Is The Moon's Orbit Elliptical

Shape of the Moon's Orbit

Kepler'southward first law implies that the Moon'southward orbit is an ellipse with the Earth at 1 focus. The distance from from the Earth to the Moon varies past virtually xiii% as the Moon travels in its orbit around us. This variation tin can be measured with a telescope; we will brand a series of measurements and combine them to study the Moon's orbit.

The near useful laws of nature can be applied in many different situations. Kepler'south 3 laws, invented to describe the orbital motion of planets about the Sun, are very useful: with minor modifications, they likewise describe the Moon's motion about the Earth, the orbits of Jupiter's satellites, and even the orbital motions of binary stars. The Moon provides a natural laboratory for orbital motion; we tin can use it to brand a uncomplicated test of Kepler's first police.

Kepler'southward three laws of planetary motion are:

A planet travels around the Lord's day in an elliptical orbit with the Sunday at one focus.
A directly line drawn from the planet to the Sun sweeps out equal areas in equal times.
The quantity P ²/a ³, where P is a planet's orbital period and a is its average distance from the Sun, is the same for all planets.

In result, the first police 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 3rd law provides a way to compare different orbits.

These same three laws tin likewise describe the Moon's orbital motion around the Globe: but substitute Earth for Sun and Moon for planet. (Of course, the Earth has only one Moon, just we could employ the third constabulary to compare the Moon's orbit with the orbit of the Infinite Station or other artificial satellite.)

THE MOON'S ORBIT

Kepler'south showtime law says that planets have elliptical orbits. As a issue, the altitude between a planet and the Sun changes rhythmically equally the planet moves in its orbit. In many cases, this rhythmic alter is rather subtle; for example, the Earth's distance from the Sun varies between 98.3% and 101.seven% of its boilerplate value. (By the way, the Sun is closest in Jan, and furthest in July, so this alter doesn't explicate the seasons!) In contrast, the ellipticity of the Moon'southward orbit is fairly dramatic; the Moon's altitude from the World varies betwixt 92.7% and 105.8% of its average value of 384,400 km.

This variation in distance produces several furnishings which nosotros can observe here on World. For example, when the Moon is closest to the Earth ( perigee ), information technology moves faster, while when information technology is furthest from the Earth ( apogee ), it moves slower. The Moon too appears to nod back and along a scrap as it orbits the World. Only the most dramatic issue is the change in the Moon's apparent diameter : when the Moon is close, it looks larger, and when the Moon is far, information technology looks smaller. We will use this effect to study the change in the Moon's distance.

OBSERVATIONS

To measure the Moon's apparent diameter, nosotros utilize a 25 mm eyepiece equipped with a measuring scale. Looking through this eyepiece, you lot can see the scale, which is something similar a ruler, superimposed on the Moon's image. The basic idea is to point the telescope at the Moon, marshal it so the scale goes right beyond the Moon at its widest point, and measure the Moon's bore 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 paradigm of the Moon's disk was v.viii mm + five.7 mm = 11.5 mm in bore.

Fig. 1 shows how the measurement is fabricated. Find that this scale, different a ruler, has its zero point in the middle. And then to make up one's mind the diameter of the Moon's epitome, you measure from the midpoint to each side of the Moon's disk, and add these two values to get the full. The scale is calibrated in millimeters, then 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 calibration had been rotated any other way, the measured diameter would have been less than the true value. It'due south always possible to turn the calibration to bridge the Moon's true diameter, no affair what the Moon'due south stage; for example, the bore of a crescent Moon is measured from ``horn'' to ``horn''.

The almost efficient procedure is to use the Earth's rotation to slowly move the calibration across the face of the Moon. Get-go, rotate the eyepiece in the holder until the scale is parallel with the widest part of the image (if the eyepiece doesn't rotate hands, loosen the screw holding it in place). 2nd, bespeak the telescope a picayune 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 outcome of the World's rotation. Effort to place the dividing line somewhere in the middle of the Moon'southward deejay, just 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 epitome falls on top of the scale. Tape the distances from the dividing line to the two sides of the Moon'due south disk separately; and then add together them and record the total.

Repeat these steps at to the lowest degree 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 likewise give y'all a fighting chance of spotting any errors you may have made.

Weather permitting, we volition brand measurements each time the Moon is visible this semester.

ANALYZING YOUR DATA

The three measurements you've made each dark give you lot iii 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.ii mm or then; that's normal measurement uncertainty. But if one value is very different from the other two, y'all probably fabricated some kind of fault while taking that measurement. You should drop whatsoever obviously wrong measurements before going on to analyze your observations.

For example, suppose y'all made 3 measurements, and found total diameters of 11.0 mm, 11.one mm, and 11.2 mm. These values are all pretty close to one another, and you can boilerplate them to get 11.1 mm. On the other hand, suppose you lot constitute diameters of 10.one mm, 11.0 mm, and 11.2 mm; while 2 of these values are reasonably close together, the other is very different. In this example, information technology'due south likely that the 10.1 mm value is wrong, while the others are reliable and can be averaged to get 11.1 mm.

For each night, boilerplate all the values you think are reliable; the result is your best measurement of the diameter of the Moon's paradigm that night. Call that boilerplate value d. Now to calculate the Moon's distance, use this equation:

D = F ÷ d .

Here F is the focal length of the telescope's master 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 bore.

An example may help make this clear. In Fig. 1, the Moon's image is d = 11.5 mm beyond. Using this value in the equation, nosotros get D = 104.iii for the Moon's distance, in units of the Moon'southward diameter. To express the Moon's altitude in units of, say, kilometers, y'all can multiply D by the Moon'southward actual diameter in kilometers (iii,476 km); the result is about 363,000 km, which is a reasonable distance for the Moon when it's about perigee. Merely for this assignment, the Moon's diameter provides a perfectly good yardstick, and then there's no need to go through the final stride 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, using the bare graph we'll hand out in class. Unfortunately, the data points you'll have won't look like a shine curve; in that location's too much time between measurements, and your graph won't include the half of each calendar month when the Moon rises belatedly at dark. So we will take photographs of the Moon at other times which yous can measure in class. With these additional measurements, your graph should show a polish variation in the Moon's distance with time.

To actually plot the Moon'south orbit as an ellipse nosotros would need more than information. It's non enough to know how far away the Moon is; we as well need to know the direction from the World to the Moon.

WEB RESOURCES

Worksheet #ane and Worksheet #2
Use these worksheets to record and organize your information.
Blank Graph for Moon's Distance: PNG file or Postscript.
Utilise this chart to make a graph of D over time.
Inconstant Moon: The Moon at Perigee and Apogee
Spider web page describing the variation in the Moon's apparent size as a result of its elliptical orbit. Created by John Walker.
Lunar Perigee and Apogee Calculator
JavaScript program to calculate dates of lunar perigee and apogee. Created by John Walker.
Diameter of the Moon: high res. (3.ix Mbyte mpeg); low res. (0.eight Mbyte mpeg)
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 xviii:00 UT). Note the rhythmic variation in the Moon'south credible diameter and the ``wobbling'' motion known every bit libration. Generated using Solar Organisation Simulator (Courtesy NASA/JPL-Caltech).

REVIEW QUESTIONS

During a solar eclipse, the Moon comes between you lot and the Sunday. In some full eclipses, the Moon completely blocks the Dominicus's light. In others it does not, fifty-fifty when the Moon is exactly in front of the Sun. Why?
Why does the Moon appear to move faster beyond the sky at perigee, and slower at apogee? (Note: a complete answer to this question besides involves Kepler'south 2nd police.)
Suppose the Moon's orbit was an ellipse with the World at the center, rather than at i focus. How many times per month would the Moon approach and recede from the Earth?