Ephemerides 2023 January 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 ( this month, not enough room for Pluto )
    for a short time-span of 32 days, i-e  2022/12/31 0h TT to 2023/02/01  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 "ECL" first for the ecliptic coordinates, with at least SIZE 031
-Then "EQ" for the equatorial coordinates ( SIZE 039 )
-And then "AZ" 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 "AZ"
 • 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
  $
-EPH2023JAN
 V
 ECL
 EQ
 AZ
 Subroutine that is called by "V"
 Section Header
 Ecliptic Coordinates of the Sun, the Moon & the Planets

 Takes day of month & time and calls "V"
 Ecliptic -> Equatorial Coordinates
 Equatorial -> Azimuthal Coordinates
  


-"ECL"  "EQ"  &  "AZ"  calculate & store the coordinates in registers R01 thru R27 as follows:

>>>   h0 is the height, corrected for refraction
 
 

      Celestial Body    Registers                  "ECL"                  "EQ"            "AZ"
            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  ( ° ' " )

 

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 2023/01/24 at 16h41m  TT


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

       24       ENTER^
    16.41     XEQ "ECL"            >>>>     L0 =   304°417981         = R01                   
                                                RDN      B0 =   -0°000175           = R02
                                                RDN      R0 =  0.98441255  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 [ 2022/12/31 0h TT , 2022/02/01   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 "ECL", use "EQ" 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 2023/01/24 at 16h41m  TT

After executing "ECLI"


       XEQ "EQ"  or simply R/S if you've just executed "ECL"

                           >>>>     RA0 =    20h27m00s53     = R01              
                            RDN    Decl 0 =  -19°09'19"04      = R02
                            RDN      eps  =     23°438235        = R31
 

-The distances in R03-R06-.....-R27  are unchanged.  
-Cf paragraph 4°) for the other results 


3°) Azimuthal Topocentric Coordinates
 

-AFTER executing "ECL" & "EQ" use "AZ" 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 ..... R27
 
 

      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 2023/01/24  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   = 130°05'58"94   = R01          
                                                        RDN         h   =  19°07'34"43     = R02
                                                        RDN         h0  =  19°10'17"05    = R03

         which are the topocentric coordinates of the Sun.
 

>>>  We also have the local sidereal time in R32 = LST = 17h07m21s83
 

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 ( "ECL" )
-Right-ascensions in hh.mnss & declinations in ° ' "  ( "EQ"   )
-Azimuths & heights in ° ' "  too   ( "AZ" )
  
-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
    304.417981
     -0.000175
    0.98441255
     20.270053
    -19.091904
    unchanged
   130.055894
    19.073443
    19.101705
               MOON       R04
      R05
      R06
    -15.700602
     -4.094316
  0.0024337174
    23.083316
    -9.571719
    unchanged
    98.103776
    -6.414644
    -6.414644
            MERCURY       R07
      R08
      R09
    280.375032
      1.940554
    0.89986523
    18.442957
   -21.055273
    unchanged
   153.274611
    30.492007
    30.505532
              VENUS       R10
      R11
      R12
    -33.004531
      -1.580754
    1.52866921
    21.590353
   -13.595310
     unchanged
   111.083520
     6.012929
     6.093515
              MARS       R13
      R14
      R15
     68.981982
      2.774501
    0.81175060
      4.271408
     24.321130
     unchanged
    10.412809
   -31.193838
   -31.193838
             JUPITER       R16
      R17
      R18
      4.728129
     -1.199670
    5.36407301
      0.191569
      0.463913
     unchanged
    79.133144
   -14.293863
   -14.293863
             SATURN       R19
      R20
      R21
    -35.005509
     -1.244114
    10.74486720
     21.504953
    -14.214250
     unchanged
   112.383559
     7.241037
     7.305792
            URANUS       R22
      R23
      R24
     44.941726
     -0.346786
    19.46280626
      2.502012
     15.591218
     unchanged 
    39.032038
   -30.484562
   -30.484562
            NEPTUNE       R25
      R26
      R27
     -6.611482
     -1.176066M2
    30.54886745
    23.373502
    -3.421776
    unchanged
    88.565248
    -8.203924
    -8.203924
  True obliquity of the ecliptic       R31
           /
    23.438235
    unchanged
      Local Sidereal Time
      R32
           /
             /
    17.072183

 
    
5°) V

 

-This subroutine may be used for itself to calculate the geocentric ecliptic coordinates
-First initialize R00 before executing "V".
 
 -With the example above,  R00 = 0.5434461806


WARNING !!!


-Unlike "ECL" , this routine does not check if R00 is between -1 and +1

 

6°)  Refraction

-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