An Explosion of Equations(Orbital Edition)

Okay, so here is where I start hitting you with equations. There won’t be any requirement to derive these from basic principles(although, sometimes, that’s how I came up with them), so if you have a good calculator you can just plug-and-chug.

Variables and Constants

a = semimajor axis: The characteristic distance between an orbiting body and its primary. In the special case of a zero-eccentricity(circular is the big technical term), this is the radius of the orbit. Distance units.

e = eccentricity: A measure of the variation in the distance between an orbiting body and its primary. No units; it’s just a ratio.

P = orbital period: The time it takes for a body to complete its orbit around the primary. For the Earth or another planet, this is called a year; for a moon, this is called a month. Time units.

G = Gravitational constant: for distances in meters and time in seconds, the value is: G = 6.67259×10^-11 m^3 * s^-2 * kg^-1

D =: the distance at a given time between an orbiting body and its primary. This is constant with a circular orbit.

Dp = periapsis distance: the closest approach between an orbiting body and its primary. A special case of D for eccentric orbits.

Da = apoapsis distance: the farthest recession of an orbiting body from its primary. Another special case of D, again, for eccentric orbits.

m1 = mass of larger body or primary.

m2 = mass of smaller body. This would be the planet or moon.

v = the planet’s orbital velocity at a given time.

ϴ = theta: the angle between the current orbital position of a body and its position at periapsis.

mSol = mass of the Sun: mSol = 1.9891 x 10^30 kg

mEarth = mass of the Earth: mEarth = 5.9758 x 10^24 kg

PEarth = Earth’s sidereal year, or orbital period around the Sun: PEarth = 1 year = 31.5569259747 x 10^6 s = 365.242198781 days.

aEarth = Earth’s orbital distance: aEarth = 1 a.u. = 149.5979 x 10^9 m = 149.6 million km.

Period

P^2 = a^3*([4*π^2] / [G*(m1 + m2)])

Use the preceding where m1 and m2 are fairly close in value. As in most binary stars or the Earth/Moon system.

In cases where m2 is much smaller than m1, as with planet/sun relationships or when trying to determine the orbital period of your starship around a planet, m2 can be neglected, leading to…

P^2 = a^3*[(4*π^2) / (G*m1)]

Simplifying further, we can use mass in Solar masses, distance in a.u.’s and period in years:

(P/PEarth)^2 = (a/aEarth)^3 / (m1/mSol).

Handling Eccentricity

Note: This isn’t about what my wife has to do every day. We’re still on orbits, folks, stay focussed. 🙂

Periapsis:

Dp = a*(1 – e)

v = [(2*π*a) / P] * [(1 + e) / (1 – e)]^(1/2)

Apoapsis:

Da = a*(1+e)

v = [(2*π*a) / P] * [(1 – e) / (1 + e)]^(1/2)

General case:

D = [a*(1 – e^2)] / [1 + e*cos ϴ]

Now, figuring out the value of ϴ(theta) at a given time in the orbit is a bit complicated. Probably worthy of its own post.

Here is the velocity in the general case:

v^2 = 2*G*m1*[(1/D) – (1/[2*a])].

Center of mass:

Massive bodies will actually orbit around the center of mass between them. When one mass is significantly larger than the other, this can be treated as the larger body being stationary and the smaller body orbiting the larger one. If the two bodies are of the same mass, they both orbit on opposite sides of the midpoint between them. Earth orbiting the Sun can be treated as the second case. The same is true, for our purposes, of Jupiter orbiting the Sun. For the Moon orbiting the Earth, the first case would be technically preferable, but, again, for our purposes the first case will probably be sufficient.

Anyway, this is always true:

m1*r1 = m2*r2 and r1 + r2 = D,

where

m1 = the mass of the first body

m2 = the mass of the second body

r1 = the distance from the first body to the center of mass

r2 = the distance of the second body from the center of mass.

That’s it for now. If anyone can think of additional useful orbital equations for planetology, let me know.