4. Lunar Motions & Eclipses
Review: Chapter 1 - particularly pages 20-22
Chapter S1 also has some useful material
The cycles of the Moon feature in many cultures of the world. In this session we will describe the Moon's apparant and actual motions as well as discuss the very special occasions when the Sun, Moon and Earth are lined up- eclipses.
The phases of the Moon are shown in Figure 1.20 and Figure 3.11. Each day the Moon shows a slightly different phase, depending on where it was in its orbit around the Earth, The lunar phases cycle takes the 29.5 days that the Moon orbits the Earth and comes back to the same place relative to the Sun.
|(1a) Where in the sky and when does the full Moon rise? |
(b) Where in the night sky is the full Moon at mid-night?(2) Why is it not possible to see the Moon when it is exactly new Moon?
|Lunar Phases Home Experiment: For this quick and easy home experiment, you will similate the changing phases of the Moon.|
Since both the Earth and Moon are also orbiting the Sun, the orientation with respect to the stars is a little different. Figure 3.12 shows the difference between the 29.5 day lunar phases (synodic) month and the 27.3 day (siderial) month relative to the stars.
As the Moon orbits the Earth, its position in the sky will appear to change night after night. In fact, it will appear to rise later by nearly an hour each day.
Why the Moon rises nearly an hour later each day
If we were to look down on the North Pole of the Earth and the Moon in its orbit about the Earth, we would see something like the diagram below. The Earth would appear to rotate counter-clockwise, while the Moon would orbit also in the counter-clockwise direction (as shown by the arrows). In this picture, the Moon is directly overhead Boulder.
What happens 24 hours later? Boulder rotates around until it is in the same position as before. However the Moon has moved in its orbit by 12° (this is from 360 ° / 29.5 days). For us on the Earth, it would appear that the Moon is appearing in the sky later, since we have to wait a little longer the second day for the Moon to appear in the same position in the sky. Wait another day and the Moon will be even further in its orbit, and it would take longer for us to "catch up" to the Moon.
How long does it take for us to catch up each day? The Moon moves in its orbit 12° each day. The Earth then has to rotate an additional 12° for us to see the Moon in the same position in the sky as the previous day. Since the Earth rotates 360° every 24 hours, then its rate of rotation is:
Rotation rate = 360° / 24 hours = 15° per hourTo find out how long it takes for the Earth to rotate 12°, we set up the problem as follows:
Time for Earth to rotate 12° is 12° / (15°/hour) = 0.85 hours = 0.85 x 60 minutes = 51 minutesThis calculation assumes we are close to the equator. It is not exactly 51 minutes - the daily delay of moonrise can vary considerably with latitude.
The motion of the Moon would appear to us on the Earth like in the below diagram, where we are standing in Boulder, and looking towards the South, with the Flatirons to the West. The rotation of the Earth gives us the appearance of the Sun and Moon rising in the East and setting in the West. Since the Moon is appearing 51 minutes later each day, it would show up further to the east each day, and would take 51 minutes for it to move westward until it was in the same position in the sky as we had seen in the day before. The Moon will appear to move 12° further east from day to day. Click on the picture for an animation of this - each snapshot is a day later.
|(3) One day you see the Moon directly overhead at sunset. What phase is the Moon? |
(4) Where will the Moon be at sunset a week later?
(6) What phase will it be a week later?
EclipsesIn the below picture, the line across the sky is the ecliptic, the path that the Sun takes in the sky. If we were to view the Solar System from the outside, the ecliptic would actually be the orbit of the Earth about the Sun. From our normal perspective from the Earth however, the ecliptic appears as the path followed by the Sun. Marked on the ecliptic are the dates that the Sun is in that particular position. We are watching the Sun and the Moon in the early morning in Boulder. (The Greenwich Mean Time in the lower left corner reads 15:15 or 3:15 pm. What time is it in Boulder?) Over the course of a year, the elevation of the Sun changes depending on whether it is summer or winter. Click on the frame below to see an animation of the changing location of the Sun, Moon and ecliptic over the year.
The Moon's orbit about the Earth is nearly aligned with the Earth's orbit about the Sun, but is tilted by 5°. Figure 3.14 in the text shows the view if you could look down on the Sun/Earth system from far away. To us, this appears as if the path of the Moon in the sky is tilted by 5° with respect to the path of the Sun. As a result at any point in time, the Moon could be as much as 5° to either side of the ecliptic as shown in the animation. In fact, in the animation, you can see that the Moon's path results in it crossing the ecliptic many times, but it is never exactly on the ecliptic for very long.
Since the Moon is only 0.5°across, a new Moon does not guarantee a solar eclipse. The Moon and Sun could appear to miss each other as they pass in the sky by as much as 4.5° (that is about 9 diameters of either the Moon or Sun) because of the tilt of the Moon's orbit.
|(7) Why are you more likely to see lunar eclipses much more frequently than solar eclipses? |
(8) What does the Earth's shadow look like on the Moon? What does the Moon's shadow look like on the Earth? Compare the sizes of the shadows.
Solar EclipseLook at the diagram below which shows the alignment of the Earth, Moon and Sun during a solar eclipse. It is a remarkable fact that the Earth just happens to have a Moon of just the right size at just the right distance that from the Earth the Moon appears to be the same angular size as the Sun (about 1/2° or 30 arcminutes). Aha! We have a pair of similar triangles!
This means that:
dmoon / Dmoon = dsun / DsunAnd
Dsun / Dmoon = dsun / dmoonYou are probably begining to realize that angles play an important part in astronomy. Because we shall be using the properties of triangles to help us work with angles, particularly very small angles, go to Triangles & Small Angle Approximation for a review of similar triangles - before you try these last few exercises.
|(9) The distance to the Sun (Dsun) is 390 times greater than the distance to the Moon (Dmoon). What is the value of Dsun/Dmoon? |
(10) How many times bigger is the diameter of the Sun (dsun) than the diameter of the Moon (dmoon)?
(11) The Moon's orbit is not exactly circular--sometimes the Moon is a little closer to or farther from the Earth than the "just right" distance to fit exactly over the Sun. Figures 3.16 & 3.17 in the text may help.
(b) What does a solar eclipse look like then the Moon is farther from the Earth?
(c) What would an eclipse look like if the Moon was exactly twice the normal distance? Drawing a diagram will probably help.
(d) What would an eclipse look like if the Moon was exactly twice the normal distance? Drawing a diagram will probably help.
|Measuring the Sizes of the Sun and Moon Home Experiment:You will measure the sizes of the Sun and the Moon using a pinhole camera made out of a cardboard tube.|
Just for amusement.........
The Moon often appears to be much bigger when it is low in the sky. This is an illusion. The circles in the diagram are exactly the same size.
|To learn more about why the Moon appears larger on the horizon, go to Carl Wenning's excellent article New Thoughts on Understanding the Moon Illusion.|
Model answers to the comprehension questions.
Most moons in our solar system are tiny relative to the planets they orbit. These planets wouldn’t miss a moon or two if one got knocked out of orbit. But Earth’s moon is relatively large. So Earth without its large nearby moon would be a very different world indeed.
Imagine … no solar or lunar eclipses.
No calendars based on a system of months. The word month, after all, stems from a word that means moon. That’s because many calendars are based on the changing phases of the moon.
With no moon, there’d be no nearby world for astronauts to visit. We might never have begun to venture out into the solar system.
The moon and sun together cause the tides. If we’d never had a moon, we’d still have tides, but they wouldn’t be as strong.
What’s more, the moon has a place in human culture. Imagine no romantic moonlight walks – no concept of moon madness, or lunacy.
But the biggest change – for us humans and for other earthly life – would be in the length of Earth’s day. Without a moon, Earth would spin faster. Our day would be shorter. Why?
It’s because, billions of years ago when Earth was young, our planet spun around on its axis much faster. Our world’s cycle of day and night was less than 10 hours long. The ebb and flow of the tides are what put the brakes on Earth’s spin. So – if you’re imagining Earth with no moon – you have to imagine our day on Earth much shorter than our present-day 24 hours.
Selected moons of the solar system, with the Earth for scale. Notice that the moon is pretty big relative to Earth. But Pluto and its moon are even closer in size. Image via NASA.
Bottom line: How our planet would be different, if Earth didn’t have a moon.