Orbit Perturbations¶
\[
\vec{\ddot{r}} = -\frac{\mu}{r^3} \vec{r} + \vec{a}_p \;,
\]
where \(\vec{a}_p\) is the acceleration due to perturbations from thrust, atmospheric drag, gravitational perturbations, or any other force.
Atmospheric drag¶
Gravitational perturbations¶
The rotationally symmetric perturbation \(\Phi (r, \phi)\) is
\[
\Phi (r, \phi) = \frac{\mu}{r} \sum_{k=2}^{\infty} J_k \left( \frac{R}{r} \right)^k P_k (\cos \phi) \;,
\]
where \(J_k\) are the zonal harmonics of the planet, \(R\) is the equatoriall radius (\(R/r < 1\)), and \(P_k\) are Legendre polynomials of order \(k\).
For the Earth, the six zonal harmonics are:
\[\begin{split}
\begin{align}
J_2 &= 0.00108263 \\
J_3 &= -2.33936 \times 10^{-3} J_2 \\
J_4 &= -1.49601 \times 10^{-3} J_2 \\
J_5 &= -0.20995 \times 10^{-3} J_2 \\
J_6 &= 0.49941 \times 10^{-3} J_2 \\
J_7 &= 0.32547 \times 10^{-3} J_2
\end{align}
\end{split}\]
The perturbing acceleration is \(\vec{a}_P = - \nabla \Phi\).
The gravitational perturbation due to \(J_2\) is
\[
\vec{a}_P = \frac{3}{2} \frac{J_2 \mu R^2}{r^4} \left[ \frac{x}{r} \left( 5 \frac{z^2}{r^2} - 1 \right) \hat{I} + \frac{y}{r} \left( 5 \frac{z^2}{r^2} - 1 \right) \hat{J} + \frac{z}{r} \left( 5 \frac{z^2}{r^2} - 3 \right) \hat{K} \right]
\]
The gravitational perturbation due to \(J_3\) is
\[
\vec{a}_P = \frac{1}{2} \frac{J_3 \mu R^3}{r^5} \left[ 5 \frac{x}{r} \left( 7 \frac{z^3}{r^3} - 3 \frac{z}{r} \right) \hat{I} + 5 \frac{y}{r} \left( 7 \frac{z^3}{r^3} - 3 \frac{z}{r} \right) \hat{J} + \left( 35 \frac{z^4}{r^4} - 30 \frac{z^2}{r^2} + 3 \right) \hat{K} \right]
\]
Then, solving the equation of motion yields the perturbed orbit:
\[
\vec{\ddot{r}} = -\frac{\mu}{r^3} \vec{r} + \vec{a}_P
\]