# Hohmann Transfer & Plane Changes

The Shuttle typically deploys a satellite into an orbit with an altitude of about 300 km and an inclination of 28.5 degrees. To be in a geosynchronous orbit, the satellite must move to an equatorial orbit (inclination = 0 degrees) with an altitude of 38,000+ km. This means changing two separate aspects of the initial deployment orbit, radius (or altitude) and inclination.

In general, changing a spacecraft's orbit (size, shape, or inclination) involves firing an engine to change the magnitude or the direction of the spacecraft's velocity. These firings have to occur at the right time, fire in the right direction and produce the right amount of velocity change to create the correct new orbit. The transfer from "LEO to GEO" (Low Earth Orbit to Geosyncronous Orbit) requires a variation of a "Hohmann transfer".

### Changing Altitude

A Hohmann transfer is a fuel efficient way to transfer from one circular orbit to another circular orbit that is in the same plane (same inclination), but a different altitude.

To change from a lower orbit (A) to a higher orbit (C), an engine is first fired in the opposite direction from the direction the vehicle is traveling. This will add velocity to the vehicle causing its trajectory to become an elliptic orbit (B). This elliptic orbit is carefully designed to reach the desired final altitude of the higher orbit (C). In this way the elliptic orbit or transfer orbit is tangent to both the original orbit (A) and the final orbit (C). This is why a Hohmann transfer is fuel efficient. When the target altitude is reached the engine is fired in the same manner as before but this time the added velocity is planned such that the elliptic transfer orbit is circularized at the new altitude of orbit (C).

### Changing Planes

A plane change requires an engine firing in the out-of-plane direction. Since the point in an orbit where the engines are fired automatically becomes a point in the new orbit (or the burn point becomes the intersection of the old and new orbits), this firing must occur where the current orbit and the desired orbit intersect.

So, if the satellite is in an orbit inclined 28.5 degrees to the equator the firing must occur at one of two points during each orbit revolution where the spacecraft is directly over the equator.

We calculate the change in velocity with vector mathematics.

1. Velocity in inclined orbit: 7.726 km/sec at 28.5 degrees to equator.
2. Velocity in equatoral orbit 7.726 km/sec at 0 degrees (Note: Same speed since only the angle is changing.
3. Change in velocity (Delta V) needed is (1) + (3) = (2) so (3) = 3.801 km/sec at -75 degrees.

### Changing both Altitude and Planes

The two types of changes can be done at the same time if we do our vector mathematics correctly. If you are in an inclined orbit and you want to make it equatorial, you must produce a change in the velocity and add it to the original velocity vector, which will generate the desired velocity vector.

There are only a few more considerations. Do we do the plane change with the first Hohmann transfer burn or the second one? Or do we change the plane a little with the first and a little with the second burn? As it turns out, if you tried to do all of the plane change at the lower altitude you would have to use a ridiculous amount of fuel. Plane changes are fuel-expensive anyway you do them so you definitely want to do it the smartest way!

It turns out that the optimum solution dictates that the plane be shifted by about 2 deg. at the lower altitude and the remaining 26 deg. at the geostationary altitude. The only consideration left is one of precise timing. Geostationary satellites have a particular destination in orbit, a certain longitude on the equator that it is to "hover" over. Therefore, the scheme that we have arrived at has to occur such that when the satellite reaches the desired altitude it is above the right point on the equator. This involves picking the right equator crossing point of the satellite at which to initiate the transfer.

For the TDRS-G (a communications satellite) delivery initiated by STS-70, they wished to deposit the satellite at about 180 deg. E longitude. If the Earth did not rotate, the LEO-GEO transfer (which takes 180 deg. of orbit travel) would need to occur at the equator crossing at 0 deg longitude. Of course the Earth does rotate. In fact it will rotate about 80 deg. in the time it takes for the transfer to happen. Therefore, for this mission, the first burn took place at about 80 deg. E longitude. It so happens that there was an equatorial crossing at 80 deg. E on the 6th full orbit after Discovery launched. This became the planned time of the first transfer burn.

Even without discussing the specific equations it is easy to see that planning a space mission boils down to a problem of geometry, timing, the physics of orbital motion and a lot of common sense. With Discovery and the IUS up there making all of this look like an effortless dance in space, hopefully you now have an idea of some of the choreography involved!

Updated September 21, 1995 Contacts