4.1.2 Gravity model formulation
Without compromising the general validity
of underlying theories, all subsequent gravity model discussions are focused on
the Earth. The gravity acceleration acting on a point mass, which is external
to the central body, is the gradient of the potential function U of
that body. The corresponding geopotential surface satisfies the so
called
The corresponding perturbing acceleration can be determined from equation (4‑3) by means of computationally efficient recursion algorithms (e.g. as in [RD.1]).
|
(4‑3) |
where
is the 2nd
time derivative of the position vector.
The solution U of the partial differential equation (4‑2) is typically written in the form of a series expansion, in terms of so-called surface spherical harmonic functions, for a location defined in spherical coordinates r, λ, ϕ.
|
(4‑4) |
GM = μ is the gravity constant of the Earth (M being its mass);
μ = 3,98604415×1014 m3s-2 for the EIGEN-GL04C model
ae is the mean equatorial radius of the Earth;
ae = 6 378 136,460 m for the EIGEN-GL04C model
r is the radial distance from centre of the Earth to satellite
N is the maximum degree of the expansion
l is the degree of a certain harmonic function
m is the order of a certain harmonic function
Clm, Slm are coefficients that determine amplitude and phase of a certain harmonic function
λ is the geodetic longitude of the sub-satellite point
ϕ is the geodetic latitude of the sub-satellite point
Plm are associated Legendre functions of the first kind, of degree l and order m; recurrance relations for these functions are available in the literature (e.g. [RD.1]).
A gravity model consists of adopted values for GM, ae, and a set of model coefficients Clm, Slm. Practical implementations of gravity models, e.g. for numerical integration of a satellite orbit, are typically interested in the gravity acceleration resulting from the potential function U in (4‑4). Corresponding partial derivatives of (4‑4) in Cartesian coordinates of an Earth-fixed system x, y, z can be computed recursively (see [RD.1]).
The model coefficients Clm, Slm are typically provided in their normalized versions, according to (4‑5) in order to limit their numerical range for higher degrees and orders.
|
(4‑5) |
The Legendre functions Plm (sin ϕ) in this case are normalized by the inverse of the square root in equation (4‑5).