Deviations of various kinds can be caused if the central gravitational force is not the only one acting on the satellite. It may also deviate if the satellite does not move in the equatorial plane of the rotating central body, or if the latter is not spherical but oblate. All of these cause periodic disturbances in the satellite's motion.
The period \(P_+\) of a satellite which is slightly disturbed from its elliptic path can be calculated from its major semi axis \(a_+\), using equation similar to that of \(T_0\) for the unperturbed motion.
$$T_0 = 2\pi\sqrt{\frac{a^3}{Gm}}$$
Here \(a\) is the major semi axis of the unperturbed motion and \(T_0\) is the corresponding time of revolution. \(P_+\) is related to \(a_+\) by
$$P_+ = 2\pi\sqrt{\frac{a_+^3}{Gm}}=T_0\sqrt{\frac{a^3}{a^3_+}}=T_0 \left( \frac{1+e'}{1+e} \right)^{3/2}$$
where \(e'\) is the eccentricity of the disturbed motion and \(e\) that of the unperturbed motion.
The satellite's position will precess, meaning that the major axis will turn slowly in the orbit plane from what would be the major axis of the unperturbed motion. The speed of that rotation is given by
$$\omega_a=\frac{2\pi}{P_+}-\frac{2\pi}{P_e}=\frac{2\pi}{T_0}\left(\frac{3}{2}e \cos i \sqrt{\frac{a}{GM_e}} + \frac{3n_e R_E^2 a cos i}{2GM_e a}\right)$$
Where:
- \(\omega_a\) is the precession angular velocity.
- \(P_e\) is the period of the Earth's rotation: \(P_e=24\) hours.
- \(G\) is the gravitational constant: \(G=6.67\cdot 10^{-11}\text{ m}^3\text{ kg}^{-1}\text{s}^{-2}\).
- \(a\) is the semi-major axis.
- \(M_e\) is the mass of the Earth: \(M_e=5.98\cdot 10^{24}\text{ kg}\).
- \(R_e\) is the Earth radius: \(R_e=6.38\cdot 10^6\text{ m}\).
- \(i\) is the inclination of the orbit with respect to the equatorial plane.
