ELIT: How to calculate ridgid-body orbital velocity.

[Whirligig Girl]

Explain Like I'm a Teenager and have yet to take calculus: (oh, wait...)

I am in the early process of creating a game, and we need to know how to find the orbital velocity of a moon of a planet that is in orbit of the parent body. One of the features in the game is the small size of the planets. Still much larger than SoTS planets, but much smaller than Kerbal or Simple Rockets planets. The Earth-Analog is smaller than Gilly. (Let that sink in for a moment)

The Planet's stats:

Diameter: 12 km

Atmosphere Height: 2.5 km

Atmospheric Pressure ASL: 1

GeeASL: 1 (9.81 m/s²)

Semi-Major Axis: Irrelevant as of now.

The Moon's stats:

Diameter: 3.4 km.

No Atmosphere

GeeASL: 0.166 (1.622 m/s²)

Semi-Major Axis (Circular Orbit): 38.44 km

Let's assume the laws of gravitation are the same in this universe, but densities just happen to be much higher, for the sake of the simulation. I know the mass is needed, so how do I calculate the Mass of each planet and then (most importantly) the orbital rigid-body velocity of the moon.

Edited August 12, 2014 by GregroxMun

[Z-Man]

No need to go over the mass of the bodies! You only need to know two laws. The first is that gravity acceleration goes down with the square of the distance:

I : a_G® = C / r²

With some constant C. The second law is that of the centripetal acceleration of circular motion:

II : a_ctp® = v² / r

From the surface gravity of Earth and equation I, you extract C:

C = surface acceleration * (diameter/2)²

Then, for r = orbital radius R of the moon, you set a_G and a_ctp equal, because that is the state of equilibrium (faster or slower movement would yield an elliptic orbit, escape or crash)

C / R² = v² / R

=> v = sqrt(C / R)

And you'll probably find that it's hilariously fast.

[N_las]

First we need to calculate the mass of the Planet. We don't actually caluclate the numbers, we just need the Formula.

G*M*m/r² = m*g

G*M/r² = g

M = g *r^2 / G

G = Gravitational konstant

M = Mass Planet

m = probe body on the planet surface

r = Radius Planet

g = gravitational acceleration at Sea Level (for the Planet)

The centrifugal force on the moon must be the same as the gravitational pull from the planet:

M_moon v^2/R = G*M*M_moon/R²

v^2 = G*M/R

M_moon = Mass of the moon

v = orbital velocity of the moon

R = orbital radius of the moon

If we substitute M for the expression of the planet mass, we get:

v = (g*r²/R)^0.5

We see that the Masses of the Planet or the Moon, and even the gravitational constant of your universe doesn't matter for this problem.

v = ((9.81 m/s²) * (6km)^2 / (38.44 km))^0.5

v = 95.85 m/s

So a month in your Universe would be 42 minutes long.

Edited August 12, 2014 by N_las

[Whirligig Girl]

Alright, thanks for the great help N_las and Z-man. I used Universe Sandbox to help with some calculations and visualizations. Funny, eh?