Planets - Coordinates
In 1987, Bretagnon and Francou make the VSOP87 Theory (Variations Séculaires des Orbites Planétaires) that contains, in three different series, all the terms necessaries to the calculation of the heliocentric positions of the planets::
L, the ecliptic longitude;
B, the ecliptic latitude;
R, the radius vector (distance to the Sun).
In the table regarding to the VSOP87 Theory, we find: name of the planet; the series identifier (L, B or R) and the number of the term of the series. These series contains many terms and is not possible to show all of them, so a few of them are bellow:
|
Planet |
Series |
Nº |
A |
B |
C |
|
Earth |
L0 |
1 |
175347046 |
0 |
0 |
|
|
|
2 |
3341656 |
4.6692568 |
6283.07585 |
|
|
|
3 |
34894 |
4.62610 |
12566.1517 |
|
|
|
... |
... |
... |
... |
|
|
|
64 |
25 |
3.16 |
4690 |
|
|
L1 |
1 |
628331966747 |
0 |
0 |
|
|
|
2 |
206059 |
2.678235 |
6283.075850 |
|
|
|
3 |
4303 |
2.6351 |
12566.15717 |
|
|
|
... |
... |
... |
... |
|
Planet |
Series |
Nº |
A |
B |
C |
|
Venus |
R2 |
1 |
1407 |
5.0637 |
10213.28555 |
|
|
|
2 |
16 |
5.47 |
20426.57 |
|
|
|
3 |
13 |
0 |
0 |
|
|
R3 |
1 |
50 |
3.22 |
10213.29 |
|
|
R4 |
1 |
1 |
0.92 |
10213.29 |
The L0, L1,... series are necessaries to calculate the heliocentric ecliptic longitude L, the series B0, B1,... are needed to the ecliptic latitude B determination and the series R0, R1,... are for the calculation of the radius vector R.
Each line of the table represents a periodical term and contains four members: the number of the series term and three numbers at we call A, B and C.
Let JDE be the Julian Ephemeris Day corresponding to a given instance. First we should calculate the time t measured in julian millenniums since the epoch J2000.0:
t = (JDE - 2451545.0)/365250
The value of each term are:
Acos(B + Ct)
In the table, the values for B
and C are in radian, and the value for A it's in 10-8
radians in the case of the longitude and the latitude and 10-8
astronomical units in the case of the radius vector.
To obtain the Heliocentric ecliptic longitude L of a planet at a given instance referred to the equinox of date, it's necessary to follow the next steps: First we calculate the sum L0 of the terms of the L0 series, the sum L1 of the L1 series, ... So we have the longitude as the result of the following expression:
L = (L0 + L1t + L2t2 + L3t3 + L4t4 + L5t5)/108
For the heliocentric latitude and the radius vector we should apply the same procedure.
However, these longitude L and latitude B, are referred to the dynamic mean ecliptic and the equinox of date, this reference are a quiet different of the one of FK5. So we must make the conversion between the two systems, taking T=10t and calculating L':
L' = L - 1º.397T - 0º.00031T2
The corrections for L and B are:
DL = -0''.09033+0''.03916(cosb + sinL')tanB
DB = +0''.03916(cosL' + sinL')
Note: These corrections are only necessaries when a higher accuracy is required!