Ephemerides 2026 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  2026/08/31 0h TT to 2026/10/02  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 "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 )
 
 

-"ECL"  "EQ"  &  "AZ"  calculate & store the coordinates in registers R01 thru R30 as follows:

>>>   h0 is the height, corrected for refraction
 
 

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

    Where  L = Longitude   B = Latitude   R = radius vector

Example:    Calculate the apparent geocentric ecliptic coordinates of the Sun, the Moon and the planets on 2026/09/24 at 16h41m  TT

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

       24       ENTER^
    16.41     XEQ "ECL"            >>>>     L₀ =  181.654489          = R01
                                                RDN      B₀ =    0°000052            = R02
                                                RDN      R₀ =  1.00309853  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 [ 2026/08/31 0h TT , 2026/10/02   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 "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 )

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

  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 2026/09/24 at 16h41m  TT

After executing "ECLI"

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

                           >>>>     RA₀ =     12h06m04s34     = R01              
                            RDN    Decl ₀ =    -0°39'28"67       = R02
                            RDN      eps  =      23°438143        = R31
 

-The distances in R03-R06-.....-R30  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 ..... R30
 
 

                  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 2026/09/24  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"

         which are the topocentric coordinates of the Sun.
 

>>>  We also have the local sidereal time in R32 = LST = 9h06m30s58
 

Notes:

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

->  h₀   is often meaningless when  h <   0
 

 
 
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 !!!

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 about 0"12 if  -0°32'58"0 <= h <=  90°
 

    h₀  ~  h + 1° / 62.93951 / Tan ( h + 4°80017 / ( h + 6°90263 / ( h +10°06891 / ( h + 31°76812 / ( h + 8°87360 ) ) ) ) ) 

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