Relationships of the Geometry, Conservation of Energy and Momentum

of an object in orbit about a central body with mass, M.
G = gravitational constant = 6.674x10-11N.m2/kg2
A very fundamental constant in orbital mechanics is k = MG

More convenient units to use in Solar System Dynamics are AU for distance and years for time
1 AU = 1.4960x1011 metres
1 year = 3.1558x107 seconds


Central
Body
Mass k = MG
Sun MS = 1.9891x1030 kg 1.3275x1020 m3/s2
39.487 AU3/yr2
887.4 AU km2/s2
Earth 5.974x1024 kg
1/332960 MS
3.987x1014 m3/s2
3.987x105 km3/s2
0.00011860 AU3/yr2
Jupiter 1.8987x1027 kg
1/1047 MS
1.2672x1017 m3/s2
1.2672x108 km3/s2
0.05648 AU3/yr2

The equations below enable one to determine the motion and timing of an object in orbit. The time for travel along an orbit can be determine by calcuation of the difference of the mean anamolies, M, of the two points.

TYPE ORBIT Circular[1] Elliptical Parabolic[2] Hyperbolic
Eccentricity, e e = 0 0 < e < 1 e = 1 e > 1
Semi-major Axis, a a > 0 a > 0 a = undefined
see peri-helion/-gee
a < 0
Periapsis
Distance, q
q = a q = a (1 - e) q
defining parameter
q = a (1 - e)
Parameter, p
Semi-latus rectum
p = a p = a(1-e2) p = 2q p = a(1-e2)
Total Energy, E E = - k/2a E = - k/2a E = 0 E = -k/2a
Distance from
Central Body
r = a r = p/[1 + e cos(θ)] r = 2q/[1 + cos(θ)] r = p/[1 + e cos(θ)]
Velocity, v
θ = angle from periapsis
(true anamoly)
v2 = k(1/a)
v = constant
v2 = k(2/r - 1/a)
v2 = [k/p](1+e2-2 cos(θ))
v2 = k(2/r)
v2 = (k/q)[1 + cos(θ)]
v2 = k(2/r - 1/a)
v2 = [k/p](1+e2-2 cos(θ))
Angle of Velocity, φ
relative to the
perpendicular to
the radial direction
φ = 0 tan(φ) =
[e sin(θ)/(1 + e cos(θ))]
φ = θ/2 tan(φ) =
[e sin(θ)/(1 + e cos(θ))]

θmax = 1/cos(e)
Periapsis
Velocity, vq
vq = [k/a]1/2 vq = [(k/a)(1+e)/(1-e)]1/2 vq = [2k/q]1/2
This is the 'escape' velocity
vq = [(-k/a)(1+e)/(e-1)]1/2
Areal Velocity, A A = [ka]1/2 A = [ka(1+e)/(1-e)]1/2 A = [kq/2]1/2 A = [-ka(1+e)/(e-1)]1/2
Orbit Period, P P = 2π[a3/k]1/2 P = 2π[a3/k]1/2 undefined undefined
Eccentric
Anomaly, E, D, F
Units = radians
E = θ [e + cos(θ)]
cos(E) = -------------------
[1 + e cos(θ)]
D = tan(θ/2) [e + cos(θ)]
cosh(F) = ------------------
[1 + e cos(θ)]
Mean Anomaly, M M = E M = E - e sin(E) M = D +D3/3 M = e sinh(F) - F
Time from Periapsis, t - to t - to = M P/(2π)
t-to = [a3/k]1/2 M
t - to = M P/(2π)
t-to = [a3/k]1/2 M
t - to = [2 q3/k]1/2 M t - to = [(-a)3/k]1/2 M

Note 1: Circular Orbits are a special case of Elliptical orbits
The relationships can be determined from the Elliptical orbit equations by subsituting:
r = a and e = 0.
I provide them here for comparison.

Note 2: The Parabolic Orbit is very long stretched Elliptical Orbit and cannot be characterized by a semi-major axis or eccentricity. It is determined only by its periapsis distance from the central body.


Reference: "Fundamentals of Astrodynamics"
by R.R.Bate, D.D.Mueller, and J.E. White
Dover Publications (1971)

Note: I have used a different definition for the Parabolic Eccentic Anamoly, D, than the standard references so that the Parabolic Mean Anamoly is dimensionless.


Larry Bogan
25 Jan, 2004