A very fundamental constant in orbital mechanics is

More convenient units to use in Solar System Dynamics are AU for
distance and years for time

1 AU = 1.4960x10^{11} metres

1 year = 3.1558x10^{7} seconds

CentralBody |
Mass |
k = MG |

Sun |
M_{S} = 1.9891x10^{30} kg |
1.3275x10^{20} m^{3}/s^{2}39.487 AU ^{3}/yr^{2}887.4 AU km ^{2}/s^{2} |

Earth |
5.974x10^{24} kg^{1}/_{332960} M_{S} |
3.987x10^{14} m^{3}/s^{2}3.987x10 ^{5} km^{3}/s^{2}0.00011860 AU ^{3}/yr^{2} |

Jupiter |
1.8987x10^{27} kg^{1}/_{1047} M_{S} |
1.2672x10^{17} m^{3}/s^{2}1.2672x10 ^{8} km^{3}/s^{2}0.05648 AU ^{3}/yr^{2} |

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-e^{2}) |
p = 2q | p = a(1-e^{2}) |

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) |
v^{2} = k(1/a)v = constant |
v^{2} = k(2/r - 1/a)v ^{2} = [k/p](1+e^{2}-2 cos(θ)) |
v^{2} = k(2/r)v ^{2} = (k/q)[1 + cos(θ)] |
v^{2} = k(2/r - 1/a)v ^{2} = [k/p](1+e^{2}-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))] θ _{max} = cos^{-1}(1/-e) |

Periapsis Velocity, v _{q} |
v_{q} = [k/a]^{1/2} |
v_{q} = [(k/a)(1+e)/(1-e)]^{1/2} |
v_{q} = [2k/q]^{1/2}This is the 'escape' velocity |
v_{q} = [(-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[a^{3}/k]^{1/2} |
P = 2π[a^{3}/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 +D^{3}/3 |
M = e sinh(F) - F |

Time from Periapsis, t - t_{o} |
t - t_{o} = M P/(2π)t-t _{o} = [a^{3}/k]^{1/2} M |
t - t_{o} = M P/(2π)t-t _{o} = [a^{3}/k]^{1/2} M |
t - t_{o} = [2 q^{3}/k]^{1/2} M |
t - t_{o} = [(-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.

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.

25 Jan, 2004