By L. Salvadori
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6, 1987) . There exists a vast amount of mathematical theory associated with this equation, especially for the case where is constant. x/ at t D 0. How, then, does this initial temperature profile evolve in time? x; t/ for t > 0. x; t/ approaches the limit solution T D constant after sufficient time. 32) where d is a characteristic length over which the (initially given) temperature variation changes. Obviously, only temperature profiles with variations over small distances can change rapidly in time.
1). 6), a D 7:57 10 15 erg cm 3 K 4 is the radiation density constant. 10) represents the coefficient of conduction for this radiative transport. 11) After transformation to the independent variable m (as in Sect. 12) Of course, this neat and simple equation becomes invalid when one approaches the surface of the star. ). Fortunately, however, we have then left the stellar-interior regime with which this book deals, and we happily leave the complicated remainder to those of our colleagues who feel the call to treat the problem of stellar atmospheres.
39) has only one time-independent solution: T D Ts . What is the timescale of this adjustment of T ? 39) give the same estimate for ; indeed a change of h occurs only as a consequence of, and together with, the change of T . 41) which is the time it takes the gas to adjust its temperature to that of the surroundings. This timescale for our piston model plays a role similar to the Kelvin–Helmholtz timescale in stars. For sufficiently small (sufficiently large adj ), we have hydr adj , similar to the situation in stars, where hydr KH .
Bifurcation theory and applications by L. Salvadori