moon stuff (WARNING: Math ahead!)


Miller, Mark C <mark.c.miller@...>
 

You can compute the angle taken up by the moon by knowing how wide it is and
how far away it is, then using a bit of basic trigonometry. Start by finding
the angle that half the moon makes. The almanacs say that the radius of the
moon is 1737.4 km, and the mean distance from Earth is 384400 km. If you
imagine a right triangle with base 384400 km and height 1737.4 km, the
desired angle is the small one opposite the right angle. The sine of that
angle is the ratio of the height to the base, which is 0.00452, and the
angle is the arc sine of that, or 0.25896 degrees. The full angle is twice
that (imagine two such triangles touching at their bases), or 0.5179
degrees. If 100 birds pass over a square 0.5179 degrees wide by 0.5179
degrees long every 70 minutes, and that density is uniform across the whole
sky, then the number of birds passing through an arc 0.5179 degrees wide and
180 degrees long in that time is 100 * (180/0.5179) or 34750, which matches
Greg's number to 4 significant figures. Even if the distribution of birds
isn't quite uniform, that's a lot of thrushes, warblers, vireos, and
flycatchers zipping by.

Mark Miller


Steve Sosensky
 

At 12:53 PM 5/12/00 -0700, Miller, Mark C wrote:
The almanacs say that the radius of the moon is 1737.4 km, and the mean distance from Earth is 384400 km. If you imagine a right triangle with base 384400 km and height 1737.4 km, the
desired angle is the small one opposite the right angle. The sine of that angle is the ratio of the height to the base, which is 0.00452, and the angle is the arc sine of that, or 0.25896 degrees. The full angle is twice that (imagine two such triangles touching at their bases), or 0.5179 degrees. If 100 birds pass over a square 0.5179 degrees wide by 0.5179 degrees long every 70 minutes, and that density is uniform across the whole sky, then the number of birds passing through an arc 0.5179 degrees wide and 180 degrees long in that time is 100 * (180/0.5179) or 34750, which matches Greg's number to 4 significant figures. Even if the distribution of birds isn't quite uniform, that's a lot of thrushes, warblers, vireos, and flycatchers zipping by.


I have a small quibble with the underlined statement above. The ratio discussed is the tangent, not the sine. The sine is the ratio between the height and the hypotenuse, not the height and the base. However, for very small angles (and this qualifies), the numbers are the same for tangent and sine.


Good birding,

Steve Sosensky, photographer                        www.sosensky.com
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