The Distance and Orbit of the Moon
We made measurements of the diameter of the Moon at the telescope two weeks ago,
getting an average value for the diameter of the Moon’s image in our telescopes of 11.4
mm. Last week, you measured diameters of photographs of the Moon taken over almost 3
orbits and plotted these on a chart. Tonight, we are going to use these measurements to
estimate a distance for the Moon and something about the nature of the Moon’s orbit.
In order to get a distance for the Moon in physical units (e.g., kilometers), we need to
know its diameter in those units. By about 100 BC, Greek astronomers correctly estimated
the size of the Moon by observing the curvature of the Earth’s shadow on the Moon during
an eclipse. They already had determined the size of the Earth, so they could figure out
from the size of the Earth’s shadow the actual size of the Moon. We will give more details
on how this was done later.
First, take the charts on which you have plotted your measurements of the diameter of the
Moon pictures and try to sketch a smooth curve through the points (do this in pencil until
you think it looks good). Don’t put in wiggles to make it go through the center of each
point, and your curve might even miss going through some of the lines you have drawn
through your points; what you want is a curve that represents the broad distribution of
your points. This curve is your estimate of the actual change in size of the Moon around
the 2+ orbits covered by the pictures.
Of course, the actual numbers you measured are arbitrary, depending on the size the
images were printed. We now want to relate them to the actual physical diameter of the
Moon. This is a three-step process:
1. On the date that you measured the Moon’s diameter at the telescope (10/09/08 HST),
the Moon was at almost exactly the same distance as it was on 10/20/05. Find the point
on the curve you have drawn that corresponds to this date, and go over to the left to get
the Moon diameter from the photograph for the date. Divide this value into our average
telescope measurement (11.4 mm). You should get a number that should be between 0.15
and 0.19; if you get something else, call for help. Note that this is a pure number, since
the units of mm divide out. We will call this number s. If you were to multiply the values
of your graph by s, you would have a graph showing what you would have expected to
measure for the size of the Moon at the telescope on any of the dates shown on the graph.
2. We need now to relate what you measured at the telescope (in mm) to an angular
measurement on the sky (in degrees). Our telescopes have focal lengths of 1200 mm. This
means that the image that our eyepiece is looking at lies on a small part of a sphere with
this radius centered on the center of the telescope mirror. A great circle on this sphere will
have a circumference of 2πr, where r = 1200 mm, or about 7540 mm. Since there are
360° in a circle, each mm is about 0.048°. So now if you were to multiply the values for
your graph by 0.048s, your graph would give the size of the Moon in degrees in the sky
