The Moon’s Size and Distance

Size of the Moon

Around 300 B.C. Aristarchus of Samos determined the size and distance to the Moon.  The diameter of the moon was compared to the diameter of the Earth by examining the size of earth’s shadow on the moon during an eclipse.  Note that the Earth’s shadow at the moon is not the same diameter as the Earth itself, due to the cone effect (see drawing below).  This has to be taken into account when estimating the size.

The first order of business is to find the diameter of the earth’s shadow compared to the size of the moon.  Call that number D_S.  In other words, if the shadow appears to be twice the diameter of the moon, then D_S=2.  Then the diameter of the earth compared to the diameter of the moon is given by: 

D_E = (1 + D_S) D_M 

D_E = diameter of the Earth

D_S = diameter of the Earth’s shadow

D_M = diameter of the Moon

Aristarchus estimated that D_S=2 (i.e., the shadow of the Earth was about twice the diameter of the Moon), so he concluded that the Earth is about three times (1+2) the diameter of the Moon. (if you want to see how the above equation is derived see Crowe’s Theories of the World, p. 29)

Distance to the Moon

Consider the relationship between the diameter (D), radius (R) and circumference (C) of a circle:

The relationship between the segment S, the radius R and the angle A is :

S/C = A/360;  S/(2piR) = A/360;  2piR = Sx360/A;  R= (Sx360)/(Ax2pi)

[pi = 3.1415]

Now, considering the sky as a circle, the moon’s diameter as a segment of that circle (S) and the amount of sky the moon occupies as the angular diameter of that segment (A) we have:

S = Moon Diameter  R= Distance to Moon   A = angular size of Moon = ½ degree

Distance to Moon = (Moon diameter x 360)/ (Ax2pi) = (Moon diameter x 360)/( ½x2pi) =

(Moon diameter x 360)/ pi

Aristarchus (circa 300 B.C.) found the moon’s diameter to be 1/3 that of the Earth.  Thus,

 Distance to the Moon = (Earth diameter x 1/3 x 360)/ pi=  (Earth diameter x 120)/ pi

Homework

Now you try it.  Here is a picture of the moon during an eclipse.  The image here is a negative to make it easier to work with.  Estimate how much bigger the earth’s shadow is than the diameter of the moon.  I would suggest that you first complete the circle of the moon, then try to fit a larger circle to the shadow curve. You can draw a circle however you like, but the use of a compass is  highly recommended.  Once you have the diameter of the shadow (D_S), estimate the diameter of the earth (D_E) using the equation:

D_E(Earth diameter) = (1+D_S) D_M (Moon diameter)

or

D_M=D_E/(1+D_S)

 This is the diameter of the moon relative to the diameter of the Earth

Now determine the distance to the moon in terms of Earth diameters using the formula above.  Remember Aristarchus’ estimate given above is not right.  Try to improve on his figure.