By Edmund C. Berkeley, Illustrated
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Here is the essential complication. e. ni = n∗i (T, ρ), so that the thermal absorption and emission coeﬃcient are functions of only temperature and density. However, in many cases of interest, the LTE approximation breaks down, and one has to determine the atomic level populations by a set of statistical equilibrium (rate) equations. For a given level i, the rate equation reads (for simplicity, we consider here the time-independent form; the general form would contain additional terms (1/c)(∂ni /∂t) + ∇ · (vni )).
The preconditioning method consists in taking another matrix, P , which is close enough to A, and P −1 A is close to the identity matrix, so that the eigenvalues of I − P −1 A are small. The iteration process (26) is modiﬁed to y(k+1) = y(k) + P −1 r(k) , (27) Obviously, the method is advantageous if the matrix P is easy and cheap to invert. , y(k+1/2) = y(k) + P1−1 r(k) , (28) y(k+1) = y(k+1/2) + P2−1 r(k+1/2) . (29) A choice of suitable preconditioner(s) depend on the problem. 30 I. 2 Solution of Large Linear Systems Let us consider here a linear scattering problem, but a time-dependent one, with generally anisotropic scattering, and in several spatial dimensions, 1 ∂I(n) ˆ + D(n)I(n) = −χ I(n) + S(n) , c ∂t (30) ˆ where the operator D(n) is a general “spatial transport” operator, S(n) is the source term, in which we split the thermal and scattering part, S = S th +S sc .
20 I. Hubeny In the simplest case of pure radiative equilibrium, which reads ∞ [χ(ν, T )I(ν, n) − η(ν, n, T )] dν dΩ = 0 , (13) 0 we see that the temperature structure depends on the radiation ﬁeld in a complex way. 4 Hierarchies of Approximations There is a wide range of diﬀerent radiative transfer problems considered in astrophysics, depending on a degree of sophistication in treating the interaction of radiation and matter. I will brieﬂy summarize the basic types of problems below, although the list is by no means exhaustive.
Giant Brains or Machines That Think by Edmund C. Berkeley, Illustrated