Effect of Moon's Orbital Speed on Frequency of Solar Eclipses

[jdlawlis]

A question from Chaisson & McMillan's "A Beginner's Guide to the Universe" is as follows:

If the Moon orbited Earth twice as fast, but in the same orbit, the frequency of solar eclipses would (a) double; (b) be cut in half; (c) stay the same.

The answer key states that the correct answer is (c), but I don't see how this can be correct. Setting aside the violation of Kepler's 3rd law in the statement above, the synodic month would be 14.8 days if the Moon orbited twice as fast. Under normal circumstances, at least two eclipses (solar or lunar) can occur during an eclipse season, which is a period of 31-37 days occurring every six months, where the Moon's orbit and Sun's path (differing by 5º) overlap. At least one new moon and one full moon will occur during an eclipse season, since a synodic month (29.5 days) is shorter than an eclipse season. This means that at least one solar eclipse (formed during a new moon) and one lunar eclipse (formed during a full moon) will occur during an eclipse season. Now, if the Moon orbits twice as fast, shouldn't there be twice as many eclipses? (There would be at least 2 full moons and 2 new moons occurring in 31-37 days). Am I missing something?

 

[jdlawlis]

I guess I'll answer my own question. According to the authors of the textbook, they subcontract the review questions/conceptual self-test questions at the end of each chapter to other people. It looks likely that there may be an error in the solution set.

 

[Drakkith]

No idea, sorry man.

 

[256bits]

I assume it would depend upon what they mean by frequency.

You are using the frequency to mean the number of eclipses per number of Earth units ie days, and the answer would be a) if the moon revolved twice as fast.

If you calculate the number of eclipses per number of revolutions of the moon, then yes this frequency would stay the same and the answer is c)

A bit of a crappy question from the book, if they did not explain what interpretaion of frequency the reader is to use.

 

[Vanadium 50]

Let's look at this another way - if the moon moved twice as fast, eclipses would last half as long, right? The total fraction of time the Earth spends in eclipse doesn't depend on the speed: that just depends on geometry. So if the duration is halved and the time is constant the frequency must double.

 

[256bits]

Let's look at this another way - if the moon moved twice as fast, eclipses would last half as long, right? The total fraction of time the Earth spends in eclipse doesn't depend on the speed: that just depends on geometry. So if the duration is halved and the time is constant the frequency must double.

And another way in case someone is asking how the moon could rotate twice as fast while in the same orbit:
Go out in space with a camera and take a video of the moon orbiting the Earth for a set time frame, say a year. Count the number of eclipses.
Then replay the video at twice the speed. The moon revolves at twice the speed. Count the number of eclipses in the video. It will be the same as before.

 

[Drakkith]

The moon cannot orbit twice as fast and stay in the same orbit. The orbital radius would have to be greatly reduced for the moon to stay in orbit at that velocity. So the effect on eclipses would greatly depend on the details of its new orbit.

 

[Vanadium 50]

Then replay the video at twice the speed. The moon revolves at twice the speed. Count the number of eclipses in the video. It will be the same as before.

Yes, for half as long. Which means you need to play the video twice.

 

[256bits]

Yes, for half as long. Which means you need to play the video twice.

Which means there would be twice as many eclipses counted in twice as many revolutions so the frequency of eclpises per revolution remains constant.

 

[Redbelly98]

There are usually 2 solar eclipses per year, at least judging from the 2001-2010 occurrences shown http://eclipse.gsfc.nasa.gov/SEdecade/SEdecade2001.html" . The sun has to be within about 1/2 degree of the plane of the moon's orbit; it passes through this plane twice per year.

For the frequency of solar eclipses to increase, the moon would have to orbit fast enough to get in between the Earth and sun twice while the sun remains within 1/2 degree of this plane. Don't know offhand if halving the moon's orbital period would accomplish this; the problem authors apparently say that it wouldn't.

 

[Nik_2213]

For a shorter 'lunar month', the Moon must orbit much nearer, and subtend a larger solid angle in sky. Not only will you get more total eclipses, but you will get a disproportionate increase in the number of partial eclipses...

This is the 'reverse problem' to exo-planet hunting by watching for transits...

FWIW, given tidal dissipation, the Moon is steadily retreating from Earth so, in distant future, Earth can only get partial solar eclipses...