ORBITAL MECHANICS

It will help the reader to understand the concepts behind rendezvous if a short diversion is taken into the field of orbital mechanics. At first glance, this topic seems arcane and, if studied rigorously, it is. Additionally, it can appear counter-intuitive but the basic concepts behind the subject are easy enough to grasp, and are really an extension of the orbital lessons discussed in Chapter 4.

To lay down the groundwork for this we need to establish some basic ideas. Unless some kind of propulsion is being used, all movement in space is governed by the gravitational attraction of the bodies (stars, planets, moons, asteroids, etc.) among which things move. In general, the gravity of the nearest large body dominates, so for the purposes of this explanation we shall ignore the pull from other bodies. Any spacecraft in orbit moves around the central body in an ellipse. Even a perfectly circular orbit is treated as a special form of ellipse whose eccentricity value is zero.

There are three principles to bear in mind with orbital motion. First, a spacecraft in a higher orbit takes longer to go around than one in a lower orbit. At first glance, this appears obvious because there is a longer circumference to travel, but that is only part of the story. The more important point to grasp is that it really is a slower orbit. The spacecraft is moving at a slower linear speed because the pull of gravity from the central body becomes weaker with distance, and hence a lower speed can maintain the perpetual fall that is orbital motion. As an illustration, the Saturn V inserted the Apollo spacecraft into an orbit only 170 kilometres above

Earth, taking only an hour and a half to go around at a linear speed of 7.8 kilometres per second. Geostationary satellites, which are the mainstay of global communications and televi­sion satellite broadcasting, orbit 35,800 kilometres above Earth’s equator, take 24 hours to get around once and travel at only 3.1 kilometres per second.

With this in mind, we can see a method by which one spacecraft can manoeuvre with respect to another, assuming that both are travelling in the same orbital plane. If the target ship is ahead, a pursuer can catch up with it by manoeuvring into a lower orbit, which is achieved by firing

Orbital mechanics 397

against the direction of travel, as if trying to get away from the target. We said it was counter-intuitive. The burn will cause the pursuer to fall into a lower orbit, which will have a shorter period and a higher linear speed. This will allow it to catch up with the target. The difficulty lies in choosing the exact moment to start climbing back into the original orbit, which we shall deal with later.

The reverse is also true. If the pursuer is ahead in the orbit, it can ‘slow down’ by accelerating forward, which causes it to rise to a higher and therefore slower orbit. It can then drop down again when the target has caught up.

The concept of changing from one orbit to another is a common requirement in space­flight and is embodied by our second principle which we have already met in Chapter 4 as the Hohmann transfer orbit. It is the most efficient and simplest way to change an orbit whereby firing a spacecraft’s engine along the direction of motion at one point in the orbit will increase its speed and thereby raise the altitude that will be reached on the opposite side of the orbit. Firing against orbital motion will slow the space­craft and lower the altitude of the opposite side of the orbit. Control of the total impulse from the burn allows control of the altitude that will be reached at the opposite side. We have met this already in the way the CSM and LM made burns around the Moon’s far side to raise and lower their near-side altitude.

To move from a lower circular orbit to a higher one, a burn must be made in the direction of motion until it is calculated that Diagram of basic rendezvous the point in the orbit opposite the spacecraft, techniques, now the apogee, is at the height of the

intended circular orbit. Once the spacecraft has coasted around in its orbit to its apogee, another burn must be made along the direction of motion to raise the perigee until it equals the apogee’s altitude.

So far we have dealt with two spacecraft within the same orbital plane. The third principle behind orbital mechanics deals with the situation when the two objects are

in different orbital planes. This is a common requirement, since few launch sites are located cqualorially yet many satellites need to reach a geostationary orbit above the equator. For example, a spacecraft launched from Cape Canaveral will necessarily have an orbital inclination of at least 28 degrees: this being the latitude of that site. The most efficient way for a spacecraft to move from one orbital plane to another is for it to make a burn at the point in the orbit where the two planes intersect, known as a node. Unfortunately, the physics of the situation dictate that all but the smallest of plane changes will be expensive in propellant – indeed to move a communications satellite from a 28-degree low Earth orbit to its geostationary outpost requires almost as much energy as would be required to send it to the Moon! For Apollo, it was vital to minimise plane change manoeuvres, especially for the LM’s ascent stage where propellant margins were very Light.