Kepler's first law implies that the Moon's orbit is an ellipse with the Earth at one focus. The distance from from the Earth to the Moon varies by about 13% as the Moon travels in its orbit around us. This variation can be measured with our telescopes; we will make a series of observations and combine them to study the Moon's orbit.
The most useful laws of nature apply in many different situations. Kepler's three laws, invented to describe the orbital motion of planets about the Sun, also give a fairly accurate description of the Moon's motion about the Earth, the orbits of Jupiter's satellites, and even the orbits of double stars. The Moon provides a natural laboratory for orbital motion; we can use it to test Kepler's first law.
Kepler's three laws of planetary motion are:
In effect, the first law describes the shape of a planet's orbit, the second says how a planet's speed varies at each point on its orbit, and the third law compares average speeds of different planets.
These three laws also describe orbital motion around the Earth: substitute Earth for Sun and Moon for planet. (Of course, the Earth has only one Moon, but we could use the third law to compare the Moon's orbit with the orbit of the Space Station or an artificial satellite.)
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. (By the way, the Sun is closest in January, and furthest in July, so this change doesn't explain the seasons!) In contrast, the ellipticity of the Moon's orbit is fairly dramatic; the Moon's distance from the Earth varies between 92.7% and 105.8% of its average value of 384,400 km.
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 angular diameter, a measure of how big the Moon appears as seen from Earth. 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.
To measure the Moon's angular diameter, we use a 25 mm eyepiece equipped with a measuring scale. Looking through this eyepiece, you can see the scale, which is something like a ruler, superimposed on the Moon's image. The basic idea is to point the telescope at the Moon, align it so the scale goes right across the Moon at its widest point, and measure the diameter of the Moon's image using the scale.
Fig. 1. Measurement of Moon's angular diameter on 20-Feb-2003 at 06:55 HST (16:55 UT). At this date and time, the image of the Moon's disk was 5.8 mm + 5.7 mm = 11.5 mm in diameter.
Fig. 1 shows how the measurement is made. Notice that this scale, unlike a ruler, has its zero point in the middle. So to determine the diameter of the Moon's image, you measure from the midpoint to each side of the Moon's disk, and add these two values to get the total. The scale is calibrated in millimeters, so your result should be expressed in millimeters. Also, notice that the eyepiece has been rotated so the scale crosses the disk of the Moon at widest point. If the scale had been rotated any other way, the measured diameter would have been less than the true value. It's always possible to turn the scale to span the Moon's true diameter, no matter what the Moon's phase; for example, the diameter of a crescent Moon is measured from `horn' to `horn'.
The most efficient procedure is to use the Earth's rotation to slowly move the scale across the face of the Moon. First, rotate the eyepiece in the holder until the scale is parallel with the widest part of the image (if the eyepiece doesn't rotate easily, loosen the screw holding it in place). Second, point the telescope a little to the west of the Moon — you can easily tell which way is west since the Moon appears to move west as a result of the Earth's rotation. Try to place the dividing line somewhere in the middle of the Moon's disk, but don't worry about centering it exactly. Third, wait while the Moon's image drifts past the scale, and make a measurement when the widest part of the image falls on top of the scale. Record the distances from the dividing line to the two sides of the Moon's disk separately; then add them and record the total.
Repeat these steps at least three times, using three different telescopes! Repeated measurements yield better accuracy; they also give you a fighting chance of spotting any errors you may have made. If you notice that the total diameters you get with different telescopes don't agree, you may want to try your measurement again.
Weather permitting, we will make measurements each time the Moon is visible for the rest of this semester.
The three measurements you've made each night give you three independent (and probably different) values for the total diameter of the Moon's image. Don't worry if these values differ by 0.1 or 0.2 mm or so; that's normal measurement uncertainty. But if one value is very different from the other two, you probably made some kind of mistake while taking that measurement. You should drop any obviously incorrect measurements before going on to analyze your observations.
For example, suppose you made three measurements, and found total diameters of 11.0 mm, 11.1 mm, and 11.2 mm. These values are all pretty close to one another, and you can average them to get 11.1 mm. On the other hand, suppose you found diameters of 10.1 mm, 11.0 mm, and 11.2 mm; while the last two values are reasonably close together, the first is very different. In this case, it's likely that the 10.1 mm value is incorrect, while the others are reliable and can be averaged to get 11.1 mm.
For each night, average all the values you think are reliable; the result is your best measurement of the diameter of the Moon's image that night. Call this average value d. Now to calculate the Moon's distance R, use this equation:
Here F = 1200 mm is the focal length of the telescope's main mirror, and D = 3,476 km is the Moon's diameter. Since d and F both have units of millimeters, the ratio F/d is a pure number — the units of d and F cancel out. In fact, this ratio is also the ratio of the Moon's distance to its diameter. To get the Moon's actual distance, just multiply this ratio by the Moon's diameter.
An example may help make this clear. In Fig. 1, the Moon's image is d = 11.5 mm across. Using this value, we get F/d = 104.3 (look Mom, no units!). To find the Moon's distance in kilometers, you multiply F/d by the Moon's actual diameter D in kilometers; the result is R = 363,000 km, which is a reasonable distance for the Moon near perigee.
Once you've calculated R for each night, you can make a plot showing how the Moon's distance varies with time, using the blank graph we'll hand out in class. Unfortunately, the data points you'll have won't look like a smooth curve; there's too much time between measurements, and your graph won't include the half of each month when the Moon rises late at night. So we will take photographs of the Moon at other times throughout the semester, which you can measure in class. With these additional measurements, your graph should show a smooth variation in the Moon's distance with time.
To actually plot the Moon's orbit as an ellipse we would need more information. It's not enough to know how far away the Moon is; we also need to know the direction from the Earth to the Moon. However, the graph you've made can still be used to test Kepler's 1st law.
Use these worksheets to record and organize your data.
Use this chart to make a graph of R over time.
Animation showing the Moon as seen from the Earth from 01-Aug-2011 at 00:00 UT to 31-Dec-2011 at 18:00 UT (31-Jul-2011 at 14:00 HST to 31-Dec-2011 at 08:00 HST). Note the regular variation in the Moon's angular diameter and the `wobbling' motion known as libration. Generated using Solar System Simulator (Courtesy NASA/JPL-Caltech).
Another way to measure the Moon's angular diameter. Do you think this will work in practice?
Joshua E. Barnes
(barnes at ifa.hawaii.edu)
13 September 2011