# 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 mass E = - k/(2a) Angular Momentum, L, per mass L = √[k a (1 - e2)] Period of planet, P(years)with a in AU P = a3/2 Period √[a3/k]/(2π) Momentum per mass L = r x v = √[k a(1 - e2)] Tisserand Parameterfor 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 Velocity v = √(k/a)*(2a/r -1) mean velocity (circular) in orbit vc = √(k/a) = 2πa/P Eccentric Anomaly, E cos(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 InfluencePlanet, 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)]