Thermal X-ray radiation (PDF) - SRON

More documents

Recommendations

Info

of computing time. Therefore one uses a trick. First, the eigenvalues (λ) of A are determined, and also the corresponding eigenvectors ⃗ V . These are put into a matrix V. We now transform the concentrations ⃗n to the eigenvector basis; the concentrations on the eigenvector basis will be denoted by ⃗m. By definition, we then have ⃗n = V⃗m. (3.61) We now define a diagonal matrix Λ which has the eigenvalues on its diagonal. We then have AV = ΛV. (3.62) Using this we rewrite (3.60) as 1 d V⃗m = ΛV⃗m. (3.63) n e dt For small steps in T the transition matrix A does not vary much. Therefore, for any time interval where T changes by only a small amount we can assume A and thereby Λ and V to be constant. In that case, one can interchange the order of matrix multiplication and differentiation in the left hand side of (3.63). It is then simple to solve this equation. One finds as a solution ⃗m(t) = V −1 PV⃗m(t 1 ) (3.64) where the solution is valid for t ≥ t 1 and as long as T(t) − T(t 1 ) is small. Here P is a diagonal matrix with elements e λ∆u where ∆u ≡ ∫ t t 1 n e (t)dt. (3.65) For each interval in T one can now determine the eigenvectors and eigenvalues once and for all and store these for example in the computer memory. By systematically subdividing the range of T(t) in small intervals ∆T, one finds the final solution simply by applying repeatedly simple exponentiation and matrix multiplication. The time evolution of the plasma can be described in general well by the parameter ∫ U = n e dt. (3.66) The integral should be done over a co-moving mass element. Typically, for most ions equilibrium is reached for U ∼ 10 18 m −3 s. We should mention here, however, that the final outcome also depends on the temperature history T(t) of the mass element, but in most cases the situation is simplified to T(t) = constant. 3.7.3 Photoionisation Equilibrium (PIE) The third case that we treat are the photoionised plasmas. Usually one assumes equilibrium (PIE), but there are of course also extensions to non-equilibrium photoionised situations. Apart from the same ionisation and recombination processes that play a role for plasmas in NEI and CIE, also photoionisation and Compton ionisation are relevant. Because of the multiple ionisations caused by Auger processes, the 45
equation for the ionisation balance is not as simple as (3.55), because now one needs to couple more ions. Moreover, not all rates scale with the product of electron and ion density, but the balance equations also contain terms proportional to the product of ion density times photon density. In addition, apart from the equation for the ionisation balance, one needs to solve simultaneously an energy balance equation for the electrons. In this energy equation not only terms corresponding to ionisation and recombination processes play a role, but also several radiation processes (Bremsstrahlung, line radiation) or Compton scattering. The equilibrium temperature must be determined in an iterative way. A classical paper describing such photoionised plasmas is Kallman & McGray (1982). 46
Page 1 and 2: High Energy Astrophysics Jelle Kaas
Page 3 and 4: 3.1 Introduction Thermal X-ray radi
Page 5 and 6: Figure 3.2: The observed X-ray spec
Page 7 and 8: Figure 3.6: X-ray spectrum of the S
Page 9 and 10: series of alkaline spectra). The no
Page 11 and 12: • The first atomic shell (n = 1)
Page 13 and 14: Z El 1s 2s 2p 3s 3p 3d 4s Ground te
Page 15 and 16: m l1 s 1 m l2 s 2 M L M S +1 + +1
Page 17 and 18: 3.3 Energy levels We have seen the
Page 19 and 20: 3.4 Transitions between different l
Page 21 and 22: Figure 3.10: Transitions between a
Page 23 and 24: 3.5 Abundances of the elements With
Page 25 and 26: 3.6 Basic processes In order to be
Page 27 and 28: leading to Here the constant S 0 is
Page 29: 3.6.3 Radiative transition probabil
Page 32 and 33: Figure 3.12: Fluorescence yield as
Page 34 and 35: The formula of Younger While the Lo
Page 36 and 37: 3.6.6 Excitation-Autoionisation In
Page 38 and 39: 3.6.7 Photoionisation Figure 3.20:
Page 40 and 41: 3.6.9 Radiative recombination Radia
Page 42 and 43: 3.6.10 Dielectronic recombination T
Page 44 and 45: 3.7 The ionisation balance In order
Page 48 and 49: 3.8 Continuum emission processes Af
Page 50 and 51: decays back to the 1s orbit by a ra
Page 52 and 53: Inner-shell ionisation The third li
Page 54 and 55: 3.9.3 Line width For most X-ray spe
Page 56 and 57: Table 3.6: The strongest emission l
Page 58 and 59: Table 3.8: Some important iron line
Page 60 and 61: with τ(λ) = τ 0 ϕ(λ) (3.83) wh
Page 62 and 63: Table 3.9: The most important X-ray
Page 64 and 65: where the hydrogen column density N
Page 66 and 67: source itself. Fortunately, with hi
Page 68 and 69: Figure 3.35: Stability for the ioni
Page 70: Table 3.10: Wavelengths in Å of th

Thermal X-ray radiation (PDF) - SRON

Create successful ePaper yourself

Delete template?

Save as template?