All Experts Redux: Getting the Actual Diameter Of An Astronomical Object From Its Apparent (Angular) Diameter

 

Question -
If two objects have the same angular diameter, what does that tell us about their actual diameter?

Answer:

It doesn't actually tell you anything unless you also know the distance (r) to the objects. Only then can you infer the actual diameters.

For example, the Moon and the Sun have the same angular diameter in the sky, which is why total solar eclipses are possible. (Moon blocking out the Sun for certain parts of Earth).

In terms of actual diameter, however, the Moon is barely 2160 miles across, while the Sun is around 864,000 miles across. The Sun is actually 400 times larger than the Moon in diameter.  But, the only way this could be found out was by first finding the distances to each body.

To see how this would work, let 's' be the actual diameter, and the angle q ('theta') be the angular diameter (in radians) . You don't know 's' but do know the distance r.

Then you can find s from:

s = r x (q).

This equation merely makes use of the well-known relation for the length of an arc along the circumference of a circle (s), to its radius (r) and the angle (theta) subtended by the arc.

For example, we know the Moon subtends an apparent diameter of 1/2 degree.

We need to first convert this to radians to be able to use the equation.

One radian (1 rd) can first be converted into arcsec as follows, given there are 3600 arcsec per degree.:

1 rd = 57.3 degrees = 57.3 deg/rad x (3600"/ deg)= 206 280 "

But 1/2 degree  =   1800 arcsec = 1800"

Then: q = 1800"/ 206 280"/ rad = 0.009 rad

The distance r of the Moon from Earth in meters is:

r =  3.84 x 10 ⁸    m

Then the actual diameter of the Moon is:

s = r x (q) =  3.84 x 10 ⁸    m  (0.009 rd) = 3.45 x  10 ⁶  m

There are about 1600 meters in a mile so to convert the above to miles we simply divide to get:

3.45 x  10 ⁶  m/   1600 m/ mile =  2.155 x 10 ³  miles

Or, about 2, 160 miles as per our early estimate.