A&A 408, 1197-1203 (2003)
DOI: 10.1051/0004-6361:20030969
A. Grosjean^{1} - M.-L. Dubernet^{2} - C. Ceccarelli^{3}
1 - Laboratoire d'Astrophysique, UMR CNRS 6091,
Observatoire de Besançon, Université de Franche-Comté,
41 bis avenue de l'Observatoire, BP 1615, 25010 Besançon Cedex, France
2 -
LERMA FRE 2460, Observatoire de Paris, Section de Meudon, 5 place Janssen,
92195 Meudon Cedex, France
3 -
Laboratoire d'Astrophysique de l'Observatoire de Grenoble,
414 rue de la Piscine, BP 53, 38041 Grenoble Cedex 9, France
Received 13 March 2003 / Accepted 12 June 2003
Abstract
We present the results of
close coupling calculations of pure rotational excitation rate
coefficients of H_{2}O by ortho-H_{2}, performed between 5 K and 20 K.
At 20 K there is a good agreement with
those results obtained previously by Phillips et al. (1996).
Fits of our de-excitation collisional rate coefficients are provided.
These new rate coefficients are used in the modelling of cold interstellar
clouds to determine the influence of the precision of
the rate coefficients on predicted line intensities.
Key words: molecular data - molecular processes - ISM: molecules
This paper is the second part of a thorough theoretical study of the collisional excitation rates of H_{2}O by H_{2}. The study is motivated both by the intepretation of the data recently acquired by the Infrared Space Observatory (Kessler et al. 1996) and the Submillimeter Wave Astronomy Satellite (hereinafter SWAS; Melnick et al. 2000), and by the future ESA mission, the Herschel Space Observatory (hereinafter HSO). Specifically, the Heterodyne Instrument for the Far-Infrared on board HSO will allow observations of several water lines with unprecedented sensitivity in different environments, from the interstellar medium to stellar or planetary atmospheres.
In Paper I Dubernet & Grosjean (2002) calculated the pure rotational (de)-excitation rate coefficients of ortho/para H_{2}O by para H_{2} at very low temperatures ( K). The other available excitation rate coefficients of H_{2}O by H_{2} are those calculated by Phillips et al. (1996,1995), using the close-coupling and the coupled states methods with a potential energy surface (PES) calculated by Phillips et al. (1994). Those authors provided data between 20 K and 140 K for a number of ortho/para H_{2}O - ortho/para H_{2} pure rotational transitions. The results presented in Paper I (Dubernet & Grosjean 2002) were significantly different from those presented earlier by Phillips et al. (1996) at 20 K. In particular, the excitation rate coefficient for the 1_{01}-1_{10} transition of water was more than 50% larger than the Phillips et al. (1996) result. This was a consequence of an inadequately fine energy grid used by Phillips et al. (1996) in integrating the cross-sections over a Maxwellian distribution of kinetic energies.
An analysis of the results of Paper I (Dubernet & Grosjean 2002) shows that reasonable extrapolation of the high temperature data gives up to 50% error at 5 K for the 1_{01}-1_{10} transition of water. By reasonable extrapolation, we mean using a power law fit for the de-excitation rate coefficients of the 1_{01}-1_{10} transition, with our calculated points at 16 K and 20 K, and obtaining the excitation rate coefficients by detailed balance.This type of extrapolation may be justified because of the low dependence of the de-excitation rate coefficients on temperature.
Both the differences found with earlier results of Phillips et al. (1996) and the inadequacy of rate coefficient extrapolation in the particular case of Paper I (Dubernet & Grosjean 2002) motivated the present determination of pure rotational excitation rate coefficients of ortho/para H_{2}O by ortho H_{2} at very low temperatures ( 20 K) where no data have yet been calculated. We use the same PES (Phillips et al. 1994) and the same description of the water molecule as Phillips et al. (1995), which allows us to compare the collision rate coefficients we obtain at 20 K with those of Phillips et al. (1996,1995). However we use improved dynamical calculations compared to the earlier calculations of Phillips et al. (1996). This implies using a larger rotational basis set, a finer energy grid and close coupling calculations on the whole energy range. The description of the quantum calculation and the analysis of the rotational excitation rate coefficients are given in Sect. 2.
Table 1: Propagation parameters of MOLSCAT used in our calculations (see the MOLSCAT documentation, Hutson & Green 1994, for the meaning of the parameters).
Section 3 investigates the use of rate coefficients in a specific LVG model of cold interstellar clouds, in order to provide a simple test of the influence of the precision of rate coefficients on the predicted water line intensities.
The state-to-state rotational inelastic rate coefficients are the Boltzmann
thermal averages of the inelastic cross sections:
We carefully spanned the energy ranges above the inelastic channels and
added more points in the presence of resonance structures. We found
that the resonances do not play an important role once the
cross-sections are averaged over energy, contrary to what was found
for the excitation of water by para-H_{2} (Dubernet & Grosjean 2002).
This is due to the large
magnitude of the excitation cross-sections of water by ortho-H_{2},
compared to the magnitude of the resonance structures, as is shown in
Fig. 1. The energy grid used is therefore sparser than for para-H_{2}, but
still much finer than the one used by Phillips et al. (1996).
Figure 1: Inelastic cross-section (in Å^{2}) as a function of kinetic energy (in cm^{-1}) of the water transition (with j_{2}=1). | |
Open with DEXTER |
The effective rotational inelastic rate coefficients are given by the sum of the inelastic rate coefficients (Eq. (1)) over the final j_{2}' states for a given initial j_{2}:
Table 2: Effective rate coefficients of Eq. (2) (in cm^{3} s^{-1}) of ortho-H_{2}O with ortho-H_{2} (j_{2} = 1). The levels are labelled with j_{K-1K1}. The column "Points'' gives the number of energy points used in the Boltzmann average. The last column gives the data of Phillips et al. (1996). The levels are organized according to increasing energy as in the paper of Phillips et al. (1996).
For collisions of H_{2}O with ortho-H_{2}, Phillips et al. (1996) concluded that a B(j_{1}=5, j_{2}=1) basis set is sufficient for obtaining an accuracy of better than 10% in the temperature range from 20 K to 140 K; it is not necessary to use a B(j_{1}=5, j_{2}=3) basis set (the values of j_{1} and j_{2} indicated are the maximum rotational quantum numbers of H_{2}O and H_{2} used included in the basis). We performed close coupling calculations with both B(5, 1) and B(5, 3) basis sets in the energy range from the opening of the lowest inelastic channel to a total energy of 750 cm^{-1}. The corresponding effective rate coefficients have absolute relative differences ranging from 2% to 30%, the B(5, 3) rate coefficients being generally but not always larger than B(5, 1) rates. The biggest differences correspond to the smallest rate coefficients. Tables 2 and 3 give the effective rotational inelastic rate coefficients at several temperature calculated with a B(5, 3) basis set, for ortho and para H_{2}O respectively, together with the values obtained by Phillips et al. (1995) at 20 K with a B(5, 1) basis set. The discrepancies between our rate coefficients and the Phillips et al. (1996) rate coefficients at 20 K range between 2% and 25%. Phillips et al. (1996) used a smaller basis set, a sparser energy grid and coupled states calculations above 450 cm^{-1}. The various small errors introduced in their calculations sometimes compensate for one another and sometimes add to one another. Therefore the effect on the data is neither homogeneous among different transitions nor among different temperature, and is difficult to predict. Nevertheless it should be noted that the largest differences correspond to the smallest rates. We believe that most of our effective rate coefficients have a maximum error of 3% for all given transitions and temperatures, and for the given potential energy surface. The smallest effective rate coefficients might have a higher error, but these rates are unimportant.
Table 3: Effective rate coefficients of Eq. (2) (in cm^{3} s^{-1}) of para-H_{2}O with ortho-H_{2} (j_{2}=1). The levels are labelled with j_{K-1K1}. The column "Points'' gives the number of energy points used in the Boltzmann average. The last column gives the data of Phillips et al. (1996). The levels are organized according to increasing energy as in the paper of Phillips et al. (1996).
Table 4: Coefficients (n = 0 to 4) of the polynomial fit (Eq. (3)) to the effective de-excitation rate coefficients of ortho-water in Table 2. The effective excitation rate coefficients can be obtained by detailed balance. The first column gives the final state and the polynomial order n, the following columns give the fitting coefficients for various initial states. The levels are labelled with j_{K-1K1}. We emphasize that these fits are only valid in the temperature range from 5 K to 20 K.
Table 5: Coefficients (n = 0 to 4) of the polynomial fit (Eq. (3)) to the effective de-excitation rate coefficients of para-water in Table 3. The effective excitation rate coefficients can be obtained by detailed balance. The first column gives the final state and the polynomial order n, the following columns give the fitting coefficients for various initial states. The levels are labelled with j_{K-1K1}.We emphasize that these fits are only valid in the temperature range from 5 K to 20 K.
For astrophysical use, our de-excitation effective excitation
rate coefficients
may be fitted by
the analytical form used by Balakrishnan et al. (1999) and used in
Paper I (Dubernet & Grosjean 2002):
A fourth-order polynomial (n =4) is required to cover the whole range of temperature and to provide a fitting error better than 0.02% on all rate coefficients. We emphasize that these fits are only valid in the temperature range from 5 K to 20 K.
It is interesting to note that the extrapolation method described in the introduction works very well for the present results with ortho dihydrogen. We recall that it does not work well for results with para dihydrogen. It is therefore difficult to judge the precision of an extrapolation procedure before the calculations are actually performed. It raises the question of the required precision of rate coefficients in the modelling of astrophysical media.
Motivated by this problem we checked the influence of the precision of the rate coefficients on the predicted line intensities of the ground state transition observed by SWAS (Melnick et al. 2000) at 557 GHz. The predicted intensities were obtained by means of a LVG model to solve self-consistently the non-LTE population equilibrium, and whose details are reported in Ceccarelli et al. (2002,1998). H_{2} densities of cm^{-3} and cm^{-3} were used. The results were not very sensitive to the change in density.
We compared the predicted line intensities and obtained respectively with the fully converged rate coefficients corresponding to the large basis set B(5, 3), and with the less accurate rate coefficients calculated with a B(5, 1) basis set. We found a maximum difference of 2% in the predicted line intensities, which corresponds to the difference in the corresponding rate coefficients for the transition. We performed extra tests using arbitrary rate coefficients with higher differences and found the same linear effect. More work should be carried out on the subject, but it is not the purpose of the present paper.
This paper is a continuation of the work of Dubernet & Grosjean (2002) and of the work of Phillips and co-workers (Phillips et al. 1996; Green et al. 1993; Phillips et al. 1994,1995) on the excitation rates of H_{2}O by H_{2}. Phillips et al. (1996) gave excitation rates in the range from 20 K to 140 K; we have calculated these rates in the range from 5 K to 20 K. Unlike Paper I (Dubernet & Grosjean 2002), our improved dynamical calculations confirm the results of Phillips et al. (1996) at 20 K. We have also provided fits of the de-excitation rates that are valid in the range from 5 K to 20 K. The results of the present paper, the excitation rates of other transitions and the associated cross sections will be made available on our web site^{}. We are currently investigating the temperature range up to 1500 K.
We gave a first estimation of the effect of the rate coefficient precision on the water ground transition at 557 GHz observed by SWAS. We found a linear correlation between differences in rate coefficients and differences in predicted line intensities. We hope that this simple test will motivate more general studies on the influence of the rate coefficient precision on predicted quantities extracted from modelling, in particular in the case of the excitation of H_{2}O. The outcome of such studies would be a very useful guide for calculations of rate coefficients for astrophysical applications.
Acknowledgements
Most scattering calculations were performed at the CINES under project aob2271. This work is supported by the PCMI National Program.