On Saturday, April 11, 1970 at 13:13 CST, a Saturn-V rocket blasted off from Pad 39-A of the Kennedy Space Center, beginning the now-famous Apollo-13 mission to the moon. The mission was not to go as expected. At about 56 hours into the mission, the No. 2 oxygen tank in the Service Module exploded with a loud bang. The first word came to Mission Control in the deceptively calm tone that astronauts are known for: "...Houston, we've had a problem..." The explosion had actually blown out one whole side of the service module, as can be seen in film taken later by the astronauts. It also caused a cascade of other mechanical and electrical damage, ending the chances for a moon landing and forcing the mission to end early. The story has been told in riveting style by Director Ron Howard in the movie Apollo 13.

When the explosion occured, the three Apollo 13 astronauts (James A. Lovell, Jr., John L. Swigert, Jr., and Fred W. Haise, Jr.) were 200,000 miles from earth! With the mission plans scrapped, and a whole new flight plan needed to save the astronauts' lives, how was it possible to navigate them home? If you're in a spaceship, moving in some arbitrary direction and at some arbitrary speed, simply pointing your spaceship at the earth and firing the rockets is probably the last thing that will get you back. To show how such tricky navigation is possible, we're going to look at a more basic (but very important) problem, the problem of how spacecraft orbit the earth (moon not considered). To a large extent, understanding orbital dynamics was the reason calculus was invented!

The first fact we need is this: If a planet like the earth has a spherically symmetric mass distribution (which it approximately does), and the center of the earth is located at the origin of a 3-dimensional coordinate system, then the force F exerted by the earth on a particle of unit mass with position vector r is

F = (-GM/r) u

(1)

where G is a positive constant called the universal gravitational constant, M is the mass of the earth, r is the magnitude of r (that is, r=||r||), and u is a unit vector in the direction of r (that is, u = r/r). This law is known to be true for the attractive force between particles, or point masses, and, relative to the distances under consideration, a spacecraft is small enough to be idealized as a point. However, for space missions near the earth (or the earth-moon system), the earth is too large to be regarded as a point, and this law needs to be proved. For a proof which uses the divergence theorem, see The Earth's Gravitational Field.

Using equation (1) it can be shown (see your calculus textbook) that a particle, or a relatively small object like a spacecraft, orbits the earth in a plane curve given in polar coordinates by

r = p / (1 + e cos)

(2)

where p and e are constants. You may recognize this as the general equation of a conic section with eccentricity e. The conic (the orbit) is

• a circle if e = 0

• an ellipse if 0 < |e| < 1

• a parabola if |e| = 1

• a hyperbola if |e| > 1.

It is known that the constants p and e in equation (2) are given by

p = (rv)

and

e = (rv / GM) - 1.

(If you carefully follow the derivation of the orbit equation (2) in your calculus textbook, and use a little ingenuity, you can actually deduce these values of p and e.) Thus we see that we can control the orbit by controlling v, which in turn controls the eccentricity. Also, if we are already in one orbit, like the blue circle, we can increase our velocity (by firing rockets) when = 0 and boost ourselves into, say, the big green elliptical orbit. This kind of manuever is used all the time by astronauts as the first step in going from a small circular orbit to a large one. The second step would be to fire rockets again at the left end of the green ellipse to insert the spacecraft into a large circular orbit (not pictured). At either step, the all-important change in velocity is referred to as delta v and the big green ellipse would be called a Hohmann transfer ellipse.

To plan a mission to the moon, of course, more factors must be taken into consideration, such as the gravity of the moon, entering and leaving the earth's atmosphere, and the fact that the earth is not truly spherical. However, even in this setting, the above formulation continues to play an important role in making preliminary time and fuel estimates. It's pretty clear that vector calculus plays an important role in bringing astronauts (such as the Apollo 13 crew) safely back to earth!

For on-line information on Apollo 13 and the other Apollo missions, see NASA's web page on The Apollo Missions, and the Kennedy Space Center's page on Project Apollo. For more information on the mathematics of space flight, including moon missions, I recommend the book, Fundamentals of Astrodynamics by Roger R. Bate, Donald D. Mueller, and Jerry E. White (Dover, New York, 1971). This excellent softcover book sells for about $7 and can be read by anyone who has taken Math 261 and has a basic knowledge of matrix algebra.

Finally, lest you think that earth-to-moon navigation is a dead subject, have a look at the Clementine Mission web page!