Geonsynchronous insertion

As mentioned earlier, a geosynchronous orbit where the orbital period of the satellite is such that it equals the rotation period of Earth and hence the satellite is always in the same location in the sky when seen from the ground is very suitable for communication satellites. However, this condition can only be reached over the equator, i.e. at zero orbital inclination - whereas typically launch sites are at higher inclinations. Thus, one can not launch a satellite directly into a geosync orbit, it has to be maneuvered there- which needs extra propellant.

Let's study how this works and how much propellant is needed.

The initial orbit

Typically, a satellite destined for a geosync orbit is delivered into a parking orbit by the launcher rocket and pushed from there into geosync orbit by means of a so-called apogee kick motor. The minimal inclination the parking orbit can have is the latitude of the launch site --- so part of the task is to lower inclination by out of plane burns.

The launch can however be designed such that the apogee of the initial orbit is already at the geosync radius (42.164 km) and is placed over the equator. In that way, the apogee kick motor (AKM) can be ignited for simultaneous circularization and inclination lowering at the apogee.

Since we want to start the simulation right after cutoff of the launcher rocket, how do we get a suitable state vector? This is in fact reasonably easy by solving the problem backward. We start with a position init at the equator with a course corresponding to the inclination and just follow the orbit to its periapsis, note the time at periapsis, re-run the simulation to that time and request a list of the final state vector.

Following this procedure, you can derive a state vector like the following (in SI units) which corresponds to a suitable initial orbit.

x 252234.81
y -10771694
z 4572309.
vx 5880.5259
vy -3817.935
vz 1620.616

The spacecraft

If we want to compute propellant consumption, we need to tell the simulation something about the spacecraft we want to simulate. Say out satellite (and the AKM) has mass of 800 kg. Some additional propellant is needed to move this to the desired orbit, so the total mass of the package is 1500 kg.

A typical liquid propellant AKM might deliver a thrust of 1.5 kN, and the specific impulse of such an engine is in the range of 300 s. These parameters are specified in the spacecraft section of the configuration file, and using the keyword 'rocket_engine' for the burn model activates the computation of acceleration and propellant usage from these parameters. Thus, add the following sections to the configuration file (or simply copy it from the config directory).

units SI
gravity_model J3
earth_model IERS
burn_model rocket_engine
plot2d_resolution 5000
plot3d_resolution 5000
max_time 300000.

mass 1500.0
propellant_mass 700.0
isp 300.0
thrust 1500.0

As far as the spacecraft is concerned, we can specify either total mass and propellant mass or dry mass and propellant mass - the code will compute the missing one on its own. Whenever we now specify burns, the propellant mass will be reduced as a result, and acceleration given the engine thrust is recomputed accordingly. When propellant runs out, no acceleration can be produced and an on-screen warning is written. Let's see this in action.

Insertion burns

Add the following sequence of burns to your file.

name Apogee-kick-1
time 17400.0
dx 500.0
dz 0.0
dy -300.0

name Apogee-kick-2
time 62750.0
dx 500.0
dz 0.0
dy -300.0

name Apogee-kick-3
time 121400.0
dx 470.0
dz 0.0
dy -320.0

They have been determined iteratively by studying at which time the apogee is reached and distributing the delta v requirements about evenly across the three burns. The sequence of three burns is chosen because trying to do a single burn means that the burn gets longer and less precisely just around the apogee, and hence it gets more wasteful and less precise. Using a liquid rocket engine we can in fact do this - using a solid motor, a single burn would have to be used. You can explore what happens when you try a single burn on your own.

If you run the file, you should end up with about 30 kg of remaining propellant, i.e. the insertion is fairly costly!

Now add the following blocks to the output section of your file:

file geosync_insertion_3d.dat
x x
y y
z z

file geosync_insertion_groundtrack.dat
x longitude
y latitude

The first one creates the full 3d trajectory which can be visualized with any 3d plotting tool - for the following picture, gnuplot has been used):

The perspective takes some getting used to, but the apoapsis is to the upper right where all the lines intersect, and with each burn it can be seen that the orbit moves more into the equatorial plane and that the periapsis (close to the viewer) moves further away from Earth.

Geosynchronous insertion using three apogee kick motor burns.

The second set of instructions generates the groundtrack - which for this orbit is really interesting.

For the specified parameters, it starts out somewhere over Mexico, first moves eastward and as the orbit reaches further out the motion slows down and goes westward, then moves out to India during the passage through the periapsis, has another apoapsis over Indonesia and finally creeps to a halt out over the pacific ocean (the last point really contains a full orbit with the specified parameters, so the satellite has indeed become geostationary as expected).

Groundtrack of the insertion solution.

From this point on, the problem transforms to a stationkeeping task, i.e. regularly execute burns that the satellite compensates for the various perturbations. So far from the surface of Earth, J2 and J3 are no longer the dominant perturbation, rather Sun and Moon are the dominant factors.

By varying the spacecraft properties, you can now test what propellant mass is needed dependent on what kind of AKM is assumed - just note that for the liquid rocket motor, about half the mass delivered into transfer orbit by the launcher rocket is again needed to get into geosync orbit - we'll look at a much cheaper alternative next.

Continue with Ion thrusters.

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