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, re-observe it from another location, 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!
An Occultation by the Moon provides a convenient opportunity to determine the Moon's position with respect to the stars. Just before the Moon occulted the star σ Sgr on the night of 06-Oct-2008, you could observe the Moon and the star next to each other, 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 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 almost due south of Oahu; this makes the situation easier to visualize.
But there's one catch; we need an observation of the Moon's position from Tahiti made at the same time we made our observations from Oahu. (The time must be the same because the Moon moves with respect to the stars.) Since we don't have observations from Tahiti, I've used the Stellarium program to calculate what someone there would have seen. The chart with this handout shows the Moon's position from Tahiti at the same time we made our observations. A glance at this chart shows that observers in Tahiti would not have seen an occultation on 06-Oct-2008; from Tahiti, the Moon was too far to the north to cover σ Sgr as it moved east with respect to the stars. The difference in the Moon's position between Oahu and Tahiti is the parallax angle θ we need to measure.
The first step in this measurement is to lay the chart for Tahiti on top of the one for Oahu, carefully line up the stars in both charts, and trace the Moon's position from Oahu 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. Use a ruler to measure the shift in the Moon's position in centimeters. The charts have a scale of 1 cm = 0.5°; thus a shift in the Moon's position of x cm represents a parallax angle θ = x × 0.5°.
To compute the Moon's distance, you need the baseline b from Oahu to Tahiti. The straight-line distance is 4324 km (of course, this line passes deep under the Earth's surface, so the sailing distance is greater). This would be the appropriate value for b if the Moon was overhead as seen from a point midway between Oahu and Tahiti (roughly, the location of Kiritimati). But on the evening of 06-Oct the Moon was about 30° from the zenith as seen from this point; the correct baseline is approximately b = cos(30°) × 4324 km = 3745 km. (This is not exact, but it's good enough for our purposes; the math required for an accurate calculation is tedious and not very instructive.)
Using the parallax angle θ and the baseline b, the parallax equation yields the Moon's distance, D:
Note that D will be in the same units as b; since b is given in kilometers, you will get D in kilometers as well.
Do the analysis described above, and write a report on your work. This does not have to be very long — a couple of pages is quite sufficient — and a literary style is not necessary. Your report should have four sections, in this order:
What is the goal of this measurement? Explain, in your own words, (1) what parallax is, (2) what an occultation is, and (3) why an occultation is good opportunity to measure the Moon's distance.
Describe the tools and equipment used. Remember to turn in your charts showing the Moon's position as seen from Oahu and Tahiti.
Give numerical values for every measurement. Try to guess the size of any errors made in measuring θ.
What have you learned? Look up the distance to the Moon in Stars & Planets (see p. 310) and compare it with your result for D.
Joshua E. Barnes
(barnes at ifa.hawaii.edu)
17 November 2008