# Distance to the Moon

How far away is the Moon? One way to find out is by using parallax: observe the Moon from two points on the Earth's surface, and measure the shift in its position with respect to the background stars.

This measurement of the Moon's distance uses the same approach used in Parallax in the Lab. In particular, we line the Moon up with a star, observe it from two different locations, and use the apparent shift in the Moon's position to get its distance. The only real difference is one of scale; the baseline stretches from here to the island of Tahiti, and the distance we want to measure is several hundred thousand kilometers!

A lunar occultation provides a convenient opportunity to determine the Moon's position with respect to the stars. Just before the Moon occulted the star TU Gem on the night of March 11th, you were able to observe the Moon and the star through the telescope, and draw the Moon's position on the chart we handed out in class.

To make a parallax measurement, we need to combine this observation with another made from a reasonably distant location. Tahiti is a good place for the second observation; it's far enough away that the Moon's position with respect to the stars is noticeably different. In addition, Tahiti is basically South of Oahu; this simplifies the calculations since the baseline from here to Tahiti is roughly perpendicular to the line from here to the Moon.

There's one drawback, however, to using Tahiti; we need someone there to make an observation of the Moon's position at the same time we made our observations here on Oahu. (The observations must be made at the same time because the Moon is moving with respect to the stars.) Since we didn't actually have anyone observing from Tahiti, we just have to pretend we did! Attached to this handout is a chart, much like the one we handed out in class, which shows the Moon's position as seen from Tahiti at about the same time we made our observations. You can combine that chart with your own to figure out the Moon's distance.

(Before going ahead with the measurement, however, you may want to check the Moon's position on your chart. Everyone drew the Moon very close to TU Gem, which is correct, but some people got confused about the chart's orientation, and drew the Moon to the wrong side of the star. The trick is to realize that the sunlight on the Moon is coming from the West, and if you hold this chart up in the sky so the arrow points to the North, the West side of the chart is toward the right. Note that on the direction of the sunlight falling on the Moon is the same from Oahu and Tahiti; on the chart from Tahiti, dark is shown as light and light as dark, following the usual ink-saving convention.)

The first step in this measurement is to lay the chart from Tahiti on top of the one you used, carefully line things up so the stars in both charts match, and trace the lunar position that you measured onto the chart from Tahiti. The Tahiti chart now has two circles; one shows the Moon's position from Tahiti, the other shows its position from Oahu. Now use a ruler to measure the shift in the Moon's position in centimeters. These charts have a scale of 2 cm per degree; thus a 1 cm shift on the chart represents a parallax angle  = 0.5°.

To compute the Moon's distance, you need the baseline b from here to Tahiti. The straight-line distance between the islands of Oahu and Tahiti is 4324 km; you should use this distance for b. (Of course, this line passes deep under the Earth's surface; the sailing distance to Tahiti is greater.) A more accurate value for the baseline would take into account the fact that the triangle formed by Oahu, Tahiti, and the Moon is not a right triangle, but the math involved is tedious and not very instructive. However, you should be aware that distance to the Moon you compute using 4324 km for b will probably be about 20% too large.

Once you have the parallax angle and the baseline b, you can use the parallax equation to compute the Moon's distance:

As before, D will be in the same units as b; since b is expressed in kilometers, you will get D in kilometers as well.

### WEB RESOURCES

• Chart of Moon's position from Tahiti: GIF file or Postscript.

The GIF file should be printed at 100 dpi to get a scale of 2 cm per degree.

• Astronomy On-Line: The Moon Parallax

Detailed presentation of the calculations needed for an accurate parallax measurement of the Moon's distance; fairly technical.

### REVIEW QUESTIONS

• Why does the straight-line distance of 4324 km from Oahu to Tahiti yield an overestimate for the Moon's distance? (Hint: the Moon was somewhat to the North of the Zenith point as seen from Oahu, and even further to the North as seen from Tahiti; make a sketch showing the shape of the triangle formed by Oahu, Tahiti, and the Moon. Is it a right triangle?)

### REPORT: DISTANCE TO THE MOON

Do the analysis described above, and write a report on your work. This report should include, in order,

1. the general idea of the measurement,
2. the equipment you used for this work,
3. a summary of your experimental results, and
4. the conclusions you have reached.

In somewhat more detail, here are several things you should do in your lab report:

• Explain, in your own words, what an occultation is and why an occultation provides a good opportunity to measure the Moon's distance.

• Try to guess the size of any error you might have made in measuring .

• Turn in your chart showing the Moon's position as seen from Oahu.

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