User:Pt/Formulae

Celestial mechanics

If a planet's perihelion is $q$ and aphelion $Q$, then the orbit's eccentricity is:

$e=\frac{Q-q}{Q+q}$

The longer semi-axis:

$a=\frac{Q+q}{2}$

And the shorter semi-axis:

$b=\sqrt{Qq}$

The total length of the orbit:

$\begin{array}{ccc}l& =& 2\underset{-a}{\overset{a}{\int }}\sqrt{1+\frac{b^2}{a^2}\cdot \frac{x^2}{a^2-x^2}}dx=\\ & =& 2\underset{-\frac{Q+q}{2}}{\overset{\frac{Q+q}{2}}{\int }}\sqrt{1+\frac{16Qq}{(Q+q{)}^2}\cdot \frac{x^2}{(Q+q{)}^2-4x^2}}dx=\\ & =& 2(Q+q)E\left(e^2\right)\end{array}$

(because the equation of the ellipse is $y=\pm b\sqrt{1-\frac{x^2}{a^2}}$ and $l=2\underset{-a}{\overset{a}{\int }}\sqrt{1+y{\text{'}}^2}dx$; $E$ is the complete elliptic integral of the second kind, EllipticE in Mathematica)

Mean orbital speed ($P$ is the planet's orbital period):

$\overline{v}=\frac{l}{P}$

If a planet has a satellite at distance $r_s$ with orbital period $P_s$, then the planet's mass is:

$M=\frac{4{\pi }^2{r}_s^3}{G{P}_s^2}$

Or, written another way:

$P_s=2\pi \sqrt{\frac{r_s^3}{GM}}$

The escape velocity from a planet with radius $R$ and mass $M$:

$v_{\mathrm{I}\mathrm{I}}=\sqrt{\frac{2GM}{R}}$

Its average density:

$\rho =\frac{3M}{4\pi R^3}$

Gravitational acceleration on the surface:

$g=\frac{GM}{R^2}$