Orbital Mechanics

This section presents a brief overview of the concepts and basic equations necessary to understand the process of sending a spacecraft from Earth to Mars and achieving a successful orbit. Brief reviews of units and scientific notation are also included and these may also be accessed by the students during the exercise. The information presented here has been gleaned from several text and web-based sources, which are listed at the end of the section.

The drawings below simplify the physics of orbiting Earth. We see Earth with a huge, tall mountain rising from it. The mountain, as Isaac Newton first envisioned, has a cannon at its summit. When the cannon is fired, the cannonball follows its ballistic arc, falling as a result of Earth's gravity, and it hits Earth some distance away from the mountain. If we put more gunpowder in the cannon, the next time it's fired, the cannonball goes halfway around the planet before it hits the ground. With still more gunpowder, the cannonball goes so far that it just never touches down at all. It falls completely around Earth. It has achieved orbit. Note that the shape of the orbit is not in the shape of a circle but of an ellipse

If you were riding along with the cannonball, you would feel as if you were falling. The condition is called free fall. You'd find yourself falling at the same rate as the cannonball, which would appear to be floating there (falling) beside you. You'd just never hit the ground. Notice that the cannonball has not escaped Earth's gravity, which is very much present--it is causing the mass to fall. It just happens to be balanced out by the speed provided by the cannon.

In the third drawing in the figure, you'll see that part of the elliptical orbit comes closer to Earth's surface that the rest of it does. This is called the periapsis of the orbit. It also has various other names, depending on which body is being orbited. For example, it is called perigee at Earth, perijove at Jupiter, periselene or perilune in lunar orbit, and perihelion if you're orbiting the sun. In the drawing, the mountain represents the highest point in the orbit. That's called apoapsis (apogee, apojove, aposelene, apolune, aphelion). The time it takes, called the orbit period, depends on altitude. At space shuttle altitudes, say 200 kilometers, it's 90 minutes.

The cannonball provides us with a pretty good analogy. It makes it clear that to get a spacecraft into orbit, you need to raise it up (the mountain) to a high enough altitude so that Earth's atmosphere isn't going to slow it down too much. You have to accelerate it until it is going so fast that as it falls, it just falls completely around the planet. In practical terms, you don't generally want to be less than about 150 kilometers above the surface of Earth. At that altitude, the atmosphere is so thin that it doesn't present much frictional drag to slow you down. You need your rocket (or cannon) to speed the spacecraft up to the neighborhood of 30,000 kilometers (about 19,000 miles) per hour. Once you've done that, your spacecraft will continue falling around Earth. No more propulsion is necessary, except for occasional minor adjustments. These very same mechanical concepts apply whether you're talking about orbiting Earth, the moon, the sun, or anything. Only the terms and numbers are different. The cannonball analogy is good, too, for talking about changes you can make to an orbit. Looking at the third drawing, imagine that the cannon has still more gunpowder in it, sending the cannonball out a little faster. With this extra speed, the cannonball will miss Earth's surface by a greater margin. The periapsis altitude is raised by increasing the spacecraft's speed at apoapsis.

This concept is very basic to space flight. Similarly, decrease the speed when you're at apoapsis, and you'll lower the periapsis altitude. Likewise, if you increase speed when you're at periapsis, this will cause the apoapsis altitude to increase. Decelerating at periapsis will lower the apoapsis.

HOHMANN TRANSFER ORBITS

To launch a spacecraft to an outer planet such as Mars, using the least propellant possible, first consider that the spacecraft is already in solar orbit as it sits on the launch pad. Its existing solar orbit must be adjusted to cause it to take the spacecraft to Mars. In other words, the spacecraft's perihelion (closest approach to the sun) will be Earth's orbit, and the aphelion (farthest distance from the sun) will intercept the orbit of Mars at a single point. This is called a Hohmann Transfer Orbit. The portion of the solar orbit that takes the spacecraft from Earth to Mars is called its trajectory.

To achieve such a trajectory, the spacecraft lifts off the launch pad, rises above Earth's atmosphere, and is accelerated in the direction of Earth's revolution around the sun to the extent that it becomes free of Earth's gravitation, and that its new orbit will have an aphelion equal to Mars' orbit. After a brief acceleration away from Earth, the spacecraft has achieved its new orbit, and it simply coasts the rest of the way. To get to the planet Mars, rather than just to its orbit, requires that the spacecraft be inserted into the interplanetary trajectory at the correct time to arrive at the Martian orbit when Mars will be at the point where the spacecraft will intercept the orbit of Mars. This task might be compared to throwing a dart at a moving target. You have to lead the aim point by just the right amount to hit the target. The opportunity to launch a spacecraft on a transfer orbit to Mars occurs about every 25 months.

To be captured into a Martian orbit, the spacecraft must then decelerate relative to Mars (using a retrograde rocket burn or some other means). To land on Mars, the spacecraft must decelerate even further (using a retrograde burn, or spring release from a mother ship) to the extent that the lowest point of its Martian orbit will intercept the surface of Mars. Since Mars has an atmosphere, final deceleration may be performed by aerodynamic braking, and/or a parachute, and/or further retrograde burns.

To launch a spacecraft to an inner planet such as Venus using the least propellant possible, its existing solar orbit must be adjusted so that it will take it to Venus. In other words, the spacecraft's aphelion will be on Earth's orbit, and the perihelion will be on the orbit of Venus. As with the case of Mars, the portion of this orbit that takes the spacecraft from Earth to Venus is called a trajectory. To achieve an Earth to Venus trajectory, the spacecraft lifts off of the launch pad, rises above Earth's atmosphere, and is accelerated opposite the direction of Earth's revolution around the sun (decelerated) to the extent that its new orbit will have a perihelion equal to Venus's orbit. Of course the spacecraft will end up going in the same direction as Earth orbits, just a little slower. To get to Venus, rather than just to its orbit, again requires that the spacecraft be inserted into the interplanetary trajectory at the correct time to arrive at the Venusian orbit when Venus will be at the point where the spacecraft will intercept the orbit of Venus. Venus launch opportunities occur about every 19 months.