3. Stellar Atmospheres: Opacities

Transcription

1 3. Stellar Atmospheres: Opacities 3.1. Continuum opacity The removal of energy from a beam of photons as it passes through matter is governed by o line absorption (bound-bound) o photoelectric absorption (bound-free) o inverse bremstrahlung (free-free) o photon scattering Stimulated emission acts as negative opacity by creating photons that add to the beam. Figure 4.1: A summary of possible transitions in an atom or molecule. Stellar atmospheres are predominantly hydrogen (90% by number, 75% by mass), while helium makes up almost all the rest. These two elements provide most of the opacity over most wavelengths for most stars. Figure 4.2: A simplified energy level diagram for hydrogen. S.V. Berdyugina. Theoretical Stellar Astrophysics. Stellar Atmospheres: Opacities 1

2 Inelastic processes Processes leading to change of the particle energy Bound-free transitions Photoionisation from a given level, with threshold frequency 0. Hydrogen-like atoms (Kramers' formula): bf Figure 4.3: H I bound-free extinction coefficient bf per hydrogen atom in level n against wavelength. The Lyman, Balmer, Paschen, Brackett and Pfund edges are marked by the quantum number n of the ionizing level. Their amplitudes increase with n and have not been added up in this figure. Lyman edge (n=1) is at 91.2 nm, Balmer (n=2) at nm, Paschen (n=3) at nm, Brackett (n=4) at m, and Pfund (n=5) at m. The metals Al I, Mg I, Si I, CI and Fe I cause a sequence of edges in the near to middle ultraviolet. The H I and He I Lyman continua dominate the extinction/emission at wavelengths below their edges. In the X-ray regime, the edges of highly ionized metal ions contribute continuum extinction/emission. S.V. Berdyugina. Theoretical Stellar Astrophysics. Stellar Atmospheres: Opacities 2

3 Figure 4.4: Top: observed solar disk-center continuum intensities in the far ultraviolet. Bottom: corresponding brightness temperatures. Note that the wavelength scale along the x axis is reversed. The data points represent continuum windows between the many emission lines in this part of the spectrum; the only line shown is Ly = nm. Bound-free threshold wavelengths: H I Lyman continuum (1c) at 91.2 nm, He I at 50.4 nm, CI at 110 nm, Si I at nm. From Vernazza et al. (1981). S.V. Berdyugina. Theoretical Stellar Astrophysics. Stellar Atmospheres: Opacities 3

4 Free-free transitions Photon-absorbing encounters between ions and free electrons. No threshold frequency. Wien limit: Rayleigh-Jeans limit:,, Free-free transitions play an important role at longer wavelengths (mm to m) in solar-type stars. At these wavelengths the radiation comes from the chromosphere, transition region and corona, above the height of the classical temperature minimum. The strength of the absorption depends on the electron velocity, because a slow encounter increases the probability of a photon passing by during the collision. In B, A, and F stars, neutral hydrogen is a dominant continuous absorber. But in cooler stars the absorption from neutral hydrogen diminishes and the absorption from the negative hydrogen increases. Negative hydrogen H I + e H one bound state with ev ionization energy, i.e. photons with < m can ionize H into neutral hydrogen. Number of negative hydrogen ions in the solar atmosphere is not high: where = 5040/T e, but number of neutral hydrogen in the n = 3 state responsible for optical continuum (Paschen continuum) is even smaller: S.V. Berdyugina. Theoretical Stellar Astrophysics. Stellar Atmospheres: Opacities 4

5 So, there are / = 40 times more H ions available to cause continuous extinction than H I atoms in the n = 3 state. Thus, the solar photosphere consists nearly exclusively of neutral hydrogen atoms (about cm 3 at 500 = 1) and visible-wavelength photons (roughly cm 3 ). At any moment, only about one in a hundred million of the atoms may experience the presence of the photons, either by having caught a rare free electron which it may loose through extincting a passing photon, or, even less likely, by sitting excitedly in its n = 3 level, a briefly occupied perch from which it may also extinct a passing photon. Both free-free and bound-free transitions occur with H. - Figure 4.5. The absorption coefficient of the negative hydrogen ion. Left: bound-free transitions peak at near 850 nm. Right: free-free transition coefficient increases with wavelength. These are usually approximated by polynomials. Bottom: a combined effect of the two opacities with the minimum at about 1.6 m. From Gray (1992). S.V. Berdyugina. Theoretical Stellar Astrophysics. Stellar Atmospheres: Opacities 5

6 Molecular ions Molecular hydrogen ions: o H2 is more abundant than H I in stars cooler than ~M5 o H2 + : a significant absorber (bf) in the UV, < 400 nm o H2 : ff absorption is significant in the infrared, can be comparable to that of H Other molecular ions: o CN, C2, H2O are likely to be important absorbers in very cool objects Helium Bound-free and free-free absorption of helium is important in the hottest stars. Qualitatively the helium continuum has much in common with that of hydrogen. Ionization energy of neutral helium is 24.6 ev ionizing photons with < 50.4 nm Second ionization energy is 54 ev ionizing photons with < 22.7 nm Negative helium, He : one bound state with 19 ev ionisation energy o bound-free absorption is negligible in normal stars o free-free absorption can be significant at long wavelengths in cool stars. Line haze Lines are formally not a source of continuum extinction/emission, but in practice they represent one in the solar violet and ultraviolet where the line haze is so crowded that it acts as quasicontinuous extinction Elastic processes No particle energy change Thomson scattering Coherent (monochromatic) scattering on free electrons (low-energy) Induced emission processes are generally not included in elastic electron scattering High-energy photons Compton scattering High-energy electrons inverse Compton scattering, Doppler shift effects (incoherent scattering) S.V. Berdyugina. Theoretical Stellar Astrophysics. Stellar Atmospheres: Opacities 6

7 Tomson scattering is important in O and B stars. In cooler stars, it becomes important at lower pressure, e.g. in cool supergiants. It is also important for stellar coronae, e.g. it causes the solar K corona. Rayleigh scattering Coherent scattering on bound electrons with a characteristic bounding energy h0 (~resonance transition): Angular redistribution, in principle averages to zero over the symmetrical geometry of the star and multiple scattering in the atmosphere. Rayleigh scattering is important in sunspot umbrae which are cool enough to contain many molecules. Also appreciable in the near UV due to the 4 dependence of the cross-section, especially in cool components of the chromosphere where hydrogen is not ionized. Scattering on macro-particle In very cool atmospheres (brown dwarfs, planets) dust and condensates form. Coherent scattering on bound electrons in macro-particles should be considered, which is extremely complicated. One particular case is Mie scattering, on spherical particles Total opacity S.V. Berdyugina. Theoretical Stellar Astrophysics. Stellar Atmospheres: Opacities 7

8 Figure 4.6: Continuous extinction coefficients from hydrogen and helium, per neutral hydrogen atom and per unit electron pressure, for the optical depth = 1 at 500 nm in the photospheres of three dwarf stars. The coefficients are here measured per neutral hydrogen atom in whatever state of excitation, assuming Boltzmann population ratios, and normalized by the electron pressure because the H /H density ratio scales with Pe. The cross-sections are in units of cm 2, not cm 2 as speci_ed in the y- axis labels. Panel (a) is for the Sun, panel (b) for a late A dwarf, panel (c) for a late B dwarf. The curves do not extend beyond the Balmer edge at left where the neglected metal edges become important. From Gray (1992). References: Rutten, R.J. 2003, Radiative Transfer in Stellar Atmospheres, Utrecht University lecture notes, 8th edition, Chapters 2 & 8. Gray, D. 1992, Observation and Analysis of Stellar Photospheres, Chapter 8. S.V. Berdyugina. Theoretical Stellar Astrophysics. Stellar Atmospheres: Opacities 8

9 3.2. Stellar atmospheres: Line Opacities Bound-bound transitions between the lower and upper energy levels of a discrete electromagnetic energy-storing system such as an atom, ion or molecule may occur as: o radiative excitation o spontaneous radiative deexcitation o induced radiative deexcitation o collisional excitation o collisional deexcitation Figure 4.7: A simplified energy level diagram for hydrogen Einstein coefficients Radiative bound-bound processes are described by coefficients proposed by Albert Einstein in The Einstein coefficient is a measure of the probability of occurring of that particular process. Spontaneous emission Spontaneous emission is the process by which an electron "spontaneously" (i.e without any outside influence) decays from a higher energy level to a lower one. The process is described by the Einstein coefficient Aik which gives the probability per unit time that an electron in state i (upper level) with energy Ei will decay spontaneously to state k (lower level) with energy Ek, emitting a photon with an energy Ei Ek = hν. If nk is the number density of atoms in state k, then the change in the number density of atoms or molecules in state k per unit time due to spontaneous emission will be: dnk An ik i dt S.V. Berdyugina. Theoretical Stellar Astrophysics. Stellar Atmospheres: Opacities 9

10 8 e g 64 ( e a ) 64 ( e a ) A f M M S k 0 ki 2 0 ki ik D 3 ki ki 3 k k i i 3 ki mc gi 3hc gi M 3 km hc g i i Induced emission Induced emission is the process by which an electron is induced to jump from a higher energy level to a lower one by the presence of electromagnetic radiation at (or near) the frequency of the transition. The process is described by the Einstein coefficient Bik which gives the probability per unit time per unit energy density of the radiation field, that an electron in state i with energy Ei will decay to state k with energy Ek, emitting a photon with an energy Ei Ek = hν. The change in the number density of atoms in state k per unit time due to induced emission will be: dn dt k Bikni I( vik ), where I(νik) is the spectral intensity of the radiation field at the frequency of the transition. This coefficient (as well as Bki) can also be defined via mean intensity Jν (see, e.g., lecture notes by R. Rutten). Then the difference between the definitions is 4. There is a simple relation between Aik and Bik coefficients (see proof below): B ik 2 c Aik. 2h 3 ki Photoabsorption Absorption is the process by which a photon is absorbed by the atom, causing an electron to jump from a lower energy level to a higher one. The process is described by the Einstein coefficient Bki which gives the probability per unit time per unit energy density of the radiation field, that an electron in state k with energy Ek will absorb a photon with an energy Ei Ek = hν and jump to state i with energy Ei. The change in the number density of atoms in state k per unit time due to absorption will be: dn dt k Bkink I( vik ) Again, there is a simple relation between Bki and Bik coefficients (see proof below): g B g B i ik k ki S.V. Berdyugina. Theoretical Stellar Astrophysics. Stellar Atmospheres: Opacities 10

11 Emission and absorption coefficients When the radiation passes through a media along path dr, the intensity changes due to emission and absorption processes: di( ) di ( ) di ( ) abs em di em ( ) j ds v di abs ( ) Ivds, where j is the emission coefficient and is the absorption coefficient. For a spectral line both coefficients can be expressed via Einstein coefficients: j h n A 4 v i ik h h ng i k v ( nk Bki ni Bik ) nk Bki 1, 4 4 ng k i where the expression in the parenthesis corrects for induced emission, taken into account as negative extinction. In LTE it is expressed via Boltzmann law. The ratio of the emission coefficient to the absorption coefficient is the source function: S v j n A A B h / 1 / 1, 3 v i ik ik / ik v nk Bki ni Bik nk Bki ni Bik c nk gi ni gk where the latter is obtained using the relations between the Einstein coefficients. In LTE, when using Boltzmann law, this becomes Planck function Thermodynamic equilibrium The relations between the Einstein coefficients can be easily found under the equilibrium: di abs Thus, ( ) di ( ), em I v j v Sv. v The radiation is blackbody radiation, and the source function is given by Planck s law: S.V. Berdyugina. Theoretical Stellar Astrophysics. Stellar Atmospheres: Opacities 11

12 S 3 2h 1 i ik 2 c exp( hv / kt ) 1 nk Bki ni Bik B ( T) na. Using Boltzmann distribution for the number of excited atoms or molecules: ni n g e Q i E i / kt, Q is a partition function. we obtain the relations between the Einstein coefficients: A 2h g, 3 ik k 2 Bki c gi B B ik ki g g k. i These are also valid for a non-equilibrium case because they do not depend on any medium properties. Collisional excitation and deexcitation These are introduced analogously to the Einstein coefficients. For collisional deexcitation: dn dt k Cikni, where Cik is the probability per unit time that an electron in state i (upper level) will decay due to a collision to state k (lower level). For collisional excitation: dn dt k Cn ki i where Cki is the probability per unit time that an electron in state k will be excited by a collision to state i. For example, for electron collisions, which are often the most important type of collisions, the transition rate is expressed in general as follows: S.V. Berdyugina. Theoretical Stellar Astrophysics. Stellar Atmospheres: Opacities 12

13 C N (v)v f (v) dv, ik e ik vo where Ne the electron density, ik(v) the electron collision cross-section, f(v) the area-normalized velocity distribution (usually Maxwellian), and v0 the threshold velocity with (1/2)mv0 2 = h0. The collision cross-section ik(v) is a property of each transition that is independent of external state parameters except the velocity v (difference to the Einstein coefficients). The relation between the collisional excitation and deexcitation coefficients: C C ik ki gk e E g i / ik kt It can be deduced from equating the upward and downward collisional rates in thermodynamic equilibrium. It holds also outside the equilibrium if the Maxwell distribution holds Spectral line broadening Spectral lines profiles are defined by the height dependent line extinction profiles. We now specify the shape of the line extinction coefficient. The basic aspects are: o natural broadening or radiation damping, due to the limited lifetimes of excited states o collisional broadening, due to collisions with or perturbations by other particles o Doppler broadening by thermal motions o Doppler broadening by non-thermal motions; often split between microturbulence and macroturbulence for scales that are small or large compared to photon mean free paths o rotational Doppler broadening, of lines in the flux spectrum of a non-resolved star o partial frequency redistribution, a situation between the limits of coherent scattering and complete redistribution (NLTE) Radiation broadening In the absence of collisions and of any other transitions than the ik one, the mean lifetime of particles in state i is 1 t [ s ] A ik The corresponding spread in energy is (Heisenberg): h E 2t or rad with 2 rad 1, the radiative damping constant. t S.V. Berdyugina. Theoretical Stellar Astrophysics. Stellar Atmospheres: Opacities 13

14 This natural" broadening process defines an emission probability distribution around the line center at = 0 that is given by the area-normalized Lorentz profile: ( ) rad /4 0 2 rad 2 ( 0) ( / 4 ). Then the probability for spontaneous emission at a given frequency is Aik ( 0). In the absence of other broadening agents this is the emission profile function. This line broadening is called natural broadening or radiative damping ( natural means that a line is broadened even in the absence of other particles and damping" comes from the classical description of a spectral line as a damped driven harmonic oscillator). Figure 4.8: A comparison between Lorentzian and Gaussian. From Tissue (2000). An estimate of its width: 2 rad 0 fki 7 10 c 2 4 Å The classical value for the damped harmonic oscillator is 2 rad 8 e 3mc e independent of wavelength. This broadening is very small. The other broadening agents are usually more effective. However, radiation damping does not depend on location and therefore exceeds collisional damping in the low-density outer layers of stellar atmospheres. S.V. Berdyugina. Theoretical Stellar Astrophysics. Stellar Atmospheres: Opacities 14

15 In real atoms and ions the lower level of a given line may also have finite lifetime (when it isn't the ground state) and there may also be multiple downward transitions from each level. Multiple transition probabilities add up as rad i E E k i A ik because they measure deexcitations per second, an additive quantity. In other words, the convolution of two Lorentz profiles with half-width parameters 1 and 2 delivers a new Lorentz profile with half-width parameter = The total natural damping width is therefore given by: A A. rad rad rad i k ij kj E j Ei E j Ek In this case extinction and stimulated emission out of each level are automatically taken into account Collisional broadening Collisional broadening or pressure broadening results from other particles in the neighborhood. They may be electrons, ions, atoms or molecules. Their charge affects the radiating or extincting atom or ion through the Coulomb interaction and therefore affects the frequency of a boundbound transition between perturbed levels. These collisional encounters are often termed elastic although the energies are slightly changed momentarily; the term inelastic is reserved for collisions involving bound-bound transitions between different energy levels. Neutral atoms take part to some extent as perturber because they are polarizable, i.e. having a net electric field at close quarters. Of these, neutral H I atoms have the largest polarizability due to the bad shielding of the proton by the single electron. They are therefore important spectral line broadeners in cool atmospheres in which hydrogen is not ionized and which contain few free electrons. The classical classification is to split various interactions by their schematic dependence on the separation r between the absorbing atom or ion and the perturber in the form E C h r n n with Cn the interaction constant. The power index n defines the name and type of the interaction. The interaction of the atom with a perturber effectively deexcites it (shortening its lifetime). The resulting line profile is Lorentzian (as with natural broadening) with a width of S.V. Berdyugina. Theoretical Stellar Astrophysics. Stellar Atmospheres: Opacities 15

16 1, t where t is the time between collisions. Linear Stark effect (n = 2). The lowest-order broadening is called the linear Stark effect. It is important for H I lines and explains their very large width in spectra from hot stars. It is also important for hydrogenic lines such as He II lines and Rydberg lines (lines with high principal quantum number n obeying the Rydberg formula for hydrogen levels) of other elements. These interactions are with protons and electrons as perturbers. Figure 4.9: Example of linear Stark broadening in early B stars increased width for increased pressure: top spectrum from a B III star (giant), two lower spectra are from B V stars (dwarfs). Resonance broadening (n = 3). The n = 3 decay describes the interaction scale for collisions between neutral hydrogen atoms themselves. In order to see Balmer lines the atmosphere should be not too cool. It shouldn't be too hot or neutral hydrogen is not the major perturber. It seems to be important for the solar H line. Quadratic Stark effect (n = 4). Most lines other than H I lines are broadened by electron and ion impacts with spatial extent r 4 because they arise from systems without dipole moment. Electron S.V. Berdyugina. Theoretical Stellar Astrophysics. Stellar Atmospheres: Opacities 16

17 broadening dominates for non-hydrogenic atoms and ions in the atmospheres of hot stars where the electron density is high and the neutral hydrogen density small. Van der Waals broadening (n = 6). The overwhelming number of neutral hydrogen atoms makes them the dominant broadener of spectral lines in cool stellar atmospheres. Figure 4.10: An example of collisional and radiative damping in a stellar atmosphere. From Carroll et al. (2008) Doppler broadening The motion of a radiating particle along the line of sight produces a Doppler shift given by For purely thermal motions the distribution of velocities in the line of sight is given by the component form of the Maxwell distribution: which is an area-normalized Gaussian distribution with variance We thus obtain the Gaussian extinction line profile which is normalized to unity: S.V. Berdyugina. Theoretical Stellar Astrophysics. Stellar Atmospheres: Opacities 17

18 with Doppler width or Voigt profile Assuming that collisional broadening and thermal Doppler shifting are independent processes, the Lorentz shape function must be convolved with the Gaussian for thermal broadening which results in the Voigt function: where is the total damping constant: = rad The Voigt function is not normalized but has area in v units. The shape of the Voigt function approximates a Gaussian near line center but possesses 2 damping decay in far wings. Figure 4.11: The Voigt function H(a;v) for different values of the damping parameter a. From Unsöld (1955). S.V. Berdyugina. Theoretical Stellar Astrophysics. Stellar Atmospheres: Opacities 18

19 Other broadening Rotational broadening: macroscopic broadening of the lines in a stellar flux spectrum caused by the rotation of the whole star. If the relative line profile of the emergent intensity does not vary across the stellar disk, the rotational broadening of the relative flux profile can be described as a convolution of the Voigt profile with a Gaussian. It has the shape of a half ellipse. For details see Gray (1982). Turbulent broadening: in addition to microscopic (thermal) and macroscopic (rotation) motions, there are other motions in stellar atmospheres which are introduced, operating on microscopic (microturbulence) and macroscopic (macroturbulence) scales, via convolutions with Gaussian velocity distribution. The microturbulence is simply entered by redefining the Doppler width Adding macroturbulence is done by convolving the computed emergent intensity profile numerically with a Gaussian velocity distribution: Hyperfine structure: usually negligible, with the component separations well below the thermal Doppler width, except in some optical spectra (Mn I, VI, Co I, Cu I). Details are in the lecture on atomic transitions. Isotope splitting: different isotopes have different nuclear mass and therefore slightly different term energies. The effect is most outspoken for the light elements (H I, Li I) but also evident in Ba II. Zeeman splitting: At optical wavelengths the splitting is seen as line broadening. Towards the infrared the Zeeman splitting becomes more noticeable. This will be considered in more detail in subsequent lectures. S.V. Berdyugina. Theoretical Stellar Astrophysics. Stellar Atmospheres: Opacities 19

20 3.4. Emission and absorption coefficients with profiles In general, profiles of spontaneous emission, induced emission and absorption are different. For emission: h jv ni Aik ( v vki ), 4 where ( v v ki ) is a Lorentz profile in case of natural broadening or a Voigt profile (area normalzed) in case when deexcitation is independent on preceding exciting process ( complete redistribution ): ( vv ) ki Ha (, v) D The emission profile is more complex when the frequency redistribution over the line profile is incomplete ( partial redistribution ), which is the case if the photon that is emitted per deexcitation has some correlation with the photon that previously excited the atom in a scattering up-down sequence. Coherent scattering, without frequency change, is the other extreme. For absorption: h h ng i k ( ki ) v [ nk Bki ( ki ) ni Bik ( ki )] nk Bki 1, 4 4 ng k i( ki ) where profiles ( v v ki ) and ( v v ki ) are area-normalized. Then, the source function: S v j n A v v A B h. ( ) ( ) / / / / 3 v i ik ( ki ) ik / ik 2 / 2 v nk Bki v vki ni Bik v vki nk Bki ni Bik c nk gi ni gk The line source function may vary strongly with frequency across the line when the profile shapes are not equal due to coherent scattering or partial frequency redistribution. The profiles become equal ( v vki ) ( v vki ) ( v vki ) when complete redistribution holds in which every process has no memory for any preceding process. References: Rutten, R.J. 2003, Radiative Transfer in Stellar Atmospheres, Utrecht University lecture notes, 8th edition, Chapters 2 & 3. Gray, D. 1992, Observation and Analysis of Stellar Photospheres, Chapter 11. S.V. Berdyugina. Theoretical Stellar Astrophysics. Stellar Atmospheres: Opacities 20