- Collisional excitation of water in warm astrophysical media
- 1 Introduction
- 2 Collisional rate coefficients
- 3 Astrophysical implications
- 4 Conclusions
- References

A&A 492, 257-264 (2008)

DOI: 10.1051/0004-6361:200810717

**A. Faure ^{1} -
E. Josselin^{2}**

1 - Laboratoire d'Astrophysique de Grenoble (LAOG),
Université Joseph-Fourier, UMR 5571 CNRS, BP 53, 38041
Grenoble Cedex 09, France

2 -
Groupe de Recherche en Astronomie et Astrophysique du
Languedoc (GRAAL), Université Montpellier II, UMR 5024 CNRS,
34095 Montpellier Cedex 05, France

Received 30 July 2008 / Accepted 25 September 2008

**Abstract**
*Context.* The interpretation of water line emission from infrared and submillimetre observations requires a detailed knowledge of collisional rate coefficients over a wide range of levels and temperatures.

*Aims.* We attempt to determine rotational and rovibrational rate coefficients for H_{2}O colliding with both H_{2} and electrons in warm, molecular gas.

*Methods.* Pure rotational rates are derived by extrapolating published data using a new method partly based on the information (phase space) theory of Levine and co-workers. Ro-vibrational rates are obtained using vibrational relaxation data available in the literature and by assuming a complete decoupling of rotation and vibration.

*Results.* Rate coefficients were obtained for the lowest 824 ro-vibrational levels of H_{2}O in the temperature range 200-5000 K. Our data is expected to be accurate to within a factor of 5 for the highest rates (10^{-11} cm^{3} s^{-1}). Smaller rates, including the ro-vibrational ones, should be generally accurate to within an order of magnitude. As a first application of this data, we show that vibrationally excited water emission observed in evolved stars is expected to be at least partly excited by means of collisions.

**Key words: **molecular data - molecular
processes - ISM: molecules

Water molecules are found in a large variety of astronomical environments, ranging from interstellar molecular clouds to stellar photospheres, circumstellar envelopes, and comets. In these media, they represent a major reservoir of oxygen. Furthermore, water emission may dominate the cooling of the warm part (a few 100 K) of these regions (Neufeld & Kaufman 1993). The water molecule is also one of the main sources of radio (from the submillimetre to the centimetre domain) maser emission, which has the potential to enable detailed studies of physical conditions, source dynamics, and magnetic fields (Humphreys 2007). A quantitative modelling of the excitation and line formation processes is therefore essential to understand the nature of these media. Besides the approximations introduced into radiative transfer calculations, such analyses are hampered by the lack of reliable collisional rate coefficients, especially for high-lying rotational and vibrational levels.

Despite progress in both laboratory and theoretical studies of H_{2}O
collisional excitation (see Barnes et al. 2004; Faure et al. 2007, and references therein), there is still a
crucial lack of state-to-state rate coefficients for water in
vibrationally excited states. Current rotational data (Green et al.
1993 for H_{2}O-He; Faure et al. 2004 for
H_{2}O-electron; Dubernet et al. 2006; and Faure et al.
2007 for H_{2}O-H_{2}) is also restricted to levels of
energy below the first excited vibrational state, which lies at 1594.7 cm^{-1} above the ground state. The main difficulty in
computing rate coefficients for higher levels relates to the fact that
quantum scattering calculations become computationally prohibitive as
soon as the number of coupled channels to be included in the (time
independent) Schrödinger equation exceeds a few thousands. This
imposes severe limitations especially when the projectile has internal
degrees of freedom such as H_{2}. The quantum wave packet approach
(see e.g. Otto et al. 2008) could provide an efficient
alternative to fully coupled channel calculations but to our knowledge
it has yet to be developed for polyatomic targets. There is currently
therefore no quantum calculations for H_{2}O-H_{2} that include
vibrational motion and/or rotationally ``hot'' molecules (
K). At the quasi-classical level, hot molecules can be handled
but other limitations occur, such as the inability of classical
mechanics to estimate low probability processes (see Faure et al.
2007, and references therein). As a result, to extend the
collisional data of water to kinetic temperatures at which it becomes
vibrationally excited ( K), a new approach is necessary
and needs to be developed. This is the objective of the present
article.

The method of computation of collisional rate coefficients, and its
application to H_{2}O-H_{2} and H_{2}O-electron rotational and
rovibrational excitation is exposed in Sect. 2. First astrophysical
implications, including cooling rates and critical densities, are
discussed in Sect. 3.

Faure et al. (2007) computed quasi-classical rate
coefficients for rotational de-excitation in the lowest 45 para and 45 ortho levels of H_{2}O colliding with both para- and ortho-H_{2} in
the temperature range 20-2000 K. This corresponds to a total of 990 para and 990 ortho H_{2}O downward rates for each temperature. We note
that in these calculations para- and ortho-H_{2} were assumed to be
(separately) thermalized at the kinetic temperature. The objective of
this section is to estimate the rotational rates for thousands of
transitions involving states above the lowest 45.

Our analysis is inspired by the *information theory* approach
developed by Levine and co-workers (see the review by Levine
1978). This approach was originally introduced as a
procedure to compact and correlate the vast amounts of data, both
theoretical and experimental, on molecular collisions. For the
rotational excitation of a linear rotor by a structureless particle,
the following expression for the rate coefficient was suggested by
Procaccia & Levine (1976):

In the above expression,

Following Eq. (1), the expression below can be assumed for
the rotational rates of water:

where specify the initial and final H

Figure 1:
Downward rotational rate coefficients of Faure et al.
(2007) (divided by the degeneracy in the final
rotational state) for ortho-H_{2}O colliding with para-H_{2} as a
function of the energy gap. The kinetic temperature is 2000 K. The
dashed line corresponds to a linear least squares fit of the
logarithm of the data. |

The scatter of data points in Fig. 1 reflects a complicated
competition between different dynamical effects, including the energy
gap but also specific dynamical constraints. In particular, although
no rigorous selection rules exist for inelastic collisions, * propensity* rules are generally observed. For water-H_{2}collisions, the ``preferred'' rotational transitions can be defined to
be those for which (Faure et al. 2007):

The above rules are of course not accounted for by the information theory approach.

Our analysis of the data of Faure et al. (2007) demonstrated
that for a given set of
,
,
,
rates are approximatively constant, that is, do not strongly depend on
*J*, *K*_{a} and *K*_{c}. The sum of downward rates out of any upper state
and restricted to transitions that do not obey Eq. (3):

is also approximatively constant (in Eq. (4), the

computed using the corresponding rates within the lowest 990 transitions. For the latter, we compute the average of for the lowest 45 levels, , and we define the low propensity downward rates to be:

where

(7) |

It should be noted that Eq. (6) has similarities with the extrapolation formula used by Neufeld & Melnick (1987) and Neufeld & Kaufman (1993).

In Fig. 2, we compare rates calculated from the combination
of Eqs. (5) and (6) with the original data of Faure et al. (2007). The high propensity rates are easy to identify
because they correspond to ``peaks'' in the plots (see in particular
the lower panel). It can be noticed that Eq. (5) does
successfully reproduce the largest peaks. Equation (6) is also
found to reproduce the general trend of the low propensity rates,
although the lowest rates (those below 10^{-13} cm^{3} s^{-1})
are generally overestimated. These low rates are expected to play only
a minor role in the radiative transfer equations. As a result, the
present method based on the combination of Eqs. (5) and (6) is found to provide a good, quantitative estimate of the
highest rotational rates with a typical accuracy of factors of 2-3.
Since the original data of Faure et al. (2007) are generally
in error by factors of 1-3 (Dubernet et al., private communication),
the present data are believed to be accurate within a factor of 5 for the highest rates (those above 10^{-11} cm^{3} s^{-1}).
For lower rates, our extrapolation method is inevitably less accurate
and some individual rates may be in error by (at most) one order of
magnitude (except for the lowest rates, for which overestimates by 2 orders of magnitude may be found). We emphasize, however, that only
comparisons with experimental data will enable the accuracy of the
original data of Faure et al. (2007) and, therefore, of the
present rates to be assessed reliably. In this context, it should be
noted that state-to-state inelastic scattering experiments on
H_{2}O-H_{2} are being performed in Nijmegen (ter Meulen, private
communication, see e.g. Moise et al. 2007). Detailed
comparisons between theory and experiment are in progress and will be
published elsewhere.

Figure 2:
Downward rotational rate coefficients for
ortho-H_{2}O-para-H_{2} at 2000 K as a function of the
transition number (1-990) corresponding to the lowest 45 levels.
The original data of Faure et al. (2007) (red stars and
dashed line) is compared with our estimate based on the
combination of Eqs. (5) and (6) (blue solid line).
The lower panel zooms in on the range 490-600, corresponding to
transitions involving initial levels 6_{61}, 7_{52}, and
9_{18}. See text for details. |

We also computed *downward summed* rates (rates from a level
summed over all possible downward transitions) for ortho-H_{2}O
colliding with para-H_{2} using both the QCT data of Faure et al.
(2007) and unpublished quantum data of Dubernet et al. (in
preparation). Relative differences between the QCT and the quantum
summed rates were found to be lower than 50% for the 10 lowest
ortho-H_{2}O levels^{} in the
temperature range 200-1500 K, indicating a far higher accuracy for
downward summed rates than for individual state-to-state rates. It
should be noted that these summed rates are used to estimate critical
densities (see Sect. 3.2).

We conclude that the information theory approach, supplemented by both
low-lying level data and the knowledge of collisional propensity
rules, provides a good framework for extrapolating H_{2}O-H_{2}rotational rates to higher-lying levels. In the present study, the
above procedure was applied to all para and ortho rotational
transitions involving levels above the lowest 45 and below a threshold
energy of 5000 cm^{-1} with respect to the ground rotational state
(corresponding to the 165th para and 164th ortho level). For
temperatures above 2000 K, a simple *T*^{1/2} temperature dependence
was assumed for the rates, as suggested by the high temperature
results of Faure et al. (2007). Finally, since we are
interested in high temperatures ( K), an H_{2} ortho-to-para
ratio (OPR) of 3:1 was assumed for all temperatures. We note that this
choice may be inadequate in some cases, for example in molecular
outflows where the OPR has been found to be well below its value at
local thermodynamical equilibrium (LTE) (Lefloch et al.
2003; Neufeld et al. 2006). However, the
dependence of the H_{2}O rates on the H_{2} OPR has been shown to be
modest above 200 K (Faure et al. 2007).

Rate coefficients for electron-impact excitation of water were
computed by Faure et al. (2004) for rotational de-excitation
among the lowest 14 para and 14 ortho rotational levels of H_{2}O
(corresponding to )
in the temperature range 100-8000 K.
This corresponds to a total of 91 para and 91 ortho H_{2}O downward
rates for each temperature. These calculations were performed by
combining the molecular *R*-matrix method with the
adiabatic-nuclei-rotation (ANR) approximation. An alternative theory,
for which state-to-state cross sections are derived from experimental
data, was also reported (Curík et al. 2006).
Both approaches were found to be in good agreement, confirming the
accuracy of the *R*-matrix calculations. Faure et al.
(2004) also found that H_{2}O-electron rotational rates are
dominated by dipolar transitions:

They also demonstrated that the accuracy of the Born approximation, which considers only the electron-dipole interaction, increases with the collision energy (see Fig. 2 of Faure et al. 2004). Since we are interested in high temperatures (

Figure 3:
Downward rate coefficients for electron-impact rotational
excitation of ortho-H_{2}O at 2000 K as a function of the
transition number (1-91) corresponding to the lowest 14 levels.
The original data of Faure et al. (2004) (red stars and
dashed line) is compared with our estimate based on a combination
of the Born approximation and Eq. (6) (blue solid line).
See text for details. |

We conclude that the information theory approach supplemented by both
low-lying level data and the Born approximation provides a good
framework for extrapolating H_{2}O-electron rotational rates to
higher-lying levels. In the present study, the above procedure was
applied to all para and ortho-H_{2}O levels above the lowest 14 and
below a threshold energy of 5000 cm^{-1}, as for H_{2}O-H_{2}.

Figure 4:
Rate coefficients for the relaxation of the first excited
bending mode of water by H_{2} as a function of temperature. QCT
results are plotted as filled squares (with error bars
corresponding to two Monte Carlo standard deviations). The solid
line indicates the interpolation of the QCT data and the
experimental point using the polynomial form of Eq. (14).
This corresponds to our recommended values. The dashed line
denotes a T^{-1/3} fit, as predicted by the Landau-Teller
classical model. Finally, values previously used in the
astrophysical literature are indicated by a dotted line (see text
for details). |

Faure et al. (2005a) computed quasi-classical rate
coefficients for the relaxation of the first excited bending mode of
H_{2}O (lying at 1594.7 cm^{-1} above the ground state) by H_{2}.
Their high temperature (
K) relaxation rate coefficients
were found to be large (>10^{-11} cm^{3} s^{-1}) and compatible with
the single experimental value at 295 K (Zittel & Masturzo
1991), as illustrated in Fig. 4.
At lower temperatures (*T*<1500 K), however, the quasi-classical
trajectory (QCT) approach was shown to be questionable, due to small
number statistics. This problem is, of course, even more severe for
state-to-state ro-vibrational transitions. Quantum mechanics is
computationally prohibitive for such a system, particularly because
the standard rotational infinite-order-sudden (IOS) approximation is
unreliable for molecules with high rotational constants (Faure et al.
2005b, and references therein). As a result, the calculation
of state-to-state rate coefficients for the vibrational (de)excitation
of H_{2}O by H_{2} appears highly challenging from both a classical
and quantum point of view. We develop below an approximate method to
estimate rates for state-to-state transitions connecting two
vibrational states of water, by considering the lowest 5 vibrational
levels denoted by
*v*=(*v*1, *v*2, *v*3)^{}. The first rotational levels
of the four lowest excited vibrational levels are at 1594.7 cm^{-1}(010), 3151.6 cm^{-1} (020), 3657.1 cm^{-1} (100), and
3755.9 cm^{-1} (001) above the ground state (000) (Tennyson et al.
2001).

It was generally assumed in the astronomical literature (see e.g.
Chandra & Karma 2001) that state-to-state ro-vibrational
rates are
to be proportional to the rates for the
corresponding rotational transitions in the ground vibrational state:

where specify respectively the initial and final H

In their QCT study, Faure et al. (2005b) demonstrated that
the final distribution of the water rotational levels after a
vibrational relaxation is similar to the final distribution after a
pure rotational relaxation: it peaks at the initial *J* with an
exponential decay as a function of the rotational energy gap (see
Fig. 3 of Faure et al. 2005b). Exception occurs for low
initial *J*, in which case the rotational distribution is broad and
peaks at higher *J*. We note that similar results were observed for
simpler diatomic systems (see for example the quantum VCC-IOS
calculations of Lique & Spielfiedel 2007 for CS-He).
The QCT results of Faure et al. (2005b) therefore suggest
that Eq. (9), which corresponds to decoupling rotation and
vibration, is *qualitatively* correct. It should be noted,
however, that Eq. (9) completely neglects quasiresonant,
vibration-rotation, energy-transfer, which is expected to occur at low
energy or for high initial *J* (see Faure et al. 2005b, and
references therein). Furthermore, vibrational transitions for which
the initial and final rotational quantum numbers are identical require
special consideration: in this case, Eq. (9) does not apply
because rotationally elastic rates are typically 1-2 orders of
magnitude higher than rotationally inelastic rates. For vibrationally
inelastic transitions, this propensity is still marked but it is
significantly reduced (Faure et al. 2005b; Lique &
Spielfiedel 2007). Our recommendation for these
transitions is therefore to employ the highest rate from the
considered initial state.

To define *P*_{vv'}, our guiding principle is that of detailed
balance, which states that upward and downward ro-vibrational rates
must obey the relation:

By expressing the rotationally summed and averaged vibrational relaxation rate to be:

(11) |

and assuming

where is the vibrational energy gap, we can show that the following definition of

ensures that detailed balance is automatically fulfiled by Eq. (9). We note that

In Fig. 4, experimental and QCT relaxation rate coefficients
for the lowest vibrationally excited mode of water, (010), are
plotted as a function of temperature. We added to this figure a
previous estimate based on an empirical equation derived from
shock-tube measurements of diatomic systems (see e.g.,
González-Alfonso et al. 2002, and references therein).
It can be noticed that this empirical prediction underestimates the
experimental rate (at 295 K) by more than two orders of magnitude and
the QCT rates (at higher temperature) by typically one order of
magnitude. The vibrational (de)excitation of water by H_{2} is
therefore far more rapid than anticipated.

QCT and experimental data were interpolated using the polynomial form:

which was found to provide an accurate representation of the rate in the range 200-5000 K, as shown in Fig. 4. The fitting coefficients are the following:

For higher vibrational modes of water, the only available data is
provided by the measurements of Zittel & Masturzo (1991)
at 295 K. In addition to the (010) bending level, these authors
measured rate coefficients for vibrational relaxation of the (100) and
(001) stretching level reservoir and the (020) bending overtone. Since
the relaxation between the nearly resonant (100) and (001) stretching
levels is far more rapid than relaxation from the stretching levels,
(100) and (001) were experimentally found to reach equilibrium
immediately and to relax as a single reservoir. Rates for relaxation
of the stretching level reservoir are therefore the Boltzmann-weighted
averages of the rates for the individual (100) and (001) levels.
Zittel & Masturzo (1991) measured *total* relaxation
rate constants and only some information was deduced about the
individual, state-to-state vibrational rates. As a result, the
following assumptions were made in the present work:

and

(16) |

where

From Tables 1, 2 of Zittel & Masturzo (1991), vibrational
relaxation rates at 295 K among the lowest 5 vibrational levels were
determined and are summarized in Table 1. We note that the
parameter
(see Eqs. (1)-(3)
of Zittel & Masturzo 1991) is taken to be 0.15. We also
assumed that the rate for relaxation from the *v*_{1, 3} reservoir to
the ground state equals the rate for relaxation to the (010) level
(see discussion in Zittel & Masturzo 1991). Finally,
vibrational excitation of H_{2} was neglected, in agreement with
experimental results (see Table 2 of Zittel & Masturzo
1991).

Initial level | Final level | k (cm^{3} s^{-1}) |

010 | 000 | 1.30(-12) |

020 | 010 | 2.30(-12) |

020 | 000 | 0.60(-12) |

100 | 020 | 0.64(-12) |

100 | 010 | 0.08(-12) |

100 | 000 | 0.08(-12) |

001 | 020 | 0.64(-12) |

001 | 010 | 0.08(-12) |

001 | 000 | 0.08(-12) |

The rate constants presented in Table 1 were extrapolated
from 200 to 5000 K, assuming the same temperature dependence as the
rate for the
transition (see Eq. (14)).
This hypothesis should be reasonable, especially at high temperatures
( K) where all rates are expected to follow a
Landau-Teller law. The coefficients
*P*_{vv'}(*T*), represented by
Eq. (13), were then computed for each vibrational transition
using the above defined rates and the experimental level energies
taken from Tennyson et al. (2001). The coefficients
*P*_{vv'}(*T*) were found to be similar for ortho and para-H_{2}O,
typically to within 5%, and to have values between 10^{-4} and 10^{-1}, as illustrated in Fig. 5. The coefficients
*P*_{vv'}(*T*) were found to vary significantly (over an order of
magnitude) in the temperature range 200-5000 K, with a minimum at 400 K. This minimum reflects the competition in Eq. (13)
between the temperature dependences of
and the
sums in the numerator and denominator.

Figure 5:
Temperature dependence of the
P_{vv'}(T) coefficients for
different vibrational transitions, as defined in Eq. (13),
for (ortho-)H_{2}O-H_{2} collisions. Note that only five over the
nine transitions are plotted for clarity in the figure. |

Finally, we note that Eq. (13) can be applied to both
excitation and de-excitation transitions. We recommend using the
equation for de-excitation (relaxation) transitions for which rates
have been measured. However, we computed
*P*_{vv'}(*T*) as a test for
excitation transitions. The resulting state-to-state rate constants
were found to satisfy detailed balance to within typically 20%. This
indicates that Eq. (12) does not introduce major
inacurracies.

In the present study, the above procedure was applied to all para and
ortho ro-vibrational de-excitation transitions involving levels below
a threshold energy of 5000 cm^{-1} with respect to the ground state.
This corresponds to 108 para and 107 ortho rotational levels within
(010), 56 para and 55 ortho levels within (020), 44 para and 44 ortho
levels within (100) and, finally, 41 para and 40 ortho levels within
(020). This results in a total of 824 levels, including those in the
ground vibrational state (000). The complete set of de-excitation
rates among all levels between 200 and 5000 K are provided as online
material (Tables A.1, A.2). Excitation rates can be obtained from the
detailed balance relation, Eq. (10). Tables A.1, A.2 will be
also made available in the BASECOL database^{} and at the CDS^{}.

Figure 6:
Rate coefficients for the relaxation of the three normal
modes of water by electron-impact, calculated from the cross
sections of Nishimura & Gianturco (2004). |

Nishimura & Gianturco (2004) computed quantum
vibrationally inelastic cross-sections for the water molecule
colliding with low-energy electrons. Results were obtained to low
energies for the three normal modes of H_{2}O (010, 100, and 001) and
good agreement with existing experiments was found. From their cross
sections (tabulated in Table 3 of their paper), we computed thermal
relaxation rate coefficients for the temperature range 200-5000 K.
These rates are plotted in Fig. 6.
In contrast to H_{2}O-H_{2} relaxation rates, we note that
electron-impact relaxation rates decrease with increasing temperature.
This reflects the energy dependence of the excitation cross-sections,
which show strong peaks close to thresholds, especially the *v*_{1} and *v*_{2} modes (see Nishimura & Gianturco 2004). As a
result, excitation rates decline slowly to zero at low temperature.
This behaviour is generally attributed to the dominant long-range
dipole potential terms at low impact energies. Furthermore, the fact
that the *v*_{1} and *v*_{2} modes have significantly higher rates than
*v*_{3} is due to much higher polarizability and quadrupole gradients
(Nishimura & Gianturco 2004). Finally, as in the case
of rotational excitation, we note that these rates are several orders
of magnitude higher than those for H_{2}O-H_{2} (see discussion on
cooling rates in Sect. 3.1).

The relaxation rates plotted in Fig. 6 were interpolated
using the polynomial form of Eq. (14). The fitting
coefficients are given in Table 2.

Initial level | Final level | a_{0} |
a_{1} |
a_{2} |
a_{3} |

010 | 000 | -21.54 | 57.26 | -254.2 | 368.0 |

100 | 000 | -22.17 | 58.31 | -271.4 | 398.8 |

001 | 000 | -19.57 | -64.22 | 629.4 | -1794 |

To our knowledge, neither theoretical nor experimental data are
available in the literature for other transitions. This is true in
particular for those involving the *v*_{2}=2 mode for which we have
employed the polynomial fit of the
(010)-(000) transition
(Table 2) and applied the corresponding ratios observed by
Zittel & Masturzo (1991) for H_{2}O-H_{2} rates, deduced
from Table 1. Rates for transitions from (100) and (001)
to (010) were also assumed to be identical to those for (000), as for
H_{2}O-H_{2}. In other words, for transitions not included in
Table 2, we assumed the same propensity rules as in the
case of H_{2}O-H_{2} (see Table 1). Equation (15)
was also assumed.

The corresponding coefficients
*P*_{vv'}(*T*) have values between a few
10^{-4} and a few 10^{-2}, as illustrated in Fig. 7. In
contrast to the H_{2}O-H_{2} case, however, the coefficients
*P*_{vv'}(*T*) are found to decrease with increasing temperature and
depend only weakly on the temperature (in the range 200-5000 K). This
of course reflects the temperature dependence of the rates, as plotted
in Fig. 6.

Figure 7:
Temperature dependence of the
P_{vv'}(T) coefficients for
different vibrational transitions, as defined in Eq. (13),
for (ortho-)H_{2}O-electron collisions. Note that only six over
the nine transitions are plotted for clarity in the figure. |

As for H_{2}O-H_{2}, Eq. (13) was applied to all para and
ortho ro-vibrational de-excitation transitions involving levels below
a threshold energy of 5000 cm^{-1} with respect to the ground state.
The complete set of de-excitation rates for all levels between 200 and
5000 K are provided in online material at the CDS (Tables A.3, A.4). Excitation
rates can be obtained from the detailed balance relation in
Eq. (10). Tables A.3, A.4 will also be made available in the
BASECOL database^{} and at the
CDS^{}.

Detailed analysis of the implications of our collisional rates requires radiative transfer calculations, which will be presented in a forthcoming paper (Josselin & Faure, in preparation). We restrict ourselves to basic considerations, by comparing cooling rates and critical densities with those derived from previously assumed hypotheses, such as a constant (independent of temperature) proportionality between rovibrational and rotational rates.

To illustrate the overall difference between the present new rates and
those previously employed in the literature, we calculated the cooling
rate in the optically thin low-density limit:

For both rotational cooling and vibrational cooling,

Figure 8:
Rotational cooling rates in the optically thin low-density
limit, Eq. (17), as a function of temperature. The present
H_{2}O-H_{2} and H_{2}O-electron rates are compared to the
H_{2}O-H_{2} results of Neufeld & Kaufman (1993). |

In Fig. 9, it can be noticed that the new H_{2}O-H_{2}vibrational rates increase the cooling rate by one to three orders of
magnitude with respect to the value of Neufeld & Kaufman
(1993). The vibrational cooling rates are still lower than
the rotational rates except at *T*>3000 K where they are comparable.
We note, however, that Neufeld & Kaufman (1993)
demonstrated that cooling due to vibrational transitions of H_{2}O may
dominate over rotational transitions at high temperature and H_{2}density when opacity and non-LTE effects are taken into account (see
Fig. 6 of their article). We also note that the vibrational cooling
rate is essentially dominated by the lowest vibrational mode (010) at
these temperatures. Finally, the H_{2}O-electron vibrational cooling
rate exceeds the H_{2}O-H_{2} rate by two to four orders of
magnitude.

Figure 9:
Vibrational cooling rates in the optically thin low-density
limit, Eq. (17), as a function of temperature. The present
H_{2}O-H_{2} and H_{2}O-electron rates are compared with the
H_{2}O-H_{2} results of Neufeld & Kaufman (1993).
Present results including only the lowest v_{2}=1 level are also
plotted. |

The concept of a critical density, i.e. that above which collisions dominate over radiative processes in populating energy levels and ensure local thermodynamic equilibrium (LTE) conditions, is poorly defined in the case of a multi-level system. For a 2-level system, we have: , where denotes the Einstein A-coefficient and the collisional rate coefficient, for the transition from the upper (u) to the lower (l) level. In the case of a multi-level, these coefficients have to be summed over all possible downward transitions. For radiative rates, we use the computed line list published by Barber et al. (2006).

As an illustration, we consider the (010)
6_{61}-7_{52} water line,
observed for the first time by Menten et al. (2006)
towards the red supergiant star (RSG) VY CMa. The H_{2} critical
density for the 6_{61} level as a function of temperature is shown
in Fig. 10, with the location of typical conditions for a
RSG atmosphere (MARCS spherical model, with
K,
,
;
Gustafsson et al. 2008),
the circumstellar conditions for which this submillimetre line may be
formed (*T* = 1000 K,
cm^{-3}), and the
H_{2} critical density derived by Menten et al. (2006):
cm^{-3}. These authors assumed
that rate coefficients for transitions from the ground to the
state are identical to those for transitions within the
ground state with identical quantum numbers multiplied by a * constant* factor of 0.02. This is about two orders of magnitude
larger than the critical density we derive from our collisional rates,
mainly because Menten et al. (2006) did not consider
vibrationally elastic (i.e. pure rotational) rates within (010). Our
new value (about
cm^{-3} at 1000 K) still implies
that collisions with H_{2} molecules cannot thermalize the observed
lines, but their contribution cannot be neglected. In other words,
purely radiative excitation now appears questionable for this line
(and the other submillimetre maser lines). Furthermore, the gas
density rises above the value of the critical density in the
transition region between the outer stellar atmosphere and the
circumstellar region where the maser lines are supposed to be formed.
A detailed, multi-zone, radiative-transfer model is therefore
essential to interpret these observations. We finally note that the
maser lines are believed to be collisionally-pumped (Humphreys
2007, and references therein). Adopting our new
collisional rates, we propose to test the efficiency of maser pumping with radiative transfer calculations (Josselin & Faure,
in preparation).

In this paper, we have presented new collisional rate coefficients for
the (de)excitation of the lowest 824 ro-vibrational levels of water by
both H_{2} and electrons in warm and hot molecular gas
(
200<*T*<5000 K). Pure rotational rates have been obtained by
extrapolating the data of Faure et al. (2004) and Faure et al. (2007) using a new method partly based on the
information (phase space) theory of Levine and co-workers.
Ro-vibrational rates have been obtained using vibrational relaxation
data available in the literature and by assuming complete decoupling
between rotation and vibration. Since the published data that we have
employed are believed to be accurate to within a factor of 3, our
extrapolated data should be accurate to within a factor of 5 for
the highest rates (those above 10^{-11} cm^{3} s^{-1}). Lower
rates, including the ro-vibrational ones, should be generally accurate
to within an order of magnitude. Despite these modest accuracies, we
note that summed rates (see Sect. 2.1.1) and cooling rates (see
Sect. 3.1) have a higher precision and the present data should be
adequate for estimating density and temperature conditions from
emission spectra. The prediction of physical conditions required to
produce non-LTE effects should also be reasonable. We have therefore
shown that H_{2}O *v*_{2}=1 lines observed in RSG stars are expected to
be at least partly excited by collisions. Forthcoming radiative
transfer calculations will investigate this point in more detail
(Josselin & Faure, in preparation). Finally, we would like to
emphasize that the present study provides the most extensive
collisional data to date for an asymmetric rotor.

E.J. thanks for their hospitality the Uppsala observatory where part of this work was initiated. This research was supported by the CNRS national program ``Physique et Chimie du Milieu Interstellaire'', by the FP6 Research Training Network ``Molecular Universe'' (contract number MRTN-CT-2004-512302), and by the French National Research Agency (ANR) through program number ANR-06-BLAN-0105.

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