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Unformatted text preview: 23. The evolution of non-thermal systems 23.1 Equilibrium situations In this chapter we want to study only the total particle spectrum, i.e. only the energy distri- bution of the particles regardless of their spatial distribution. Assuming a homogeneous sytem that should suffice to calculate the volume-integrated emission spectra or total flux of emission. Unlike thermal systems, for which we would only need to consider the heating/cooling balance of the temperature as a parameter of a known energy distribution of particles, the Maxwellian, for nonthermal system we need to determine the energy distribution that is shape by the balance of effects that depend on the particle energy. We can write down a continuity equation for the particle spectrum, that contains terms for continuous energy losses and gains, functions that describe catastrophic losses like radioactive decay or escape from the system, and a source term. Then the total number spectrum of particles, N ( E ) = dN/dE , in steady state follows ∂ ∂E ˙ E N ( E ) = Q ( E )- N ( E ) T ( E ) (23 . 1) Here Q ( E ) is the source rate, T ( E ) the timescale for catastrophic losses, and ˙ E the rate of continuous energy changes, which is positive for energy gains and negative for losses. What would a general solution of 23.2 look like? It would certainly be practical to know a general solution, in which we only have to insert the actual coefficients for a variety of astrophysical problems. Assuming this factor is not zero, we may multiply the entire continuity equation with ˙ E and solve for the function F ( E ) = ˙ E N ( E ). ˙ E ∂ ∂E F ( E ) = ˙ E Q ( E )- F ( E ) T ( E ) = ∂ ∂τ F ( E ) mit τ = Z E dE ˙ E (23 . 2) where we have introduced the timescale τ for continuous energy changes. Obviously the functionfor continuous energy changes....
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