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:

- 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.

- 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?)

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

- the general idea of the measurement,
- the equipment you used for this work,
- a summary of your experimental results, and
- 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)

Last modified: March 17, 2003

`http://www.ifa.hawaii.edu/~barnes/ASTR110L_S03/lunardist.html`