Lecture 6:

1. Lecture 6: Self-consistency ofthe RT solutions

2. Self-consistentsolution of RT • Take into account the intensity part of the source function (scattering) • Construct self-consistent solution for the intensity and the source function(Λ-iterations) • Efficiency of the process in different situations • Speeding things up

3. Source function again Source function is a ratio of emission to absorption:The emission consists of two parts: stimulated( to the local intensity) and spontaneous (function of the local conditions):

4. RT for a b-b transition A complete set of equations for a single b-b transition: Now we can write the formal solution:

5. Formal solutions: coordinate convention • Source function: • Intensity:

6. Now we can combine formal solutions in one self-consistent integral equation: We can re-write this in operator form also known as Λ (lambda) operatorΛ -operator is linear:

7. Λ-iterations Recurrence relation:convergence conditions: –

8. Convergence etc. • Convergence is slow for small e and τ>1. • Convergence is monotonous (corrections have always the same sign). • Note that: • We could also substitute the expression for the source function to the expression for intensity. This is useful when we need to include the radiative energy transport to hydrodynamics. • Home work: write the expression forΛ-operator for angle averaged intensityJν

9. 1D semi-infinite medium Given one selected direction:Now we can write the formal solution:

10. Now we can write a self consistent equation for the source function in 1D:

11. Accelerated Λ-iterations • Λ-operator is a convolution with non-negative kernel in frequency and spatial domains. Numerically it can be approximated with summation: • Newton-Raphson acceleration (Cannon) • Operator splitting (Olson, Scharmer et al.)

12. The recurrence relation for lambda iterations (Cannon):

13. Operator splitting (Scharmer, Olson):

14. More general assessment of what was done: • Λ-iterations are used to find source function in case of radiation-dependent absorption/emission • The direct iteration scheme for the source function can be used but the convergence is slow, especially at high optical depths • Lambda integral in n and t can be replaced with a quadrature formula using a discrete grid • We have replaced the true lambda operator with an approximate operator that has better convergence properties (higher order convergence) and can be easily inverted • The freedom in choice of the accelerated lambda operator helps to optimize it for special cases

15. Lambda iterations and statistical equilibrium • Even with the accelerated techniques we need to iterate • Iterations in S can be combined with the solution for the level populations • It is more convenient to re-formulate the Lambda-iterations for J instead of S • The procedure is trivial for a two bound level atom but gets quickly very complicated for multi-levels • Combining levels in “LTE” groups is one solution

16. Further reading Scharmer Global Operator: • Scharmer: 1981, Solutions to Radiative Transfer Problems Using Approximate Lambda Operators, Astrophys. J. 249, 730 • Scharmer and Carlsson: 1985, A New Approach to Multi-Level Non-LTE Radiative Transfer Problems, J. Comput. Phys. 59, 56 Diagonal or tridiagonal ALI: • Olson, Auer and Buchler: 1986, A Rapidly Convergent Iterative Solution of the Non-LTE Line Radiation Transfer Problem J. Quant. Spectrosc. Radiat. Tansfer. 35, 431  • Rybicki and Hummer: 1991, An accelerated lambda iteration method for multilevel radiative transfer. I. Non-overlapping lines with background continuum, Astron. Astrophys. 245, 171 ALI review: • Hubeny: 1992, Accelerated Lambda Iteration, in Heber and Jeffery (Eds.), • The Atmospheres of Early-Type Stars, Lecture Notes in Physics 401, Springer, p. 377