Ptolemy sizes and distancesplease refer to ptolemys...

Ptolemy: Sizes and Distances Please refer to Ptolemy's planetary model (geocentric) in an astronomical and historical context. Question 1: The absolute distance to the moon, at closest approach, is 33 earth radii (e.r.). The absolute distance to the sun, at closest approach, is 1100 e.r. You may need to refer to a calculator. (a) Use the formula for the circumference of a circle to work out the circumferences of the circles corresponding to the closest positions of the moon and sun, in earth radii. (b) If the moon and sun both seem to be half a degree wide when viewed from the earth, how much larger, in earth radii, is the diameter of the sun than the diameter of the moon?

Transcript

00:01 So let's say we have earth here, and the moon is orbiting in an orbit like this.

00:07 And we're told this distance, we'll call it d, is 33 times the radius of earth.

00:13 We use a capital letter r from that.

00:16 And then the earth's orbit from the sun, we'll call this distance little r, is 1 ,100 times the radius of earth.

00:28 And so we want to find the circumference.

00:34 Corresponding to these positions in terms of the earth's radius.

00:38 So the circumference corresponding to, i guess, the moon, is just going to be 2 pi times the distance d, which is going to be 2 pi times 33 times the earth's radius.

00:52 So we could write this as 66 pi times the earth's radius.

00:58 And then doing this for the sun, this is going to be 2 pi times little r, which is going to be 2 ,200 times pi times capital r.

01:10 I guess that's both part a.

01:12 Part b asks us to find how much larger in terms of the earth's radius is the diameter of the sun than the diameter of the moon, given that they have the same angular diameter.

01:25 So what that really means is like if we think about the angular size, the diameter of the moon, let's call it, we'll call it d with the subscript for the moon.

01:37 This is going to be basically the angular diameter of it, we'll call this theta, times this distance, d.

01:44 And we're told that if they have the same angular diameter, then that means theta also equals the diameter of the sun divided by this distance, right? and so what we have is the diameter of the sun, which is this symbol, divided by its distance from earth, should be equal to the diameter of the moon divided by its distance from earth.

02:05 And and so if we're looking for the difference between these numbers, like the diameter of the sun minus the diameter of the moon, right, is just going to be equal to, this should just be equal to r over d minus one times the diameter of the moon.

02:30 And we're told r is 1100...

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