15. Moon Diameter and Orbit Lab

The Moon has a rather eccentric orbit. In this lab you'll analyze data to learn something about how it moves around the Earth.

 

THE MOON'S ORBIT

Kepler's first law says that planets have elliptical orbits. As a result, the distance between a planet and the Sun changes rhythmically as the planet moves in its orbit. In many cases, this rhythmic change is rather subtle; for example, the Earth's distance from the Sun varies between 98.3% and 101.7% of its average value. In contrast, the ellipticity of the Moon's orbit is fairly dramatic.

This variation in distance produces several effects which we can observe here on Earth. For example, when the Moon is closest to the Earth (perigee), it moves faster, while when it is furthest from the Earth (apogee), it moves slower. The Moon also appears to nod back and forth a bit as it orbits the Earth. But the most dramatic effect is the change in the Moon's apparent diameter: when the Moon is close, it looks larger, and when the Moon is far, it looks smaller. We will use this effect to study the change in the Moon's distance.

SIMULATED LUNAR OBSERVATIONS

I've included as links to this lab several "observations" of the Moon. The first two images (Fig. 1, JPG or PDF and Fig. 2, JPG or PDF) are a collection of 12 photographs of the Moon obtained near full Moon for the next year (with the help of "Starry Night"). Each panel shows a patch of sky 1 degree by 46 arcminutes, centered on the Moon. Using a ruler, determine a convenient scale factor for these data (e.g. arcminutes/inch or arcminutes/cm or arcseconds/cm, etc.).

Determine how precisely you can measure the diameter of a lunar image, by measuring one image several times (more than 5) in various directions and computing the standard deviation. This can be taken as the uncertainty in a single measurement.

Now make a table of measurements that shows the date (all observations were obtained at approximately 8pm from Honolulu), the measured diameter and the derived angular diameter. Indicate the units used for both the ruler measurement and the angle. Somewhere on this table describe your estimate of the measurement errors.

On a graph that shows time on the horizontal axis make a plot of the time variation of the apparent lunar diameter. Be sure to include error bar estimates on your plotted points. You'll want to use this data plot to answer some questions below.

In a third figure (Fig. 3, JPG or PDF) are images of what the Moon and background stars look like on April 23, 2005 at 8:00pm, but from two locations. The first is an image from Honolulu. The Moon's position in the sky is azimuth 112°, altitude 17°. The second image shows the Moon at the same time as it appears (from a boat) about 110km north of Honolulu. These two images can be used, with the technique presented in our Parallax lab, to determine the actual distance to the Moon on the evening of April 23. Hint: look at the images. They are different.
 

LAB WRITEUP

Include your data tables, plots, and any calculations you do in order to answer these questions. By now I don't have to remind you to write actual paragraphs, not just answer the questions as if you were taking a quiz.
  1. From you graph, what are the dates when are we closest and furthest from the Moon (include your estimate of the uncertainty in these dates)?
  2. We know the Moon orbits the Earth in about 28 days. Why does it appear that the Moon is closest to the Earth only once per year? Explain the geometry of these observations and why we obtain them from one year of measurements.
  3. What would your graph look like if the Moon's orbit were circular? By what percentage does the radius of the Moon's orbit deviate from a circle?
  4. Why doesn't the graph show a complete cycle in a 12 month period? The Earth is back in the same location in its orbit (that's the definition of a year), and since the Moon is full, it's in the direction away from the Sun, same as last year. Try a drawing to help think about this. What happens between the full Moon observations?
  5. Using the parallax data (above) calculate approximately what the Earth-Moon distance is. A drawing, looking down from very far above Honolulu and showing the two observing sites, the Moon and stars, will help to see how to do this. Using the Earth-Moon distance, and the angular diameter of the Moon from your measurement for April 23, what is the approximate diameter of the Moon in kilometers? As always, include your estimates of the errors in these numbers.

 

mickey@ifa.hawaii.edu

Last modified: May 13, 2005
http://www.ifa.hawaii.edu/users/mickey/ASTR110L_S05/moondialab.html