Weighing the Moon

How would we go about determining the mass of the Moon?  The most 
direct way of determining the mass of an astronomical body is 
examining the radius and period of a satellite in orbit around that 
body.  Fortunately the Moon has a natural satellite, namely, the 
Earth.  Actually the two bodies revolve about their common center 
of mass, which is about 4670 km from the center of the Earth, i.e., 
about 3/4 the Earth's radius.

The Earth and Moon both revolve around this point every 27.3 days 
as the point revolves around the Sun.  This "wobble" in the Earth's 
orbit causes nearby objects such as the Sun and planets to exhibit 
a periodic variation in their expected longitudes, and this 
variation is not hard to detect with careful measurements.  It 
may even have been noticed in ancient times.  Anyway, these 
fluctuations in observed longitudes were the basis of our best 
estimates of the Moon's mass, right up until the Ranger 5 lunar 
orbit mission in 1962.

If R_e and R_m are the distances of the Earth and Moon, respectively, 
from their common center of mass, and if M_e and M_m are their masses, 
then we obviously have 

              R_e M_e =  R_m M_m

Since we know the distance between the Earth's center and the 
Moon's center is about 384,400 km from parallax measurements,
(as the Earth's rotation takes us from one vantage point to
another relative to the Moon each day) and "wobble" of the Earth 
is about 4670 km from observed solar longitude fluctuations, it 
follows that the mass of the Moon is about 4670/(384400-4670) = 
1/81.3 times the mass of the Earth.  Also, we can estimate 
the Earth's mass from the equation

    M_e  =  [ 4pi^2 (R_m + R_e)^2 R_m ] / GT^2

where T is the period 27.3 days and the gravitational constant
G is determined from ordinary terrestial measurements.  If we
take the values G = 6.67E-11 Nm^2/kg^2, T = 2.358E+06 sec, and
R_m = 3.797E+08 meters, R_e = 4. 670E+06 this gives 

           M_e  =  5.973E+24 kg
and so
           M_m   =   (M_e)/81.3  =  7.346E+22 kg

which agrees pretty nearly.  Of course, this all relies on the
precision of our parallax and longitude measurements, but people 
who pay close attention to the sky have been able to make remarkably 
precise observations of this kind, even back in ancient times, 
noting things like the occassional apparent retrograde motions of 
certain planets, and the precession of the equinoxes, and so on.