# Half Question/Half-Challenge: Dimensions of of the Earth-Moon System

1. May 16, 2012

### Vorde

I've been considering this problem for about a year now (whenever I remember about it, that is), and I've come to the conclusion that I can't figure out a way to do what I want to see is possible, and so I've decided to ask it here to see if anyone else can.

The original question I considered was: could a classical astronomer determine the size of the moon to an arbitrary level of accuracy? Since then I've sort of refined the question, and here are my qualifications:

1) The process has to be able to be theoretically done from a single spot on the Earth (so triangulation is allowed, but only one point of the triangle can be on Earth).
2) You are only allowed one-way measurements, so you can throw something at the moon and measure that, or measure something coming from the moon (or somewhere else), but you can't involve the measurements of one object going to and from the moon (including light).
3) An arbitrary amount of accuracy is allowed with imaginary telescopes and the like.
4) You don't know any distance or velocity values for other solar system objects.

With that in mind, the question is: What is the radius of the moon?

My solution is to, using an extremely accurate telescope, measure the gravitational lensing of the moon, thereby getting its mass. Using a tide-measuring system, and knowing the moons mass, you could figure out the acceleration/force you are experiencing from the moon, and thereby figure out the distance to the moon, and then the radius.

However to me this fails on two counts. 1) I'm not positive the gravitational lensing test will produce a value for M instead of an M/D ratio, and 2) It feels like the use of GR goes against the original purpose of the question.

Does anyone have a more elegant solution? Or knowledge that there isn't one?

2. May 16, 2012

### Steely Dan

Joke answer: send an astronaut to the moon with a meter stick to measure the distance, and have him beam back a radio signal with the answer. One object went, and a different one came back :)

Serious answer: The closest method that comes to mind is of course to use the parallax method. You've thrown in the monkey wrench of not being able to use triangulation from different points on the Earth, but you could avoid that by using the parallax occurring from the Earth's rotation (for example, measure the angular position on the sky at sunset and then again at sunrise). Unfortunately, the Moon will move through space during that several hour period (though it works better for more distant objects whose proper motions are smaller in that period -- David Gill used this to measure parallaxes from a single spot to high accuracy, for his time anyway). If you're allowed to know the velocity of the moon (and surely one can measure that using classical techniques) then one might be able to account for that proper motion during the interim period, but I haven't given it much thought as to exactly how you would do it.

3. May 16, 2012

### Vorde

That would work, I think, if you knew the velocity of the moon, but how would that be obtained?

4. May 16, 2012

### Steely Dan

Well, the moon's orbit is eccentric. Suppose one could figure out the degree of eccentricity of the orbit using the change in apparent angular size of the moon from perigee to apogee. Then, one could in principle use the Doppler effect to figure out what the non-circular component of the velocity is, and perhaps from there work out what the total velocity must be? Just a guess, I haven't even worked it out to see if this is enough information.