PHYS 633: Introduction to Stellar Astrophysics Spring Semester 2006 Rich Townsend ([email protected])

PHYS 633: Introduction to Stellar Astrophysics Spring Semester 2006 Rich Townsend ([email protected]) Opacity The equation of radiative diffusion, which is one of the governing equations of stellar structure, relates the radiative flux of energy,FR, to the local temperature gradient, 4ac dT F = . (1) R 3κρ dr Playing a very important role in this equation is the Rosseland mean opacity κ (also known as the mass absorption coefficient; and note that some authors use the term ‘opacity’ actually to refer to the product κρ). The opacity charac- terizes the degree to which the passage of radiation is retarded by the presence of matter, due to scattering and absorption of photons. For a sample of stellar material with mass dm, the total cross section presented by the particles constituting the sample is given by κdm. If the volume of the sample is dV , then we have κdm = κρdV , and we can see that the term κρ can be recognized as the cross section per unit volume. Since κρ has dimensions of inverse length, we can see that we can multiply it by a distance dz and obtain a dimensionless number. What does this number represent? Well, if dz is infinitesimal, then κρdz is the probability that a photon will be absorbed when traveling through dz, and 1−κρdz is the complementary probability that the photon won’t be absorbed. To work out the probability of an ‘average’ photon (in the Rosseland mean sense) traveling a macroscopic distance, let P (z) be the probability that the photon traverses distance z without being absorbed. Then, the probability that the photon travels a distance z + dz unabsorbed is given by P (z + dz) = (1 − κρdz)P (z), (2) which is the product of the individual probabilities for each leg of the journey. Thus, we have P (z + dz) − P (z) = −κρP (z), (3) dz and in the limit of vanishing dz, dP = −κρP (z). (4) dz Integrating, with the boundary condition of P (0) = 1, we therefore obtain P (z) = e−τ , (5) 1 where we have introduced the (dimensionless) optical depth τ as Z z τ(z) = κρ dz. (6) 0 This latter equation can be used to define the mean free path ` of an average photon, such that τ(`) = 1. So far, we have been dealing with the Rosseland mean opacity κ. We recall from out treatment of the diffusion equation that for material in thermodynamic equilibrium with temperature T (by which we mean the radiation field has a black-body spectrum with temperature T ), the Rosseland mean is defined by R ∞ 1 ∂Bν 1 0 κ ∂T dν ≡ ∞ , (7) κ R ∂Bν 0 ∂T dν where 2π ν3 B = . (8) ν c2 ehν/kT − 1 is the Planck function. Clearly, we need to know the frequency-dependent opacity κν to calculate κ. Unfortunately, there is no simple recipe for this quantity; calculation of κν — and thus κ — is one of the most complex tasks of stellar structure and evolution, and in reality any physically-realistic opacity model is far too complex to incorporate directly in a stellar evolution code. In fact, the standard approach nowadays is to store the results from de- tailed opacity calculations in table form, so that the evolution code can simply look up/interpolate appropriate values. Typically, the Rosseland mean opacity κ is tabulated as a function of temperature T and density ρ, or sometimes 3 6 temperature and R ≡ ρ/T6 (here, T6 is the temperature in units of 10 K). The present-day state-of-the-art tabulations are by two main groups: OPAL, based at Lawrence Livermore National Laboratory, and the international Opac- ity Project (OP). Of course, even though these tabulations are available for download by any- one who wants to use them, we still should have an idea about where the values come from. So, let’s briefly discuss the physical processes contributing toward stellar opacity. Electron Scattering When an electromagnetic wave is impingant on a free electron, it causes the electron to oscillate. The electron itself then emits a new electromagnetic wave, with the net effect being that the original wave has been scattered through some 2 angle. If the original wave has a frequency ν such that hν mec (which occurs typically for blackbody radiation at temperatures cooler than ∼ 108 K), then the scattering is coherent: the outgoing wave (photon) has the same frequency (energy) as the incoming wave (photon). Such coherent electron scattering is known as Thomson scattering, and has a cross-section per electron of 8π σ = a2 ≈ 6.652 × 10−25 cm2 (9) T 3 0 2 where e2 a0 = 2 (10) mec is the classical radius of the electron (in cgs units; SI units require an extra factor of 4π0 in the denominator). To calculate the opacity due to Thomson scattering, we multiply σT by the number of free electrons per unit mass, n κ = σ e . (11) ν T ρ Since the mean molecular weight per electron is defined by ρ µemu = , (12) ne we also have σT κν = . (13) µemu Note that, although we indicate that the opacity κν can in principle depend on the frequency, in the case of Thomson scattering it is frequency independent. In the limit where the stellar material is completely ionized, µe is given by 2 µ = (14) e 1 + X (see eqn. 20 of the notes on the Equation of State), where X is the hydrogen mass fraction. Hence, the Thomson scattering opacity becomes σT 2 −1 κν = (1 + X) ≈ 0.20(1 + X) cm g (15) 2mu For typical hydrogen abundances X = 0.7, we therefore obtain a value 0.34 cm2 g−1, independent of the temperature, density or frequency. 2 If the scattered radiation has a frequency such that hν > mec (typically occurring for blackbody radiation at temperatures hotter than ∼ 108 K), then the electron recoils during the scattering by a non-negligible amount. To con- serve energy, the recoil means that the scattered radiation (photon) must have a frequency (energy) less than that of the incoming radiation (photon); the scattering is no longer coherent. In this regime, we speak of Compton scattering. The cross section for Compton scattering is obviously frequency dependent, and is always smaller than the Thomson cross section. Free-Free Absorption When a free electron passes an ion, they temporarily form an (unbound) sys- tem capable of absorbing electromagnetic radiation of any frequency. The opacity associated with such free-free absorption1 was originally characterized by Kramers, and has the form −3 1/2 κν ∝ ν neT . (16) 1The reverse process of free-free emission occurs when a charged particle decelerates in the electrostatic field of another particle, and is known as Bremsstrahlung (radiation braking). 3 −3 The ν term comes from classical electrodynamics. The ne term appears because, for a given amount of mass, the number of possible free-free absorption systems depends on the number of available (free) electrons. Finally, the T 1/2 term appears because the time that electrons and ions spend together scales inversely with their relative velocity due to thermal motions, which itself scales as ∼ T 1/2. The constant of proportionality in Kramers’ expression above depends on a number of things, including the charge on the ion, and a factor g, known as the Gaunt factor, that corrects for quantum mechanical effects (remember, Kramers’ expression is derived from classical electrodynamics). One of the most important corrections is for degeneracy; if the density is very large or the temperature very low, then there is a scarcity of quantum states for free electrons to move into, and the free-free opacity is found to be smaller than in the corresponding non- degenerate limit. If free-free opacity is the only (or dominant) opacity source in the star, then we can go ahead and calculate the Rosseland mean opacity as R ∞ 3 ∂Bν 1 1/2 −1 0 ν ∂T dν ∝ T ne ∞ (17) κ R ∂Bν 0 ∂T dν. With a little bit of algebra, we find that −7/2 κ ∝ neT , (18) a characteristic temperature and density dependence often termed ‘Kramers opacity’ (irrespective of whether the opacity source is due to free-free absorption or not). Note that there may be an additional temperature dependence hidden in this equation, because the electron number density ne itself depends on the (temperature-sensitive) ionization state of the stellar material. Clearly, free-free absorption is most effective at high densities and low temperatures — typically, the sorts of conditions found in lower-mass stars. Bound-Free Absorption Bound-free absorption is photoionization from the point of view of the radiation field: a photon is removed from circulation, being used to eject an electron from a bound state of an atom or ion. Bound-free absorption only works for photons with frequencies satisfying hν > χ, where χ is the ionization potential – the amount of energy required to remove the electron from the bound state and place it at rest at infinity. The excess energy of the photon over χ goes into the electron kinetic energy. For sufficiently energetic photons, the Rosseland mean opacity for bound- free absorption exhibits the same basic temperature and density dependence as free-free absorption (see eqn. 18), and thus is also often referred to as ‘Kramers opacity’. However, as with the free-free case, there are additional temperature dependencies hidden away, relating in this case to the number of bound (un-ionized) atomic systems. If the temperature is too high (typically, above 4 106 K), then all elements will be completely ionized, and there is no bound-free absorption.

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