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 2023/04/30 0h TT to 2023/06/01
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 |
S -EPH2023APR 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" & "S" Ecliptic -> Equatorial Coordinates Equatorial -> Azimuthal Coordinates |
-"ECL" "EQ" & "AZ"
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 2023/05/24 at 16h41m TT
-Enter the
day of the month and the time expressed in
Terrestrial Time ( TT )
24
ENTER^
16.41 XEQ "ECL"
>>>>
L0 = 63°267284
= R01
RDN B0 =
0°000199
= R02
RDN R0 =
1.01264765 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 [ 2023/04/30 0h TT , 2023/06/01
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 2023/05/24 at 16h41m
TT
After executing "ECLI"
XEQ "EQ" or simply
R/S if you've just executed "ECL"
>>>>
RA0 =
4h04m56s40 = R01
RDN Decl 0 = 20°48'31"38
= R02
RDN eps = 23°438270
= 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 |
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 2023/05/24 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
which are
the topocentric coordinates of the Sun.
>>> We also have the local sidereal time in R32 = LST
= 1h00m28s40
Notes:
-Cf paragraph 4°) for the other results.
-The difference TT - UTC = 69.184 seconds.
-> h0 is often meaningless
when h < 0
Celestial Body | Registers | "ECL" | "EQ" | "AZ" |
SUN | R01 R02 R03 |
63.267284 0.000199 1.01264765 |
4.045640 20.483138 unchanged |
95.030593 47.261669 47.270901 |
MOON | R04 R05 R06 |
121.040807 5.199496 0.0026964117 |
8.180311 24.594346 unchanged |
58.455268 -1.594821 -0.562971 |
MERCURY | R07 R08 R09 |
39.237321 -3.570764 0.74433325 |
2.320036 11.110366 unchanged |
131.170075 59.292922 59.300280 |
VENUS | R10 R11 R12 |
108.223142 2.784289 0.79652070 |
7.203657 24.572413 unchanged |
66.185303 9.313369 9.365808 |
MARS | R13 R14 R15 |
122.315522 1.539451 1.94199924 |
8.194956 21.083144 unchanged |
61.331140 -3.472073 -3.472073 |
JUPITER | R16 R17 R18 |
31.797953 -1.076931 5.79076500 |
2.000278 11.051527 unchanged |
145.024463 63.525068 63.531862 |
SATURN | R19 R20 R21 |
336.738476 -1.493097 9.80743593 |
22.360894 -10.252834 unchanged |
-135.262412 34.214256 34.230576 |
URANUS | R22 R23 R24 |
49.791589 -0.311233 20.63143323 |
3.094377 17.230409 unchanged |
110.484051 56.552667 56.560379 |
NEPTUNE | R25 R26 R27 |
357.316963 -1.201253 30.30744511 |
23.520378 -2.100896 unchanged |
-152.102665 50.585531 50.594149 |
PLUTO | R28 R29 R30 |
300.243352 -2.563046 34.21202701 |
20.120633 -22.360290 unchanged |
-118.302077 1.284274 1.472709 |
True obliquity of the ecliptic | R31 |
/ |
23.438270 |
unchanged |
Local Sidereal
Time |
R32 |
/ |
/ |
1.002840 |
-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 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