Tutorial: Basic Orbiting (Math)

Target Speed for Stable Orbit

In the basic orbiting tutorial, you were introduced to the concept of orbiting, and basic orbit stabilization, as well as an orbital table to help you along. Now, what if you want an orbit that isn't on that table? What if you want to have an orbit with a specific period? That's where these formulae come in.

The relation between orbital speed and acceleration is given by the formula:

a = v² / r,

where a is the acceleration due to gravity, v is the horizontal speed, and r is the radius of orbit.

Of course, gravity varies depending on your distance from the planet, so we also need the following formula to determine a based on your altitude:

a = g * (R / (R + h))²,

where g is the acceleration due to gravity at sea level (9.807 m/s²), R is the radius of Kerbin (600 km), and h is the altitude of your orbit.

Note: From here, we will substitute R + h for r, since the radius of your orbit is equal to the radius of Kerbin plus your altitude.

Substituting for a and simplifying, we get:

g * (R / (R + h))² = v² / (R + h)

g * R² / (R + h) = v²

v = R * sqrt(g / (R + h))

Finally, substituting known values for g and R,

v = 600 000 m * sqrt(9.807 m/s² / (600 000 m + h)) ne. From the basic mechanics formula: d = v * t We know v from the above, and d is simply the circumference of a circle with a radius equal to your orbital altitude plus the radius of Kerbin: t = 2π * (600 000 m + h) / v

From these two formulas, you can easily find orbital speed, as well as orbital period, if you know your way around a calculator.