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14.4: Lagrange's Planetary Equations

Last updated

Mar 5, 2022
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- 14.3: The Poisson Brackets for the Orbital Elements
- 14.5: Motion Around an Oblate Symmetric Top

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We now go to Equation 14.2.8 to obtain Lagrange’s Planetary Equations, which will enable us to calculate the rates of change of the orbital elements if we know the form of the perturbing function:

$\begin{align} \dot{a} &= - \frac{2a^2}{GMm} \frac{\partial R}{\partial T} , \label{14.4.1} \\[5pt] \dot{e} &= - \frac{a(1-e^2)}{GMme} \frac{\partial R}{\partial T} , \label{14.4.2} \\[5pt] i &= - \frac{1}{\sqrt{GMm^2 a (1-e^2) \sin i}} \frac{\partial R}{\partial \Omega} - \frac{1}{me} \sqrt{\frac{1 - e^2}{GMa}} \frac{\partial R}{\partial ω} , \label{14.4.3} \\[5pt] \dot{ω} &= \frac{1}{me} \sqrt{\frac{1 - e^2}{GMa}} \frac{\partial R}{\partial e} - \frac{1}{\sqrt{GMm^2 a (1 - e^2)} \tan i } \frac{\partial R}{\partial i} , \label{14.4.4} \\[5pt] \dot{Ω} &= \frac{1}{\sqrt{GMm^2 (1 - e^2) \sin i}} \frac{\partial R}{\partial i} , \label{14.4.5} \\[5pt] \dot{T} &= \frac{2a^2}{GMm} \frac{\partial R}{\partial a} + \frac{a(1 - e^2)}{GMme} \frac{\partial R}{\partial e}. \label{14.4.6} \end{align}$

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