252 ANNUAL REPORT SMITHSONIAN INSTITUTION, 1960 



THE SHAPE OF THE MOON 



Measurements of the elevations of the lunar surface have been made, 

 but the methods of observation are difficult and not very precise : reli- 

 able evidence that it has an irregular shape comes from its dynamical 

 motions. Textbooks on celestial mechanics give a formula for com- 

 puting the difference in the moment of inertia about the polar axis 

 {C) and that about the axis pointing toward the earth (A). From 

 observed data {O — A)/C=^.000 629, whereas the theoretical value 

 deduced from consideration of the moon's shape imder its own gravi- 

 tational field, the earth's gravitational field, and the centrifugal forces 

 of rotation, is only 0.000 037 5. The ratio of these numbers is 16.7 : 1. 

 If the moon has a uniform density, the observations require that the 

 radius toward and away from the earth should be larger than that to- 

 ward the poles by about 1 km., whereas theory requires a difference 

 of only about 60 m. This irregular shape must result in a difference 

 in stress at the center of about 20 atm., and this requires considerable 

 strength of the material at the moon's deep interior if the density is 

 uniform throughout. 



If the density of the moon is not uniform, but varies according to 

 latitude and longitude by a small amount, the irregular shape can be 

 explained by a theory recently advanced [2a]. This requires the 

 density near the poles to be larger than that about the axis pointing 

 toward the earth. In this case, the strength of the deep interior need 

 not be very great, and the center could be at a high temperature. But 

 this variation of density with latitude and longitude could hardly 

 have been preserved if the moon had e\'er been generally molten. 



But could the moon have been molten even if it has a uniform 

 density? So large an object can cool only very slowly, even in a 

 time as long as the moon has existed, which we believe is about 4.5 

 eons (an eon being defined as 10^, or 1 billion, years). This age is as- 

 sumed to be the same as that of the meteorites. Calculations of the 

 loss of heat sliow that the center of the moon would lose little heat even 

 in this length of time. If the moon were ever molten, its center would 

 still be very close to its melting point and hence would not have the 

 required strength to support the irregular shape [3]. 



The presence of the radioactive elements potassium, uranium, and 

 thorium would increase the internal temperature, the effect depending 

 on the amounts of these elements and their distribution. We know 

 nothing about the distribution of these elements in the moon and have 

 difficulty in estimating their distribution in the earth. However, 

 if the heat being lost from the earth is assumed to be entirely due to 

 radioactivity, we find that the total amounts of these elements in the 

 earth may be very similar to that of the meteorites and that possibly 

 less than one-half of them are in the crust. In the moon, a similar 



