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Rayleigh Scattering in the Atmospheres of Hot Stars J A&A 590, A95 (2016) Astronomy DOI: 10.1051/0004-6361/201628291 & c ESO 2016 Astrophysics Rayleigh scattering in the atmospheres of hot stars J. Fišák1; 2, J. Krtickaˇ 1, D. Munzar1, and J. Kubát2 1 Masarykova Univerzita, Prírodovˇ edeckᡠfakulta, Kotlárskᡠ2, Brno, Czech Republic e-mail: [email protected] 2 Astronomický ústav, Akademie vedˇ Ceskéˇ republiky, Fricovaˇ 298, 251 65 Ondrejov,ˇ Czech Republic Received 10 February 2016 / Accepted 30 March 2016 ABSTRACT Context. Rayleigh scattering is a result of an interaction of photons with bound electrons. Rayleigh scattering is mostly neglected in calculations of hot star model atmospheres because most of the hydrogen atoms are ionized and the heavier elements have a lower abundance than hydrogen. In atmospheres of some chemically peculiar stars, helium overabundant regions containing singly ionized helium are present and Rayleigh scattering can be a significant opacity source. Aims. We evaluate the contribution of Rayleigh scattering by neutral hydrogen and singly ionized helium in the atmospheres of hot stars with solar composition and in the atmospheres of helium overabundant stars. Methods. We computed several series of model atmospheres using the TLUSTY code and emergent fluxes using the SYNSPEC code. These models describe atmospheres of main sequence B-type stars with different helium abundance. We used an existing grid of models for atmospheres with solar chemical composition and we calculated an additional grid for helium-rich stars with N(He)/N(H) = 10. Results. Rayleigh scattering by neutral hydrogen can be neglected in atmospheres of hot stars, while Rayleigh scattering by singly ionized helium can be a non-negligible opacity source in some hot stars, especially in helium-rich stars. Key words. atomic processes – scattering – stars: chemically peculiar – stars: atmospheres – stars: early-type 1. Introduction only a small fraction in the neutral state and second, abundances of the other elements are low. In addition, the Rayleigh scattering The interaction of photons with bound electrons is very impor- cross section is smaller for atoms with a higher atomic number. tant in stellar atmospheres. Processes of excitation and ionization However, the population of singly ionized helium is much are very well known and have been studied in detail. However, larger than the population of neutral hydrogen in hot main se- photons may interact with bound electrons even when photon quence stars with solar composition. In addition, there are stars energy is not equal to the energy difference between any bound with helium overabundant atmospheres, which means they have levels. Rayleigh scattering is one of these processes. even larger population of singly ionized helium. Rayleigh scattering is a special kind of scattering process. The aim of this paper is to investigate the possible effect According to Loudon(1983), the scattering processes by bound of Rayleigh scattering by ionized helium on radiation emerging electrons are divided into three cases: fluorescence, Raman scat- from model atmospheres of hot stars. tering, and Rayleigh scattering. In the case of Rayleigh scattering an ion is excited by a photon and transits to a virtual state. This state is very unstable and the electron transits to the original state 2. Rayleigh scattering immediately. If the initial state is different from the final state we Rayleigh scattering is a type of interaction of radiation with call this process Raman scattering. bound electrons. The energy of an interacting photon is not equal Rayleigh scattering was studied by John William Strutt (Lord to any energy difference between an excited state of the atom and Rayleigh, see Rayleigh 1870). This effect causes the blue colour the ground state. An electron is excited to an unstable virtual of the sky and the red colour of sunsets. Scattering centres are state (in contrast to the line transitions), from which it transfers oxygen and nitrogen molecules because they have the largest immediately back to its original state. This is shown in Fig.1. abundance in the Earth’s atmosphere. Rayleigh scattering is also The process of Rayleigh scattering is nicely described by the an important opacity source in the atmospheres of other planets time dependent perturbation theory. Using an electric-dipole ap- (Buenzli et al. 2009). proximation and the Krammers-Heisenberg formula for the dif- Rayleigh scattering by H2 was detected in exoplanet atmo- ferential cross-section we obtain in SI units (see Loudon 1983) sphere HD 209458b (Lecavelier des Etangs et al. 2008b) and it is suspected that Rayleigh scattering is responsible for the blue dσ(!) e4! = colour of exoplanet HD 189733b (Lecavelier des Etangs et al. dΩ 16c4π22~2 2008a). Rayleigh scattering is an important opacity source in 0 ! 2 cool stars because of large neutral hydrogen atom population X ("ks D1n)("kDn1) ("kD1n)("ks Dn1) × + ; (1) (see Hubeny & Mihalas 2015). ! − ! ! ! n n n + Scattering by bound electrons is often neglected in hot star atmospheres because of the low abundance of Rayleigh scatter- where D ji = j r i is the matrix element of the sum of the ing atoms. First, both hydrogen and helium are ionized leaving electron position vectors (of the electron position vector for Article published by EDP Sciences A95, page 1 of6 A&A 590, A95 (2016) and σe is the Thomson scattering cross section Excited state e4 σ = = 0:665 × 10−24 cm2: (8) e 2 4 2 6"0c m π Excited state 2.1. Cross section for Rayleigh scattering on hydrogen First, we consider only hydrogen atoms. In this case, the re- Ground state duced matrix elements can be calculated analytically. According Fig. 1. Scheme of the Rayleigh scattering process. The bound electron to Lee & Kim(2004), the total cross section for Rayleigh scat- is excited by an incoming photon to a virtual state which is unstable and tering by neutral hydrogen is equal to the electron transits immediately to the original state. !4 σ( f ) 5:5758 × 10−25 ! ≈ 2 2 1 cm 1 cm !L hydrogen-like atoms). Furthermore, "k and "k are the polariza- s !6 !8 tion vectors for the incoming photons with wave vector k and 1:8567 × 10−24 ! 4:9480 × 10−24 ! + + k 2 2 for the scattered photons with wave vector s, respectively, and 1 cm !L 1 cm !L !n represents the angular frequency of the transition between the −23 !10 −23 !12 ground state (l) and the nth state 1:274 × 10 ! 2:656 × 10 ! + 2 + 2 1 cm !L 1 cm !L En − El ! = · (2) !14 n ~ 5:396 × 10−23 ! ··· ; + 2 + (9) The sum runs over all intermediate states. Here e is the electron 1 cm !L charge, " is the vacuum permittivity, c the speed of light, and m 0 f ! the electron mass. where = 2π . After some calculations described in Loudon(1983) and Lee & Kim(2004), and after some mathematical arrangements 2.2. Cross section for Rayleigh scattering on hydrogen-like (Zettili 2009; Bethe & Salpeter 2008; and Landau et al. 1982) atoms we obtain the following equation for Rayleigh scattering cross section in the case of one-electron atoms, The radial wavefunction of hydrogen-like atoms is given by (see Landau & Lifshitz 1977) 4 4 2 2 4 ! 2 2e ! X n; 1 r 1; 0 ! ! s σ(!) = 1 + + + ::: ; !3=2 ! !l 4 2 2 ! !2 !4 1 (n + l)! 2Z Zr 2Zr 3c π0 ~ n n n n R (r) = exp − Z;nl (2l + 1)! 2n(n − l − 1)! na na na (3) 0 0 0 ! 2Zr where n; 1 r 1; 0 is the reduced matrix element, 1; 0 rep- × F l + 1 − n; 2l + 2; ; (10) na0 resents the 1s state, and n; 1 represents the np state, n is the principal quantum number and p is the azimuthal quantum where n is the principal quantum number, l is the orbital quan- number (both bound states and continuum states are included). tum number, Z is the atomic number, a0 is the Bohr radius, and The reduced matrix element is given by the Wigner-Eckart 2Zr F l + 1 − n; 2l + 2; na is the hypergeometric function. It can be theorem valid for any vector operator Aˆq (see Eq. (7.326) in 0 Zettili 2009): seen that 0 0 0 0 0 0 0 3=2 n ; j ; m Aˆq n; j; m = j; 1; m; q j ; m n ; j A n; j : (4) RZ;nl(r) = Rnl(Zr)Z ; (11) The first matrix element on the right side of Eq. (4) is the where Rnl(r) are radial wavefunctions of hydrogen. For the ma- Clebsch-Gordan coefficient. It is possible to rewrite Eq. (3) in trix element we obtain terms of dimensionless angular frequency! ˜ n and dimensionless position vector r˜ (see Lee & Kim 2004), Z2π Zπ Z1 2 3=2 !4 2 npm z 1s0 = d' dθ sin θ dr r Z R10(Zr)Y00(θ; ') ! X n; 1 r˜ 1; 0 Z σ(!) = σ e ! ! 0 0 0 L n ˜ n 3=2 1 0 1 2 × Z Rn1(Zr)Y1m(θ; ')r cos(θ) = npm z 1s0 : (12) B 1 !2 1 !4 C Z Z=1 × B1 + + + ::: C ; (5) @B !˜ 2 2 !˜ 4 4 AC n !L n !L Using this result, the Wigner-Eckart theorem, and the fact that 2 where the energies of the excited states scale with the factor Z we obtain !n r !˜ n = ; r˜ = ; (6) ! a !4 2 L 0 ! X np r 1s σ(!) = σe the angular frequency of Lyman limit !L and Bohr radius a0 ! Z4! L n ˜ n are 0 1 2 4 2 4 2 B 1 ! 1 ! C e m 4π0~ × B1 + + + ::: C ; (13) !L = ; a0 = ; (7) @ 4 2 2 8 4 4 A 3 2 2 2 Z !˜ n !L Z !˜ n !L 32~ 0 π me A95, page 2 of6 J.
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