Hydrogen Stark broadened Brackett lines

Stark broadened lines of the hydrogen Brackett series are computed for the conditions of stellar atmospheres and circumstellar envelopes. The computation is performed within the Model Microfield Method, which includes the ion dynamic effects and makes the bridge between the impact limit at low density and the static limit at high density and in the line wings. The computation gives the area normalized line shape, from the line core up to the static line wings.


Introduction
Hydrogen is the most abundant element in the Universe. Its broad lines give noticeable features in the spectra of stellar atmospheres [1,2,3,4]. These lines are very sensitive to the interaction between hydrogen radiating atoms and the surrounding charges, electrons and ions (mostly protons), which is connected to the random electric eld generated by these charges. The electric eld has two components, with dierent time scales: the rapidly varying electronic eld and the slowly varying ionic electric eld. The net eld induces a strong mixing of the atomic states with the same principal quantum number, from which the Stark broadening originates. Astrophysical applications need to know the full line shape, from line centre up to the line wings, for a wide range of plasma conditions (i.e. electron density N e and temperature T ).
Whereas hydrogen line shapes have been widely used in stellar physics for the determination of gravity and/or temperature in the visible and UV ranges, the recent instrumental developments on various telescopes, such as AMBER [5] and CRIRES [6] on VLTI and VLT, require the availability of precise hydrogen lines in the infrared (i.e. between 1 and 5 µm). The latter are badly known [7]. This paper thus aims at providing a coherent description of the line shapes of Brackett α, β, γ, which connects the levels of principal quantum numbers n equal to 4, for the lower state of the transition, and n equal to 5, 6 and 7, for the upper state, and have central wavelengths of 4.05, 2.63, and 2.12 µm. The choosen plasma conditions are relevant to stellar photospheres and circumstellar environments (electron densities between 10 10 and 10 19 cm −3 and temperatures between 10 3 and 10 7 K). The plasma charges are assumed to be electrons and protons. This is a standard approximation for this type of study, although some improvements may be possible by including the eect of ionization of He (which is, by number of atoms, only 10% as abundant as hydrogen) and traces of heavier elements.
We suppose that plasma collective eects are included in the Coulomb interactions between the hydrogen bound electron and the plasma charges by Debye screening. This requires that the Debye length λ d , is larger than the mean distance r 0 between the protons, or that the parameter a = r 0 /λ d is smaller than unity. Our tables will thus be limited to the values of temperatures and densities satisfying the condition: which is fullled for standard stellar atmospheres. We assume also that the proton/electron density is small enough to ensure that each line 4-n' (n'= 5, 6 or 7) remains distinguishable from the subsequent line 4-(n'+1) of the Brackett series. Using the Inglis-Teller [8] criterium, the upper limit to the electron density N e is thus log(N e,max ) = 22 − 7.5 log(n ) (cm −3 ).

(2)
This respectively gives log(N e,max ) = 16.75, 16.16 and 15.66 for the n (= 5, 6 and 7) values considered in this paper. However, in order to allow interpolation within the tables for astrophysical applications, the line shapes for higher values of N e , up to a decade, have been computed. Dierent methods can be used to generate high quality spectral line shapes for hydrogen lines perturbed by protons and electrons: Molecular Dynamics for describing precisely the ion dynamics eects in the line center [9,10], quantum theory for the electron contribution to the line wings [11], short range H-H+ molecular interactions also for the line wings, leading to the apparition of quasi-molecular satellites, which are observed in the atmospheres of white dwarfs [12]. However, they are limited either to part of the prole, or to restricted plasmas conditions, or to simple lines like Lyman or Balmer lines. Thus, as for astrophysical purposes, the tabulations go from the line centre up to the line wings, it is necessary to nd a compromise between accuracy and description of the whole prole.
In this context, the tabulations of Vidal et al. [13], using Unied Theory for the electrons and static approximation for the ions, have been used for stellar atmospheres, despite the intrinsic lack of accuracy in the line center due to the neglect of ion dynamics eects. The tabulations of Stehlé et al., for the Lyman , Balmer [14,15] and Paschen lines [16], using Model Microeld Method, which was initially developed by Brissaud et Frisch [17,18] brought an important improvement by taking into account the ion dynamics eects. They are now used for atmospheres and for the computation of radiative diusion processes in the radiative stellar envelopes [19]. In the case of partly ionized plasmas, like for the atmospheres of cool stars, the contribution of neutral broadening by hydrogen has to be included in the line shape, especially in the line wings [20,21]. This eect will be neglected in the following.
In this paper we shall present the Stark broadened proles of Brackett lines. They will be computed within the formalism of Model Microeld Method, hereafter denoted by MMM. We shall neglect the ne structure eects, which play a role, at low density, in the core of the lines with low n quantum mumbers, like Lyα or Hα [22]).

Method
The broadening of spectral lines results from the interactions between the radiating hydrogen atom and the free ions and protons. These two contributions can be described in terms of interaction potentials, with the corresponding electronic and ionic plasma microelds F el and F ion . Neglecting quadrupolar and other contributions that play a role at high densities [23], the dipolar potential of interaction between the bound electron and the microelds may be written as: The spectral line prole I(ω) (with area normalized to unity) is thus dened in the Liouville space [24,25], spanned by the states |i, f (which stands for |n i , l i , m i ; n f , l f , m f ) as where < U(ω) > el,ion is the Fourier transform of the evolution operator of radiating Hydrogen atom in the Liouville space, averaged over the realizations of the stochastic dynamic electronic and ionic microelds F el and F ion . The term d i,f .d i ,f is the product of dipole operator elements between initial low states (denoted by i, i ) and nal upper states (denoted by f, f ), of the hydrogen bound electron. As ne structure and inelastic eects are neglected, The two microelds are stochastic processes. It is thus possible to dene two distribution functions P (F) [26,27], respectively associated to the slowly varying ionic and rapidly varying electronic microelds. In order to take into account the dynamic eects of these microelds, a model for the dynamical statistics of eld uctuations is necessary. In MMM, the microeld uctuations are handled with a statistical process model, where the microeld (electronic or ionic) is assumed to be constant during a given time interval. The microeld then jumps instantaneously to another constant value for the next time interval. The jumping times are assumed to follow a Poisson law, with a eld dependent frequency ν(F ). The jumping frequency ν(F ), is chosen to reproduce the true eld autocorrelation function [17], [18], [28]. This method has been tested against asymptotic impact and quasi-static limits and has been proved to lead to very good results for hydrogen [29] and hydrogenic ion lines [28].
The method has been already described in Stehlé and Hutcheon [15], and we refer the reader to this paper for the details. An important point is that it is possible to disentangle the contributions of ions and fast electrons by introducing a frequency dependent electronic relaxation operator γ el (ω), which is independent from ionic elds, and thus may be computed separately. The Fourier transform of the evolution operator, < U(ω) > el , averaged over all the realisations of the electronic elds, may be written as where ∆ω = ω − ω 0 is the detuning from line center and I is the identity operator in the Liouville space.
Thus, this electron damping is rst computed to account for average eect of the electronic elds. Then, the static Fourier transform of the evolution operator, < U(ω) > el,ion , averaged over the realisations of the electronic and ionic elds, may be written as : The MMM expression is more complex ( [17,18]) than this one, which corresponds to the usual Unied Theory with static ionic elds.
In order to reduce the dimensions of the Liouville space, (16 × 25 = 400 states for Brα, for instance), we use the formalism of the reduced Liouville space, which takes advantage of the invariance of the dierent operators, like d.d, by angular average over all the orientations of the electric elds and of the fact that the dipole tensor d in equation 4 is of rank 1 (see for instance [28,29]). However, the number of reduced states |n i , l i ; n f , l f (with |l i − l f | ≤ 1) remains important (i.e. 10 reduced Liouville states of rank 1 for Brα). Thus, we use another approximation, already called isotropic approximation in [15], which uses the diagonal form γ iso el (ω), with equal diagonal matrix elements, instead of γ el (ω). This scalar tensor is deduced from the pure electronic prole I(ω) el by the relation, or The Stark prole is thus obtained after computing the average over electronic eld values, which gives γ iso el (ω), and then the average over the ionic elds, both using the formalism of Model Microeld Method. The third step is then the Doppler convolution, which gives the nal line shape.

Results
The three Brackett α, β, γ line shapes have been studied in a wide range of stellar conditions, but we report hereafter only a selection of results, relative to the line center and the line wings.

Line center
We present in gures 1, 2 and 3 the variations of the half-width (HWHM) of Brα, Brβ and Brγ lines for dierent values of the electron density. This quantity is the detuning from the line center, at the point where the prole reaches half the maximum value of the line prole (which is not necessarily at the line center, as will be discussed below). The gures show the half width of the MMM prole with and without Doppler eect and also the value of the impact half-width in its own validity range.
Hence, impact limit has been proved to be valid, for both electron and ion contributions, in the line center and at low densities. Moreover, it has been proved that the value of the impact width is analytical and that the corresponding prole is Lorentzian in the line center [30]. The validity condition of the impact limit is that the half-width value is smaller than the ion plasma frequency. Let us take the example of of Brα. At 10 12 cm −3 and 2500 K the ion plasma frequency is equal to 2.6 10 9 rd.s −1 ,whereas the ion impact contribution to the HWHM is equal to 1.8 10 9 rd.s −1 . At 10 14 cm −3 , they are respectively equal to 2.6 10 10 and 3.9 10 10 . The impact limit should thus be reached gradually as the density decreases below 10 13 cm −3 .
The gure 1, relative to Brα, shows indeed that the half-width of the Stark prole (black), tends to converge towards the impact analytical limit (red) at these low densities. However, the convolution with the Doppler prole increases the half-width value. As a consequence, the half width is dominated by the Doppler broadening at low densities. Similar behaviour occurs for the other lines, as may be on gures 2 and 3 for Brβ and Brγ. At higher densities, the ion dynamic eects become smaller in the line center (they are negligible in the line wings, as will be seen below), and the line shape departs from the Lorentzian shape. We found that, depending on the temperature conditions, the Brackett β line may present a small dip in the line center at moderate densities of 10 15 cm −3 and for the largest temperatures as seen in gure 4. This eect was well known in earlier tabulations for the Lymanβ and Hβ lines. The dependence versus the temperature is a consequence of the the electron impact broadening, which varies in T −1/2 , and which may ll (at low T values) or not (at large T values) the central dip.

Line shapes
Another typical behaviour of hydrogen lines is the convergence towards the (Holtsmark) static limit in the line wings, which scales as |∆ω| −5.2 when the line shape is expressed in angular frequency units (which is the appropriate unit for the line shape computations). However, traditionally, the line intensity was expressed in units of ∆α = ∆λ/F 0 , where F 0 (esu) = 1. 25 10 9 (N e ) 2/3 is the normal Holtsmark eld. In these units, the Holtsmark limit is given by : where λ 0 is the central wavelength, and K α a constant, which depends on the transition, K α = 1.512 for Brα, 2.401 for Brβ and 2.926 for Brγ (10) This variation in ∆α of eq. 9 introduces, at large detunings, a "trivial" asymmetry between I(∆α) and I(−∆α) (which does not exist between I(∆ω) and I(−∆ω)). The gures 5-7 show the proles I(∆α) of Brα, Brβ and Brγ lines at 10 12 cm −3 and various temperatures. The proles, including Doppler eects are reported in red color, and the pure Stark proles in blue.
The proles are area normalized, i.e. : As a consequence, broad proles have small values of I(∆α = 0). For the density considered in the gures 5-7, the Doppler prole dominates the Stark prole in the line center and becomes indistinguishable, as expected, in the line wings, where they follow the asymptotic limit of eq. 9. At higher densities (not reported here), the Doppler width is smaller than the Stark width and Doppler convolution is no longer necessary.

Conclusions
This study shows that Stark broadened infrared Brackett lines of hydrogen follow the same trends as the lines of lower series which are more known theoretically and experimentally. This study will allow missing absorption in the spectra of stellar atmospheres in the infrared due to the lack of data to be lled in a next future. Dedicated experiments and comparisons with other methods, for instance FFM [32], would be helpful to test these theoretical results, which will be also constrained by observational data. The corresponding tables will be accessible at http://amrel.obspm.fr/stark-h.

Acknowlegdments
We would like to thank Pauline Béghin, Edrice Bouteldja, Aurélie de Paz and Cécile Turc, students at the University of Pierre et Marie Curie for their contribution in checking the data.