High-temperature (1000–7000 K) collision-induced absorption of H₂ pairs computed from the first principles, with application to cool and dense stellar atmospheres

Abstract

The collision-induced absorption (CIA) spectra of H₂–H₂ and H₂–He are known to play an important role for modelling of low-metallicity cool and dense stellar atmospheres. In this paper we present collision-induced absorption spectra of H₂–H₂ complexes in the rototranslational (Δv=0), the fundamental (Δv=1), the first (Δv=2) and the second (Δv=3) overtone bands in the temperature range from 1000 to 7000 K, and in the frequency region from 0 to 20 000 cm⁻¹. The translational spectral density functions are computed quantum mechanically, based on: (1) the newly developed ab initio collision-induced H₂–H₂ dipole functions of Zheng (Computational study of collision induced dipole moments and absorption spectra of H₂–H₂. Ph.D. thesis, Michigan Technological University, 1997), which account for the short-range H₂–H₂ intermolecular distances (as small as 2.5 a.u.) and for larger H₂ internuclear distances (as large as 2.15 a.u.); (2) semiempirical isotropic H₂–H₂ potential (Ross et al, J Chem Phys 1983;79(3):1487) suitable for high temperatures. We include the collision-induced absorption coefficient of the vibrational transitions as v₁,v₂,v′₁,v′₂≤3 which we computed rigorously. We also give our estimate for the collision-induced absorption coefficients of single vibrational transitions such as $\text{v}{}_{\mathrm{i}}\text{<3,}\text{v′}{}_{\mathrm{i}}\text{>3}$ in the first and second overtone bands. The dependence of CIA spectra on rotational states of H₂ molecules is accounted for in our computations. We have previously (Borysow et al, Astronom Astrophys. 1997;324:185–95) studied the effect of CIA for stars of a wide range of fundamental stellar parameters (effective temperature, gravity, and chemical composition), and determined for which combinations of these parameters it is necessary to include CIA in the model and spectrum computation. These calculations showed that CIA from H₂–H₂ plays an important, and often even a dominating role for stellar atmospheres of a wide range of stars. The approximate character of the estimates of the H₂–H₂ absorption coefficient we used in our previous work combined with the large effect CIA had on the stellar atmospheres, were the main inspirations to initiate the more accurate computations of the absorption coefficient we present here. The absorption coefficient we compute in the present analysis is in qualitative agreement with our preliminary estimates (Borysow et al, Astronom Astrophys 1997;324:185–95), but in some spectral regions of high importance for the stellar structure, our computed absorption coefficient is up to a factor of 3 larger than our preliminary estimates. We therefore fully confirm our previous suspicion that H₂–H₂ CIA will have a pronounced effect on the atmosphere for a wide range of stars. In this paper we therefore quantify the effect the new data have on a typical cool dense stellar atmosphere, and compare our new results with our previous estimates.

Introduction

Inert molecules and atoms without intrinsic permanent electric dipole moment, like hydrogen molecules H₂ and helium atoms He, do not absorb infrared dipole radiation on their own, but the transient collision-induced dipole moment, arising in small complexes of atoms or nonpolar molecules during the intermolecular interaction, gives rise to collision-induced absorption (CIA). The collisional complex could be a binary complex (like H₂–He), or a ternary complex (like H₂–H₂–He), or a many-body complex. Only the collision-induced spectra of binary complexes are of concern in this work. The induced dipole moments are usually extremely weak, when compared with usual values of the intrinsic dipole moments of polar molecules, and are due to the electronic overlap at short intermolecular distances, the induction by the electric multipole field of a collisional partner, and the dispersion forces at large range.

The atmospheres of low-metallicity stars such as, for example, globular cluster stars, halo stars, and primordial stars are composed predominantly of H₂ and He. At the low atmospheric temperatures and high densities (thus: pressure) prevailing, for example, in brown dwarfs, M dwarfs, and cool white dwarfs, CIA of H₂–H₂ and H₂–He may become the dominant source of opacity [1], [2], [3]. Recent studies on modelling low-metallicity stars [4] and on modelling zero-metallicity stars [5], [6], [7] show that the models, which account for the CIA bands due to H₂–H₂ and H₂–He, give dramatically different results from those with the CIA bands neglected, especially at the lowest temperatures. CIA may be important also at temperatures up to 7000 K, when the existing gas densities are sufficiently high.

Recent historical reviews on the importance of CIA in cool, stellar atmospheres are given in [3], [4], [5], [6], [7], [8]. In this paper we present new CIA spectra in the rototranslational (RT), and the three lowest rotovibrational (RV) bands of H₂–H₂ pairs at a wide temperature range, from 1000 to 7000 K.

Experimental measurements of CIA spectra at high temperatures are extremely difficult (if not impossible) to carry out. Therefore, accurate and reliable theoretical models of CIA spectra become very important. Historically, Linsky [1] proposed the first model of CIA spectra of H₂–H₂ pairs for the RT band (in the temperature range from 600 to 5000 K), the fundamental and the first overtone bands (at temperatures from 600 to 3000 K). No hot bands were included in his model. Also, Patch [9] and Tsuji [2], [10] modeled the fundamental band of CIA spectra of H₂–H₂ pairs at high temperatures.

At temperatures below 300 K accurate measurements of CIA exist. As well, rigorous quantum mechanical computations for H₂ molecular systems with improved interaction potentials and the ab initio collision-induced dipole moments surfaces have been performed. The theoretical computations of CIA of H₂–H₂ pairs agree with laboratory measurements well within the experimental uncertainties at temperatures from 20 to 300 K in the RT band [11], at the fundamental band, and the first overtone band [12], [13]. These obvious successes indicated that the same theory might be used successfully also to predict reliable, accurate spectra at temperatures higher than room temperature where laboratory measurements are not available.

The high-temperature computations of CIA of H₂–H₂ pairs, based on the collision-induced dipole moments [12], were computed by Zheng and Borysow [14] at temperatures from 600 to 7000 K for the RT band; by Borysow and Frommhold [15] from 600 to 5000 K for the fundamental band; and by Borysow et al. [4] at temperatures from 1000 to 7000 K in the first overtone band. Borysow et al. [4] tried to predict CIA of H₂–H₂ pairs at 1000 to 7000 K also for the second overtone band, to be applied to model atmospheres of cool, low-metallicity stars. These high-temperature H₂–H₂ CIA computations were less rigorous, for several reasons. To name just one, the necessary values of the collision-induced dipole moments at the short range of intermolecular distance (H₂–H₂) and at large internuclear (H–H) distance where, at that time, not available. The work [4], nevertheless, provided the most complete estimate of the high-temperature CIA intensities possible under the given conditions. These data were then applied to model stellar atmospheres, with varying T_eff, gravity and metallicity. These CIA spectra have been deposited, for public use, at http://www.astro.ku.dk/∼aborysow/programs/.

The newly developed database of induced dipole moments [16] extends the existing one [12]² by including the short range of intermolecular distance $\text{(2.5}\text{a.u.}\text{≤R≤3.5}\text{a.u.}\text{)}$ and long range of internuclear distance (r_i up to 2.150 a.u.).

The theoretical computations of CIA of H₂–H₂ pairs for the second overtone band, based on the new induced dipole matrix elements derived from the extended dipole database, agree with laboratory measurements within maximal deviations up to 30% over the frequency range from 11,500 to 13,800 cm⁻¹ at temperatures of 77.5, 85 and 298 K [18], [19]. The discrepancy between the theoretical and experimental data comes mainly from the extremely weak collision-induced dipole moment in this band.

With the improved induced dipoles of the H₂–H₂ pairs [16], [17], [18], and the semiempirical isotropic potential [20] of the H₂–H₂ complex suitable for high temperatures, it became possible, for the first time, to compute the CIA spectra of the H₂–H₂ complex at a high-temperature range in the RT band, Δv=(v₁′−v₁)+(v₂′−v₂)=0, the fundamental band (Δv=1), the first (Δv=2), and the second overtone (Δv=3) bands, where $\text{v}{}_{\mathrm{i}}\text{(i=1,2)}$ is the initial vibrational state of ith hydrogen molecule and v_i′ denotes the final state. The single and double vibrational (v₁,v₂)→(v₁′,v₂′) transitions have been taken into account for all bands of concern here. In the sections below, we briefly sketch the line shape theory used here, and present the input data for CIA computation at high temperatures. Next, we show our final, high-temperature CIA results and a comparison with the ones which have been previously used in the studies of stellar atmospheres. In the end, we show an example of a model atmosphere obtained with the new CIA intensities and conclude about the differences with our previous computations [4].