Keplerian Orbit Formulae Summary

Range of Parameters

		Eccentricity	Semi-major Axis, a	Energy
Ellipse		0 <= e < 1	a > 0			E < 0		(Includes Circular)
Parabola	e = 1		not defined		E = 0
Hyperbola	e > 1		a < 0			E > 0

Constants

G = 6.67x10-11 N.m2/kg2 
Masses in the Solar System
		Mass (Solar)	Mass (kg)	k = MG (AU (km/s)2
M(Sun)		1		1.989x1030	887.21
M(Jupiter	1/1047				0.84738
M(Earth)	1/332,946	5.974x1024	0.0026647

Orbit Energy and Momentum

Energy per massE = - k/(2a)
Angular Momentum, L, per massL = √[k a (1 - e2)]
Period of planet, P(years)
with a in AU		P = a3/2
Period√[a3/k]/(2π)
Momentum per massL = r x v = √[k a(1 - e2)]
Tisserand Parameter
for Planet, p		Tp = ap/a + 2*√[a/ap(1 - e2)]
Orbital Radius, r versus true anomaly
True Anomaly = angle, θ, from perihelion		r = a (1 - e2)/(1 - e cos(θ))
Orbital Velocityv = √(k/a)*(2a/r -1)
mean velocity (circular) in orbitvc = √(k/a) = 2πa/P
Eccentric Anomaly, Ecos(E) = (e + cos(θ))/(1 + e cos(θ))
Travel time in orbit (from perihelion) t - T = P/(2π)(E - e sin(E))
Radius of Sphere of Influence
Planet, p, relative to the Sun, S		RSOI = ap √[kp/(3 ks)]
Angle, φ, of the velocity vector
relative to circular 		tan(φ) = e sin(θ)/[1 + e cos(θ)]
where E is in radians (radian = 57.3 degrees)

Created on ... May 29, 2005 - L.Bogan