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:

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

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

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

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

If a planet has a satellite at distance $r_{s}$ with orbital period $P_{s}$, then the planet's mass is:

$M={\frac {4\pi ^{2}r_{s}^{3}}{GP_{s}^{2}}}$

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 $M$:

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

Its average density:

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

Gravitational acceleration on the surface:

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