Mon. Not. R. Astron. Soc. 406, 1745–1758 (2010)
doi:10.1111/j.1365-2966.2010.16834.x
Excitation and emission of H2 , CO and H2 O molecules in interstellar
shock waves
D. R. Flower1 and G. Pineau des Forêts2,3
1 Physics
Department, The University, Durham DH1 3LE
d’Astrophysique Spatiale (UMR 8617 du CNRS), Bâtiment 121, Université de Paris Sud, F-91405 Orsay, France
3 LERMA (UMR 8112 du CNRS), Observatoire de Paris, 61 Avenue de l’Observatoire, F-75014 Paris, France
2 Institut
Accepted 2010 April 9. Received 2010 March 16; in original form 2010 January 20
ABSTRACT
The dissipation of kinetic energy that occurs in interstellar shock waves is accompanied by the
emission of radiation. In the case of shocks that are propagating into mainly molecular gas,
the emission occurs principally in lines of the species H2 , H, O, CO and H2 O. The relative
intensities of these emission lines are indicative of the type and speed of the shock wave and
of the physical conditions in the ambient gas. We present the results of computations of the
intensities of these lines, for small grids of models of C- and J-type shock waves, and compare
with the results of previous calculations. Our results should serve to aid the interpretation of
observations made with the Herschel and other satellites.
Key words: molecular processes – shock waves – ISM: jets and outflows – infrared: ISM –
submillimetre: ISM.
1 I N T RO D U C T I O N
The diagnostic potential of molecular emission lines, formed in
interstellar shock waves, has long been recognized. Early work
(Hollenbach & Shull 1977; Kwan 1977; London, McCray & Chu
1977) concentrated on molecular hydrogen, whose rovibrational
emission lines provide valuable information on physical conditions,
such as temperature and density, in shock waves. More recently,
other, less abundant species, such as CO, H2 O and SiO, have been
observed from the ground and with satellite-borne instruments, providing complementary information on both physical and chemical
conditions.
Observations of the water molecule are a major mission objective of the Herschel satellite, whose instruments, HIFI and PACS,
will enable the intensities of rotational transitions of both orthoand para-H2 O to be measured. From these observations, the determination of the physical and chemical conditions in the emitting
regions will pass inevitably via a modelling process. In the case
of shock waves, C- and J-type and, in some cases, non-stationary
structures will have to be considered. Early studies of molecular
emission from interstellar shock waves, by Hollenbach & McKee
(1979) and Draine, Roberge & Dalgarno (1983), were followed
by the more specifically water-orientated studies of Kaufman &
Neufeld (1996a,b); henceforth 1996b is mentioned as KN96b. Subsequently, Harwit et al. (1998) detected H2 O in Orion by means of
the Infrared Space Observatory (ISO) satellite. More recently, Giannini et al. (2006) interpreted ISO observations of H2 , H2 O, CO and
E-mail: [email protected]
C
C 2010 RAS
2010 The Authors. Journal compilation also OH in the Herbig–Haro object HH54 in terms of emission from
a J-type shock with a magnetic precursor. Snell et al. (2005) and
Franklin et al. (2008) have attempted to model Submillimetre Wave
Astronomy Satellite (SWAS) observations of the 557-GHz transition
of ortho-H2 O from the supernova remnant IC443 and from molecular outflows, respectively. Melnick et al. (2008) observed several
lines of H2 O in the outflow from NGC 2071, by means of the infrared spectrometer on the Spitzer satellite. In order to interpret
these and forthcoming (Herschel) observations of molecular line
emission from such sources, there is a need for a systematic study
of molecular emission from shock waves in the interstellar medium
that incorporates the most recent developments in both the models
and the molecular data pertaining to collisional excitation by H2 .
In Section 2, we describe the model that we have used to compute
a grid of C- and J-type shocks, for typical interstellar conditions.
The results of these calculations are summarized in Section 3, where
they are also discussed. We use Spitzer observations of the outflow
from NGC 2071 as a test case in an illustrative model procedure,
described in Section 4. Finally, in Section 5, we make our concluding
remarks.
2 THE MODEL
Since the work of Kaufman & Neufeld (1996a) and KN96b, one of
the key developments in the modelling of interstellar shock waves
has been the treatment of gas–grain interactions. It had been recognized since the early work of Draine (1980) that the grain inertia
may be significant in determining the structure of C-type shock
waves. However, the importance of the variation of the grain charge
with position in the shock wave was established by later models that
1746
D. R. Flower and G. Pineau des Forêts
incorporated the processes determining the grain charge, in addition
to those responsible for momentum transfer between the charged
and the neutral fluids (Flower & Pineau des Forêts 2003). The grains
are predominantly (negatively) charged within such structures, and
their inertia can have a significant effect not only on the ion magnetosonic speed in the pre-shock fluid, but also on the strength of ionneutral coupling within the shock structure (ambipolar diffusion).1
The enhanced coupling between the ionized and neutral fluids results in narrower C-type shock structures, with higher maximum
temperatures. At the higher temperatures, the collisional dissociation of H2 (and other molecules) proceeds more rapidly. Given that
H2 is the main coolant of the medium, owing to collisional excitation of its rovibrational spectrum, stationary C-type structures can
exist only up to limiting values of the shock speed, which depend
on the conditions in the pre-shock gas, notably the density and
the transverse magnetic field strength. Thus, the domain of C-type
shock waves is restricted, and, within this domain, their thermal and
density profiles differ substantially from those predicted when grain
inertia is neglected (see Le Bourlot et al. 2002; Ciolek, Roberge &
Mouschovias 2004).
The shock model that we have used is essentially the same as
that employed by Gusdorf et al. (2008a,b), with the elemental abundances and their initial distribution between gas and solid phases
taken from Flower & Pineau des Forêts (2003). The main modification involves the treatment of the molecular line transfer. Although
we continue to use the large velocity gradient (LVG) approximation, which is well adapted to the conditions in shock waves, where
the flow velocity changes rapidly, we have incorporated this treatment into the shock code itself. Furthermore, the molecular energylevel populations are computed in parallel with the dynamical and
chemical rate equations, following Flower & Gusdorf (2009). It
follows that the so-called ‘statistical equilibrium’ approximation is
not made when evaluating the level populations of the molecules –
specifically, H2 , CO and ortho- and para-H2 O. Abandoning this approximation has the additional advantage that the iteration of the
calculations of the level populations and the line escape probabilities, inherent in LVG calculations2 for CO and H2 O that assume
statistical equilibrium, is rendered redundant (see Flower & Gusdorf
2009).
Substantial progress has been made in recent years in determining
the cross-sections, and hence the thermal rate coefficients, for the
excitation of H2 O and CO in collisions with molecular hydrogen.
Quantal calculations for H2 O–H2 collisions have been performed
by Dubernet & Grosjean (2002), Grosjean, Dubernet & Ceccarelli
(2003) and Dubernet et al. (2009), who extended earlier work by
Phillips, Maluendes & Green (1996). However, these demanding
calculations are not yet complete, from the viewpoint of their application to modelling interstellar shocks, as they do not cover all four
of the ortho and para combinations to sufficiently high temperatures. The corresponding classical-trajectory Monte Carlo (CTMC)
calculations (Faure et al. 2007) are more complete in this respect,
although they are unlikely to be as accurate as quantal results. We
adopted the CTMC rate coefficients, as provided on the website of
Leiden Observatory, together with the rate coefficients for excita-
tion of H2 O by He (Green, Maluendes & McLean 1993), available
on the same site (http://www.strw.leidenuniv.nl/).
Our data set extends to energy levels of H2 O that are approximately 2000 K above the ground level, which is less than the maximum temperatures attained in some of the models considered below.
However, the large values of the radiative (electric dipole) transition probabilities of H2 O ensure that the population densities of the
levels approaching the upper boundary of those included remain
negligibly small. It follows that the omission of more highly excited
levels is not a significant source of uncertainty for the intensities of
observable lines.
In the case of CO, the Leiden website lists rate coefficients for
excitation by H2 which derive from the quantal calculations of
Wernli et al. (2006) at low temperatures and from Flower (2001)
for 100 ≤ T ≤ 400 K; the results of these calculations were extrapolated to T = 2000 K. Extrapolations over a temperature range
400 < T ≤ 2000 K are inevitably uncertain.3 Rate coefficients
for the excitation of CO by He and H have been computed by
Cecchi-Pestellini et al. (2002) and Balakrishnan, Yan & Dalgarno
(2002), respectively. Whilst the excitation by He always plays a
secondary role, excitation by H dominates in J-type shocks that are
sufficiently energetic to dissociate molecular hydrogen. Because
the calculations of Balakrishnan et al. (2002), for excitation by H,
extend only to the rotational state J = 16 of CO, whereas those of
Flower (2001), for excitation by H2 , extend up to J = 20, there appears a non-physical discontinuity in the computed line intensities,
between J = 16 and 17, in dissociative shock models. Accordingly, we have preferred to adopt the same rate coefficients for the
excitation of CO by H as for excitation by ortho-H2 . The discontinuity is then suppressed, whilst the total energy radiated by the
CO molecule remains the same, to within a few percent. A similar
prescription was followed in the case of H2 O, where there exist no
data for excitation by H.
Information on the sources of other molecular and atomic collision data is given by Flower et al. (2003).
3 R E S U LT S A N D D I S C U S S I O N
We have computed small grids of steady-state C- and J-type models,
for the following values of the shock speed, v s , and for pre-shock
densities nH = n(H) + 2n(H2 ) = 2 × 104 and 2 × 105 cm−3 :
(i) C-type vs = 10, 20, 30, 40 km s−1 ,
(ii) J-type vs = 10, 20, 30 km s−1 .
The transverse magnetic field strength in the pre-shock gas was
1
taken to be B(μG) = b[nH (cm−3 )] 2 , with b = 1 in the C-type
and b = 0.1 in the J-type models. The initial conditions were
computed in physical and chemical equilibrium, adopting the distribution of the elements between the gas and solid phases given
in table 1 of Flower & Pineau des Forêts (2003) and the grainmantle composition given in their table 2. The cosmic ray ionization rate ζ = 5 × 10−17 s−1 . In the pre-shock gas, the hydrogen is in
molecular form but becomes at least partly dissociated behind the
J-discontinuity, when the gas temperature T 104 K.
3
1
Charged polycyclic aromatic hydrocarbons, on the other hand, are neutralized within C-type shock waves, and their inertia is negligible compared
with that of the charged grains.
2 The rovibrational emission lines of H , which has no permanent electric
2
dipole moment and a quadrupole spectrum, are optically thin.
C
We have recognized recently that the discrepancies between the steadystate and statistical equilibrium calculations of CO level populations, reported for the C-type shock model of Flower & Gusdorf (2009), were
attributable partly to different assumptions having been made in the two
calculations regarding the behaviour of the rate coefficients for T > 400 K,
i.e. beyond the upper limit for which they had been computed.
C 2010 RAS, MNRAS 406, 1745–1758
2010 The Authors. Journal compilation Emission of H2 , CO and H2 O molecules
104
106
107
n
T
n
T
H
H
n
101
106
102
-3
-3
102
103
101
101
102
104
103
101
105
105
107
108
n
T
n
T
H
n
105
103
102
time (yr)
time (yr)
H
n
104
103
-3
-3
105
102
107
density (cm )
103
density (cm )
106
temperature (K)
104
temperature (K)
density (cm )
105
density (cm )
103
temperature (K)
n
temperature (K)
104
1747
106
102
101
101
-7
10
-5
10
-3
10
-1
10
time (yr)
1
10
104
103
-7
10
-5
10
-3
10
-1
10
1
10
105
103
time (yr)
Figure 1. Temperature (thick lines) and density (thin lines) profiles of the C- and J-type models in our grid; the independent variable is the flow time (of the
neutral fluid in the C-type shocks). Upper panels: C-type; lower panels: J-type. Left-hand panels: pre-shock nH = n(H) + 2n(H2 ) = 2 × 104 cm−3 ; right-hand
panels: nH = 2 × 105 cm−3 . Colour coding: vs = 40 km s−1 in black; vs = 30 km s−1 in blue; vs = 20 km s−1 in red; vs = 10 km s−1 in green.
3.1 Physical and chemical properties
In Fig. 1, we show the temperature and density profiles for both the
C- and J-type grids. In the case of the C-type models, which are
multifluid, the temperature of the neutral fluid is plotted. The independent variable is the flow time through the entire shock structure,
from the pre- to the post-shock gas (the flow time of the neutral
fluid, in the case of the C-type models). As is well known from
previous calculations, a J-type shock wave has a shorter duration, a
higher maximum temperature, and a larger compression factor than
a C-type shock wave of the same speed, propagating into gas of a
given density.
3.1.1 C-type
The structure of the C-type shock wave with vs = 40 km s−1 and
nH = 2 × 105 cm−3 , plotted in Fig. 1, may be compared with the
equivalent model of KN96b. As might have been anticipated from
the remarks in Section 2, our calculations predict a structure that is
C
narrower – by a factor of approximately 3 – and that attains a higher
maximum neutral fluid temperature – approximately 5000 K, compared with 3000 K (KN96b, fig. 1). On the other hand, the chemical
profiles through the shock wave are qualitatively similar to those
computed by KN96b. In particular, the fraction of hydrogen that is
atomic increases with the kinetic temperature to approximately 2
per cent before decreasing as H2 reforms on grains in the cooling
flow. We adapted the expression for the H-atom sticking probability
of Hollenbach & McKee (1979, equation 3.7) for use in our shock
models. KN96b appear to have neglected the reformation of H2 :
their fig. 1 shows n(H)/nH remaining constant as the gas cools.
Gas-phase oxygen that is not bound in CO is converted into H2 O as
the temperature of the gas rises, driving the reactions O(H2 , H)OH
and OH(H2 , H)H2 O.4 Thus, we expect the main consequences of
4
The reaction O(H2 , H)OH is endothermic and has a barrier, and its rate
coefficient is proportional to exp(−2980/T ); the reaction OH(H2 , H)H2 O
has a barrier, and its rate coefficient is proportional to exp(−1490/T ).
C 2010 RAS, MNRAS 406, 1745–1758
2010 The Authors. Journal compilation 1748
D. R. Flower and G. Pineau des Forêts
Table 1. Flux of energy in the emission lines of H2 O, H2 , CO, [O I] and H I Ly α, relative to the initial flux
of energy in the flow, ρvs3 /2, and expressed as a percentage. The results for C-type shocks are listed first,
followed by the results for J-type shocks.
nH
(cm−3 )
vs
(km s−1 )
H2 O
(per cent)
H2
(per cent)
CO
(per cent)
O
(per cent)
H
(per cent)
Total
(per cent)
2 × 104
2 × 104
2 × 104
2 × 104
10
20
30
40
1.9
11.0
6.8
4.3
34.2
57.0
72.7
80.6
17.0
4.9
2.1
1.1
0.7
0.01
–
–
–
–
–
–
53.8
72.9
81.6
86.1
2 × 105
2 × 105
2 × 105
2 × 105
10
20
30
40
4.9
17.1
9.6
5.9
26.2
48.8
69.3
79.5
19.6
3.9
1.4
0.7
0.1
–
–
–
–
–
–
–
50.8
69.8
80.4
86.1
2 × 104
2 × 104
2 × 104
10
20
30
3.2
15.0
18.7
75.5
72.5
3.3
6.8
9.0
0.8
–
0.01
6.6
–
–
41.1
85.5
96.6
80.4
2 × 105
2 × 105
2 × 105
10
20
30
5.7
50.7
23.3
64.5
15.1
3.0
3.6
5.8
0.1
–
1.8
2.6
–
–
48.2
73.8
73.3
77.3
the changes in the shock structure to be lower molecular line intensities, owing to the reduced column density of shock-heated gas,
and enhanced fractional populations of excited levels, owing to the
higher maximum temperature.
The consequences of the differences in the physical structure of
the shock wave for the energy radiated by the principal coolant
molecules, H2 , H2 O and CO, can be assessed by comparing their
integral emission line fluxes (summed over all transitions) with the
total kinetic energy flux in the flow, ρvs3 /2, where ρ is the mass
density of the pre-shock medium. This comparison was made by
KN96b (table 1), for C-type shocks. Our results, for both C- and
J-type shocks, are given in Table 1.
At the lowest shock speed, vs = 10 km s−1 , only about a half of
the total energy of the shock wave is radiated by these molecular
coolants; the remaining energy is used to compress the magnetic
field (with a small fraction being radiated by other molecules and
atoms). As v s increases, the fraction of the energy that is radiated by
H2 also increases, whereas the fraction radiated by CO decreases.
These trends reflect the rising maximum kinetic temperature of the
gas and the fact that the rotational constant of H2 is much larger than
that of CO. For the highest shock speed, vs = 40 km s−1 , and the
higher pre-shock gas density, nH = 2 × 105 cm−3 , KN96b predicted
that over 99 per cent of the energy of the shock wave is radiated
by H2 , H2 O and CO. We predict a smaller percentage (86 per cent),
owing to the enhanced contribution of cooling through the collisional dissociation of H2 .5 With this proviso, the trends exhibited
by our own calculations are qualitatively and quantitatively similar
to those predicted by KN96b. By way of illustration, we show, in
Fig. 2, the rates of cooling by the main molecular coolants through
a C-type (and also a J-type) shock wave with vs = 20 km s−1 and
nH = 2 × 104 cm−3 initially.
Non-thermal sputtering of the mantles of the (mainly negatively)
charged grains by the abundant neutral species, H, H2 and He, can
enhance the gas-phase abundances of species such as H2 O, which is
present with relatively high initial abundance (of the order of 10−4 )
5
Part of the dissociation energy of 4.48 eV is recovered when molecular
hydrogen reforms on grains and is ejected into the gas phase.
C
in the mantles. In the models, this process is a major contributor to
the gas-phase abundance of H2 O at all but the lowest shock speed
(vs = 10 km s−1 ).
3.1.2 J-type
The chemical response of J-type shock waves can differ considerably from that of C-type shocks of the same speed, particularly
when the speed is sufficient to give rise to the dissociation of H2 .
Thus, whilst emission in the lines of H2 is the main radiative cooling
mechanism in C-type models, cooling by H2 O and CO can be more
important in J-type shocks; see Table 1.
In Fig. 3, we compare the profiles of selected oxygen-containing
species through J- and C-type shock waves with vs = 20 km s−1 and
nH = 2 × 104 cm−3 . As may be seen from this figure, the temperature attains approximately 2 × 104 K behind the J-‘discontinuity’
(which has a small but finite width, commensurate with the artificial
viscosity in the model). At the higher preshock density of 2 × 105
cm−3 , such a high temperature is sufficient for complete collisional
dissociation of H2 to occur. The presence of atomic hydrogen in
the hot gas leads to the chemical destruction of abundant gas-phase
species, such as CO and H2 O, in endoergic reactions with H. This
net destruction of CO and H2 O proceeds in spite of their release
from the mantles of the grains, by thermal sputtering. As the gas
cools and endoergic reactions with H begin to slow, CO and H2 O
reform before being adsorbed finally on to the grain surfaces, where
H2 is produced and ejected into the gas phase.
At the highest J-type shock speed considered (30 km s−1 ), the
temperature behind the J-discontinuity is sufficiently high for not
only dissociation of H2 to occur but also excitation and ionization of
H; see Table 1 and Fig. 4. Dissociation of H2 is attributable mainly
to collisions with neutral particles of comparable mass (H, H2 , He),
whereas excitation and ionization of H are induced by collisions
with electrons. When H+ becomes the dominant ion in the medium,
ionization of H is the main provider of free electrons, and the process of electron collisional ionization becomes self-perpetuating.
At a shock speed vs = 30 km s−1 , the fractional electron density,
ne /nH , exceeds 10−3 behind the J-discontinuity; this is six orders of
magnitude higher than in the pre-shock gas. Under these conditions,
C 2010 RAS, MNRAS 406, 1745–1758
2010 The Authors. Journal compilation Emission of H2 , CO and H2 O molecules
10-16
10-3
T
C-shock
n
-17
T
103
H
-3 -1
10-18
HO
2
CO
-19
102
10
10-20
temperature (K)
cooling (erg cm s )
2
10-21
10
n
-4
H O
103
2
CO
10-5
H O
*
2
CO
10
*
-6
102
10-7
10-8
101
1
10
10-22
102
3
10-9
1 1016
4
10
t (yr)
10
2 1016
H
T
J-shock
2
10
HO
2
10-16
CO
5 1016
6 1016
3
10
10-18
2
10
10-20
H O
n
*
2
10-4
104
CO
10
*
-5
CO
103
10-6
H O
2
10-7
102
temperature (K)
-3 -1
4
temperature (K)
cooling (erg cm s )
10-14
fractional abundance : n(X) / n
H
n
4 1016
10-3
105
T
3 1016
z (cm)
n
10-12
temperature (K)
fractional abundance : n(X) / n
H
10
10-8
101
101
-9
10
-2 1014 0 100 2 1014 4 1014 6 1014 8 1014 1 1015
-22
10
10-6
1749
10-4
10-2
100
102
104
z (cm)
t (yr)
n
Figure 2. Rates of cooling of the principal molecular coolants through
C- and J-type shock waves of speed vs = 20 km s−1 and initial density
nH = n(H) + 2n(H2 ) = 2 × 104 cm−3 . The independent variable is the
flow time of the neutral fluid (in connection with the C-type model, which
is multifluid).
Figure 3. Profiles of selected oxygen-containing species through C-type
(upper panel) and J-type (lower panel) shock waves of speed vs = 20 km s−1
and initial density nH = n(H) + 2n(H2 ) = 2 × 104 cm−3 . An asterisk on
a chemical symbol denotes a species in the grain mantles (broken lines).
The distance, z, is measured along the direction of propagation of the shock
wave.
electron collisional excitation of atomic hydrogen is a significant
contributor to the cooling of the gas behind the discontinuity. Furthermore, it is necessary to consider whether the H I Lyman series
photons that are generated contribute significantly to dissociating
or ionizing other species, either within the shock itself or in the preshock gas. The effects of the Lyman series photons produced in fast
(vs ≥ 60 km s−1 ), dissociative J-type shocks have been considered
by Neufeld & Dalgarno (1989).
In the case of the J-type model with vs = 30 km s−1 and
nH = 2×105 cm−3 , the flux of H I Ly α photons produced by electron
collisional excitation is of the order of 1011 cm−2 s−1 , which may be
compared with the integral ultraviolet flux of the mean interstellar
radiation field (of the order of 108 cm−2 s−1 ). The Ly α flux is of the
same order as the flux of H2 molecules (3 × 1011 cm−2 s−1 ) entering
the shock wave in this model. This criterion suggests that the Ly
α radiation may have a significant effect on the composition of the
pre-shock gas (cf. Shull & McKee 1979). Because the photodissociation of H2 proceeds through absorption in discrete transitions of
the Lyman and Werner bands, for example, followed by a radiative
transition into the vibrational continuum of the X1 g+ ground electronic state, the absence of wavelength coincidences with the lines
in the H I Lyman series ensures that this process may be neglected.
On the other hand, other, less abundant species may be significantly
dissociated or ionized.
In order to make a rough estimate of the possible significance of
the H I Lyman series photons, we included, in the J-type model with
vs = 30 km s−1 and nH = 2 × 105 cm−3 , the photodissociation of
H2 O, OH and O2 , and the photoionization of C, Si S and Fe, with a
radiation amplification factor (relative to the mean interstellar field)
χ = 103 , which probably overestimates the influence of the Lyman
series photons. We found that their main effects are
C
C 2010 RAS, MNRAS 406, 1745–1758
2010 The Authors. Journal compilation 1750
D. R. Flower and G. Pineau des Forêts
100
104
100
H
H
2
n
103
10-6
102
10-8
+
H
10-10
10
10
-6
10
-4
10
-2
10
0
10
2
10
T
10-2
n
H
10-6
102
10-8
+
H
10-10
1
101
10-12
10-8
4
10-6
10-4
time (yr)
10
H
5
10
H
n
104
102
103
10-6
-8
+
H
102
10-10
0
105
n
10-2
104
10-4
H
103
H
10-6
2
+
10
H
-8
102
10-10
101
10
101
-12
10-8
10
10-6
10-4
10-2
100
102
104
-12
10-8
10-6
10-4
time (yr)
H
n
10
4
103
10-6
H
+
H
2
102
10-10
105
10-2
n
104
H
10-4
103
10-6
+
10-8
H
H
2
102
10-10
101
10
101
-12
10-8
10-6
104
10-4
10-2
100
102
temperature (K)
H
fractional abundance : n(X) / n
H
102
T
T
10-8
100
100
temperature (K)
fractional abundance : n(X) / n
105
10-4
10-2
time (yr)
100
10-2
104
temperature (K)
H
10-4
10
100
T
T
temperature (K)
fractional abundance : n(X) / n
H
2
10
-2
10-2
time (yr)
0
fractional abundance : n(X) / n
10
103
10-4
temperature (K)
H
10-4
fractional abundance : n(X) / n
H
T
temperature (K)
fractional abundance : n(X) / n
H
2
10-2
10-12
10-8
104
10
104
-12
10-8
time (yr)
10-6
10-4
10-2
100
102
104
time (yr)
Figure 4. Illustrating the dissociation and ionization of hydrogen in J-type shocks with increasing shock speed, v s . Top panels: vs = 10 km s−1 ; middle panels:
vs = 20 km s−1 ; bottom panels: vs = 30 km s−1 . Left-hand panels: nH = 2 × 104 cm−3 ; right-hand panels: nH = 2 × 105 cm−3 .
C
C 2010 RAS, MNRAS 406, 1745–1758
2010 The Authors. Journal compilation Emission of H2 , CO and H2 O molecules
(i) To reduce the fraction of molecular and increase the fraction
of atomic oxygen in the pre-shock gas. As a consequence, more
O is available for conversion to H2 O in the hot gas behind the
J-discontinuity, and the maximum fractional abundance of H2 O
increases by a factor of approximately 4;
(ii) To increase the equilibrium temperature of the gas, to approximately 200 K. The process that is predominantly responsible
for this increase is the photoelectric effect on grains. The increase in
the gas temperature is more significant for CO than for H2 O because
CO has a smaller rotational constant and its levels are more accessible at low temperatures.6 The integral intensities, T dV K km s−1 ,
of the CO lines increase by approximately an order of magnitude
owing to the enhancement of the equilibrium temperature.
However, given the preliminary nature of these estimations, and
that a much more careful and complete treatment of the photoprocesses would be necessary for a reliable determination of the
influence of the Lyman series photons, they have been neglected in
the computations of line intensities reported below. We emphasize
that these effects are significant only for the J-type shocks with
vs = 30 km s−1 . In these cases, the computed CO line intensities
are likely to be lower limits.
Electron collisional excitation of H2 O might also be expected to
be significant in the region where the electron fraction is high (cf.
Faure, Gorfinkiel & Tennyson 2004). However, as the fractional
abundance of H2 O is negligible in this same region, this process
may be neglected.
45
log (N/g)
40
35
6
For the same reason, the CO line intensities are more sensitive to assumptions regarding the rates of adsorption of molecules on to grains in the
cooling flow.
C
J-shock
30
C-shock
25
20
0
5000
10000
15000
20000
energy (K)
Figure 5. Comparing the H2 excitation diagrams of C- and J-type shock
waves with vs = 20 km s−1 and nH = 2 × 104 cm−3 ; the initial transverse
magnetic field scaling parameter b = 1 for the C-type model, and b = 0.1
for the J-type model. The overlap of rotational levels in the v = 0 and 1
vibrational manifolds is evident in the results for the C-type shock.
3
10
C-shock
2
10
J-shock
-1
TdV (K km s )
3.2 Predicted line intensities
We have computed the emission line spectra of H2 , CO and H2 O.
In all three cases, the level populations were calculated by integrating the time-dependent rate equations and allowing for the optical
depths in the lines (of CO and H2 O) by means of the LVG method
(see Flower & Gusdorf 2009).
In view of the differences in the structures of C- and J-type shock
waves, discussed in Section 3.1, we expect the emission from J-type
shocks to be skewed towards transitions from more highly excited
levels. In the case of H2 , this trend is illustrated in Fig. 5, where the
column density per magnetic substate, NvJ /gJ , is plotted against the
excitation energy, EvJ , of the emitting rovibrational level, (v, J ).
Results are given for C- and J-type models with vs = 20 km s−1 and
nH = 2 × 104 cm−3 . In the case of CO, the corresponding rotational
spectra are shown in Fig. 6, where the integral line intensities, T dV ,
are plotted against the rotational quantum number, J, of the emitting
level. In order for plots of this kind to realize their diagnostic potential, it is necessary to compare with observations covering a wide
range of excitation energy. The wavelengths of rovibrational transitions of H2 involving given changes, v and J , in vibrational and
rotational quantum numbers are similar, owing to the approximately
equal energy separation of successive vibrational states (for low v)
and the similarity of the rotational structure within the vibrational
manifolds. Thus, it is possible to observe with the same instrument,
in the near infrared, transitions from levels whose excitation energies are well separated. The dipolar rotational spectrum of CO, on
the other hand, involves transitions, J → J − 1, whose frequencies
are approximately proportional to J. It follows that observations of
1751
1
10
0
10
-1
10
-2
10
0
5
10
15
J
20
up
Figure 6. Comparing the CO rotational line intensities, T dV , of C- and
J-type shock waves with vs = 20 km s−1 and nH = 2 × 104 cm−3 ; the initial
transverse magnetic field scaling parameter b = 1 for the C-type model, and
b = 0.1 for the J-type model.
transitions from high and low J levels are made with different telescopes and beam sizes, and with different receivers, which renders
their meaningful intercomparison more difficult to achieve.
In Table 2, we compare the H2 O line fluxes predicted by our
C-type models with vs = 20 km s−1 and 40 km s−1 , for a pre-shock
density nH = 2 × 105 cm−3 , with the corresponding results of
KN96b. It may be seen that our computed fluxes are systematically
smaller than those of KN96b, by factors of typically 3 (but considerably more for certain transitions). Furthermore, the discrepancies
tend to increase with the shock speed, v s . We have alluded already,
in Section 3.1, to the main reason for the systematic difference between our results and those of KN96b: our model predicts narrower
C-type shock waves, with higher maximum kinetic temperatures.
C 2010 RAS, MNRAS 406, 1745–1758
2010 The Authors. Journal compilation 1752
D. R. Flower and G. Pineau des Forêts
Table 2. A comparison of the ortho- and para-H2 O line fluxes (erg cm−2 s−1 ) predicted by our C-type models with vs = 20 and 40 km s−1 , for a pre-shock
density nH = 2 × 105 cm−3 , with the corresponding results of KN96b.
20 km s−1
Ortho
40 km s−1
Upper
Lower
Eup (K)
ν (GHz)
λ (μm)
Present
KN96
KN/pr
Present
KN96
KN/pr
110
212
303
432
414
221
505
616
725
707
818
909
10 1 10
532
10 2 9
523
101
101
212
423
303
110
414
505
634
616
707
818
909
423
918
414
61
114
197
550
324
194
468
643
1126
843
1071
1324
1604
732
1861
642
557
1670
1717
2463
2640
2774
3013
3655
4000
4167
4734
5277
5827
6249
6432
6646
538.3
179.5
174.6
121.7
113.5
108.1
99.49
82.03
74.94
71.95
63.32
56.82
51.45
47.97
46.61
45.11
1.1 (−3)
3.0 (−2)
2.5 (−2)
5.4 (−4)
2.9 (−2)
3.1 (−2)
1.5 (−2)
7.8 (−3)
1.7 (−4)
4.2 (−3)
2.6 (−3)
1.4 (−3)
7.8 (−4)
3.6 (−3)
3.5 (−5)
4.2 (−3)
1.2 (−3)
3.7 (−2)
3.1 (−2)
5.2 (−3)
5.4 (−2)
5.6 (−2)
4.7 (−2)
2.8 (−2)
5.1 (−4)
1.4 (−2)
2.2 (−3)
3.7 (−3)
2.1 (−3)
5.1 (−3)
7.2 (−4)
4.7 (−3)
1.1
1.2
1.3
9.6
1.9
1.8
3.2
3.5
3.1
3.3
0.9
2.7
2.7
1.4
21
1.1
2.1 (−3)
6.1 (−2)
5.3 (−2)
2.2 (−3)
7.7 (−2)
7.1 (−2)
4.6 (−2)
2.4 (−2)
4.8 (−4)
1.3 (−2)
7.5 (−3)
4.1 (−3)
2.3 (−3)
1.0 (−2)
1.1 (−4)
1.2 (−2)
2.6 (−3)
8.5 (−2)
8.3 (−2)
1.3 (−2)
1.7 (−1)
1.3 (−1)
1.6 (−1)
1.2 (−1)
4.1 (−3)
8.1 (−2)
1.6 (−2)
3.3 (−2)
2.3 (−2)
3.4 (−2)
8.3 (−3)
3.1 (−2)
1.2
1.4
1.6
5.9
2.2
1.8
3.5
5.1
8.6
6.4
2.2
8.2
10
3.3
73
2.7
20 km s−1
Para
40 km s−1
Upper
Lower
Eup (K)
ν (GHz)
λ (μm)
Present
KN96
KN/pr
Present
KN96
KN/pr
313
404
515
313
606
717
10 0 10
440
202
313
404
211
515
606
919
331
205
319
470
297
643
844
1604
702
2164
2392
3135
3331
3600
4191
5826
6083
138.5
125.4
95.63
89.99
83.28
71.54
51.46
49.28
1.7 (−2)
8.1 (−3)
3.9 (−3)
4.3 (−3)
2.1 (−3)
8.5 (−4)
3.0 (−4)
2.8 (−4)
3.2 (−2)
2.4 (−2)
1.6 (−2)
2.2 (−2)
8.0 (−3)
4.2 (−3)
6.4 (−4)
1.4 (−3)
1.8
3.0
4.2
5.1
3.9
5.0
2.1
5.0
4.2 (−2)
2.5 (−2)
1.2 (−2)
1.9 (−2)
6.2 (−3)
2.5 (−3)
9.0 (−4)
9.6 (−4)
7.9 (−2)
6.9 (−2)
6.1 (−2)
6.1 (−2)
3.7 (−2)
2.4 (−2)
7.4 (−3)
5.9 (−3)
1.9
2.8
5.0
3.2
6.0
9.9
8.2
6.1
Note. The levels are identified by J K+ K− , where J is the rotational quantum number and K is its projection on the symmetry axis of the molecule; the ‘+’
and ‘−’ subscripts refer to the oblate and prolate symmetric top limits, respectively. Eup is the excitation energy of the upper level of the transition, relative to
the 000 ground level of para-H2 O, ν is the frequency and λ is the wavelength of the transition. Numbers in parentheses are powers of 10.
The complete list of computed CO and H2 O line intensities, for our
grids of C- and J-type models, is given in Appendix A.
The forbidden lines of atomic oxygen, at 63 and 147 μm, fall
within the wavelength range of the PACS instrument on the Herschel
satellite. The fractional abundance of atomic oxygen increases when
H2 is collisionally dissociated, as a consequence of the chemical
dissociation of H2 O, OH, O2 , and even CO in reactions with H,
which become rapid at high kinetic temperatures. The requisite
temperatures are attained in J-type shock waves of sufficiently high
speed, as discussed in Section 3.1, but not in C-type shock waves.
Consequently, the relative intensities of the O I and H2 forbidden
lines tend to increase with the shock speed in J-type shocks, but
to decrease in C-type shocks. These trends are illustrated in Fig. 7
and are seen to be similar when the 63 μm line of O I is compared
with lines of H2 of both high and low excitation. The computed
intensities of the O I and of selected H2 forbidden lines are given in
Tables 3 and 4 for the grids of C- and J-type models, respectively.
The relative intensities of the O I and H2 lines depend, to some
degree, on the rate of reformation of H2 on dust in the cooling flow
and hence on the probability that hydrogen atoms stick to grains.
3.3 Predicted line profiles
In Fig. 8 are presented the computed profiles of the CO J = 5 → 4
520.23 GHz and ortho-H2 O 110 → 101 556.94 GHz lines, both
of which should be observable by the HIFI instrument on the Her
C
schel satellite. The profiles are shown for the C-type models of our
grid.
The line temperatures, T L K, which are plotted in Fig. 8, are
evaluated from
TL =
λ3 β L
(Eu − El )Au→l nu ,
8πdv/dz
where λ is the wavelength of the transition, in cm, βL is the line escape probability and dv/dz s−1 is the velocity gradient; Eu , El K are
the energies of the upper and lower levels of the transition, Au→l s−1
is the spontaneous radiative transition probability and nu cm−3 is
the population density of the emitting level. However, rather than
simply equating the velocity gradient to dvn /dz, where v n is the
flow velocity of the neutral fluid, we took
dv
=
dz
dvn
dz
2
+
cs
z + zmin
2 12
,
where cs is the sound speed and zmin = 1013 cm, in practice. This
expression makes some allowance for broadening arising from the
thermal velocity gradient of the gas, although its effect is small for
the C-type shocks, which have lower maximum kinetic temperatures
than J-type shocks of the same speed. We note that the sense of
the velocity shift between the maxima of the CO and H2 O line
temperatures changes between vs = 10 and 20 km s−1 , owing to the
onset of sputtering of water ice from the grain mantles.
C 2010 RAS, MNRAS 406, 1745–1758
2010 The Authors. Journal compilation Emission of H2 , CO and H2 O molecules
OI / 1-0S(1)
C-shocks
nH= 2 104 cm-3
C-shocks
nH= 2 105 cm-3
OI / 1-0S(1)
-1
line flux (erg cm s sr )
103
-1
101
101
-2
-2
-1
-1
line flux (erg cm s sr )
103
10-1
10-3
OI / S(1)
OI / S(3)
10-1
10-3
OI / S(1)
OI / S(3)
10-5
10-5
10
20
30
-1
v (km s )
40
10
20
40
OI / S(1)
103
103
-1
OI / S(3)
OI / S(1)
OI / S(3)
-1
line flux (erg cm s sr )
OI / 1-0S(1)
-1
-1
101
OI / 1-0S(1)
101
-2
-2
30
-1
v (km s )
s
s
line flux (erg cm s sr )
1753
10-1
10-3
J-shocks
nH= 2 104 cm-3
10-5
10
20
30
10-1
10-3
J-shocks
nH= 2 105 cm-3
10-5
40
10
-1
20
v (km s )
30
40
-1
v (km s )
s
s
Figure 7. The relative intensities of the [O I] 63 μm line and the specified lines of H2 , as functions of the shock speed, v s . Upper panels: C-type; lower panels:
J-type. Left-hand panels: pre-shock nH = n(H) + 2n(H2 ) = 2 × 104 cm−3 ; right-hand panels: nH = 2 × 105 cm−3 .
Table 3. [O I] and selected H2 line intensities, in erg cm−2 s−1 sr−1 , for our grid of C-type shock models, identified by the shock speed, in km s−1 , and the
pre-shock density, in cm−3 (e.g. v10n2e4 signifies vs = 10 km s−1 and nH = n(H) + 2n(H2 ) = 2 × 104 cm−3 ). Numbers in parentheses are powers of 10.
Transition
λ (μm)
v10n2e4
v20n2e4
v30n2e4
v40n2e4
v10n2e5
v20n2e5
v30n2e5
v40n2e5
O I 2p4 3 P1 → 2p4 3 P2
O I 2p4 3 P0 → 2p4 3 P1
63.1
147
1.1 (−5)
1.0 (−6)
1.4 (−6)
1.3 (−7)
1.3 (−6)
9.8 (−8)
4.6 (−7)
4.0 (−8)
1.5 (−5)
7.4 (−7)
4.5 (−6)
2.0 (−7)
2.8 (−6)
1.1 (−7)
3.4 (−6)
1.2 (−7)
H2
H2
H2
H2
H2
H2
H2
H2
H2
28.22
17.04
12.28
9.66
8.02
6.91
6.11
5.51
2.12
5.4 (−6)
1.6 (−4)
1.3 (−4)
3.1 (−4)
2.8 (−5)
8.9 (−6)
1.3 (−7)
1.8 (−8)
1.3 (−9)
8.1 (−6)
3.8 (−4)
5.3 (−4)
3.3 (−3)
1.3 (−3)
2.5 (−3)
3.3 (−4)
2.4 (−4)
4.5 (−6)
7.8 (−6)
4.0 (−4)
6.7 (−4)
5.6 (−3)
3.4 (−3)
1.2 (−2)
3.5 (−3)
5.9 (−3)
3.6 (−4)
7.3 (−6)
3.7 (−4)
6.4 (−4)
5.6 (−3)
3.7 (−3)
1.6 (−2)
6.1 (−3)
1.5 (−2)
3.1 (−3)
1.0 (−5)
3.6 (−4)
4.0 (−4)
2.0 (−3)
6.7 (−4)
1.2 (−3)
1.4 (−4)
8.5 (−5)
4.2 (−7)
8.7 (−6)
4.4 (−4)
7.5 (−4)
6.4 (−3)
4.1 (−3)
1.8 (−2)
6.5 (−3)
1.7 (−2)
9.9 (−4)
7.9 (−6)
4.1 (−4)
7.7 (−4)
7.3 (−3)
5.4 (−3)
2.8 (−2)
1.3 (−2)
4.4 (−2)
1.3 (−2)
8.3 (−6)
4.1 (−4)
7.2 (−4)
6.5 (−3)
4.5 (−3)
2.2 (−2)
1.0 (−2)
3.7 (−2)
4.5 (−2)
C
0−0 S(0)
0−0 S(1)
0−0 S(2)
0−0 S(3)
0−0 S(4)
0−0 S(5)
0−0 S(6)
0−0 S(7)
1−0 S(1)
C 2010 RAS, MNRAS 406, 1745–1758
2010 The Authors. Journal compilation 1754
D. R. Flower and G. Pineau des Forêts
Table 4. [O I] and selected H2 line intensities, in erg cm−2 s−1 sr−1 , for our grid of J-type shock models, identified by the shock speed,
in km s−1 , and the pre-shock density, in cm−3 (e.g. v10n2e4 signifies vs = 10 km s−1 and nH = n(H) + 2n(H2 ) = 2 × 104 cm−3 ).
Numbers in parentheses are powers of 10.
Transition
λ (μm)
v10n2e4
v20n2e4
v30n2e4
v10n2e5
v20n2e5
v30n2e5
O I 2p4 3 P1 → 2p4 3 P2
O I 2p4 3 P0 → 2p4 3 P1
63.1
147
9.0 (−9)
2.7 (−10)
1.6 (−6)
4.5 (−8)
3.0 (−3)
8.4 (−5)
5.0 (−8)
1.0 (−9)
2.4 (−3)
6.7 (−5)
8.5 (−3)
2.3 (−4)
H2
H2
H2
H2
H2
H2
H2
H2
H2
28.22
17.04
12.28
9.66
8.02
6.91
6.11
5.51
2.12
2.3 (−7)
9.9 (−6)
1.4 (−5)
1.0 (−4)
5.9 (−5)
2.4 (−4)
9.2 (−5)
2.6 (−4)
2.5 (−5)
1.4 (−6)
4.5 (−5)
3.9 (−5)
1.5 (−4)
5.2 (−5)
1.6 (−4)
6.1 (−5)
2.0 (−4)
3.0 (−4)
9.1 (−7)
2.0 (−5)
1.3 (−5)
4.5 (−5)
2.4 (−5)
1.0 (−4)
5.2 (−5)
1.7 (−4)
1.5 (−4)
4.6 (−7)
2.3 (−5)
3.7 (−5)
3.0 (−4)
1.8 (−4)
7.7 (−4)
3.1 (−4)
9.8 (−4)
6.0 (−4)
1.3 (−6)
3.8 (−5)
3.3 (−5)
1.5 (−4)
7.6 (−5)
3.4 (−4)
1.8 (−4)
6.1 (−4)
8.0 (−4)
1.1 (−6)
2.4 (−5)
1.6 (−5)
5.9 (−5)
3.9 (−5)
2.1 (−4)
1.4 (−4)
4.8 (−4)
4.2 (−4)
0−0 S(0)
0−0 S(1)
0−0 S(2)
0−0 S(3)
0−0 S(4)
0−0 S(5)
0−0 S(6)
0−0 S(7)
1−0 S(1)
4 A T E S T C A S E : T H E O U T F L OW
F RO M N G C 2 0 7 1
5 CONCLUDING REMARKS
In order to illustrate the applicability of our models to the interpretation of interstellar observations of molecular spectral lines, we
consider the outflow from NGC 2071, which was observed with the
Spitzer satellite by Melnick et al. (2008). In tables 2 and 3 of their
paper, these authors list the fluxes of several rotational transitions of
H2 and of (sometimes blended) lines of H2 O. In Fig. 9, we compare
their observed line fluxes, in a 15-arcsec beam, with the two C-type
models from our grid that provide the best fits to the observed H2
line fluxes. The J-type models underestimate the flux of the S(0)
line at 28.2 μm by more than an order of magnitude.
The best overall fit, of both the H2 and H2 O line fluxes,7 is
provided by the C-type model in which vs = 20 km s−1 and nH =
2 × 104 cm−3 initially. As may be seen from Fig. 9, increasing the
shock speed, to vs = 30 km s−1 , or increasing the pre-shock density,
to nH = 2 × 105 cm−3 , yields an improved fit to the and H2 O lines;
but the quality of the fit to the H2 lines is degraded. More refined
modelling of outflow sources requires the temporal evolution of the
shock waves to be considered, as these sources are often too young
for the shock waves to have reached a steady state (McCoey et al.
2004; Giannini et al. 2006). Furthermore, the observed structures
appear to be bow shocks, rather than planar shocks, in which the line
of attack and the physical conditions vary along the bow. It is not our
intention here to engage in bespoke modelling of individual outflow
sources but merely to confirm the viability of shock-modelling of
such objects.
In addition to the lines of H2 and H2 O, Melnick et al. (2008)
list the intensities of the [S I] 25.249 μm and [Fe II] 25.988 μm
transitions. Our best-fitting C-type model (vs = 20 km s−1 and nH =
2×104 cm−3 ) underestimates the intensities of these lines, by factors
of approximately 3 and 80, respectively. The discrepancy for [Fe II]
is unsurprising: similar discrepancies were found in protostellar
jets studied by Giannini et al. (2006), who suggested that the ionic
and molecular lines arise in different regions. The much smaller
discrepancy for [S I] indicates that this atomic line and the molecular
lines are emitted from the same gas.
7
The large flux of the feature observed at 34.55 μm has been remarked by
Melnick et al. (2008) and may be attributable to an unidentified blend.
C
We have modelled the spectral line emission by shock waves in
the interstellar medium. Small grids of C- and J-type shocks were
considered, with initial conditions that are believed to be relevant
to molecular outflow sources. Consideration has been given to the
physical and chemical processes occurring in the medium that is
perturbed by the passage of the shock wave. The spectral line emission is related to the thermal balance of the gas, given that the energy
radiated by the shock wave derives from the kinetic energy of the
gas. The LVG approximation has been used to allow for the effects
of finite optical depths in the lines.
In the case of C-type shock waves, comparison can be made with
the previous calculations of KN96b. We find that the structure of
the shock waves is modified significantly by the inclusion, in the
present models, of dynamical effects associated with the inertia of
charged grains. As a consequence, our C-type models are narrower,
by a factor of typically 3, than those of KN96b and have correspondingly higher maximum temperatures; these differences have
repercussions for the line emission from the medium. At low shock
speeds, vs 20 km s−1 , the rate of sputtering of water ice from
grain mantles falls rapidly with decreasing shock speed, and this
process is negligible by vs = 15 km s−1 . For vs 10 km s−1 , the
formation of water, from atomic oxygen, in the gas phase, becomes
negligibly slow. As a consequence, the fractional abundance of gasphase water n(H2 O) n(CO) at low shock speeds, and the H2 O
emission lines are relatively weak; this trend may be seen in Fig. 8.
The transfer of momentum between the charged grains and the
neutral fluid is determined by the collision frequency, and hence by
ng σg vd , where ng is the grain number density, σg is the geometrical
cross-section of the grain and vd = |vi − vn | is the ion-neutral drift
speed. The value of ng σg depends on the grain size distribution,
for which we have adopted the result of Mathis, Rumpl & Nordsieck
(1977), derived from observations of the diffuse interstellar medium.
However, there is some indirect evidence (Flower, Pineau des Forêts
& Walmsley 2006) that, in the molecular clouds that are the sites of
low-mass star formation, the mean grain size may be considerably
larger than predicted by the distribution of Mathis et al. In this case,
the coupling between the charged grains and the neutrals would be
correspondingly weaker, and the shock structure would revert to
something more akin to the calculations of KN96b.
Our discussion of the line emission refers to H2 , CO and H2 O, but
we list the forbidden infrared lines of atomic oxygen also. These
species are the main coolants of the medium, with their relative
C 2010 RAS, MNRAS 406, 1745–1758
2010 The Authors. Journal compilation Emission of H2 , CO and H2 O molecules
50
100
4
n = 2 10 cm
-3
40
5
CO
H
n = 2 10 cm
-3
CO
H
v = 10 km s
-1
80
s
line intensity (K)
line intensity (K)
1755
30
20
v = 10 km s
-1
s
60
40
HO
2
10
20
HO
2
0
0
2
4
6
-1
v - v (km s )
s
8
0
10
0
8
10
n
60
4
n = 2 10 cm
-3
5
n = 2 10 cm
H
40
CO
v = 20 km s
-1
v = 20 km s
s
30
20
10
HO
-3
HO
CO
2
H
50
line intensity (K)
line intensity (K)
4
6
-1
v - v (km s )
s
50
-1
s
40
30
20
10
2
0
0
0
5
10
15
-1
v - v (km s )
s
20
0
5
10
15
-1
v - v (km s )
n
s
20
n
80
50
4
n = 2 10 cm
5
-3
40
v = 30 km s
n = 2 10 cm
70
CO
H
s
30
HO
2
20
-3
HO
2
H
-1
line intensity (K)
line intensity (K)
2
n
v = 30 km s
60
-1
CO
s
50
40
30
20
10
10
0
0
0
5
10
15
20
-1
v - v (km s )
s
25
0
30
10
15
20
-1
v - v (km s )
s
50
25
30
n
100
4
n = 2 10 cm
-3
5
40
v = 40 km s
n = 2 10 cm
CO
H
80
-1
s
30
-3
HO
H
line intensity (K)
line intensity (K)
5
n
HO
2
20
10
2
v = 40 km s
-1
s
CO
60
40
20
0
0
0
5
10
15 20 25 30
-1
v - v (km s )
s
35
40
n
0
5
10
15 20 25 30
-1
v - v (km s )
s
35
40
n
Figure 8. Computed profiles of the CO J = 5 → 4 520.23 GHz and ortho-H2 O 110 → 101 556.94 GHz transitions, in C-type shock waves. From top to
bottom panels: vs = 10, 20, 30 and 40 km s−1 . Left-hand panels: nH = 2 × 104 cm−3 ; right-hand panels: nH = 2 × 105 cm−3 . The independent variable,
vs − vn , is the velocity of the neutral fluid, relative to the pre-shock gas. Note that the integration was terminated at the point in the post-shock gas at which
the temperature of the neutral fluid had fallen to 15 K.
C
C 2010 RAS, MNRAS 406, 1745–1758
2010 The Authors. Journal compilation 1756
D. R. Flower and G. Pineau des Forêts
will prove useful for the interpretation of forthcoming observations
with the Herschel satellite.
4
AC K N OW L E D G M E N T S
3
10
We thank Malcolm Walmsley, for his comments on an earlier version of this paper, and David Neufeld, for helpful correspondence
regarding the formation of CO lines in shock waves.
2
10
REFERENCES
1
10
Observed (Melnick et al. 2008)
4
-3
-1
n = 2 10 cm ; v = 20 km s
H
s
2
H Flux (10
-21
-2
W cm ) in 15 arcsec beam
10
5
-3
-1
n = 2 10 cm ; v = 10 km s
H
s
0
10
5
10
15
20
25
30
(μm)
1
0
10
2
H O Flux (10
-21
-2
W cm ) in 15 arcsec beam
10
10
-1
Observed (Melnick et al. 2008 )
4
-3
-1
n = 2 10 cm ; v = 20 km s
10
H
-2
s
4
-3
-1
n = 2 10 cm ; v = 30 km s
H
29
30
31
32
s
33
34
35
36
37
(μm)
Figure 9. The outflow from NGC 2071. Crosses: the fluxes of rotational
transitions of H2 and of H2 O, observed through a 15-arcsec beam by Melnick
et al. (2008). The corresponding fluxes for the models of the grid that provide
the best fits to the H2 line fluxes are plotted also. Circles: vs = 20 km s−1
and nH = 2 × 104 cm−3 ; squares: vs = 10 km s−1 and nH = 2 × 105 cm−3
(upper panel) or vs = 30 km s−1 and nH = 2 × 104 cm−3 (lower panel).
importance depending on the type of shock and its initial conditions. In J-type shock waves with vs = 30 km s−1 , the highest speed
considered for this type of shock, molecular hydrogen is dissociated and atomic hydrogen is ionized behind the J-discontinuity. In
these models, emission in the Lyman lines of H I, particularly Ly α,
becomes the major cooling process. Given the large energy of these
photons (10.2 eV) and the high radiation intensity, it becomes necessary to consider their effects on the ambient and shock-perturbed
gas, in which photodissociation of some molecules and photoionization of some atoms can occur. Our investigation has shown that,
whilst the effects of this radiation are significant, notably for the
CO emission, they are not as dramatic as might have been expected.
In so far as H2 and CO are not dissociated by this radiation, the
structure of the shock wave remains essentially unchanged.
We have listed in Appendix A the computed values of the intensities of CO and H2 O emission lines. We anticipate that these results
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APPENDIX A: PREDICTED LINE
INTENSITIES
In Tables A1 and A2, we list the computed intensities of the emission
lines of CO, as predicted by our C- and J-type shock models. The line
C 2010 RAS, MNRAS 406, 1745–1758
2010 The Authors. Journal compilation Emission of H2 , CO and H2 O molecules
1757
Table A1. CO line intensities, T dV K km s−1 , for our grid of C-type shock models, identified by the shock speed, in km s−1 , and the pre-shock density,
in cm−3 (e.g. v10n2e4 signifies vs = 10 km s−1 and nH = n(H) + 2n(H2 ) = 2 × 104 cm−3 ).
J up
J low
ν (GHz)
λ (μm)
Eup (K)
v10n2e4
v20n2e4
v30n2e4
v40n2e4
v10n2e5
v20n2e5
v30n2e5
v40n2e5
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
115.27
230.54
345.80
461.04
576.27
691.47
806.65
921.80
1036.9
1152.0
1267.0
1382.0
1496.9
1611.8
1726.6
1841.3
1956.0
2070.6
2185.1
2299.6
2600.7
1300.4
866.96
650.25
520.23
433.55
371.65
325.22
289.12
260.24
236.61
216.93
200.27
186.00
173.63
162.81
153.27
144.78
137.20
130.37
5.5300
16.600
33.190
55.320
82.970
116.16
154.87
199.11
248.88
304.16
364.97
431.29
503.13
580.49
663.35
751.72
845.59
944.97
1049.8
1160.2
4.3 (+1)
9.1 (+1)
1.4 (+2)
1.6 (+2)
1.6 (+2)
1.3 (+2)
9.6 (+1)
6.1 (+1)
3.4 (+1)
1.8 (+1)
9.5 (+0)
4.8 (+0)
2.4 (+0)
1.2 (+0)
5.4 (−1)
2.7 (−1)
1.3 (−1)
6.4 (−2)
3.0 (−2)
1.2 (−2)
5.7 (+1)
1.1 (+2)
1.8 (+2)
2.3 (+2)
2.5 (+2)
2.3 (+2)
1.8 (+2)
1.3 (+2)
8.4 (+1)
5.1 (+1)
3.0 (+1)
1.7 (+1)
9.2 (+0)
5.0 (+0)
2.5 (+0)
1.4 (+0)
7.3 (−1)
3.7 (−1)
1.8 (−1)
7.9 (−2)
5.6 (+1)
1.1 (+2)
1.7 (+2)
2.3 (+2)
2.7 (+2)
2.7 (+2)
2.3 (+2)
1.8 (+2)
1.2 (+2)
8.1 (+1)
5.0 (+1)
3.0 (+1)
1.7 (+1)
9.8 (+0)
5.2 (+0)
2.9 (+0)
1.6 (+0)
8.3 (−1)
4.1 (−1)
1.8 (−1)
5.3 (+1)
1.1 (+2)
1.6 (+2)
2.2 (+2)
2.6 (+2)
2.7 (+2)
2.5 (+2)
2.0 (+2)
1.5 (+2)
1.1 (+2)
6.9 (+1)
4.3 (+1)
2.6 (+1)
1.5 (+1)
8.2 (+0)
4.6 (+0)
2.6 (+0)
1.4 (+0)
6.8 (−1)
3.0 (−1)
6.3 (+1)
1.1 (+2)
1.5 (+2)
1.9 (+2)
2.2 (+2)
2.5 (+2)
2.7 (+2)
2.7 (+2)
2.6 (+2)
2.3 (+2)
2.0 (+2)
1.6 (+2)
1.2 (+2)
8.5 (+1)
5.5 (+1)
3.6 (+1)
2.2 (+1)
1.3 (+1)
6.7 (+0)
3.0 (+0)
6.3 (+1)
1.1 (+2)
1.6 (+2)
2.0 (+2)
2.4 (+2)
2.7 (+2)
3.0 (+2)
3.1 (+2)
3.0 (+2)
2.9 (+2)
2.6 (+2)
2.3 (+2)
1.9 (+2)
1.5 (+2)
1.1 (+2)
7.6 (+1)
5.1 (+1)
3.3 (+1)
1.8 (+1)
8.7 (+0)
6.2 (+1)
1.1 (+2)
1.5 (+2)
1.9 (+2)
2.3 (+2)
2.6 (+2)
2.9 (+2)
3.0 (+2)
3.0 (+2)
2.9 (+2)
2.8 (+2)
2.5 (+2)
2.2 (+2)
1.8 (+2)
1.4 (+2)
1.1 (+2)
8.1 (+1)
5.5 (+1)
3.3 (+1)
1.6 (+1)
7.3 (+1)
1.3 (+2)
1.8 (+2)
2.2 (+2)
2.6 (+2)
2.9 (+2)
3.2 (+2)
3.4 (+2)
3.4 (+2)
3.3 (+2)
3.1 (+2)
2.8 (+2)
2.5 (+2)
2.2 (+2)
1.8 (+2)
1.4 (+2)
1.0 (+2)
7.5 (+1)
4.6 (+1)
2.4 (+1)
Note. The upper and lower levels of the transition are identified by the values of J, the rotational quantum number. Eup is the excitation energy of the upper
level, ν is the frequency and λ is the wavelength of the transition. Numbers in parentheses are powers of 10.
Table A2. CO line intensities, T dV K km s−1 , for our grid of J-type shock models, identified by the shock speed, in km s−1 , and the
pre-shock density, in cm−3 (e.g. v10n2e4 signifies vs = 10 km s−1 and nH = n(H) + 2n(H2 ) = 2 × 104 cm−3 ).
J up
J low
ν (GHz)
λ (μm)
Eup (K)
v10n2e4
v20n2e4
v30n2e4
v10n2e5
v20n2e5
v30n2e5
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
115.27
230.54
345.80
461.04
576.27
691.47
806.65
921.80
1036.9
1152.0
1267.0
1382.0
1496.9
1611.8
1726.6
1841.3
1956.0
2070.6
2185.1
2299.6
2600.7
1300.4
866.96
650.25
520.23
433.55
371.65
325.22
289.12
260.24
236.61
216.93
200.27
186.00
173.63
162.81
153.27
144.78
137.20
130.37
5.5300
16.600
33.190
55.320
82.970
116.16
154.87
199.11
248.88
304.16
364.97
431.29
503.13
580.49
663.35
751.72
845.59
944.97
1049.8
1160.2
1.3 (+0)
3.8 (+0)
5.2 (+0)
5.8 (+0)
6.0 (+0)
6.3 (+0)
6.5 (+0)
6.6 (+0)
6.6 (+0)
6.3 (+0)
5.7 (+0)
5.0 (+0)
4.1 (+0)
3.2 (+0)
2.3 (+0)
1.7 (+0)
1.1 (+0)
7.0 (−1)
3.9 (−1)
1.9 (−1)
4.6 (+0)
1.4 (+1)
2.1 (+1)
2.6 (+1)
2.9 (+1)
3.3 (+1)
3.6 (+1)
3.8 (+1)
3.9 (+1)
4.0 (+1)
3.9 (+1)
3.7 (+1)
3.4 (+1)
3.1 (+1)
2.7 (+1)
2.3 (+1)
1.9 (+1)
1.5 (+1)
1.0 (+1)
6.0 (+0)
1.2 (+0)
3.4 (+0)
5.2 (+0)
6.2 (+0)
7.0 (+0)
7.7 (+0)
8.4 (+0)
9.0 (+0)
9.3 (+0)
9.5 (+0)
9.4 (+0)
9.2 (+0)
8.8 (+0)
8.2 (+0)
7.4 (+0)
6.6 (+0)
5.7 (+0)
4.8 (+0)
3.6 (+0)
2.3 (+0)
1.9 (+0)
5.4 (+0)
7.4 (+0)
8.0 (+0)
8.4 (+0)
8.9 (+0)
9.6 (+0)
1.0 (+1)
1.1 (+1)
1.1 (+1)
1.2 (+1)
1.2 (+1)
1.2 (+1)
1.2 (+1)
1.2 (+1)
1.2 (+1)
1.1 (+1)
1.0 (+1)
8.3 (+0)
5.7 (+0)
3.9 (+0)
1.2 (+1)
2.0 (+1)
2.6 (+1)
3.4 (+1)
4.2 (+1)
5.2 (+1)
6.2 (+1)
7.3 (+1)
8.3 (+1)
9.4 (+1)
1.0 (+2)
1.1 (+2)
1.2 (+2)
1.3 (+2)
1.4 (+2)
1.4 (+2)
1.5 (+2)
1.4 (+2)
1.1 (+2)
1.7 (+0)
5.0 (+0)
7.4 (+0)
8.6 (+0)
9.4 (+0)
1.0 (+1)
1.1 (+1)
1.2 (+1)
1.2 (+1)
1.3 (+1)
1.3 (+1)
1.2 (+1)
1.2 (+1)
1.2 (+1)
1.1 (+1)
1.1 (+1)
1.0 (+1)
9.9 (+0)
9.2 (+0)
8.0 (+0)
Note. The upper and lower levels of the transition are identified by the values of J, the rotational quantum number. Eup is the excitation
energy of the upper level, ν is the frequency and λ is the wavelength of the transition. Numbers in parentheses are powers of 10.
intensities, T dV , are given in K km s−1 , and the models are identified
by the shock speed, v s , in km s−1 , and the pre-shock gas density,
nH , in cm−3 . The numerical integration was terminated in the postshock gas when the temperature of the neutral fluid had fallen to
15 K. The corresponding results for H2 O are given in Tables A3
to A6, samples of which are given here; the complete tables are
available in the online journal (see Supporting Information). The
intensities of the infrared transitions of [O I] and of selected lines of
C
H2 are to be found in Tables 3 and 4. The relationship between T dV ,
in K km s−1 , and the emergent line flux, F, in erg cm−2 s−1 , is
F =
8 × 105 πkB
T dV ,
λ3
where λ is the wavelength of the transition, in cm. The influence
of the 2.7 K background radiation field, which is negligible for the
wavelengths of relevance here, has been neglected when deriving
C 2010 RAS, MNRAS 406, 1745–1758
2010 The Authors. Journal compilation 1758
D. R. Flower and G. Pineau des Forêts
Table A3. Ortho-H2 O line intensities, T dV K km s−1 , for our grid of C-type shock models, identified by the shock speed, in km s−1 , and the pre-shock density,
in cm−3 (e.g. v10n2e4 signifies vs = 10 km s−1 and nH = n(H) + 2n(H2 ) = 2 × 104 cm−3 ). The frequency range of the lines is 22 GHz ≤ ν ≤ 23 THz.
Upper
Lower
Eup (K)
ν (GHz)
λ (μm)
v10n2e4
v20n2e4
v30n2e4
v40n2e4
v10n2e5
v20n2e5
v30n2e5
v40n2e5
...
...
...
...
...
...
...
...
...
...
...
...
...
Note. The levels are identified by J K+ K− , where J is the rotational quantum number and K is its projection on the symmetry axis of the molecule; the ‘+’
and ‘−’ subscripts refer to the oblate and prolate symmetric top limits, respectively. Eup is the excitation energy of the upper level of the transition, relative
to the 000 ground level of para-H2 O, ν is the frequency and λ is the wavelength of the transition. Numbers in parentheses are powers of 10. The full table is
available in the online version of the journal (see Supporting Information).
Table A4. Para-H2 O line intensities, T dV K km s−1 , for our grid of C-type shock models, identified by the shock speed, in km s−1 , and the pre-shock density,
in cm−3 (e.g. v10n2e4 signifies vs = 10 km s−1 and nH = n(H) + 2n(H2 ) = 2 × 104 cm−3 ). The frequency range of the lines is 22 GHz ≤ ν ≤ 23 THz.
Upper
Lower
Eup (K)
ν (GHz)
λ (μm)
v10n2e4
v20n2e4
v30n2e4
v40n2e4
v10n2e5
v20n2e5
v30n2e5
v40n2e5
...
...
...
...
...
...
...
...
...
...
...
...
...
Note. The levels are identified by J K+ K− , where J is the rotational quantum number and K is its projection on the symmetry axis of the molecule; the ‘+’
and ‘−’ subscripts refer to the oblate and prolate symmetric top limits, respectively. Eup is the excitation energy of the upper level of the transition, relative
to the 000 ground level of para–H2 O, ν is the frequency and λ is the wavelength of the transition. Numbers in parentheses are powers of 10. The full table is
available in the online version of the journal (see Supporting Information).
Table A5. Ortho-H2 O line intensities, T dV K km s−1 , for our grid of J-type shock models, identified by the shock speed, in km s−1 , and the pre-shock density,
in cm−3 (e.g. v10n2e4 signifies vs = 10 km s−1 and nH = n(H) + 2n(H2 ) = 2 × 104 cm−3 ). The frequency range of the lines is 22 GHz ≤ ν ≤ 23 THz.
Upper
Lower
Eup (K)
ν (GHz)
λ (μm)
v10n2e4
v20n2e4
v30n2e4
v10n2e5
v20n2e5
v30n2e5
...
...
...
...
...
...
...
...
...
...
...
Note. The levels are identified by J K+ K− , where J is the rotational quantum number and K is its projection on the symmetry axis of the molecule; the ‘+’
and ‘−’ subscripts refer to the oblate and prolate symmetric top limits, respectively. Eup is the excitation energy of the upper level of the transition, relative
to the 000 ground level of para-H2 O, ν is the frequency and λ is the wavelength of the transition. Numbers in parentheses are powers of 10. The full table is
available in the online version of the journal (see Supporting Information).
Table A6. Para-H2 O line intensities, T dV K km s−1 , for our grid of J-type shock models, identified by the shock speed, in km s−1 , and the pre-shock density,
in cm−3 (e.g. v10n2e4 signifies vs = 10 km s−1 and nH = n(H) + 2n(H2 ) = 2 × 104 cm−3 ). The frequency range of the lines is 22 GHz ≤ ν ≤ 23 THz.
Upper
Lower
Eup (K)
ν (GHz)
λ (μm)
v10n2e4
v20n2e4
v30n2e4
v10n2e5
v20n2e5
v30n2e5
...
...
...
...
...
...
...
...
...
...
...
Note. The levels are identified by J K+ K− , where J is the rotational quantum number and K is its projection on the symmetry axis of the molecule; the ‘+’
and ‘−’ subscripts refer to the oblate and prolate symmetric top limits, respectively. Eup is the excitation energy of the upper level of the transition, relative
to the 000 ground level of para-H2 O, ν is the frequency and λ is the wavelength of the transition. Numbers in parentheses are powers of 10. The full table is
available in the online version of the journal (see Supporting Information).
this expression. The flux received at the Earth is F /4π , where
str is the solid angle subtended by the source.
Table A5. Ortho-H2 O line intensities, T dV K km s−1 , for our grid
of J-type shock models.
Table A6. Para-H2 O line intensities, T dV K km s−1 , for our grid of
J-type shock models.
S U P P O RT I N G I N F O R M AT I O N
Additional Supporting Information may be found in the online version of this article:
Table A3. Ortho-H2 O line intensities, T dV K km s−1 , for our grid
of C-type shock models.
Table A4. Para-H2 O line intensities, T dV K km s−1 , for our grid of
C-type shock models.
C
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C 2010 RAS, MNRAS 406, 1745–1758
2010 The Authors. Journal compilation