220 GEOPHYSICAL THEORY UNDER THE PLANETESIMAL HYPOTHESIS. 



gible; for example, in the case of a planetoid a mile in diameter, composed 

 of rock similar to the earth's surface strata and with a = 2X10~^, it would 

 be only two parts in a billion; but for a sphere having the same mass 

 and dimensions as the earth, with the same value of a, it would be over 

 one-fourth; in the latter case the treatment of the conduction of heat 

 independently of its mechanical effects could hardly give more than a crude 

 approximation. It seems conceivable that a planet considerably larger 

 than the earth, even though practically soHd, might exhibit the phenomenon 

 described by Lane as occurring in gaseous bodies, of contraction accompanied 

 by rise of temperature. 



As compared, however, with the simple case just mentioned, there is 

 an essential contrast shown with a distribution of temperature like that 

 described above as characteristic of the mode of origin postulated by the 

 planetesimal hypothesis. Here not only are the initial temperatures near 

 the surface small in comparison with those developed in the interior, but 

 the changes are widely different at different depths, the interior steadily 

 shrinking as the heat is conducted outwards, while the outer strata tend at 

 first to expand under the rise of temperature which continues until the 

 maximum surface-gradient is reached. In the early stages the surface- 

 gradient is slight; the thermal energy is for the most part simply redis- 

 tributed within the earth, while comparatively little is lost through the 

 surface. Thus any gain of heat from potential energy on the whole could 

 come only through a preponderance of the internal shrinkage. 



To estimate the nature of these movements let it be supposed that there 

 is continual accommodation of the density of each portion of the mass to its 

 temperature, while the accompanying variations of the pressure affecting a 

 given particle are relatively negligible, so that there may be considered to 

 be a definite coefficient of expansion a, a function of r. 



If then d denote variations in time, the adjustment for equiUbrium is 

 determined by the conditions 



* 



ddv=advdd dv= ATtr^dr 



where 5r is the change in the central distance r of a given particle, so that 



r 



— At: I a 

 J a 



dv=A.Ti I addr^dr 

 J a 



The equation of expansion may also be written 



8dv ^^ a 



dv op 



in which a is in effect the coefficient of specific volume expansion referred 

 to the variation in the heat content Q instead of the temperature d. Then 

 the rate of radial motion is 



^=1 rnl(lr^-'l\dr (128) 



dt 



r^ J dr\ dr / 



