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Radiative Transfer in Stellar Atmospheres RADIATIVE TRANSFER IN STELLAR ATMOSPHERES R.J. Rutten Sterrekundig Instituut Utrecht December 1, 2000 ii Copyright c 1995 Robert J. Rutten, Sterrekundig Instuut Utrecht, The Netherlands. Copying permitted for non-commercial educational purposes only. R.J. Rutten asserts the moral right to be identified as the author of these notes. In no way can he be held responsible for any liability wit respect to these notes. First Utrecht edition: March 7, 1995. Based on “Stellar Atmospheres” lecture notes by C. Zwaan. First WWW edition: June 1, 1995 for the 1995 Oslo Summer School. Figures scanned by Sake Hogeveen. Corrections from Louis Strous, Bart-Jan van Tent, Guus Oonincx, Dan Kiselman, Mats Carlsson. Second WWW edition: June 22, 1995. Corrections from Carine Briand, Kees Dullemond, Martijn Smit. Third edition: March 11, 1996. Corrections from Ferdi Hulleman. Section “Exercises” started. Fourth WWW edition: October 1, 1997. Corrections from Hans Akkerman, Thijs Krijger, Nils Ryde, Bob Stein. Fourth SIU edition: January 6, 1998. Corrections from Oliver Ryan; new figures from Thijs Krijger. Fifth SIU/WWW edition: January 4, 1999. Corrections from Mark Gieles, Jorrit Wiersma, Marc van der Sluys, Niels Zagers. Sixth edition: May 20, 1999, for the ESMN Summer School at Oslo. Corrections from Wouter Bergmann Tiest. Seventh edition: December 1, 2000. Corrections from Karin Jonsell, Torgny Karlsson, Hans van Rijn, Louis Strous. These lecture notes are freely available as a service of the European Solar Magnetome- try Network (http://www.astro.uu.nl/∼rutten/tmr/). There are also corresponding equation viewgraphs for classroom display. Details on how to get printable files are given at http://www.astro.uu.nl/∼rutten and in the README file in directory pub/astronomy/rutten available per anonymous ftp from ftp.phys.uu.nl. These lecture notes still evolve; the WWW and README information contains an update on their status. Major renewals are announced to those who request to be put on the notification email list. Corrections and additions are very welcome; please send them to R.J.Rutten @ astro.uu.nl. You are also most welcome to cite these lecture notes. Please do so as: Rutten, R.J., 2000, Radiative Transfer in Stellar Atmospheres, lecture notes Utrecht Univ., 7th edition. Cover: a stellar atmosphere is where photons leave the star, a dramatic transition from warm dense comfort in near-thermal enclosure to bare isolation in the cold emptiness of space — sufficiently traumatic to make stellar atmospheres highly interesting to astro- physicists. On average, photons get scarcer, longer, and more outwards directed further out in the atmosphere until they escape. Copied from Mats Carlsson’s poster for the 1995 Oslo “Intensive Summer School on Radiative Transfer and Radiation Hydrodynamics”. ii Contents Preface xv Bibliography xvii 1 Brief History of Stellar Spectrometry 1 Fraunhofer lines . 1 Linesaselementencoders................................ 2 Stellarclassification................................... 2 Abundance determination . 4 Reversing-layerlineformation.............................. 5 LTElineformation.................................... 5 NLTElineformation................................... 5 Numericallineformation................................ 5 Diagnosticlineformation................................ 5 2 Basic Radiative Transfer 9 2.1Radiation......................................... 9 2.1.1 Localamount................................... 9 Intensity ................................... 9 Mean intensity ................................ 10 Flux ...................................... 10 Density .................................... 11 Pressure .................................... 12 Moments of the intensity .......................... 12 2.1.2 Localchange................................... 12 Emission ................................... 12 Extinction .................................. 13 Source function ................................ 13 2.2Transportequation.................................... 14 2.2.1 Transportalongaray.............................. 14 Discussion .................................. 14 Optical length and thickness ........................ 14 Homogeneous medium ............................ 15 2.2.2 Transportthroughanatmosphere....................... 17 Optical depth ................................. 17 Standard plane-parallel transport equation ................ 17 Formal solution ................................ 17 Eddington-Barbier approximation ..................... 17 2.3Linetransitions...................................... 18 2.3.1 Einsteincoefficients............................... 19 Spontaneous deexcitation .......................... 19 Radiative excitation ............................. 21 Induced deexcitation ............................. 22 iii iv CONTENTS Collisional excitation and deexcitation ................... 22 Einstein relations ............................... 22 2.3.2 Volumecoefficients................................ 23 Extinction .................................. 23 Emission ................................... 24 Source function ................................ 24 2.4Continuumtransitions.................................. 25 2.4.1 Inelasticprocesses................................ 25 Bound-free transitions ............................ 25 Discussion .................................. 25 Free-free transitions ............................. 25 2.4.2 Elasticprocesses................................. 27 Thomson scattering ............................. 27 Rayleigh scattering .............................. 27 Redistribution ................................ 27 2.5LTE............................................ 28 2.5.1 MatterinLTE.................................. 28 Maxwell distribution ............................. 28 Boltzmann distribution ........................... 28 Saha distribution ............................... 29 Saha-Boltzmann distribution ........................ 29 2.5.2 RadiationinLTE................................. 29 Planck function ................................ 29 Wien and Rayleigh-Jeans approximations ................. 30 Stefan-Boltzmann law ............................ 30 Induced emission ............................... 30 Line extinction ................................ 30 Discussion .................................. 31 2.6NLTE........................................... 31 2.6.1 Statistical equilibrium . 31 Rate equations ................................ 31 Transport equations ............................. 32 Time-dependent transfer .......................... 32 Multi-dimensional transfer ......................... 32 2.6.2 NLTEdescriptions................................ 32 Departure coefficients ............................ 32 Bound-bound source function ........................ 32 Bound-bound extinction ........................... 33 Laser regime ................................. 33 Bound-free source function ......................... 34 Bound-free extinction ............................ 34 Bound-free emission ............................. 34 Free-free source function, extinction, emission ............... 35 Discussion .................................. 35 Warning .................................... 35 Formal temperatures ............................. 36 2.6.3 Coherentscattering............................... 37 Two-level atoms ............................... 38 Coherently scattering medium ....................... 38 Destruction probability ........................... 39 Effective path, thickness, depth ....................... 39 Source function ................................ 40 Transport equation .............................. 40 2.6.4 Multi-levelinterlocking............................. 41 CONTENTS v 2.6.5 Coronalconditions................................ 41 3 Bound-Bound and Bound-Free Transitions 43 3.1Photonictransitions................................... 43 3.1.1 Atomictransitions................................ 43 3.1.2 Moleculartransitions............................... 43 3.1.3 Two-electrontransitions............................. 43 Dielectronic recombination ......................... 43 Autoionization ................................ 44 Bound-free resonances ............................ 44 3.1.4 Charge-transfertransitions........................... 44 3.2Transitionrates...................................... 44 3.2.1 Bound-bound radiative rates . 44 3.2.2 Bound-free radiative rates . 45 Physics .................................... 45 Einstein-Milne equations .......................... 46 Photoionization ............................... 46 Spontaneous recombination ......................... 46 Discussion .................................. 47 Induced recombination ........................... 47 Total radiative recombination ........................ 47 3.2.3 Unifiedradiativerates.............................. 48 Discussion .................................. 48 3.2.4 Netradiativerates................................ 48 Net radiative recombination ......................... 48 Net radiative deexcitation .......................... 49 Discussion .................................. 49 3.2.5 Collision rates . 50 Net collision rates .............................. 51 Collisional coupling ............................. 52 Collisional LTE ................................ 52 3.3Spectrallinebroadening................................. 52 3.3.1 Radiationbroadening.............................. 53 Damping profile ............................... 53 Extinction profile ............................... 53 Derivation
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