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 



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 



where T is the period 27.3 days and the gravitational constant G is determined from ordinary terrestrial measurements. If we take the values G = (6.67)10^{11} Nm^{2}/kg^{2}, T = (2.358)10^{6} sec, and R_{m} = (3.797)10^{8} meters, R_{e} = (4. 670)10^{6} this gives 



and so 


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 occasional apparent retrograde motions of certain planets, and the precession of the equinoxes, and so on. 

Oddly enough, although Isaac Newton was well aware of the fact that the Earth and Moon revolve around their common center of gravity, he was evidently not aware of any parallax observations due to this motion, so he couldn’t use this method to estimate the mass of the Moon. Instead, he relied on the relative magnitude of the tides in the Earth’s oceans caused by the Sun and Moon. Note that a force of equal magnitude applied to all the various parts of an extended object will not cause the object’s shape to be distorted, because every part of the object will be accelerated equally in tandem. The tidal effect (i.e., the tendency for an object to be deformed into an ellipse) is proportional to the difference in the forces applied to the near and far sides of the object. Thus tidal effects are proportional to the derivative of the inversesquare force, which gives an inversecube effect, since d(r^{−2})/dr = −2r^{−3}. Based on the observed heights of the tides when the Sun and Moon are aligned, compared with when they are in opposition, and knowing the ratios of the distances, Newton estimated (in the Corollaries to Proposition 37 in Book 3) that the Moon’s mass was about 1/40 of the Earth’s mass, so his estimate was too high by a factor of 2. (As noted above, the real ratio of masses is about 1/81.) The problem is that the height of tides near various land masses is a complicated function of many different factors and resonance effects, so it can’t be used to give a simple estimate of the tidal forces. Ironically, Newton’s overestimation of the Moon’s mass implied that the center of mass of the EarthMoon system was outside the Earth, so the parallax effect would have been even more noticeable – but he never seems to have considered this. 
