Orbital Mechanics and Interplanetary Trajectories

When we fly in space, the rules of the road here on Earth don’t apply. We can’t catch up with another spacecraft by speeding up. We actually have to slow down. That’s because increasing speed at a certain point moves the craft into a higher orbit. In a higher orbit, the speed of the craft decreases. This is because the higher the orbit of a satellite, the slower it needs to move to balance the pull of gravity. The other craft will move further away.

To catch another craft, the chasing spaceship must drop down to a lower orbit—where it would speed up and catch up with the target. Then it can move into the target’s orbit and rendezvous with it.

Such are the vagaries of orbital mechanics. It’s all because orbits are conic sections—ellipses and circles. Parabolas and hyperbolas are also conic sections, but they are open to infinity and so would take a spacecraft away from the planet. Johannes Kepler defined the laws of bodies orbiting other bodies in the 17^th century, and Sir Isaac Newton refined them later. Basically, Newton explained Kepler in this way:

If two bodies interact gravitationally, they will orbit each other about the common mass of the two in an ellipse.

His other two laws describe the speed at which the smaller body will travel around the larger body. We will come back to these laws later.

Now, let’s talk about how spacecraft get into orbit and what happens then. Let’s do a mental experiment to see what is actually happening. Imagine a cannonball being fired from a very high mountain (we assume no atmospheric drag in this experiment). If it is a very strong cannon and you can put enough gunpowder in it, the cannonball will go fast enough that even though it begins to fall back to earth, as it falls it begins to speed up under the pull of gravity at 32 feet per second/per second. Half way around the Earth it reaches its lowest point, but it is now going as fast as it was when it left the cannon. It doesn’t hit the ground because it’s high enough to continue its flight. Now it begins to gain altitude again. As it does it slows, but still has sufficient speed to come back to the point from which it was fired, and continue its flight.

In other words, it was ‘falling’ around the planet.

That is what a spacecraft is effectively doing. A rocket boosts it to a speed and altitude at which it experiences little or no air resistance, and it is going fast enough to fall around the Earth without falling back to Earth. That’s why weightlessness is sometimes called ‘freefall.’ The speed required to achieve a low earth orbit (LEO) of about 186 miles is about 18,000 mph or 25,000 fps.

The point at which the spacecraft goes into orbit is generally the high point of its orbit, called the apogee. It then ‘falls’ around the earth to a low point, called the perigee. Here is where Kepler’s laws come into play again. At apogee, the spaceship’s speed is at its slowest. At perigee it is at its fastest. Very often, it is necessary to raise the perigee, and sometimes to change the elliptical orbit into a circular orbit.

How is this done?

The speed at apogee must be increased. Firing a rocket motor there will raise the perigee without affecting the apogee. By the same token, firing a motor at perigee will raise the apogee without affecting the perigee. This is how a LEO orbit can be changed into a higher orbit. Usually it is desired to raise both the apogee and perigee, and circularize the orbit. This requires two burns—one at each of the critical points in the orbit.

These burns are called delta v, or changing velocity. In those we’ve discussed, the burns are done in the direction of travel and are called posigrade burns. It is also possible to move a spacecraft from a high orbit to a lower orbit with retrograde burns. In this case the burns are done opposite to the direction of travel. This of course is how the shuttle ends its flights and reenters the atmosphere.

It is not just altitude and shape that can be changed by engine burns. A spacecraft enters orbit at a particular orbital inclination—that is, an angle to the equator. Inclination can also be changed with delta v, but it must be applied perpendicular to the direction of travel and in the direction of the desired inclination change. In low orbits, only very small inclination changes can be made. At very high orbits, inclinations can be changed as much as 30 degrees. This is possible because the orbital velocity is low, as low as around 6000 mph in some, and the pull of gravity is much less than in LEOs. Therefore, less energy is required to change the inclination.