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
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
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 |
-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