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 2025/09/30 0h TT to
2025/11/01 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 |
$ -EPH2025OCT 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 | "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 2025/10/24 at 16h41m TT
-Enter the day of the month and the time expressed
in Terrestrial Time ( TT
)
24
ENTER^
16.41
XEQ "ECL"
>>>> L0
= 211.528155
= R01
RDN
B0 = -0°000140
= R02
RDN
R0 = 0.99459216 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 [ 2025/09/30 0h TT , 2025/11/01
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 2025/10/24 at 16h41m
TT
After executing "ECL"
XEQ "EQ"
or simply R/S if you've just executed "ECL"
>>>>
RA0 =
13h57m29s59 = R01
RDN Decl 0
= -12°00'18"55 = R02
RDN eps
= 23°438429
= 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
STACK | INPUTS | OUTPUTS |
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
|
Example: Calculate the apparent topocentric
azimuthal coordinates of the Sun, the Moon and
the planets on 2024/12/24 at 16h41m
TT
at the Palomar Observatory,
Longitude = 116°51'50"4 W
Latitude = 33°21'22"4 N Altitude
= 1706 m
>>> After executing "ECL" & "EQ"
-116.51504 STO 33
which are
the topocentric coordinates of the Sun.
>>> We also have the local sidereal time in R32 = LST
= 11h05m44s25
Notes:
-This month, not enough room to compute the refraction.-Cf paragraph 4°) for the other results.
-The difference TT - UTC = 69.184
seconds.
Celestial Body | Registers | "ECL" | "EQ" | "AZ" |
SUN |
R01 R02 R03 |
211.528155 -0.000140 0.99459216 |
13.572959 -12.001855 unchanged |
130.250774 28.554062 unchanged |
MOON |
R04 R05 R06 |
246.094993 -5.028592 0.00271549423 |
16.125890 -26.162620 unchanged |
118.544432 -5.031283 unchanged |
MERCURY |
R07 R08 R09 |
234.698374 -2.602558 1.10815619 |
15.263864 -21.275763 unchanged |
121.364277 7.090089 unchanged |
VENUS |
R10 R11 R12 |
193.443445 1.530619 1.59444098 |
12.515179 -3.534320 unchanged |
140.512567 45.052276 unchanged |
MARS |
R13 R14 R15 |
232.287690 -0.321608 2.40033708 |
15.190956 -18.390570 unchanged |
120.354618 10.185045 unchanged |
JUPITER |
R16 R17 R18 |
114.625869 0.069091 4.97257418 |
7.461443 21.155623 unchanged |
-92.113348 44.304642 unchanged |
SATURN |
R19 R20 R21 |
356.165231 -2.481808 8.71192501 |
23.495233 -3.480673 unchanged |
-21.340473 -58.420984 unchanged |
URANUS |
R22 R23 R24 |
60.544926 -0.209722 18.62637441 |
3.534396 20.033258 unchanged |
-63.275248 -3.051959 unchanged |
NEPTUNE |
R25 R26 R27 |
359.947589 -1.370782 29.03503729 |
0.015934 -1.164261 unchanged |
-25.161432 -55.185729 unchanged |
PLUTO |
R28 R29 R30 |
301.393561 -3.774712 35.36282134 |
20.180848 -23.314130 unchanged |
86.054145 -52.082637 unchanged |
True obliquity of the ecliptic | R31 |
/ |
23.438429 |
unchanged |
Local Sidereal
Time |
R32 |
/ |
/ |
11.054425 |
-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
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