138 G. F. BECKER RELATIONS OF RADIOACTIVITY TO COSMOGONY 



Mr Strutt independently reached the equation of the distribution of 

 temperature in a thin radioactive shell uniformly charged with radium. '^^ 

 He computed the thickness of the crust at 45 miles and the temperature 

 of the interior at 1530° centigrade, assuming a surface temperature gra- 

 dient of 1° Fahrenheit in 42.4 feet. His constants have been in part 

 superseded, and I have therefore recomputed his equations for a gradient 

 of 1° Fahrenheit in 50 feet with the conductivity of the Calton Hill 

 trap, 0.00415, assuming that the thermal effect of 1 gram of pure radium 

 is 100 gram calories per hour. It follows that the thickness of the radio- 

 active shell needful to maintain the temperature gradient is 137 kilo- 

 meters if it contains 1.4 x 10~^^ grams of radium per gram of rock uni- 

 formly distributed, while the resulting internal temperature would be no 

 less than 2488° centigrade, which seems high. 



As is well known, Mr Barus investigated the melting point of diabase, 

 finding that it rose in simple proportion to the pressure. At a depth ol 

 137 kilometers diabase would melt at about 2100° centigrade; hence a 

 temperature of 2488° centigrade at this depth would imply a liquid 

 couche and consequently tidal instability, which is inadmissible. 



Evidently, then, there must be a limit to the effects which can be 

 attributed to radioactivity, and with the data in hand a surface gradient 

 of 1° Fahrenheit in 50 feet can not be accounted for in this manner. 

 The limitations are easily determined from Mr Strutt's equations, which 

 may be written 



where v is the temperature excess; q the quantity of heat developed in 

 the radioactive shell per cubic centimeter per second ; h the thermal con- 

 ductivity of the rock; s the thickness of the shell, and x the distance 

 from the surface. The temperature at which diabase melts may be repre- 

 sented with sufBcient accuracy by 



y = 1170° + 67.5 x IQ-^a;. 



If the V curve is tangent to this diabase line at x^, it is easy to eliminate 

 s and show that 



V 2. 



mok 



'= When the curvature of the earth is neglected, this problem is mathematically iden- 

 tical with that of the heating effect of a constant electric current on a thermally in- 

 sulated conducting wire. Of. Kohlrausch, Lehrhuch der prak. Physlli, 10th edition, 

 p. 214. 



Then with the conductivity of the Calton Hill rock, 0.00415, the density 1 



i 



