Open AccessTHE EFFECT OF CONTRACTION IN THE COOLING BY CONDUCTION OF A GRAVITATING SPHERE, WITH SPECIAL REFERENCE TO THE EARTH
Author(s) -
Lapwood E. R.
Publication year - 1952
Publication title -
geophysical journal international
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.302
H-Index - 168
eISSN - 1365-246X
pISSN - 0956-540X
DOI - 10.1111/j.1365-246x.1952.tb03027.x
Subject(s) - thermal conduction , hydrostatic equilibrium , contraction (grammar) , physics , thermal conductivity , heat equation , mechanics , circular symmetry , classical mechanics , thermodynamics , quantum mechanics , medicine
Summary When a hot gravitating sphere cools, gravitational potential energy is lost in contraction. We consider a sphere that can be taken as in continual hydrostatic equilibrium for slow changes of state, and, assuming radial symmetry, set up the modified Fourier equation for heat conduction. This contains two extra terms, one contributed by the heating effect of contraction, and another from the change of temperature gradients with shrinkage. This equation is checked by integration through time and through the sphere to give a comprehensive energy equation. For an initial parabolic temperature distribution in the Earth, the heating effect of contraction is small, and more than counteracted by the extra cooling provided by increased temperature gradients. Moreover, it is probable that the modifications to the first (uniform rigid sphere) approximation which are needed to allow for departures from uniformity in density, specific heat, and thermal conductivity in the Earth are greater than those which allow for contraction.