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General Equations

These general formulae govern all types of orbital motion in the gravitational two-body problem, including both bound and unbound orbits. More specialized formulae, valid only for certain types of orbits, can be derived from these.

Specific Specific Distance Speed Pericenter

Energy Angular Momentum & Apocenter

$E_B = - \frac{GM}{2a}$ $h = \sqrt{GMa(1 - e^2)}$ $r = \frac{a (1 - e^2)}{1 + e \cos f}$ $v = \sqrt{GM \left( \frac{2}{r} - \frac{1}{a} \right)}$ $\stackrel{\textstyle{\, q \, = a(1 - e)}}{Q = a(1 + e)}$

( & are per unit reduced mass; ; & are relative coordinates)

Derek C. Richardson 2005-09-16

Specific	Specific	Distance	Speed	Pericenter
Energy	Angular Momentum			& Apocenter
$E_B = - \frac{GM}{2a}$	$h = \sqrt{GMa(1 - e^2)}$	$r = \frac{a (1 - e^2)}{1 + e \cos f}$	$v = \sqrt{GM \left( \frac{2}{r} - \frac{1}{a} \right)}$	$\stackrel{\textstyle{\, q \, = a(1 - e)}}{Q = a(1 + e)}$