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{16 Qq}{(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:
 * $$M = \frac{4\pi^2 r_s^3}{G P_s^2}$$

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

Its average density:
 * $$\rho = \frac{3M}{4\pi R^3}$$

Gravitational acceleration on the surface:
 * $$g = \frac{GM}{R^2}$$