.

        [24] Illegal DATE requested!
             Check CALENDAR FLAG

                  You have requested an illegal date which falls either 
             in October, 1582 (Calendar Flag = 1) or September, 1752 
             (Calendar Flag = 2) and which was one of the dates abolished 
             as part of the adoption of the Gregorian Calendar. Observe 
             the calendar for the month in question using Display Mode 6 
             to see the days that were deleted. Check the CALENDAR FLAG 
             using ALT-F10.

   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 134


        [25] Illegal DATE requested!
             Req. year NOT Leap Year!

                  You have requested February 29th for a year which is 
             not a Leap Year for the calendar convention currently in use 
             by ASTROCLK. Verify the requested date and check the 
             CALENDAR FLAG using ALT-F10.

        [26] Illegal DATE requested!
             Illegal MONTH and/or DAY!

                  You have requested an illegal MONTH and/or an illegal 
             DAY. The MONTH must be from 1 to 12, and the DAY must be 
             from 0 to the maximum number of days in the MONTH. Day 0 is 
             allowed to conform with astronomical usage. Separate each 
             item with a comma: dd[.d],mm,yyyy.

        [27] Clocks must be SIMULATED/
             OFF with Julian Calendar!

                  You have requested the strict Julian Calendar using 
             ALT-F10 while the clocks are ON. The clocks are set OFF; 
             ASTROCLK cannot operate in real time with the Julian 
             Calendar. You may, however, enable SIMULATION using ALT-F4 
             to observe time/date changes with the Julian Calendar.

        [28] CALENDAR FLAG restored to
             1 = Gregorian @ 1582!

                  After setting the CALENDAR FLAG for the strict Julian 
             Calendar, you have pressed F4 to restart ASTROCLK's clocks. 
             The clocks will be set ON, but the date and time will be 
             restored to system time and the Calendar Flag set for the 
             Gregorian Calendar as of October, 1582. ASTROCLK cannot 
             operate in real time with the Julian Calendar. However, you 
             may use ALT-F4 for simulated real time with the Julian 
             Calendar.


        [30] Illegal PLANET name or
             number requested!

                  You must enter either a valid number (1,2,4-9) or at 
             least the first letter of the planet's name. Mercury and 
             Mars require at least two letters, "ME" and "MA" 
             respectively. The Earth is planet number 3, and planetary
             data are not calculated. Press RETURN and enter a valid 
             number or name.

        [31] Requested Minor Planet
             NUMBER not in file!

                  You have requested a Minor Planet number which is not 
             included in the current Minor Planet Catalog. The range of 
             available minor planets is shown in the upper portion of the 
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 135


             window.

        [32] <mpfilename>:
             File not found!

                  The Minor Planet Catalog whose path and name is shown 
                  on the first line of the error message could not be 
                  found. Check that the path and name have been correctly 
                  entered using ALT-F10.

        [33] <mpfilename> is
             not BINARY or is CORRUPT!

                  The Minor Planet Catalog whose path and name is shown 
             on the first line of the error message is not a BINARY 
             catalog OR its contents are corrupt. Check that the path and 
             name have been correctly entered using ALT-F10.

        [34] Requested Minor Planet
             NAME not found!

                  The requested minor planet NAME could not be matched in 
             the current Minor Planet Catalog. The name either does not 
             exist in the catalog or you have misspelled it. Names may be 
             entered in upper or lower case and only sufficient letters 
             are required to unambiguously identify the desired minor 
             planet(s). Do not include a trailing space in the name.

        [35] Data for this Minor Planet
             is MISSING from Catalog!

                  Although the requested Minor Planet Number is within 
             the range of this catalog, the catalog has no data for this 
             Minor Planet. (A blank record is included.)

        [40] Old version ASTROCLK.INI!
             File read and deleted.

                  ASTROCLK has read file ASTROCLK.INI and it was not in 
             the current version's format. The file was read up to the 
             point where an error was detected and then the file was 
             deleted. For most prior versions of ASTROCLK, all of the 
             local coordinate and time zone information will have been 
             read correctly; to be sure, verify this information on the 
             screen and correct any items in error. Upon exit, ASTROCLK 
             will write a new copy of ASTROCLK.INI in the correct format.

        [41] Can't delete ASTROCLK.INI!
             Disk write-protected/R.O.?

                  ASTROCLK attempted to delete an old version of the file 
             ASTROCLK.INI and the delete failed. This message will 
             immediately follow error message #22 above. The disk may be 
             write protected or full. ASTROCLK attempts to update the 
             file ASTROCLK.INI each time the program completes and the 
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 136


             program expects that the disk drive will NOT be write 
             protected. The file may be set to "Read Only" which also 
             prevents the delete and update functions. The program will 
             operate properly but local coordinates and other program 
             parameters cannot be saved from one execution to the next. 
             "Error writing ASTROCLK.INI" will also probably occur when 
             you exit ASTROCLK.

        [50] ICE and FA are disabled!
             Use ALT-F10 to enable.

                  You have used ALT-F9 to request the ICE or FA and both 
             programs are disabled. Use ALT-F10 to enable one or the 
             other and to set the proper drive and/or path.

        [51] Cannot run Floppy Almanac:
             File FA.DFT open error!

        [52] Cannot run ICE Ephemeris:
             File ICE.DFT open error!

                  You have used ALT-F9 to request the ICE or FA and 
             ASTROCLK is unable to open the file ICE.DFT/FA.DFT to write 
             the current default parameter information. Check the ICE/FA 
             and ASTROCLK paths using ALT-F10. Alternatively, your disk 
             may be full. Press RETURN to resume ASTROCLK operation.

        [53] Cannot run Floppy Almanac:
             1988 <= Year <= 1999

                  ASTROCLK's date is set prior to 15 DEC 1987 or after 
             15 JAN 2000 and you have used ALT-F9 to request the Floppy 
             Almanac. If you have a version of the Floppy Almanac which 
             will execute outside those dates, you must exit ASTROCLK 
             using F9 and run it manually. Alternatively, change to ICE 
             for dates from 1800 through 2049. ASTROCLK resumes normal 
             operation after pressing RETURN.

        [54] Cannot run ICE Ephemeris:
             1800 <= Year <= 2049

                  ASTROCLK's date is set prior to 1800 or after 2049 and 
             you have used ALT-F9 to request the ICE. ICE actually will 
             only execute for dates from December 21, 1800 through June 
             7, 2049. ASTROCLK resumes normal operation after pressing 
             RETURN.

        [60] NAVIGATION mode disabled;
             Set with F10 [F10 + F2].

                  The navigation made is currently disabled. Use Function 
             Keys F10 + F10 to set the UT ZONE OFFSET, then use Function 
             Keys F10 + F2 to set the navigation data. ASTROCLK resumes 
             normal operation after pressing RETURN.

   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 137


        [61] Must set UT ZONE OFFSET
             TIME using F10 + F10!

                  This function cannot be performed until you set the UT 
             ZONE OFFSET using Function Keys F10 + F10. ASTROCLK resumes 
             normal operation after pressing RETURN.

        [62] Invalid Navigation Data!
             Must set using F10 + F2.

                  The navigation data is not valid or has been disabled 
             by the use of Function Key F6. Use Function Keys F10 + F2 to 
             re-enable existing data or enter new data. ASTROCLK resumes 
             normal operation after pressing RETURN.

        [63] Invalid Navigation Data!
             Requires 2 Star Sights." 

                  ASTROCLK requires a minimum of 2 star sights in order 
             to calculate the position. Data for Star #1 is required, and 
             data must be entered for either Star #2 or Star #3 or both. 
             ASTROCLK resumes normal operation after pressing RETURN.

        [99] QuickBASIC 4.50  ERR = nn
             <error description>

                  An error has been detected by QuickBASIC during 
             execution of ASTROCLK. "nn" is the QuickBASIC Run-Time Error 
             Code, as described in Table D-1, Page 476, of the QB4 
             Language Reference Manual. <error description> is the plain 
             text description of the detected error. After RETURN is 
             pressed, execution of program ASTROCLK is aborted and the 
             user is returned to the DOS prompt. NOTE: All expected 
             errors have been trapped by other error routines, as 
             described above. If you receive this error message, please 
             report the circumstances to Dave Ransom either by mail or to 
             the BBS at (231) 541-7299.

        Other ASTROCLK, QuickBASIC and DOS Errors:
        ------------------------------------------

        Error writing ASTROCLK.INI

                  This error message occurs as you exit ASTROCLK and may 
             indicate that your disk is full or write protected. The disk 
             drive used for ASTROCLK must NOT be write protected since 
             updated information is written to the disk upon exit. The 
             error may also be related to a change in ASTROCLK version, 
             or the file may have been manually edited and the format 
             changed. ASTROCLK terminates but ASTROCLK.INI may not be 
             updated to reflect current data.
                  To correct the problem, delete file ASTROCLK.INI. The 
             next time you use ASTROCLK, the default coordinates (Rancho 
             Palos Verdes, CA) will appear; use Function Key F6 to re-
             enter your local coordinates.
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 138



        (DOS SHELL fails to execute)

                  No error message is displayed but when Function Key F9 
             is pressed ASTROCLK pauses momentarily and then continues 
             normal execution without displaying the DOS prompt. Either 
             insufficient memory is available to execute COMMAND.COM or 
             COMMAND.COM cannot be located. If present, remove any RAM 
             DISK from your CONFIG.SYS file and do not execute any large 
             Terminate and Stay Resident (TSR) programs when using 
             ASTROCLK. See your DOS documentation for use of the SET 
             command to verify the COMSPEC parameter (which gives the 
             location of COMMAND.COM).

        (ALT-F3 fails to set software clock)

                  The message "Bad command ..." may be seen briefly at 
             the lower left of the screen. Verify that your version of 
             DOS provides the program RTCLOCK to set the software clock 
             from the hardware clock AND that the program can be found 
             using the current DOS path. If you are using a batch file 
             called RTCLOCK.BAT to set your clock, verify its operation 
             and that it can be found using the current DOS path. See 
             also the section PROGRAM OPERATION for further information.

        (SHIFT-F3 fails to set alarm time, alarm sounds immediately)

                  The alarm must be set using LOCAL TIME and the selected 
             time may not be more than 23 hours in the future. If the 
             alarm time would have occurred within the past hour, the 
             alarm will immediately sound and the alarm window at the 
             lower right will appear then immediately disappear.

        (ALT-F9 fails to execute the USNO Floppy Almanac or ICE)

                  Insufficient memory may be available to execute the 
             Floppy Almanac or ICE. See "DOS SHELL fails to execute" 
             above. The version of FA required for the current date may 
             not be present: a different version of FA is required for 
             each calendar year named "FA88.EXE" for 1988, "FA89.EXE" for 
             1989, and so forth.
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 139



        A BRIEF EDITORIAL

             One of the first decisions that has to be made when writing 
        software is the choice of programming language. Of course, for 
        the individual who only wishes to use the end product and doesn't 
        care how it was done, it couldn't make less difference as long as 
        the software gets the job done. Each programming language has 
        its strong points and its weaknesses, and personal preference 
        usually plays a strong role in the choices that are made. For 
        ASTROCLK, my choice is Microsoft's QuickBASIC.
             I've written software professionally for many years using 
        quite a few different computers and languages and have frequently 
        encountered the attitude that "BASIC isn't real programming, it's 
        just a hobby." The people who feel that way should really check 
        out Microsoft's QuickBASIC, Version 4.50, before they are helped 
        off their soapbox. BASIC has undergone a major evolution in the 
        past several years. While it may not be suited to every job, the 
        times that a BASIC programmer must resort to assembly language or 
        some other higher level language are diminishing at an extremely 
        rapid rate. It has been a real pleasure for me to rediscover 
        BASIC and I use it frequently. Unlike "C", for example, BASIC is 
        a language that I can be away from for an extended time and not 
        have to start all over when I resume using it. For the other side 
        of the coin, however, see also the COMPILER discussion under 
        PRECISION AND ACCURACY TESTS.
             There is another factor that strongly influenced my decision 
        to use QuickBASIC to implement ASTROCLK. As has been written 
        elsewhere, BASIC in one form or another is the "lingua franca" of 
        micro-computers. If my efforts are to be instructive or useful to 
        the greatest number of interested computer users and would-be 
        programmers, they must be understandable to the majority of those 
        individuals. Writing in C or Fortran may result in "better" code, 
        but I would cut myself off from too many people who are not 
        familiar with those programming languages. BASIC, and Microsoft's 
        QuickBASIC in particular, is relatively easy to understand and 
        the software product is easily obtained, well documented and 
        inexpensive.
             One of the items on my list of pet peeves is "shareware" or 
        "userware", as it is commonly called. While I don't begrudge an 
        author the opportunity to recoup some of his or her investment in 
        a program, I am not completely convinced that our free bulletin 
        board systems are an appropriate marketplace. But even if they 
        are, some authors go far beyond a simple request for a modest 
        donation if you like and use their software. Threats of legal 
        action annoy me almost as much as "free" programs that are 
        crippled unless and until you send in money; given the quality of 
        some of this software, any amount sometimes seems exorbitant. I'd 
        rather use supported commercial software that performs as 
        advertised right out of the box. I hope the individuals who 
        practice these threats and deceptions quietly starve; in the mean 
        time, they are embarrassing honest folk everywhere.
             What ever happened to "freeware"? It's now rare indeed to 
        find software that is really free, and even more rare to find the 
        source for that software. And the source can be a terrific 
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 140


        learning tool for the interested programmer and hobbyist alike. 
        Maybe I just remember the days of CP/M too well, when you didn't 
        consider (or trust) public software unless the source was 
        available. You didn't get many unpleasant surprises when someone 
        was willing to sign his name and show you how it was done. I owe 
        a considerable debt to those authors who provided their source in 
        the past. Perhaps the recent outbreak of so-called computer 
        viruses will encourage more users to demand, and authors to 
        include, the source for their programs. I certainly hope so!
             My thanks to Dave Evers of Quincy, Illinois, for his public 
        domain WINDOW TOOLS, a version of which was adapted for use in 
        ASTROCLK. While it wouldn't have been that difficult to write the 
        simple window routines I needed, it was nice to have some 
        QuickBASIC routines already debugged, documented, and which 
        included the source code.
             A project like ASTROCLK can continue indefinitely; so far, 
        it's been going on for over two years. Being considered for 
        future versions are Lunar tracking data, times for rising and 
        setting, and various other items large and small that may or may 
        not ever happen. Portions of code to implement new features may 
        appear from time to time in ASTROCLK and are either not used or 
        are commented out. As new or improved features are contemplated, 
        I try to strike a balance between accuracy and reasonable 
        computational times -- a battle I seem destined to lose one way 
        or the other. Already, a math coprocessor is the only way to keep 
        all operations in strict real time when the clocks are running 
        and you wish to view planetary data.
             Program ASTROCLK is free for non-commercial use. Use it if 
        you like it, discard it if you don't. There are no warranties of 
        any kind. Version 8806 was the first public release of ASTROCLK in 
        February of 1988. While I don't know of any obvious or 
        catastrophic bugs after many versions, updates, and corrections, 
        I will probably never feel sufficiently confident to say there 
        aren't any. Microsoft's QuickBASIC 4.50 IS NOT included and IS 
        required to compile the source files. The compiled version, 
        ASTROCLK.EXE, is a stand-alone program and does not require 
        QuickBASIC support.
             Comments and suggestions are welcome, and any error or bug 
        reports will be greatly appreciated. Use the mail or call the 
        bulletin board system (BBS) at the number below and leave a 
        message for "SYSOP" or "Dave Ransom". The BBS has an automatic 
        power controller; if it doesn't answer by the third ring, hang 
        up, and then call back in TWO MINUTES. (It's an older computer, 
        and those two minutes are used for boot-up and BBS housekeeping 
        chores.) The BBS will always have the most recent version of 
        ASTROCLK in compressed format; ASTROCLK is located in the 
        ASTRONOMY area, File Area #5. Updated versions are posted at 
        irregular intervals, typically every four to eight weeks. Use 
        program PAK, Version 2.10 or higher (also available on the 
        BBS), to decompress the files. 

                  (213) 541-7299      [24 hours, 2400/1200 baud]

   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 141


             The BBS version of ASTROCLK is available in three compressed
        files (currently approaching a total of about 500K), and download 
        times are considerable. Individuals without access to a modem or 
        who wish to avoid toll charges for these large files may obtain a 
        complete set of ASTROCLK disks with the current version (MS-DOS 
        DS/DD, specify 5-1/4" 360K or 3-1/2" 720K) by sending US $20.00 
        to cover disks, postage and handling.

                                      David H. Ransom, Jr.
                                      7130 Avenida Altisima
                                      Rancho Palos Verdes, CA  90274
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 142



        BIBLIOGRAPHY

             The following principal sources have been consulted during 
        the preparation and/or testing of Program ASTROCLK and this text.

        ------, THE ASTRONOMICAL ALMANAC 1984. U. S. Government Printing 
        Office, Washington, DC, 1983.

        ------, THE ASTRONOMICAL ALMANAC 1988. U. S. Government Printing 
        Office, Washington, DC, 1987.

        ------, THE ASTRONOMICAL ALMANAC 1989. U. S. Government Printing 
        Office, Washington, DC, 1988.

        ------, THE ASTRONOMICAL ALMANAC 1990. U. S. Government Printing 
        Office, Washington, DC, 1989.

        ------, THE NAUTICAL ALMANAC 1989. U. S. Government Printing 
        Office, Washington, DC, 1987.

        ------, NBS TIME & FREQUENCY DISSEMINATION SERVICES, NBS Special 
        Publication 432. U. S. Government Printing Office, Washington, 
        DC, 1979.

        Acker, Agnes and Jaschek, Carlos, ASTRONOMICAL METHODS AND 
        CALCULATIONS. John Wiley & Sons, New York, NY, 1986.
        [First published in French in 1981.]

        Bretagnon, Pierre and Simon, Jean-Louis, PLANETARY TABLES AND 
        PROGRAMS FROM -4000 TO +2800. Willmann-Bell, Inc., Richmond, VA, 
        1986.

        Burgess, Eric, CELESTIAL BASIC. Sybex Inc., Berkeley, CA, 1982

        Carroll, Tim S., THE FLOPPY ALMANAC USER'S GUIDE, 2nd Edition. 
        Nautical Almanac Office, United States Naval Observatory, 
        Washington, DC, 1988.

        Danby, J. M. A., FUNDAMENTALS OF CELESTIAL MECHANICS, 2nd 
        Edition. Willmann-Bell, Inc., Richmond, VA, 1988.

        Doggett, LeRoy E. et al, ALMANAC FOR COMPUTERS 1988. Nautical 
        Almanac Office, United States Naval Observatory, Washington, DC, 
        1988.

        Duffett-Smith, Peter, ASTRONOMY WITH YOUR PERSONAL COMPUTER. 
        Cambridge University Press, New York, NY, Reprinted (with 
        corrections) 1986.

             [NOTE: The disk available from Cambridge University Press, 
             containing the programs from this text, does NOT include the 
             1986 corrections (as of mid-1988). In particular, subroutine 
             PELEMENT, Page 141, contains errors in the DATA statements 
             for Mercury and Mars, lines 3725 and 3800; see text for 
   ASTROCLK Astronomical Clock and Celestial Tracking Program        Page 143


             corrections.]

        Duffett-Smith, Peter, PRACTICAL ASTRONOMY WITH YOUR CALCULATOR, 
        2nd Edition. Cambridge University Press, New York, NY, 1981.

        Espenshade, Edward B., Jr., Editor, GOODE'S WORLD ATLAS, 17th 
        Edition. Rand McNally & Co., Chicago, IL, 1987.

        Hirshfeld, Alan and Sinnot, Roger W., Editors, SKY CATALOGUE 
        2000.0. Sky Publishing Corp., Cambridge, MA, 1982.

        Hobbs, Richard R., MARINE NAVIGATION 2, 2nd Edition. Naval 
        Institude Press, Anapolis, MD, 1987.

        Lawrence, J. L., BASIC ASTRONOMY WITH A PC. Willmann-Bell, Inc., 
        Richmond, VA, 1989. 

             [NOTE: A diskette is also available with the BASIC programs 
             for IBM-compatible PC's.]

        Meeus, Jean, ASTRONOMICAL FORMULAE FOR CALCULATORS, 4th Edition. 
        Willmann-Bell, Inc., Richmond, VA, 1988.

             [NOTE: The 4th Edition is identical to the 3rd Edition with 
             the exception of an added Chapter 43 giving formulae for the 
             position of Pluto.]

        Menzel, Donald H. and Pasachoff, Jay M., A FIELD GUIDE TO THE 
        STARS AND PLANETS, 2nd Edition. Houghton Mifflin Co., Boston, 
        MA, 1983.

        Sinnott, Roger W., monthly column "Astronomical Computing", Sky & 
        Telescope Magazine, various issues 1984 through 1989.

        Taff, Laurence G., CELESTIAL MECHANICS. John Wiley & Sons, New 
        York, NY, 1985.
