and the Earth's Thermal History. 69 



shell (thickness 6'3 km.) and the remaining two-thirds to an under- 

 lying zone of basic rock (thickness 47 km.), would give a maximum 

 temperature of nearly l,0u0° C. This value is of the order required, 

 and the conditions appear to agree fairly well with the probabilities 

 of rock distribution. 



The conditions, however, are not capable of being treated mathe- 

 matically when stated in so crude a form. We require some law of 

 decrease which can readily be stated mathematically so as to deal with 

 the problem of a cooling earth as well as with volcanic temperatures. 

 We have already seen that even in rock of uniform composition there 

 is reason to believe that the radio-activity decreases with depth. 

 Moreover, it is unlikely that the boundaries between successive zones 

 of different density are as clearly cut as assumed in the above case. 

 It is more probable that the lower portions of the acid shell are more 

 basic than its upper parts, and similarly that the lower levels of the 

 gabbroid sub-zone in turn, approximate to the composition of the 

 ultra-basic rocks beneath. It must be remembered that this variation 

 of rock-types with depth refers to averages at each level and not to 

 individual rock-masses. Igneous geology points to a natural antipathy 

 between acid, basic, and ultra-basic types, so that it ought not to be 

 assumed that there are levels at which intermediate types are 

 characteristic, but only that the proportions of each main type vary 

 at different depths. Combining the variation of rock-types with the 

 suggested decrease of radio-activity in depth within any one rock- 

 type, it would seem to be probable that the degree of heat generation 

 decreases steadily from the surface downwards. The mathematical 

 curve which can be made to agree most nearly with such a distribution 

 is the exponential curve. We shall therefore assume an exponential 

 decrease such that A x , the heat generated per cubic centimetre at 

 any depth x, is given by the equation — 



A x = Ae-<™ (5) 



in which a is the percentage decrease in heat generation per unit 

 distance. This law of decrease is such that if at a depth d the radio- 

 active generation of heat per unit volume of rock has diminished 

 to A/2, or half its surface value, then at a depth 2d the heat 

 production will be reduced to A J '4 or A/2 2 , at a depth 3d to A/8 or 

 A/2 3 , and generally at a depth nd to A/2n. 



The value of a can be calculated from the total heat, q, which 

 issues from each unit area of the earth per second. 



7-x> 

 Ae-ax dx = A/a 

 o 

 = k d6/dx 



.-. ' a= , / (G) 



k de/dx 



The half-value depth d is given bv 



l/a.= l'-443rf (?) 



The maximum temperature, 0, for the steady state is 



= A/a 2 k (8) 



and the temperature 0' at any depth x is 



0' = A/a?k(l-e-™). (9) 



