110 Arthur Holmes — Radio-activity 



This is practically the same as the result ultimately arrived at hy 

 Kelvin (twenty to forty million years, and probably nearer twenty than 

 forty). Ingersoll & Zobel discuss a case in which one-fourth of the 

 earth's heat is due to radio-activity, and show that the age is then 

 to be doubled. If it be assumed that oh the average three-fourths 

 of the heat is due to radio-activity, then the age is raised to 

 1,600 million years, as shown by equation 12. It should be noticed, 

 however, that the mathematical treatment does not take into con- 

 sideration the slow decrease of the radio-active generation of heat 

 with time. For example, 400 million years after the consolidation 

 of the crust, the gradient due to radio-activity would be about 

 15 per cent higher than now, and at the time of consolidation the 

 gradient would be about 20 per cent greater than now. It thus 

 happens that, actually, the effect of radio-activity has been to retard 

 normal cooling in the early periods of the earth's history more 

 effectively than the mathematical treatment would suggest. Taking 

 this into account, it can be shown that on the average the rate of 

 cooling has been retarded about 10 per cent more than if the heat 

 generation due to radio-activity were independent of time. Thus, 

 the age of the earth as measured from the consolidation of the crust, 

 if that event ever took place, would be 1,750 million years instead of 

 1,600 million years, or alternately, if 1,600 million years be adopted 

 as the maximum, the heat flow from radio-activity must be slightly 

 less than three-fourths of the whole, say about 27/40. 



If the proportion of heat flow due to radio-activity is assumed to be 

 higher than this, the age of the earth increases with great rapidity, 

 and we are driven to choose between two alternatives — either 



(a) that the age of the earth is much greater than 1,600 million 



years, or 



(b) that the earth has never possessed a molten or even a nearly 



molten crust. 

 (a) cannot be readily granted, and therefore the practical alternatives 

 lie between 



(b) with more radio-thermal energy than is necessary to supply 



three-fourths of the total heat loss, and 

 (<?) the conception of a cooling earth which originally had a molten 

 crust and in which radio-activity supplies three-fourths, or 

 nearly three-fourths of the present heat loss. 



12. Deep-seated Temperatures. 



The temperature at any depth x, and at any time t, is given by 



Ingersoll & Zobel's equation 85. Theoretically, the " steady state" 



in which all the heat from radio-active sources passes to the surface 



is arrived at only after an infinite time, and equation 85 then 



reduces to n ' , ", , .. ri -, 



= mx + A/a 2 k [1 - e-"*\. 



Practically, however, the time t, 1,600 million years, is sufficiently 

 long for the steady state to be assumed without appreciable error, and 

 the temperature at any depth may thus be regarded as made up of 

 two components, one, 0", due to the initial thermal condition of the 



