Theory of Pulsating Stars
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The fundamental memoir on the motion of a fluid in a field of radiation is one by Rosseland1 . To the ordinary hydrodynamical equations (the equation of continuity and the equations for the rate of change of momentum) it is necessary to add an energy rate-of-change equation, which fixes the rate of change of the temperature at any point. This is obtained by applying the principle of the conservation of energy, and the resulting equation may conveniently be regarded as determining DT/Dt, the rate of change of the temperature T of any particle, following the motion. DT/Dt is the Lagrangean operator. Rosseland’s method was to work with an atomic theory of matter and to calculate the fluxes of the various kinds of energy (kinetic, radiant, thermal, sub-atomic) into a closed surface fixed in space. It appears to the writer, however, that Rosseland left out of account the mechanical effects of radiation pressure. Forms of the DT/Dt equation have also been given by Jeans2 and Vogt3 ; their method was to calculate the output of thermal energy of all kinds (but omitting kinetic energy) across a surface moving with the matter. Vogt pointed out that Jeans omitted terms due to the change of volume of the moving surface; but to the present writer it appears that Vogt omitted the correction to the flux of radiant energy across the moving surface due to the motion of the surface. The method which is used below follows Jeans and Vogt in using a moving surface but follows Rosseland in calculating the changes of kinetic as well as of thermal energy. It differs further from Rosseland’s method in using a continuous theory of matter instead of an atomic one. This is valid in large scale phenomena such as are here in question. The resulting equation agrees with Rosseland’s when a radiation-pressure term is added. Its volume average agrees with those of Jeans and Vogt since it merely corresponds to a different allocation of radiant energy amongst the moving volume elements. Applied to “adiabatic” motions in a pulsating star it gives a slightly different equation from that used by Eddington, a term due to temperature gradient appearing.
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