# Report: Shape of the Moon's Orbit

In October and November 2003 we measured the Moon's apparent diameter and used the results to study the shape of the Moon's orbit. This page summarizes the results.

As the Moon follows an elliptical orbit about the Earth, its distance varies in a regular and periodic way. To study these changes in distance, we measured the Moon's apparent diameter whenever the Moon was visible during observing sessions. These visual measurements were supplemented by photographs taken in between observing sessions. A total of 25 measurements were made between 10/05/03 and 11/30/03.

We measured the Moon's apparent diameter using 25 mm eyepieces equipped with linear scales calibrated in millimeters. This gave the diameter of the Moon's image, d, in millimeters; the measuring technique is described in the handout for this assignment. Visual observations were made using 8 inch telescopes, while the photographic observations were made using a 6 inch telescope. However, all the telescopes had focal lengths of F = 1200 mm, so visual and photographic data could be combined directly.

Fig. 1 presents measurements of the Moon's distance D = F/d for each date we observed. Here D is given in units of lunar diameters (3,476 km). Care was taken to plot the date and time of each observation accurately; for example, the observation on 10/22 was made at 06:00, so the symbols for this date were placed one quarter of a day to the right of the mark for 10/22.

Different plotting symbols indicate different estimates of the Moon's distance: open circles show the results from the photographs, while small squares represent individual visual measurements. Because most people measured d to the nearest 0.1 mm, it was not unusual for several people to get the same result on a particular night; when that happened, the squares were placed next to each other so that the total number of people reporting a given measurement can be seen at a glance.

The smooth curve in Fig. 1 shows values for the Moon's distance from the center of the Earth obtained from the Solar System Simulator web page and expressed in units of lunar diameters. These values are extremely accurate. Comparing the smooth curve with our results, it's clear that most of our visual and photographic measurements underestimate the Moon's distance D by a few lunar diameters. While some measurement error is unavoidable, the main reason for this underestimate is that our observations were made from the Earth's surface, and not from its center. When the Moon is directly overhead, it is roughly 1.8 lunar diameters closer to us than it is to the center of the Earth. Many of the observations were made with the Moon fairly high in the sky, so it's not too surprising that our D values yield results which are one or two lunar diameters too small.

Most of the visual measurements made during observing sessions are reasonably accurate. The most accurate results were obtained on 11/04; the gibbous Moon presented a fairly easy target and people already had experience making these measurements. The measurements on 11/25 were rather rough, but reasonably good given the limited time available and the difficutly seeing the scale against the crescent Moon.

Some people noticed that apogee (maximum distance) occurred around the time of the full Moon, while perigee (minimum distance) occurred close to new Moon. This is not always true; the time between one apogee and the next is 27.6 days, while the time between one full Moon and the next is 29.3 days, so the two cycles gradually get out of step with each other. In the spring semester, for example, we had apogee near new Moon, and perigee near full Moon.

From Fig. 1 it's clear that the variation in the Moon's distance due to its elliptical orbit can be measured with the equipment we used in this lab. To prove that the orbit really is elliptical, and not just an off-center circle, however, we would need to measure the change in the Moon's apparent diameter with an error of only about 0.1%; this is probably impossible with our equipment. Thus our results are consistent with Kepler's 1st law, but we can't rule out all other theories. This is actually the normal state of affairs in science; we can test a theory and disprove it, but we can never conclusively prove that it must be correct. Kepler's 1st law could have been proven wrong by our observations, but it passed this test with flying colors.

### IMPROVEMENTS

In hindsight, here are several things which would have improved the observations:

1. Distances D could be corrected for the effect of observing from the Earth's surface instead of its center. This requires measuring the angle from the Moon down to the horizon.
2. A way to illuminate the scale in the eyepieces would make it easier to measure a crescent Moon in a dark sky.
3. Measurements of the angle between the Moon and the Sun could be included to obtain data needed for a plot of the Moon's actual orbit.
4. A similar series of measurements of the Sun, over the course of a year, could test Kepler's claim that the Earth also moves in an elliptical orbit!

Joshua E. Barnes (barnes@ifa.hawaii.edu)