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Physics LibreTexts

14.4: Lagrange's Planetary Equations

  • Page ID
    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:

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