# Summary: Shape of Lunar Orbit

In March and April 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 12 measurements were made between 03/11/03 and 04/15/03.

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 generally 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 3/20 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 3/20.

Different plotting symbols indicate different estimates of the Moon's distance: open circles show the results from the photographs, while crosses 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 crosses were slightly spread out so that the total number of people reporting a given measurement can be seen at a glance. The upward-pointing arrow for 4/08 represents a value D = 123.1 which wouldn't fit on this graph.

The filled circles in Fig. 1 show 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 filled circles 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.

While a few results are way off, the vast majority of the visual measurements made during observing sessions are reasonably accurate. The most inaccurate visual results were obtained on 4/08; as several people noted, it was quite hard to see the scale that night since the crescent Moon did not properly illuminate the scale.

Fig. 2 shows a plot of the Moon's orbit. The large filled circle is the Earth, while the smaller circles show the positions of the Moon during our observations. The dotted line shows a circle with a radius of 110 lunar diameters centered on the Earth, while the grey ellipse shows the Moon's actual orbit. The three arrows show the direction to the Sun on 3/18 (full Moon), 4/01 (new Moon), and 4/16 (full Moon).

To plot an orbit, one needs to know the direction to the Moon as well as the distance D to the Moon at each observation. In Fig. 2, the angle between the Moon and the Sun was used to calculate the Moon's direction; the angle at each observation date was provided by the Solar System Simulator web page. The Sun is not an ideal reference point since the direction from the Earth to the Sun also changes as we move in our own orbit. However, the angle between the Moon and the Sun could be measured with fairly simple equipment, as shown on the web page How to study the movement of the Moon.

Some people noticed that apogee (maximum distance) occurred around the time of the new Moon, while perigee (minimum distance) occurred close to full Moon. This is not always true; three months from now, for example, perigee will occur close to 1st quarter, while apogee will be close to 3rd quarter.

A few people wondered about the relationship between these measurements and our estimate of the Moon's distance using parallax. The technique we used here yields D in units of the Moon's actual diameter, but not in kilometers or any other standard system of units. Thus we need parallax - or some other method such as radar - if we want to know D in kilometers.

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. At the same time, however, the ellipse shown in Fig. 2 is only slightly different from an off-center circle. To show that the Moon's orbit is more accurately represented by an ellipse than by an off-center circle, 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. More frequent photographic and visual observations would yield a more accurate picture of the Moon's orbital motion. In particular, it would be good to get more observations near new Moon.
4. A longer sequence of observations - say two full months or more - would help confirm the shape of the Moon's orbit.
5. 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.
6. 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)