Orbital Calculations

Basics: A circular orbit

Before we go further, it is important to understand how a basic circular orbit works.

In a circular orbit, the orbiting object is in uniform circular motion.
Therefore, the object has a radial acceleration, A_r, of (v^2)/r.

We will have a perfectly circular orbit when our radial acceleration is equal to the acceleration due to gravity at the orbital altitude.

The force of gravity at a given radius from the center of earth, according to Newton's law of gravitation, is g=(GM)/(r^2), where G is the universal gravitation constant, and M is the mass of the parent body.
We will assume that the mass of the orbiting object is negligible compared to that of the parent body

Therefore, we can solve for the orbital velocity:

(v^2)/r = GM/(r^2)
v^2 = GM/r
v = sqrt(GM/r)

For an example, we will look at a 400km circular orbit:

Our radius is 400km + the radius of the earth, 6378km, and therefore 6738km
Radius is in meters, so this gives
r=6738*1000
r=6738000m

v = sqrt(GM/6738000)

G is given as 6.67384x10^-11 Nm^2/kg^2

M, the mass of the earth, is 5.972x10²⁴kg

Therefore, we get our velocity or our orbit, as v=7690.99 m/s

As for our other orbital parameters, a circular orbit has an eccentricity of 0

To get our period, we will take the distance of our orbit, 2πr, and divide by our velocity, to get seconds

(2*π*6738000)/7690.99 = 5504.64s, or around 91.74 minutes

Continuing: Calculating Orbital Parameters

As it is required for other calculations, we will first look at the semi-major axis:
As the semi-major axis is the average distance from the center of the parent body, we get:

SM_a= (R_{a +}R_p)/2

From our semi-major axis, we can calculate the period of our orbit. From the circular orbit calculations, the period of any orbit can be generalized as:

T=2π*sqrt((SM_a^3)/GM)

As previously mentioned in The basics of an Orbit, the orbit's eccentricity is a measure of how "circular" the orbit is.
We can calculate this as:

e=(R_a-R_p)/(R_a+R_p)

when R_a and R_p are equal, as with a circular orbit, we get e=0. As the orbit becomes more elliptic, e approaches 1, thus we say an orbit is more eccentric the less circular it gets