# VSOP87

by Phill Edwards

## VSOP Data

VSOP87 (Variations Séculaires des Orbites Planétaires) defines the analytical solutions to the motions of the planets Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus and Neptune. The Moon is so large compared to the Earth that the Moon doesn't actually orbit the Earth. Rather the Earth and Moon orbit around their common centre of mass - the Barycenter. So VSOP87 defines the motion of the Earth-Moon Barycenter (EMB).

The main version of VSOP87 defines elliptic variables for the planets as a function of time.

The VSOP data provides the following elliptic variables:

1. A the semi-major axis in Astronomic Units (AU where 1AU = 149597870.7km)
2. L the mean longitude
3. k = e cos π where e is the eccentricity and π is the longitude of the perihelion
4. h = e sin π
5. q = sin i/2 cos Ω where i is inclination and Ω is the longitude of the ascending node
6. p = sin i/2 sin Ω

The calculations are based on the time in Julian centuries after the J2000 epoch 2000-01-01 12:00:00.
T = (JDE - 2451545)/365250

The data is organised a sets of periodic and poisson series. The data in represented in two equivalent ways, each of which uses a different series. Only poisson series will be used. For each of the 6 elliptic variables there is a series corresponding to values of α the power of T between 0 and 5. For each power of α a series of values Ai, Bi and Ci are defined.
$\mathrm{variable}=\underset{\alpha =0}{\overset{5}{\Sigma }}{T}^{\alpha }\underset{i=1}{\overset{N}{\Sigma }}{A}_{i}\mathrm{cos}\left({B}_{i}+{C}_{i}T\right)$

## Orbital Elements

There are six orbital elements. Each of these, when denoted with a subscript 0, defines a measured value of the parameter at the start of the epoch. The current epoch, known as J2000, began on 2000-01-01 12:00:00

Orbital parameters are defined relative to a reference direction which lies in the plane of the ecliptic - generically defined as the reference plane. This reference direction is the position of the Vernal Equinox, often denoted by the symbol γ (X in the diagram)

.

From the six elliptic variables the main orbital parameters can be derived.

1. Semi major axis a in kilometers
2. Orbit eccentricity $e=\sqrt{{h}^{2}+{k}^{2}}$
3. Orbit inclination from the ecliptic $i={\mathrm{cos}}^{-1}\left(1-2\left({p}^{2}+{q}^{2}\right)\right)$
4. Longitude of the descending node $\Omega ={\mathrm{tan}}^{-1}\frac{p}{q}$
5. Argument of perihelion ω = π - Ω
6. Mean longitude L

The following useful parameters can be calculated from the orbital parameters:

1. Mean anomaly M = L - π, the angle between the true planet and perihelion
2. True anomaly ν = M + C, the angle between the mean body and perihelion
3. Equation of the Centre C = ν - M, the difference between the true anomaly and the mean anomaly
The true anomaly is calculated by solving Kepler's Equation.