Frank Moon Article

Frank Suits

 

Measuring the Moon's Distance from Your Backyard
By Frank Suits
Figure 1. For an observer on the equator, the overhead moon is nearly 4000 miles closer, making it appear larger in the sky - opposite to what the moon illusion suggests. Mid-latitude observers would see a reduced, but still measurable effect

Everyone knows that the moon looks dramatically larger when it is rising, and amateur astronomers know that this is entirely an illusion; but did you know the moon really does change size, and that it actually gets larger as it rises, the opposite of what the illusion would suggest? The explanation for the effect is very simple:


Figure 2. Frank Suits

The rotation of the Earth swings you around as the moon rises, bringing you up to 4000 miles closer when it is overhead. Although this produces less than a 2% change in size, it is large enough to be captured with modest equipment, and allows a stationary observer to make a direct calculation of the moon's distance.

I learned about this effect from my grandfather (figure 2) , who constructed a device solely for the purpose of measuring the change, consisting of a dime suspended in a telescoping tube. The idea was to look through the tube at moonrise and set its length so the dime just covered the moon. Repeating the process when the moon was overhead would require shortening the tube by that same fraction of around one percent.

Recently I decided to take advantage of today's technology and try a more modern approach using a small telescope and digital camera.

The technique is simple: Take a picture of the moon when it is rising and when it is overhead, and compare the diameters. If you assume a spherical Earth and circular lunar orbit, you can directly calculate the distance to the moon given the fractional change in lunar size, the two altitudes of observation, and the diameter of the Earth.

As you can see from the photographs, the giant moon that loomed over the horizon on March 6, 2004, when superimposed on the overhead moon, is noticeably smaller. With these images I measure a fractional change in diameter of 0.011 between observations at 8.5 and 41.1 degrees above the horizon, yielding a distance of 184000 miles assuming an Earth diameter of 7926 miles.
Figure 3. My Grandfather - Hollis E. Suits
Kirkwood, MO

The error from the true value of 236000 miles (on the evening of the measurement) is due to the ellipticity of the lunar orbit, since the full moon on that day was midway between perigee and apogee, when its distance from Earth was changing most rapidly. A more accurate measurement can be made when the moon is near perigee or apogee when full. Despite the slight error, it was fun to update my grandfather's technique and measure the distance to the moon, without leaving my backyard.

Figure 4. Images of the moon when rising and nearly overhead (at 8.6 and 41° altitude). The rising moon is somewhat blurred by the atmosphere, but still clear enough to show the favorable libration of crater Drygalski near the lower (south) pole. The photographs were taken with a Televue Ranger and Sony F707 digital camera coupled by a ScopeTronix MaxView 40mm eyepiece. The camera was set on manual focus and maximum zoom, to give a very repeatable image scale essential for the size comparison.
Figure 5. The rising moon "eclipsing" the overhead moon. This image was created by subtracting the image of the rising moon from that of the overhead moon, clearly showing the 1.1% increase in angular size of the moon as it rose. Note that the rising moon is smaller than the overhead moon!
Notes on the measurement
1. Photos were taken from Long Island at 6:30 and 9:36 p.m.EST on March 6, 2004.
2. At 6:30 the altitude was 8.57, and at 9:36 it was 41.1 degrees, using TheSky.
3. My measured diameters were 1579 and 1597 pixels, giving a fractional change of 1.1%
4. Using TheSky, the true angular diameter of the moon at those times was 31.567' and 31.883', giving an expected change of 1.0%. This is in very good agreement with my measurement.
5. The approximate geocentric distance to the moon, given by the formula:
Comes out to 183010. The exact value, found by solving the change in the two triangle dimensions, is 184044 (indicating that the approximation holds).
6. The same predicted distance using the exact values from TheSky yield a distance of 202351, which is still a ways off from the exact value (geocentric) of 235700.
7. The explanation for the discrepancy is that the geocentric distance of the moon changed by a fair amount between the two observations: from 235867 to 235555 (from TheSky), for a change of 312 miles due to the ellipticity of the lunar orbit.
8. For comparison, the change in distance due to Earth rotation was 2013 miles. Clearly the rotation dominates the change in size, but the ellipticity has enough effect to reduce the measured distance when the moon is heading towards perigee, and to increase the measured distance when the moon is heading towards apogee (the moon gets bigger, or smaller than it should, respectively).
9. Apogee occurred on Feb. 28 and perigee on Mar. 12. Thus the full moon on Mar. 6 was right in the middle between the two, when its radial velocity with respect to Earth is greatest.
10. In 2008 on Dec. 12, the moon will be full, at perigee, and at its highest in the sky for northern observers, making this an ideal opportunity to do the measurement.
11. There is an additional very small effect due to the moon orbiting around the center of earth/moon center of mass, but that is much smaller. The geocentric distance absorbs this effect and the measurement does not assume the moon is orbiting about the center of the earth - just that the earth-moon distance is constant.
Processing of images:
1. Each image was given an identical pass of Unsharp masking
2. The overhead moon was rotated to match the orientation of the rising moon. The rising moon was the reference so that the initial horizon line is horizontal.
3. The overhead moon was adjusted in brightness to match that of the rising moon so the difference overlay was cleaner.
4. Both images were desaturated to convert to gray scale.
5. All images were cropped to 1800x1800 at 600 dpi. Lzw (lossless) compression.
6. I increased the brightness of the overlay image to enhance the annulus.
7. I did not tweak the brightness or contrast of the individual moon images since I didn't know what would be optimal for the published version. Certainly the craters will show more clearly with contrast enhancement.

This article is published in the September 2004 issue of Sky & Telescope