Oleg Zabluda's blog
Monday, May 21, 2012

Now that the annular solar eclipse is over, we can use it to calculate the distance to the Moon, from observations...
Now that the annular solar eclipse is over, we can use it to calculate the distance to the Moon, from observations (twitter/etc).

At 6:30:00 pm PDT at 39.5957° N, 122.6594° W the Moon was exactly at the center of the Sun..

At the same time, At 6:30:00 pm PDT at 40.021° N, 119.8203° W the Moon was exactly at the center of the Sun with eclipse magnitude of 0.936.

The distance between these two points (along the surface of the Earth is 153.5 mi. Parallax of the Moon is (1-0.936)/2=0.02805 degrees. Which makes the distance to the Moon equal to 153.5*360/((1-0.936)*pi) = 275 Kmi.

That's too much by 9%. But because I approximated the angular diameter of the Sun as 1/2 degrees, but during the eclipse it was 0.5267° (5.34% more) we have only 3.4% unaccounted for. I think this is due to the the extreme sensitivity of this method to the eclipse magnitude, and exactly coinciding time.

A better method may be to find a point where the Moon just touches the Sun from the outside at the same time (there is a whole "circle" of them). If the other point is where the Moon is in the middle of the Sun, then we don't care which point it touches. For example,

At 6:30:00 pm PDT at 13.8798° N, 153.4436° W the Moon was touching the Sun. Parallax is 1/2 degree, which makes the distance 2580*360*2/(2*pi)=296 Kmi, which is 17.5% too much.

Adding more precision: 2580 mi away along the geodesic. is ~1/10 of Earth circumference. sin of the 1/2 of the central angle differs from the angle itself by sin(2*pi/10/2)/(2*pi/10/2)=0.98 [1], making the "tunnel" length 2% shorter or 2530 mi. Also parallax is not 1/2 degree, but ~0.51 [2] degree. 1.02 * (0.51/0.5)=1.04, giving us about 4% correction, so the result is 296/1.04=285 Kmi. That's still too much by 13%. I attribute it to the sensitivity to the determination of the point where the Moon just grazes the Sun at exactly the same time 6:30:00 pm. That point might have been 2530*0.07=330 miles closer.

Maybe using annular solar eclipse is not a good way to determine moon parallax at all. In contrast, the famous Jan 24, 1925 total solar eclipse in NYC, allowed the boundary of totality to be established to within one city block. It was reported that those above 96th Street in Manhattan saw a total solar eclipse while those below 96th Street saw a partial eclipse.

Let try it another way. At 5:14:35.0 pm PDT at 40.0147° N, 119.8014° W (USA), on the centerline of the eclipse, the eclipse started. At the same time, at 37.926° N, 144.237° E (Japan), also on the centerline, the eclipse ended. Distance between these points is 4882 mi, 1/5 of Earth circumference, making chord correction 6.2%, and "tunnel" distance 4580 mi. Total parallax is 0.5267+0.4961=1.0228°, and the distance 4580/sin(1.0228°) = 257 Kmi, which is too much by only 2%. Whew, this actually works.

[1] Or, 1-(pi/10)^2/6=0.98
[2] During the eclipse, at the "13N,153W" Sun was actually 0.5267° , and the Moon was 0.4961° for the total parallax of 0.5114° or 2.3% more.