# Equation of motion for Geocentric orbit

I'm simulating the motion of a body in a Geocentric orbit by integrating the equation of motion.

I'm using the following equations for the acceleration of an object in a gravitational field defined by the central mass given by $\mu$ and an axisymmetric oblateness described by $J_2$:

Motion will be a Keplerian orbit, perturbed by the effects of $J_2$.

For increased accuracy, I should also consider other effects of Earth geometry, i.e J22, J3, J4. How could I modify the equation?

I would appreciate if you give links to papers.

• +1 While I've touched on the spherical harmonics of the Geopotential in this answer this new question needs a more thorough and mathematical answer than I can easily provide. Hopefully a "gravity person" will be able to address this better. – uhoh Feb 12 '18 at 8:23
• @uhoh Do you know someone to help with this? – Tarlan Mammadzada Feb 14 '18 at 12:35
• @uhoh I mean, how to use J22, J3, J4 in equation? Edited the question – Tarlan Mammadzada Feb 14 '18 at 18:13
• @uhoh Yes, I'm learning, but checking results on real tests. Thanks, I'll look on SGP4. – Tarlan Mammadzada Feb 15 '18 at 5:03
• I just ran across this answer. You may find it very interesting and possibly helpful. The links/references are great! – uhoh Mar 8 '18 at 9:04

## 1 Answer

I would recommend picking up a basic Astrodynamics book such as Vallado's "Fundamentals of Astrodynamics and Applications". Depending on what your orbit altitude is (and what level of fidelity you are trying to get to), you will need to model many more gravity terms plus third body perturbations (sun and moon), drag, and solar radiation pressure.

• Thanks, but I'm asking only about Earth geometry effects – Tarlan Mammadzada Feb 18 '18 at 20:50