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 Re and Rm are the distances of the Earth and Moon, respectively, from their common center of mass, and if Me and Mm 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 Nm2/kg2, T = (2.358)106 sec, and Rm = (3.797)108 meters, Re = (4. 670)106 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.

 

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