3 New high temperature CIA data from 400 K to 1000 K

The current need of data for modelling objects in-between traditional stellar and planetary temperatures became obvious with the long awaited identification of a brown dwarf in 1995 and with the surprising discovery of the first extrasolar "hot jupiter'' planet during the same year. The development of this field has been described in numerous papers and proceedings, for example recently in the proceedings volume "From Giant Planets to Cool Stars'' (Griffith & Marley 2000). The variety, and the vast abundance of newly discovered stars with significantly lower effective temperature than the cool M dwarfs, calls for reevaluation of the existing opacity data. In the case of CIA intensities due to hydrogen pairs, only those at temperatures starting at 1000 K were available (Borysow & Jørgensen 2000; Borysow et al. 2001) for stellar atmospheric modelling. Numerous inquiries have been made to the author about the availability of data for temperatures from 500 K to 1000 K. Stars of special interest are those classified as L- and T-dwarfs, as well as brown dwarfs, cool M dwarfs, cool white dwarfs, and hot extrasolar planets. The new classification (L and T types) has been introduced by Kirkpatrick and was also presented at the abovementioned conference (Kirkpatrick 2000). A comprehensive review on the subject of the model atmospheres of the very cool low mass stars and brown dwarfs by Allard et al. (1997) was published recently.

Figure 1: CIA spectra of hydrogen pairs computed at temperatures 150 K (lowest solid line), 200 K, 250 K, 300 K and 350 K (uppermost line).

   
3.1 Approach

The data we start with, are the existing low temperature ones. All of them assume the initial state of the H₂ molecule to be the ground (v= 0) state. Fortunately, at the highest temperature considered here, T= 1000 K, v= 0 is populated with probability 0.997. It is therefore acceptable to ignore higher initial vibrational states even at temperatures as high as 1000 K.

We decided to create, as accurately as possible, the missing intensities, based on the original dipole functions, where possible. The data were computed using the H₂-H₂ intermolecular potential suitable for "low-temperature'' computations (Schaefer & Köhler 1989, from now on called SK). After that, a smooth rescaling procedure was designed, in order to join the data, starting from 300 K, with those based on the "high temperature'' intermolecular potential (Ross et al. 1983, from now on called RRY).

It needs to be clarified what we mean by the "low-'' and the "high-'' temperature potentials. The SK potential is an ab initio isotropic potential, which has subsequently been adjusted to fit a variety of low temperature experiments. It is therefore an extremely dependable tool for computing low temperature (roughly below T= 300 K) phenomena like CIA, in molecular hydrogen. It is, however, highly uncertain at the very short range of intermolecular interactions, corresponding to the high temperatures (high energy of collisions).

On the other hand, the RRY potential is an empirical isotropic potential. It is based on shock wave experiments with collision energies corresponding to the temperature of 7000 K. It will not be as accurate as the SK potential is at low temperatures, but it has an advantage of being both effective and isotropic, and at the same time it is adequate at very short intermolecular distances.

One needs to be aware of the fact that the real H₂-H₂interaction potential is not exactly isotropic, even though such an assumption seems to work well at low temperatures. At increasingly smaller distances the potential becomes more and more anisotropic. Until presently, even if an adequate potential existed that could be used at high temperatures, the CIA computations accounting for anisotropy of the interactions were limited to very low temperatures. At increasing temperatures the RRY potential effectively accounts for higher vibrational states of the H₂ molecule. This fact has an impact on the potential shape as well. The effective potential has thus many advantages when used in the region where there exist no reliable computations of the intermolecular interactions and when, even if such existed, computations accounting for the interaction anisotropy would not yet be possible.

It can be safely assumed that the main contributor to the CIA intensities, for each band, is the quadrupole induced term. For this reason, the ratios between the integrated intensities (G0) of the quadrupolar term for Ross et al. (1983) and for Schaefer & Köhler (1989) were computed first. These are listed in Table 1 in the second column. Next, an arbitrary function was used to rescale the intensity of the data at various temperatures. We multiplied the results obtained with the SK potential, by the following function of temperature T(K):

  $$\begin{eqnarray*} && f(T)= 1/A \times \{ a * T + b \}, \\ && {\rm with } ~~~ a ... ...-3/7, \\ && {\rm where} ~~~ A = G0({\rm SK})/G0({\rm RRY}). \\ \end{eqnarray*}$$

We assumed that the rescaling will take place beginning at temperature T= 300 K and ending at T= 1000 K. The results for f(T) are shown in the third column of Table 1. As can be seen, we take the data exact at the lowest temperature considered (300 K) and gradually rescale them to match the data computed with the RRY potential at 1000 K.

 

 

Table 1: Ratios 1/A of the integrated intensities G0 of the quadrupolar terms, at selected temperatures, based on the RRY and SK intermolecular potentials.

T(K)	G0(RRY)/G0(SK)	f(T)
300.	1.161	1.000
350.	1.166	1.012
400.	1.170	1.024
500.	1.180	1.051
600.	1.189	1.081
700.	1.197	1.113
800.	1.206	1.147
900.	1.215	1.184
1000.	1.223	1.223