## Experiment II-4

Measuring the Distance to Moon

### Method of Parallax

Observation suggests that the objects in the sky revolve daily about the earth, with the Sun and stars slowly falling behind over a year, and the Moon and other planets doing so each at their own rate.  But Aristarchos suggested that most of the appearances could alternatively be due to motions of the Earth.  The Earth could rotate daily to the East making everything in the sky appear to rotate Westward.  And if the Earth revolved around the Sun annually, part of the slower motions would be explained.  This was an early recognition of relativity:  Motions can be viewed from different perspectives, but the observations differ relative to the position of the observer.

Humans had always assumed that they were in the center of the universe.  But Aristarchos suggested that since the Sun was larger than the Earth, perhaps the Sun was in the center of heavenly motions rather than the Earth.  This is different from relativity so would cause observable differences.  The most fundamental would be parallax.  The lack of any observable parallax led to the Greek's rejecting Aristarchos' proposal that the heavenly objects orbit the sun.

If parallax is observed, careful measurements of it can be used to measure the distance to the to the observed object.  In this experiment we shall measure the distance to the Moon, but the procedure could be used in a great many situations.  When the location of the Moon simultaneously viewed from two different locations, the Moon seems to be in a different position compared to the much more distant stars,  That angle is then used with the law of sines: The lengths of sides of a triangle are proportional to the sines of the opposite angles.

### Experiment

Below are two photographs taken about 1200 miles apart by amateur astronomers when Jupiter, Venus, and the Moon were unusually close together in the sky.  Because of this conjunction the photographs were accidentally taken at the same time.  The crescent lit by the Sun was over exposed by both photographers resulting in the swollen blur.  The remainder of the Moon was illuminated by earthshine.  At first glance the photographs appear to be identical but just rotated a few degrees.  But if one photograph is superimposed on the other, the stars (indicated by purple markers) and planets are aligned, but the Moon appears to be in a different position.

Photograph by Kenneth R. Polley, 4 seconds at f5.6, 135mm, 6:00 P.M. CST, 27 December 1973 at Finley Air Force Base, North Dakota [Latitude 47.5°N, Longitude 97.9°W].

Photograph by David Farley, 4 seconds at f5.6, 105mm, 6:00 P.M. CST, 27 December 1973 at Starkville Mississippi [Latitude 33.5°N, Longitude 88.7°W].

### Procedure

1. Print the Farley photograph and the negative of Polley photograph provided below on paper thin enough to be semitransparent.

2. Align the planets and stars on the Polley negative with those on the Farley photograph.

3. Mark the apparent difference in the Moon's location and used the angular scale provided to determine the amount of parallax.

4. Use a calculator to determine the sine of this angle and that of 90°.

5. Use the ratios of the distances to the sines of these two angles to determine the distance to the Moon.

1. After finding the distance to the Moon, measure the angular size of the Moon and again use the law of sines to determine the diameter of the Moon.
Finally, record your procedures, measurements, and findings in your journal.  If you need course credit, use your observations recorded in your journal to construct a formal report

### References

• Modified from an experiment developed for teaching physics by Gerald Holton, F. James Rutherford, Fletcher Watson, directors, Project Physics, at Harvard University, funded by U.S. Dept. of Education and N.S.F., 1964-1975.
• David Cassidy, Gerald Holton, F. James Rutherford, Understanding Physics, Springer

created 4/14/2003
revised 2/27/2005
by D Trapp