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 a telescope; we will make a series of measurements and combine them to study the Moon's orbit.
The most useful laws of nature can be applied in many different situations. Kepler's three laws, invented to describe the orbital motion of planets about the Sun, are very useful: with minor modifications, they also describe the Moon's motion about the Earth, the orbits of Jupiter's satellites, and the orbital motions of binary stars. The Moon provides a natural laboratory for orbital motion; we can use it to make a simple test of 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 provides a way to compare different orbits.
These same three laws can also describe the Moon's orbital motion around the Earth: just 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 other 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 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.
To measure the Moon's apparent 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 Moon's diameter in the units on the scale.
Fig. 1. Measurement of Moon's apparent diameter on 02/20/03 at 06:55 HT (16:55 UT). At this 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 is west since that's the direction the Moon appears to move 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, making three sets of measurements! This includes the initial step of rotating the eyepiece in the holder. Repeated measurements yield better accuracy; they also give you a fighting chance of spotting any errors you may have made.
Weather permitting, we will make measurements each time the Moon is visible 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 two of these values are reasonably close together, the other 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 that average value d. Now to calculate the Moon's distance, use this equation:
Here F is the focal length of the telescope's main mirror, which is F = 1200 mm. Because d and F both have units of millimeters, D is a pure number -- the units of d and F cancel out. In fact, D is the Moon's distance in units of the Moon's actual diameter.
Where did this remarkable formula come from? It is our old friend arc = angle x distance (with the angle in radians). If you think how a telescope works, you will realize that the size of an image at the focal plane is equal to the angular size of the object (in radians) times the focal length of the objective. The measuring scale is calibrated in millimeters and is located at the focal plane. Then d/F gives us the angular size of the Moon. Now we apply distance = arc/angle. Since we want to express the distance in units of the Moon's size, arc=1. But then distance = 1/angle, or F/d.An example may help make all this clear. In Fig. 1, the Moon's image is d = 11.5 mm across. Using this value in the equation, we get D = 104.3 for the Moon's distance, in units of the Moon's diameter. To express the Moon's distance in units of, say, kilometers, you can multiply D by the Moon's actual diameter in kilometers (3,476 km); the result is about 363,000 km, which is a reasonable distance for the Moon when it's near perigee. But for this assignment, the Moon's diameter provides a perfectly good yardstick, so there's no need to go through the final step of expressing the distance in kilometers.
Once you've calculated D for each night, you should make a plot showing how the Moon's distance varies with time. One way of doing this is to use the same method we used to plot light curves of variable stars. In this case the period P is the "anomalistic" month; the time it takes for the Moon to complete one revolution around the Earth from one perigee (the point closest to the Earth) to the following perigee. This period changes with time, but its average value is about 27.6 days. So you can do the following: transform the times t of all your observations to Julian dates, using the tables you already have; adopt an arbitrary t0= 2453601.75 (you will notice later, I hope, that this t0 is not so arbitrary after all), calculate the orbital phases = decimal part of (t-t0)/P, and finally plot your lunar sizes as a function of the orbital phase. Unfortunately, the data points you'll have won't be enough; 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 which you can measure in class. With these additional measurements, your graph should show a smooth variation in the Moon's distance with orbital phase.
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.
Web page describing the variation in the Moon's apparent size as a result of its elliptical orbit. Created by John Walker.
JavaScript program to calculate dates of lunar perigee and apogee. Created by John Walker.
Animation showing the Moon as seen from the Earth from 07/31/05 at 14:00 HT to 12/31/05 at 08:00 HT (08/01/05 at 00:00 UT to 12/31/03 at 18:00 UT). Note the rhythmic variation in the Moon's apparent diameter and the ``wobbling'' motion known as libration. Generated using Solar System Simulator (Courtesy NASA/JPL-Caltech).
Roberto H. Méndez
(mendez@ifa.hawaii.edu)
Last modified: October 20, 2005 http://www.ifa.hawaii.edu/~mendez/ASTRO110LAB05/moonorbit.html |