Ephemerides 2021 September Module


 Overview
 

1°)   Ecliptic Geocentric Coordinates
2°)   Equatorial Geocentric Coordinates
3°)  Azimuthal Topocentric Coordinates
4°)  Numerical Results


-These programs compute accurate positions of the Sun, the Moon and the major planets
    for a short time-span of 32 days, i-e  2021/08/31 0h TT to 2021/10/02  0h TT

-The longitudes & latitudes and the right-ascensions & declinations are geocentric apparent,
  referred to the true equator & equinox of the date, corrected for aberration and light-time.

-The precision is about 0"01 for the longitudes & latitudes and of the order of 3 E-8 AU for the distances ( 5 E-11 AU for the Moon ).
-The distances are true distances.

-The azimuthal ( topocentric ) coordinates are also given, corrected for parallax & diurnal aberration.

-These coordinates are calculated by polynomials fitted to the JPL Ephemerides DE441
 

Notes:

-Always execute "ECLI" first for the ecliptic coordinates, with at least SIZE 031
-Then "EQUA" for the equatorial coordinates ( SIZE 039 )
-And then "AZIM" for the azimuthal coordinates with at least SIZE 041.

-The azimuths are reckoned clockwise from North.
-Longitudes are positive East.
 

Data Registers

  R00 = ( DOM - 16 ) / 16 ( from -1  to +1 )  Terrestrial Time ( TT )

  R01 thru R30 = coordinates of the Sun, the Moon, Mercury, Venus, Mars, Jupiter, Saturn, Uranus, Neptune & Pluto.

  R31 = True obliquity of the ecliptic  ( deg )
  R32 = Local Sidereal Time  ( hh.mnss )

 • R33 = Longitude of the observer ( ° ' " )   positive East
 • R34 = Latitude of the observer ( ° ' " )                                                         Registers R33-R34-R35 are to be initialized before executing "AZIM"
 • R35 = Observer altitude in meters

 ( R36 to R40:  temporary data storage )
 
 

XROM  Function  Desciption
 24,00
 24,01
 24,02
 24,03
 24,04
 24,05
 24,06
 24,07
 24,08
 24,09
 24,10
 24,11
 24,12
 24,13
-EPH2021SEP
 SUN
 MOON
 MER
 VEN
 MAR
 JUP
 SAT
 URA
 NEP
 PLU
 ECLI
 EQUA
 AZIM
 Section Header
 Ecliptic Coordinates of the Sun
 Ecliptic Coordinates of the Moon
 Ecliptic Coordinates of Mercury
 Ecliptic Coordinates of Venus
 Ecliptic Coordinates of Mars
 Ecliptic Coordinates of Jupiter
 Ecliptic Coordinates of Saturn
 Ecliptic Coordinates of Uranus
 Ecliptic Coordinates of Neptune
 Ecliptic Coordinates of Pluto
 Ecliptic Coordinates of the Sun, Planets & Moon
 Ecliptic -> Equatorial Coordintes
 Equatorial -> Azimuthal Coordinates
  


-"ECLI"  "EQUA"  &  "AZIM"  calculate & store the coordinates in registers R01 thru R30 as follows:

>>>   h0 is the height, corrected for refraction
 
 

      Celestial Body    Registers                "ECLI"                 "EQUA"          "AZIM"
            SUN       R01
      R02
      R03
    Eclipt Longitude ( deg )
    Eclipt  Latitude ( deg )
    Dist from Earth ( AU )
    Right-Ascens(hh;mnss)
      Declination ( ° ' " )
    Dist from Earth ( AU )
   Azimuth ( ° ' " )
     height  ( ° ' " )
        h0  ( ° ' " )
          MOON       R04
      R05
      R06
    Eclipt Longitude ( deg )
    Eclipt  Latitude ( deg )
    Dist from Earth ( AU )
    Right-Ascens(hh;mnss)
      Declination ( ° ' " )
    Dist from Earth ( AU )
    Azimuth ( ° ' " )
     height  ( ° ' " )
        h0  ( ° ' " )
       MERCURY       R07
      R08
      R09
    Eclipt Longitude ( deg )
    Eclipt  Latitude ( deg )
    Dist from Earth ( AU )
    Right-Ascens(hh;mnss)
      Declination ( ° ' " )
    Dist from Earth ( AU )
    Azimuth ( ° ' " )
     height  ( ° ' " )
        h0  ( ° ' " )
         VENUS       R10
      R11
      R12
    Eclipt Longitude ( deg )
    Eclipt  Latitude ( deg )
    Dist from Earth ( AU )
   Right-Ascens(hh;mnss)
     Declination ( ° ' " )
   Dist from Earth ( AU )
    Azimuth ( ° ' " )
     height  ( ° ' " )
        h0  ( ° ' " )
          MARS       R13
      R14
      R15
    Eclipt Longitude ( deg )
    Eclipt  Latitude ( deg )
    Dist from Earth ( AU )
    Right-Ascens(hh;mnss)
      Declination ( ° ' " )
   Dist from Earth ( AU )
    Azimuth ( ° ' " )
     height  ( ° ' " )
        h0  ( ° ' " )
        JUPITER       R16
      R17
      R18
    Eclipt Longitude ( deg )
    Eclipt  Latitude ( deg )
    Dist from Earth ( AU )
    Right-Ascens(hh;mnss)
      Declination ( ° ' " )
    Dist from Earth ( AU )
    Azimuth ( ° ' " )
     height  ( ° ' " )
        h0  ( ° ' " )
        SATURN       R19
      R20
      R21
    Eclipt Longitude ( deg )
    Eclipt  Latitude ( deg )
    Dist from Earth ( AU )
    Right-Ascens(hh;mnss)
      Declination ( ° ' " )
    Dist from Earth ( AU )
    Azimuth ( ° ' " )
     height  ( ° ' " )
        h0  ( ° ' " )
        URANUS       R22
      R23
      R24
    Eclipt Longitude ( deg )
    Eclipt  Latitude ( deg )
    Dist from Earth ( AU )
    Right-Ascens(hh;mnss)
      Declination ( ° ' " )
    Dist from Earth ( AU )
    Azimuth ( ° ' " )
     height  ( ° ' " )
        h0  ( ° ' " )
       NEPTUNE       R25
      R26
      R27
    Eclipt Longitude ( deg )
    Eclipt  Latitude ( deg )
    Dist from Earth ( AU )
    Right-Ascens(hh;mnss)
      Declination ( ° ' " )
    Dist from Earth ( AU )
    Azimuth ( ° ' " )
     height  ( ° ' " )
        h0  ( ° ' " )
         PLUTO       R28
      R29
      R30
    Eclipt Longitude ( deg )
    Eclipt  Latitude ( deg )
    Dist from Earth ( AU )
    Right-Ascens(hh;mnss)
      Declination ( ° ' " )
    Dist from Earth ( AU )
    Azimuth ( ° ' " )
     height  ( ° ' " )
        h0  ( ° ' " )

 

1°) Ecliptic Geocentric Coordinates of the Sun, the Moon & the major Planets


            STACK            INPUTS      OUTPUTS
                 Z                 /       R0  ( AU )
                 Y       Day of the Month       B0  ( deg )
                 X        HH.MNSS(TT)       L0  ( deg )

    Where  L = Longitude   B = Latitude   R = radius vector

Example:    Calculate the apparent geocentric ecliptic coordinates of the Sun, the Moon and the planets on 2021/09/24 at 16h41m  TT


-Enter the day of the month and the time expressed in  Terrestrial Time ( TT )

       24       ENTER^
    16.41     XEQ "ECLI"            >>>>     L0 =  181.°847757          = R01                              ---Execution time = 90s---
                                                  RDN      B0 =  -0°000099            = R02
                                                  RDN      R0 =  1.00302647  AU   = R03

Notes:

-All the angles are expressed in decimal degrees.
-Cf  paragraph 4°) for the other results.

-If you key in a date outside the interval [ 2021/08/31 0h TT , 2021/10/02   0h TT ]  you'll get a DATA ERROR message.
-However, this program may probably be used a few hours outside the prescribed interval: set F25 and R/S
-But the precision is less guaranteed and the results may even become completely meaningless several days before 00 or after 32, especially for the Moon.
 

2°) Equatorial Geocentric Coordinates
 

-AFTER executing "ECLI", use "EQUA" to get the equatorial coordinates
-The right-ascensions are expressed in hh.mnss and the declinations in ° ' "
-They replace the ecliptic longitudes & latitudes ( cf the tableau in the paragraph above )

-"EQUA" also calculates the true obliquity of the ecliptic which is returned in Z-register
-A polynomial is also used for that.
 
 

           STACK          INPUTS        OUTPUTS
               Z               /        eps   ( deg )
               Y               /       Decl0 ( ° ' " )
               X               /     RA0  ( hh.mnss )

  Where  RA = Right-Ascension   Decl = declination  eps = true obliquity of the ecliptic

Example:    Calculate the apparent geocentric equatorial coordinates of the Sun, the Moon and the planets on 2021/09/24 at 16h41m  TT

After executing "ECLI"


        XEQ "EQUA"  or simply R/S if you've just executed "ECLI"

                                  >>>>     RA0 =    12h06m46s89     = R01                         ---Execution time = 50s---
                                   RDN    Decl 0 =  -0°44'05"76        = R02
                                   RDN      eps  =   23°4377232        = R31
 

-The distances in R03-R06-.....-R30  are unchanged.  
-Cf paragraph 4°) for the other results 


3°) Azimuthal Topocentric Coordinates
 

-AFTER executing "ECLI" & "EQUA" use "AZIM" to get the horizontal coordinates
-The azimuths & heights are expressed in ° ' "

-The heights corrected for refraction are also computed and replace the distances in R03  R06 ..... R30
 
 

      STACK        INPUTS      OUTPUTS
           Z             /       h0  ( ° ' " )
           Y             /       h  ( ° ' " )
           X             /      Az  ( ° ' " )

                  Long = longitude ( positive East )       Az = Azimuth ( clockwise from North )    |
  Where       Lat  =  latitude                                   h  =  height                                             >       of the Sun
                   Alt  =  altitude in meters                   h0 =  height ( corrected for refraction )    |

Example:    Calculate the apparent topocentric   azimuthal coordinates of the Sun, the Moon and the planets on 2021/09/24  at 16h41m  TT
                    at the Palomar Observatory,   Longitude = 116°51'50"4 W   Latitude = 33°21'22"4 N   Altitude = 1706 m
 

>>>  After executing "ECLI" & "EQUA"


    -116.51504   STO 33
       33.21224   STO 34
          1706       STO 35    R/S         >>>>      Az   = 119°34'44"00   = R01                         ---Execution time = 184s---
                                                        RDN         h   =  35°47'36"53    = R02
                                                        RDN         h0  =  35°48'55"45    = R03

         which are the topocentric coordinates of the Sun.
 

>>>  We also have the local sidereal time in R32 = LST = 9h07m18s85
 

Notes:

-Cf paragraph 4°) for the other results.
-The difference TT - UTC = 69.184 seconds. 

->  h0   is often meaningless when  h <   0
 

4°) Numerical Results

-Longitudes & latitudes are expressed in decimal degrees   and the distances in Astronomical Units ( "ECLI" )
-Right-ascensions in hh.mnss & declinations in ° ' "  ( "EQUA"   )
-Azimuths & heights in ° ' "  too   ( "AZIM" )
  
-Obliquity of the ecliptic  in decimal degrees ( R31 )
-Local sidereal time in hh.mnss  ( R32 )



           Celestial Body    Registers        "ECLI"       "EQUA"        "AZIM"
                 SUN       R01
      R02
      R03
    181.847757
    -0.000099
    1.00302647
    12.064689
    -0.440576
    unchanged
   119.344400
    35.473653
    35.485545
               MOON       R04
      R05
      R06
     44.152412
     -1.686875
  0.0026869142
      2.485044
    14.282485
    unchanged
    -75.230326
      3.142116
      3.270624
            MERCURY       R07
      R08
      R09
    205.134296
      -3.677914
    0.79457317
    13.273856
   -13.084891
    unchanged
   115.115851
    12.340062
    12.380961
              VENUS       R10
      R11
      R12
    225.780031
    -1.768430
    0.93492030
    14.510609
   -18.151527
     unchanged
   107.293215
    -6.402367
    -6.402367
              MARS       R13
      R14
      R15
    186.279399
     0.750141
    2.63757021
    12.241487
    -1.481621
     unchanged
    116.580060
    31.512882
    31.530030
             JUPITER       R16
      R17
      R18
    -36.761030
    -1.188976
    4.19280430
     21.435446
    -14.533869
     unchanged
   -26.221486
   -69.454501
   -69.454501
             SATURN       R19
      R20
      R21
    -52.897757
    -0.818402
     9.32716250
     20.385294
    -19.171396
     unchanged
    26.020396
   -74.340318
   -74.340318
            URANUS       R22
      R23
      R24
     44.288374
     -0.423109
    18.98446636
     2.474997
     15.431607
     unchanged 
   -74.122900
    4.374746
    4.474554
            NEPTUNE       R25
      R26
      R27
    -8.496680
    -1.171369
    28.93512533
    23.303799
    -4.264417
    unchanged
   -56.570594
   -45.521903
   -45.521903
              PLUTO       R28
      R29
      R30
    -65.651010
    -1.648180
    33.97245484
     19.461661
    -22.520530
     unchanged
   64.430775
   -69.201783
   -69.201783
  True obliquity of the ecliptic       R31
           /
     23.4377232
    23.4377232
      Local Sidereal Time
      R32
           /
             /
      9.071885

 
    
5°) Sun-Moon-Mercury-Venus-Mars-Jupiter-Saturn-Uranus-Neptune-Pluto

 

-All these subroutines may be used for themselves to calculate the geocentric ecliptic coordinates
-First initialize R00 before executing them.
 
 

      STACK        INPUTS      OUTPUTS
           Z             /       R  ( AU )
           Y             /       B  ( deg )
           X             /       L  ( deg )

    Where  L = Longitude   B = Latitude  R = radius vector

Example:    The same one, which corresponds to R00 = 0.5434461806


  XEQ "SUN"  >>>>     L 0 =  181.°847757          = R01                              ---Execution time = 9s---
                        RDN      B0 =   -0°000099            = R02
                        RDN      R0 =  1.00302647  AU   = R03

>>> Likewise with the Moon, Mercury, ........... , & Pluto ( see above the numerical values )
 

WARNING !!!

-Unlike "ECLI" , these routines do not check that R00 is between -1 and +1
 
 

Remark:

-The apparent heights are calculated by a refraction formula which approximates the Pulkovo refraction tables
  for standard conditions of temperature & pressure ( T = 15°C , P = 1013.25 mbar, humidity = 0 , wave length = 0.59µ )

-The precision is better than 0"06 over the whole range [ -0°32'58"0 , 90° ]
 

    h0  ~  h + 1° / 62.95929 / Tan ( h + 4°8043 / ( h + 7°0822 / ( h +11°1187 / ( h + 38°2290 / ( h + 9°9098 ) ) ) ) ) 



References:

[1]  Aldo Vitagliano SOLEX  http://www.solexorb.it/
[2]  ftp://ssd.jpl.nasa.gov/pub/eph/planets/ascii/
[3]  Jean Meeus - "Astronomical Algorithms" - Willmann-Bell  -  ISBN 0-943396-61-1