Plan a Transfer From a Parking Orbit to Geostationary Orbit

Geostationary orbit is the most valuable piece of real estate in space: a satellite there hangs over one spot on the equator, so a fixed dish on the ground can point at it for years. Getting there is a three-step problem that a student, a science writer or a simulation player can work through with four of the calculators on this site. This guide follows one invented satellite from a low parking orbit to its final slot and keeps every number traceable to a formula.

The mission and its inputs

The satellite has a dry mass of 2,000 kg, meaning everything except the propellant for its own apogee engine. The launch vehicle drops it into a circular parking orbit at 185 km altitude, inclined 28.5° to the equator, which is what an eastward launch from Cape Canaveral naturally produces. The apogee engine burns hypergolic propellants with a specific impulse of 320 seconds. All calculations use Earth’s gravitational parameter of 398,600.4418 km³/s² and its equatorial radius of 6,378.137 km. In the parking orbit the satellite moves at 7.793 km/s and completes a lap in 88.19 minutes.

Step 1: find the target altitude

A geostationary satellite must orbit once per sidereal day, 23.9344696 hours. Entering that period into the orbital period calculator, solving for altitude, gives an orbit radius of 42,164.2 km and an altitude of 35,786 km, with a circular speed of 3.075 km/s. Using 24 hours by mistake would place the target about 77 km too high and the satellite would drift westward by nearly a degree a day. The difference is small on paper and large for the operator who has to buy back the drift with propellant.

Step 2: size the Hohmann transfer

With a starting altitude of 185 km and a target of 35,786 km, the Hohmann transfer calculator returns a transfer ellipse with a semi-major axis of 24,364 km and an eccentricity of 0.731. The first burn takes the satellite from 7.793 to 10.252 km/s, a Δv of 2.459 km/s. This burn is usually made by the launch vehicle’s upper stage, which leaves the satellite in what is called a geostationary transfer orbit. The coast up to apogee takes 5 hours 15 minutes, half of a 10.51-hour ellipse. At apogee the satellite is moving at only 1.596 km/s, and a second burn of 1.479 km/s would circularise it at 3.075 km/s. The coplanar total is 3.938 km/s.

Step 3: add the plane change

The parking orbit is tilted 28.5°, and a geostationary orbit must have an inclination of zero. Turning a velocity vector costs Δv = 2v sin(Δi/2). Doing it in the parking orbit, where v is 7.793 km/s, would cost 3.837 km/s on its own, almost as much as the whole transfer. Doing it separately after arriving at geostationary speed would cost 1.514 km/s. The efficient answer is to combine it with the circularising burn at apogee, where the satellite is slowest. The combined burn is the vector difference between the apogee speed of 1.596 km/s and the final 3.075 km/s at an angle of 28.5°: √(1.596² + 3.075² − 2 × 1.596 × 3.075 × cos 28.5°) = 1.837 km/s. The plane change has added only 0.359 km/s to the apogee burn, and the full budget from the parking orbit is 4.296 km/s.

Step 4: work out the propellant

The apogee engine must supply 1.837 km/s to a 2,000 kg satellite. In the rocket equation calculator, choose propellant for a target Δv, enter 320 s, a dry mass of 2 t and 1,837 m/s. The answer is 1.59 t of propellant, about 1,592 kg, so the satellite leaves the launch vehicle at about 3,592 kg. Without the plane change, the 1,479 m/s burn would need only 1,204 kg. The 388 kg difference is the price of launching from 28.5° north, and it is why operators pay attention to the latitude of the launch site and why some rockets fly a supersynchronous transfer that raises the apogee above 35,786 km to make the plane change cheaper still.

Common mistakes

• Adding the plane change and the circularising burn as separate numbers instead of combining them as vectors, which overstates the budget by more than a kilometre per second.
• Entering the orbit radius into an altitude field, which counts the Earth’s radius twice and moves the target to more than 42,000 km above the surface.
• Using the solar day of 24 hours for the target period.
• Leaving the apogee engine propellant out of the satellite’s launch mass when checking whether the rocket can lift it.
• Treating the instantaneous-burn answer as exact. A real apogee engine with modest thrust burns for tens of minutes, often split into several firings over successive apogees, and loses a little efficiency compared with the ideal figure.

Checklist

1. Fix the starting orbit: altitude, inclination and the dry mass of the spacecraft.
2. Find the target altitude from the sidereal period, 35,786 km for geostationary orbit.
3. Run the Hohmann transfer calculator for the coplanar burns and the transfer time.
4. Combine any plane change with the burn at the slowest point and compute it as a vector difference.
5. Convert the burn the spacecraft itself must make into propellant with the rocket equation, using its engine’s vacuum Isp.
6. Add a margin of a few percent for finite burns, steering and years of station keeping before quoting the final mass.