Gauss planetary equations

At any given instance in time, a perturbed planetary orbit is completely determined by six osculating orbital elements. (See Section 10.2.) Let us choose these elements to be the major radius, $a$; the mean anomaly at epoch, ${\cal M}_0 = {\cal M}(t) -\int_0^t n(t')\,dt'$; the eccentricity, $e$; the argument of the perihelion, $\omega$; the inclination to the ecliptic, $I$; and the longitude of the ascending node, ${\mit \Omega }$. In the absence of a disturbing force, $a$, ${\cal M}_0$, $e$, $\omega$, $I$, and ${\mit \Omega }$ are all constants of the motion. As is clear from the analysis of the preceding two sections, in the presence of a disturbing force of the form (I.15), the osculating elements evolve in time as follows:

$\displaystyle \frac{\skew{3}\dot{a}}{a}$ $\displaystyle = \frac{2\,h}{\mu\,(1-e^{\,2})}\left[e\,\sin\theta\,F_r + (1+e\,\cos\theta)\,F_\theta\right],$ (I.53)
$\displaystyle \skew{5}\dot{\cal M}_0$ $\displaystyle = \frac{h}{\mu}\,\frac{(1-e^{\,2})^{1/2}}{e}\left(\left[\cos\thet...
... -\left[1+\frac{1}{(1-e^{\,2})}\,\frac{r}{a}\right]\sin\theta\,F_\theta\right),$ (I.54)
$\displaystyle \skew{3}\dot{e}$ $\displaystyle = \frac{h}{\mu}\left[\sin\theta\,F_r+(\cos\theta+\cos E)\,F_\theta \right],$ (I.55)
$\displaystyle \skew{3}\dot{\omega}$ $\displaystyle = -\frac{h}{\mu}\,\frac{1}{e}\left[\cos\theta\,F_r-\left(\frac{2+...
...theta\,F_\theta \right] -\frac{\cos I\,\sin(\omega+\theta)\,r\,F_z}{h\,\sin I},$ (I.56)
$\displaystyle \dot{I}$ $\displaystyle = \frac{\cos(\omega+\theta)\,r\,F_z}{h},$ (I.57)
$\displaystyle \skew{5}\dot{\mit\Omega}$ $\displaystyle =\frac{\sin(\omega+\theta)\,r\,F_z}{h\,\sin I}.$ (I.58)

These equations are known collectively as the Gauss planetary equations.