= Gauss planetary equation for the semi-major axis
{c}
{title2=$\dot a$}
For perturbing <acceleration> $R\widehat{\boldsymbol r}+T\widehat{\boldsymbol\theta}$ on an <osculating orbital element> system with fixed central parameter $\mu$,
$$
\dot a=2\sqrt{\frac{a^3}{\mu(1-e^2)}}[Re\sin f+T(1+e\cos f)].
$$
To prove it, differentiate the <specific orbital energy> $-\mu/(2a)$, obtaining $\dot a=(2a^2/\mu)(R\dot r+Th/r)$. Substitute $\dot r=\mu e\sin f/h$, $h/r=\mu(1+e\cos f)/h$ and $h^2=\mu a(1-e^2)$. Here $f$ is the <true anomaly>; a normal perturbing <acceleration> does no instantaneous <work>.
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