# 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:

$$
\begin{align}
J_2 &= 0.00108263 \\
J_3 &= -2.33936 \times 10^{-3} J_2 \\
J_4 &= -1.49601 \times 10^{-3} J_2 \\
J_5 &= -0.20995 \times 10^{-3} J_2 \\
J_6 &= 0.49941 \times 10^{-3} J_2 \\
J_7 &= 0.32547 \times 10^{-3} J_2
\end{align}
$$

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:

$$
\vec{\ddot{r}} = -\frac{\mu}{r^3} \vec{r} + \vec{a}_P
$$

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