Preface

Working on my astrodynamics toolset software library, I faced a need to implement Earth gravity field model, for calculation of non-spherical gravity acceleration to count it during satellite orbit propagation.

Motivation

In reality the gravity field of our Earth is not uniform. Shape of the Earth and heterogeneous structure of Earth crust causes minor but measurable differences in gravity acceleration. This differences, despite their low order, affect trajectories of artificial satellites. This can be same time bad or useful (for example, gravitational effect of Earth equatorial bulge makes possible repeating SSO).

Proper and accurate gravity model, providing precise nonspherical gravity acceleration values, is necessary tool for any mission analyst.

Approach

During my studies of astrodynamics, I am following an instructions from the textbook “Fundamentals of Astrodynamics and Applications”, Fourth Edition, by David Vallado.

Same as with many other astrodynamical problems, I started from this data source.

In the book Mr. Vallado provided and explained formulations of aspherical gravity potential of the Earth and the logic of modern gravity models. The core idea of this models is a usage of pretty complex math relations, expressing non-spherical component of planet gravity in a form of spherical harmonics expansion. There are other approaches like MacCullagh’s geometric method, but spherical harmonics expansion method, as far as I understood, dominating nowadays. Geodetic and astronomical institutions typically models the gravity of Earth and other planets this way.

Difficulties

As I mentioned above, for modelling Earth gravity, modern gravity models like EGM 2008 utilize method calculating gravity through expansion of spherical harmonics. Here I faced number of troubles. Expansion of spherical harmonics expresses with Associated Legendre Polynomials. Implementing them in my software toolset, I faced difficulties with:

• Complex and hard to debug way of calculation of Associated Legendre Functions
• Computation troubles with high orders/degrees of ALF
• Non-intuitive methods of normalization of ALF

Solutions required time and deep studying of gravity model principles and associated math problems, including properties of Legendre polynomials ans associated math, algorithms of effective ALF computation and methods of spherical harmonics normalization.

And of course, during implementation I faced a number of ordinary but irritating bugs, typical for such complex software.

Result

Despite all of difficulties, I implemented fast and accurate Earth gravity model, utilizing approach of EGM 2008/2020.

Model is able to calculate planetary gravity with precision of order/degree 2190/2159, and able to operate with gravity models of other planets.

Here, on Fig. 1 presented the map of non-spherical gravity accelerations of Earth with filtered J2-J6 factors, calculated over mesh size 720×360 with order/degree 72/72. The calculation (made with mid-level gaming/gamedev laptop) required ~6.5 hours:

Note that in the beginning of this post presented other gravity map with filtered J2-J6 factors and same mesh size, but calculated with order/degree 41/41. It required about 530 sec for calculation.

Below presented 3 examples of gravity maps generated by my gravity model implementation. Here,

• Fig. 2 – calculation counting J2-J6 factors only
• Fig. 3 – calculation uses EGM2008 with order/degree 10/10
• Fig. 4 – J2-J6 factors filtered from EGM2008 10/10

Calculation time required for plotting each figure given in brackets in the captions of figures. In all pictures results rounded for better representation of gravity structure.

Further steps

During implementation of this algorithms I faced number of interesting math problems and development troubles, especially with:

• Questions of precise computation of extremely big/extremely small values.
• Implementation of complex math functions (ALF & fnALF).
• Technological limitations of modern software and hardware.

In further posts I am planning to describe interesting features of this problems.

Additionally, I am planning to make a practical tests of implemented gravity models with gravitational field of other planets.