Moon Distance

In measuring the length of a 1-meter table with a Meter stick to within 0.1 millimeter, the uncertainty in the measurement is one part in ten thousand. Meter sticks, however, are usually inconvenient for measuring the distance to the Moon. Instead, a laser light pulse can be reflected from a stationary corner reflector on the Moon similar to the reflectors on bicycles, and the total duration of the pulse from Earth to Moon and back to Earth again is timed. What do you estimate for the uncertainty in the measurement for the Moon’s distance? Which determination would you expect to have the greater distance uncertainty, the table length or the distance to the Moon?

Answer

The laser light pulse traveling from the Earth to the Moon and back will encounter the Earth’s atmosphere twice. The pulse will have an initial known rise time and decay time, but these times will be extended by passage through the air medium. We assume ideal reflection at the Moon’s corner reflector that is, no pulse spreading in angle or in time.

First, the ideal case. We assume that the laser source and the reflector on the Moon are opposite each other on the line connecting the centers of Earth and Moon and that Earth’s atmosphere does not affect the transit time. The major source of uncertainty will be the ability of the detection system to locate the half-height point on the rise time of the outgoing pulse and the same point on the incoming pulse. If the system is good to about a picosecond in detecting this point, then a transit time of 2.56 seconds for the 3.84 × 10⁸ meter distance corresponds to a timing uncertainty of better than one part in 100 billion, with an uncertainty in distance of less than 4 millimeters. That is, with the proposed laser system, one can measure the distance to the Moon to almost the same distance uncertainty as one can measure the length of a table with a meter stick!

Of course, the atmosphere will foul things up a bit. The index of refraction and the change of this index with altitude will both alter the light pulse speed and spread out the pulse rise time and rise shape. Sophisticated signal processing techniques can eliminate most of these atmospheric effects. So the final uncertainty will be determined in the electronics creating the laser pulse and detecting the arrival of the pulse’s leading edge.