14.4: Lagrange's Planetary Equations
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- 6878
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:
\[ \begin{align} \dot{a} &= - \frac{2a^2}{GMm} \frac{\partial R}{\partial T} , \label{14.4.1} \\[5pt] \dot{e} &= - \frac{a(1-e^2)}{GMme} \frac{\partial R}{\partial T} , \label{14.4.2} \\[5pt] i &= - \frac{1}{\sqrt{GMm^2 a (1-e^2) \sin i}} \frac{\partial R}{\partial \Omega} - \frac{1}{me} \sqrt{\frac{1 - e^2}{GMa}} \frac{\partial R}{\partial ω} , \label{14.4.3} \\[5pt] \dot{ω} &= \frac{1}{me} \sqrt{\frac{1 - e^2}{GMa}} \frac{\partial R}{\partial e} - \frac{1}{\sqrt{GMm^2 a (1 - e^2)} \tan i } \frac{\partial R}{\partial i} , \label{14.4.4} \\[5pt] \dot{Ω} &= \frac{1}{\sqrt{GMm^2 (1 - e^2) \sin i}} \frac{\partial R}{\partial i} , \label{14.4.5} \\[5pt] \dot{T} &= \frac{2a^2}{GMm} \frac{\partial R}{\partial a} + \frac{a(1 - e^2)}{GMme} \frac{\partial R}{\partial e}. \label{14.4.6} \end{align}\]