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 2024/02/29 0h TT to 2024/04/01
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 |
$ -EPH2024MAR 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 2024/03/24 at 16h41m TT
-Enter
the day of the month and the time expressed in
Terrestrial Time ( TT )
24
ENTER^
16.41 XEQ
"ECL" >>>>
L0 =
4°528742 = R01
RDN B0 =
-0°000002
= R02
RDN R0 =
0.99713400 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 [ 2024/02/29 0h TT , 2024/04/01
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 2024/03/24 at 16h41m
TT
After executing "ECLI"
XEQ "EQ" or
simply R/S if you've just executed "ECL"
>>>>
RA0 =
0h16m37s54 = R01
RDN Decl 0 =
1°47'59"31 = R02
RDN eps =
23°438746 = 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 2024/03/24 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
= 21h02m58s12
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 |
4.528742 -0.000002 0.99713400 |
0.163754 1.475931 unchanged |
114.214599 34.504429 34.520600 |
MOON |
R04 R05 R06 |
178.048341 1.594610 0.0027134354 |
11.552248 2.142137 unchanged |
-57.340293 -36.434618 -36.434618 |
MERCURY |
R07 R08 R09 |
23.080974 2.374185 0.90340098 |
1.214997 11.103104 unchanged |
94.245721 27.094376 27.113437 |
VENUS |
R10 R11 R12 |
345.817622 -1.376099 1.59799018 |
23.095595 -6.513987 unchanged |
137.115869 39.453058 39.463902 |
MARS |
R13 R14 R15 |
331.325199 -1.194152 2.11848143 |
22.150898 -12.070856 unchanged |
156.113972 41.224421 41.234885 |
JUPITER |
R16 R17 R18 |
45.815493 -0.839713 5.70821922 |
2.542443 15.460908 unchanged |
77.502212 10.201390 10.251435 |
SATURN |
R19 R20 R21 |
342.761534 -1.653146 10.63252423 |
22.585957 -8.174781 unchanged |
141.104967 40.031467 40.042239 |
URANUS |
R22 R23 R24 |
50.449302 -0.274828 20.28054054 |
3.122077 17.354477 unchanged |
73.571207 7.404931 7.472384 |
NEPTUNE |
R25 R26 R27 |
357.632993 -1.217730 30.89108019 |
23.531498 -2.033114 unchanged |
122.442095 36.304083 36.315770 |
True obliquity of the ecliptic | R31 |
/ |
23.438746 |
unchanged |
Local
Sidereal Time |
R32 |
/ |
/ |
21.025812 |
-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° ]
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