# The Effect of the Spherical Harmonic Gravitational Potential on Satellite Orbits

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The Earth is not a perfect sphere, as its rotation causes it to be slightly flattened at the poles. This oblateness can be modeled as a extra band of material encircling the equator. The extra material exerts torque on an orbiting satellite, causing the orbital planes to slowly precess in space rather than being fixed in an inertial frame. The oblateness torque will also cause the line of the apsides to rotate slowly in the orbital plane itself. This Demonstration shows the effect of the Earth's asphericity on the mean orbit elements.

Contributed by: Pradipto Ghosh (December 2008)

Open content licensed under CC BY-NC-SA

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## Details

The gravity potential due to the oblate Earth can be expressed as

,

where denotes the radial distance from the Earth's center, is the standard gravitational parameter, is the equatorial radius, are the zonal gravitational harmonic coefficients, and are the Legendre polynomials. The harmonic, known as the oblateness perturbation, is the dominant harmonic and causes significant precession of the near-Earth satellite orbits. The perturbation affects the orbital elements right ascension of the ascending node , inclination , argument of the periapsis , semi-major axis , eccentricity , and initial mean anomaly in the following three ways: (1) shortâperiod oscillations; (2) long-period oscillations; and (3) secular drift. For long-term investigation, the secular drift is of special importance and this is what this Demonstration studies. The effect of on the mean orbit elements can be expressed by the following differential equations:

,

,

,

,

,

.

These equations imply that for a critical orbit inclination of 63.4249 degrees, no drift in the mean argument of perigee occurs.

In this Demonstration, an orbit with the following initial conditions is considered:

, , , , , and .

The differential equations were solved for 500 orbital periods (one orbital period for this orbit is 2.76445 hours). An orbit is synthesized from the orbital elements (which are continuous time functions) at the end of each orbital period and rendered on the screen. It can be seen that for the inclined orbit, both the line of nodes and the periapsis regress, whereas for the orbit with the critical inclination, only the nodal line shows secular drift.

Reference: H. Schaub and J. L. Junkins, *Analytical Mechanics of Space Systems*, Reston, VA: American Institute of Aeronautics and Astronautics, 2003.

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