User:Pt/Formulae

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[edit] 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 rs with orbital period Ps, 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}