Stellar Atmospheres

1. Giovanni Catanzaro INAF – Catania Astrophysical Observatory Italy Stellar Atmospheres Basics processes& equations

2. We call stellar atmosphere the externallayers of a star • These are the layers where radiation created in the stellar core can escape freely into interstellar medium • In practice, the atmosphere is the only part of a star from whichwereceivephotons Working definitions Spring School of Spectroscopic Data Analysis

3. Spring School of Spectroscopic Data Analysis

4. Theory of stellar atmospheres How radiationproduced in the stellar core propagates and interacts with the externallayers Spring School of Spectroscopic Data Analysis

5. Specificintensity = the energythat flow through an element of area dA in the unit of solid angle, time and frequency Basic definitions Integrating the specificintensity over all the directionsweobtain the so-calledmeanintensity Spring School of Spectroscopic Data Analysis

6. Flux = net energy flow acrosselement of area over the unit of time and frequency Fluxcould be related with the specificintensity Lookingat a point on the physicalboundary of a radiatingsphere, we can write: K integralphysicallyrelated with radiation pressure Spring School of Spectroscopic Data Analysis

7. Processescontribuiting: trueabsorption and scattering. No emission. Since the absorption, radiationinteracts with plasma. We can saythatitseesneither nor dx alone, rather their combination, so we define: Absorptioncoefficient and opticaldepth Optical depth Spring School of Spectroscopic Data Analysis

8. It measures a characteristic of matter and radiation coupled together and for a given frequency. The opticaldepth=1 corresponds, for a given frequency and absorption coefficient, to the distance at which the intesity is reduced by 1/e >> 1 plasma optically thick << 1 plasma optically thin Plasma could be opticallythin for radiation of frequencyn1 and opticallythick for anotherfrequencyn2 Spring School of Spectroscopic Data Analysis

9. Like wedid for absorption, letdefine the increment of the radiationifthereisemission: Processescontribuiting: trueemission and scatteringintodirection. No absorption. Emissioncoefficient and Source function Source function Itcould be consideredas the specificintensityemittedat some point in a hot gas. Spring School of Spectroscopic Data Analysis

10. Letwrite the number of photonsemitted in an element of volume dV over alldirections, for frequencyn and unit time dt Integration over solid angle Energy emitted in the volume dV Transformenergy in number of photons dV Source function: physicalmeaning Spring School of Spectroscopic Data Analysis

11. Pure isotropicscattering ‘absorbed’ energy Integrating over w Source functions: 2 simplecases Spring School of Spectroscopic Data Analysis

12. Pure absorption All the absorbedphotons are destroyed and all the emittedphotons are newlycreated with a distributiongoverned by the physical state of the material. Thermodinamicequilibrium The source functionisequal to the Planckfunction, depends on frequency and temperature of the material Spring School of Spectroscopic Data Analysis

13. Along a line Differential form The transfer equation Integral form Spring School of Spectroscopic Data Analysis

14. The transfer equation Different geometries • Let consider polar coordinates with z axis along the line of sight. In this case a projection factor, cosq, should be considered. • Atmosphere is thin with respect the radius, so a plane parallel approximation could be used. • That means that cosq does not depend upon z. Polar coordinates Spring School of Spectroscopic Data Analysis

15. Elementary solutions 1) No absorption (kn = 0), no emission (jn = 0) Trivial solution: in absence of any interaction with the medium the radiation intensity remains constant 2) No absorption (kn = 0), only emission (jn > 0) Outcoming radiation from an optically thin radiating slab 3) No emission (jn = 0), only absorption (kn = 0) Spring School of Spectroscopic Data Analysis

16. 4) General case: emission (jn> 0), absorption (kn> 0) At the surface, where tn=0: Spring School of Spectroscopic Data Analysis

17. 5) Special case: linear source function Eddington-Barbier relation The values of emergent intensity for all angle p/2 < q < 0 then map the values of the source function between optical depths 0 and 1. Spring School of Spectroscopic Data Analysis

18. We can write this expression in polar coordinates Considering Sn isotropic, independent on q, we obtain: The flux integral The theoretical stellar spectrum is for tn = 0 Extinction factor Spring School of Spectroscopic Data Analysis

19. depend on physical properties of the layer T, P, ni, ne... To compute Sn we must know the distributions of these quantities with tn Computing Model Atmosphere Spring School of Spectroscopic Data Analysis

20. Hypothesis: horizontally homogeneous, plane-parallel, static Mean intensities, Jn Radiative transfer equation Hydrostatic equilibrium equation Pressure, total particle density N Radiative equilibrium equation Temperature, T Stellar atmosphere: basic equations Statistical equilibrium equation Populations, ni Charge equilibrium equation Electron density, ne Spring School of Spectroscopic Data Analysis

21. Introducing: Hydrostatic equilibrium equation Effective gravity acceleration From Gray D. (2005), chapter 9 Spring School of Spectroscopic Data Analysis

22. Ensure radiative equilibrium means ensure conservation of energy First radiative equilibrium condition is then The other 2 consitions come from transfer equation, write as: Milne equations Integrating over solid angle and over frequencies Radiative equilibrium Multiply by cosq and integrating over solid angle and frequencies Spring School of Spectroscopic Data Analysis

23. R radiative rate C collisional rate Total number of transitions out of level i Total number of transitions into level i Zi is the charge associated with level i ni population of the level i ne electron density Statistical and charge conservation equations Spring School of Spectroscopic Data Analysis

24. Eddington (1926) hemispheperically isotropic outward and inward specific intensity Milne equations The grey atmosphere Spring School of Spectroscopic Data Analysis

25. Modeling stellar spectrum means computing the flux emerging at the stellar surface • To accomplish this task we need to know the radiation specific intensity along the atmosphere • The calculation of how the radiation propagates within a stellar atmospere requires knowledge of the source function • Source function depends on emission and absorption coefficient • Both jn and kn depend on the physical condition of the stellar material: T, P, electronic density and so on • We need to resolve the equations of the model atmosphere Conclusions Spring School of Spectroscopic Data Analysis

26. I. Hubeny, «Stellar atmospherestheory: an introduction» in: Stellar atmospheres: Theory and Observations, Lecture note in physics, J.P. De Greeve, R.Blomme, H. Hensberg (Eds.), Springer D. Gray, «The observations and analysis of stellar photospheres» D. Mihalas, «Stellar Atmospheres» References Spring School of Spectroscopic Data Analysis

27. Thanks for yourattention Spring School of Spectroscopic Data Analysis

28. u Nuatoms per dV DEul = hn Aul Blu Bul Nlatoms per dV l Probability that an atom will emit is quantum energy in dt and dw Spontaneus emission The Einstein coefficients Stimulated emission Absorption 2 contributions True absorption Spring School of Spectroscopic Data Analysis