A&A 375, 827-839 (2001)
DOI: 10.1051/0004-6361:20010906
J. M. Pittard 1 - J. E. Dyson 1 - S. A. E. G. Falle 2 - T. W. Hartquist 1
1 - Department of Physics & Astronomy, The University of Leeds,
Woodhouse Lane, Leeds LS2 9JT, UK
2 -
Department of Applied Mathematics, The University of Leeds,
Woodhouse Lane, Leeds LS2 9JT, UK
Received 23 March 2001 / Accepted 22 June 2001
Abstract
We show that a cooled region of shocked supernova ejecta forms in a
type II supernova-QSO wind interaction, and has a density, an ionization
parameter, and a column density compatible with those inferred for the
high ionization component of the broad emission line regions in QSOs.
The calculations are based on the
assumption that the ejecta flow is described initially by a similarity
solution investigated by Chevalier (1982) and Nadyozhin
(1985) and is spherically symmetric. Heating and cooling
appropriate for gas irradiated by a nearby powerful continuum source is
included in our model, together with reasonable assumptions for the
properties of the QSO wind. The model results are also in agreement with
observational correlations and imply reasonable supernova rates.
Key words: hydrodynamics - shock waves - stars: mass-loss - ISM: bubbles - galaxies: active
Many theoretical explanations have been proposed for the origin of
the BELR. They include: i) magnetic acceleration of clouds off accretion
discs (Emmering et al. 1992); ii) cloud formation
in shocks produced by the interaction of an accretion disc wind with a
nuclear wind (Smith & Raine 1985); iii) the interaction
of an outflowing wind with the surface of an accretion
disc (Cassidy & Raine 1996); iv) interaction of stars with
accretion discs (Zurek et al. 1994); v)
tidal disruption of stars in the gravitational field of the BH
(Roos 1992); vi) interaction of an AGN wind with supernovae and
star clusters (Perry & Dyson 1985; Williams & Perry
1994); and vii) emission from
accretion shocks (Fromerth & Melia 2001). Models
identified as containing serious difficulties include the
formation of BELRs by the thermal instability of a hot
optically thin flow
(as e.g. discussed by Beltrametti 1981; Shlosman et al.
1985). Perry & Dyson (1985) noted that this
mechanism will not occur when
as Compton
cooling dominates over bremsstrahlung, and therefore it cannot be
responsible for the observed BELR in high luminosity QSOs.
Two-phase equilibrium models (Krolik et al. 1981) are
also not applicable to BELRs in these sources because implausibly
high values of the AGN mass-loss rate would be required
(Perry & Dyson 1985). The formation of the BELR in ionized red
giant or supergiant winds (Scoville & Norman 1988;
Kazanas 1989; Alexander & Netzer 1994) has
difficulty in reproducing the observed broad line wings
(Alexander & Netzer 1997), and models involving the ballistic
deceleration of clouds have a number of problems, as summarized by
Osterbrock & Matthews (1986). There are also concerns about
the various assumptions in the infalling and orbiting cloud models proposed
by Kwan and Carroll (Kwan & Carroll 1982; Carroll 1985;
Carroll & Kwan 1985).
One model which accounts for the LIL emission
was proposed by Collin-Souffrin et al. (1988). In this model the
LIL emission arises from the surface of the accretion disc, which is
illuminated by back-scattered X-rays from the central source. Typical electron
densities and absorption columns are inferred to be
and
respectively. Approximately
three quarters of the total luminosity of the broad-line emission is
estimated to arise in this fashion (Collin-Souffrin et al. 1988).
The HIL, therefore, contribute about
one quarter of this emission. The characteristic mass of the
BELR in high luminosity QSOs is
(Osterbrock 1993), and this is dominated by the HIL.
In this paper we look at one aspect of activity in AGN connected with stars, and which is relevant to the work of Perry & Dyson (1985). We investigate the interaction of supernova ejecta with the optically thin, low density, hot QSO wind, in the presence of intense continuum radiation. In particular we examine if shocked gas can radiate efficiently enough to cool to temperatures appropriate for the HIL. The evolution of SNRs in a high density static ambient medium has been previously studied by Terlevich et al. (1992), with particular application to the formation of BELRs in starburst models developed to obviate the existence of supermassive black holes in AGNs. Although there are similarities between this work and ours, two differences exist. First, the initial conditions for the supernova ejecta differ from those in our work. Second, these authors did not include Compton cooling or any heating processes in their calculations. These factors will influence the thermal evolution of the shocked regions.
In Sect. 2 we discuss the use of a similarity solution to specify initial conditions in the calculation and the cooling and heating rates adopted. Section 3 contains results for the calculated evolution of a remnant for each of several assumed sets of environmental conditions, showing that the formation of a cool region of shocked ejecta having a density, ionization parameter, and column density in harmony with those inferred from observations occurs for reasonable assumptions. In Sect. 4 we summarize our conclusions and describe how the work can be extended.
For ,
there must be an inner core of material with a shallower
density profile (
)
in order for the mass of
the ejecta to be finite. Such a core
can be seen in the results of explosion models (e.g. Jones et al.
1981; Suzuki & Nomoto 1995). In the simplest case,
which we adopt in this paper, a core with uniform density (
)
is
surrounded by a steep outer envelope (n=12). The speed of
the core radius
,
and the
density of the envelope
are then given by
If the explosion occurred in a pure vacuum, the structure of the ejecta would evolve according to Eqs. (1)-(2). However, when the ejecta interact with a surrounding medium, the steep envelope acts as a compressible piston, and the radius of the core relative to the reverse shock increases with time. As long as ejecta in the steep envelope continue to pass through the reverse shock the solution is self-similar, but once the core radius reaches this point the solution abruptly ceases this behaviour.
In this work we adopt the Chevalier-Nadyozhin similarity solutions
to specify our initial conditions, with the ejecta distribution being
specified by an inner core with
and an outer envelope with n=12.
Whilst this is a simplification to actual distributions obtained from
explosion models, at this stage we aim to keep our results as general
as possible. It also has the benefit of being easily reproducable and
scale-free. In all our calculations we assume a canonical explosion energy
of 1051 ergs and ejecta
mass of
,
which is typical of a SN of type II.
Figure 1 shows a typical density profile
at t=0.1 yr.
![]() |
Figure 1:
The density profile for ejecta from a type II SN explosion,
modelled as an n=12 power-law for
the envelope and a ![]() ![]() ![]() |
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Heating and cooling rates for a canonical AGN spectrum were kindly supplied by Tod Woods (cf. Woods et al. 1996) and are included in our calculation. We made use of an adaptive grid hydrodynamic code (see e.g. Falle & Komissarov 1996, 1998), which is ideally suited to this problem where regions which contain small-scale structure are located within much smoother regions.
To test the accuracy of the code we first imposed the similarity structure on the flow and saw whether it sustained itself. The ambient medium had constant density, zero velocity, and negligible pressure. In Fig. 2 we show the results of this test, which compare favourably with the results from other codes in the recent literature (e.g. Blondin et al. 2000).
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Figure 2: Comparison of evolved radial profiles of density and temperature rescaled against the similarity solution (solid). The times are measured by the homologous expansion, with t=1 corresponding to the initial mapping. The initial number of grid points in the region of shocked ejecta was four. The curves are not coincident because of start-up errors, finite numerical resolution, and unavoidable diffusion and dissipation in the numerical scheme. As the remnant expands the effective resolution of the calculation increases, and the profiles converge towards the original similarity solution. |
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The heating and cooling rates are tabulated as functions of
temperature and ionization parameter,
(= F/cp where F is the
local ionizing flux, c is the speed of light, and p is the gas
pressure), and are valid in the optically thin, low-density regime.
is effectively a measure of the ratio of radiation pressure to the
gas pressure, and is
for fully ionized gas of cosmic
abundance. The heating rates include Compton and photoionization heating of
all the ionization stages of hydrogen, helium, and some trace metals
(particularly C, N, O, Si, S, and Fe). The cooling includes
collisional excitation, recombination cooling,
Compton cooling, collisional ionization, and free-free losses.
The gas is assumed to be free of dust (Laor & Draine 1993; but
see review by Osterbrock 1993 for an excellent summary of our current
understanding of dust in AGN).
The rates were calculated using the CLOUDY photoionization
code, its standard AGN spectrum, and solar abundances.
Whilst abundances in AGN remain a very contentious issue, there is
considerable evidence that the nuclear regions are not metal poor, even
at high redshift, and also little evidence that the metallicity of the
BELR changes with redshift (see Artymowicz et al. 1993).
Though it seems certain that nitrogen is overabundant by factors of
2-9, particularly for high-redshift sources (e.g. Hamann &
Ferland 1992), most theoretical work has been based on the assumption
that the gas is of solar composition: we also apply this assumption.
More details of the heating and cooling rates can be found in
Woods et al. (1996).
In Fig. 3 we show the thermal equilibrium curve for the
assumed AGN spectrum. At low temperatures photoionization heating and
cooling due to line excitation and recombination are in near balance.
At large temperatures, the equilibrium arises from a balance of Compton
heating and cooling. The solutions with positive slope,
i.e.
,
are thermally stable (Field 1965). If the slope
is negative, solutions can still be thermally stable
if the gas cools isochorically,
whilst they are thermally unstable if the gas cools isobarically. Note that
there is a small range of ionization parameters for which
there are stable solutions at intermediate temperatures (
K).
The exact shape of this part of the thermal equilibrium curve
is poorly known and most likely varies substantially from source to source,
since it is a complicated function of the irradiating spectrum, the
assumed abundances and thermal processes (cf. Krolik et al. 1981).
For a given object it is entirely possible that there are no multi-valued
equilibrium temperatures for any
.
Therefore, in our discussion of
the following results we do not assign too much importance to the
exact behaviour of the simulations in this part of parameter space since we
are not modelling a specific object. In Sect. 3
we further show that the cooling gas often does not
pass through this region of parameter space.
![]() |
Figure 3: Thermal equilibrium curve for the standard AGN spectrum in CLOUDY (see Woods et al. 1996). The symbols refer to the thermal evolution of the cool region in Models A (diamonds), B (triangles) and C (squares) (Figs. 5-7 respectively). |
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To obtain cool gas in thermal equilibrium we require ionization
parameters
.
As noted by Perry & Dyson (1985), shocked gas cooled back to
equilibrium can have a value of
much lower than its
pre-shock value. This is because
does not change if the gas cools
isobarically and the post-shock pressure is much greater than the
pre-shock value. Therefore strong shocks can create conditions
for the gas to cool to temperatures much lower than the surrounding
ambient temperature.
The ionization parameter of the QSO ISM can be written as
The ionization parameter of shocked gas cooled back to equilibrium
with the radiation field,
,
is given by
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Figure 4:
Evolution of the radius (solid) and speed (dots) of the contact discontinuity of an adiabatic SNR as a function of time (arbitrary units).
At t=1 yr, their respective values
are 1.0264 and 0.7698. The SNR evolves in a self-similar fashion
with
![]() ![]() ![]() ![]() ![]() ![]() |
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The assumption of a static ambient medium
keeps the problem in its simplest form. For the unshocked ejecta we set
T = 104 K. The Chevalier-Nadyozhin similarity solution is derived
on the assumption of negligible thermal energy in the unshocked ejecta
and ambient medium, and our specified temperature implies thermal
pressures orders of magnitude below the relevant ram pressure.
Whilst this also implies exceedingly high absorption columns through
the ejecta core, the covering fraction is very small.
In Fig. 4 the radius and speed of the contact
discontinuity are shown as functions of time. As discussed earlier,
the solution remains self-similar until the reverse shock propagates to
the edge of the core: this occurs at
yr, and is reflected in the
structure of the curves in Fig. 4.
Although we have not included heating and cooling terms in this
calculation, it is instructive to assume a radiation field and to plot
the evolution of the ionization parameter of the equilibrium temperature
of the shocked gas. This gives
an impression of whether the shocked gas is likely to heat or cool if
these terms are included. Assuming that the remnant is 0.33 pc distant
from a central ionizing source of
,
we show
in Fig. 4 the time evolution of this parameter
in the shocked ambient medium. This position was chosen because, with the
assumption
of constant ionizing flux,
tracks the inverse of p and the profile
of p is flatter at this point. Figure 4 shows that
initially evolves as t1/2, but increases more rapidly
once the flow diverges from self-similarity. This is due to a large
increase in the volume of the shocked region (as the reverse shock
begins to move back towards the centre of the remnant in the Lagrangian
frame) at roughly constant
total energy, which leads to a reduction in the internal energy per unit
volume and hence pressure, and the corresponding increase in
.
It is therefore apparent that the formation of cool clouds in a
SNR shock is favoured at early times, when the pressure of the shocked
region is high and
is low. In particular, since
rapidly
increases once the ejecta core reaches the reverse shock, a cool
region has the best opportunity to form before this point.
Therefore we equate the dynamical timescale of the remnant,
,
with the time at which the ejecta core reaches the reverse shock, and
require that the shocked gas has a cooling time,
,
shorter than
this. We derive expressions for
and
in Appendix A.
We also require that the ionization parameter
of the post-shock gas at
is low enough to allow the gas to cool to
K. Cold gas will not form if
(where
for the
AGN spectrum used to generate the thermal equilibrium curve in
Fig. 3) whether or not
.
We derive an expression for this condition also in Appendix A. Hence for cold
gas to form we need to satisfy
and
.
Both of these conditions are evaluated
for each of the following models.
In Fig. 5 the evolution of the shocked region is shown.
We find that the shocked ejecta substantially cool with heating and
cooling rates included in the calculation.
By t=1.0 yr a cool region has formed with a temperature
less than that of the undisturbed ambient medium. At t=1.4 yr
it has a temperature of
K. Divergence from
the adiabatic self-similar solution can be seen in the time evolution
of the
,
and
profiles. In the temperature profiles,
this is first manifested as a change in the slope of
the region of shocked ambient material from
to
.
The formation of the cool region at later times is of course
a much larger divergence from the self-similar solution.
In the density profiles, the formation of the cool region is
apparent as a sharp coincident growth in density. The profiles of
ionization parameter show the gradual temporal increase
in
expected
from the evolution of the self-similar solution. However,
at t=1.4 yr, the value of
associated with the cool material is
substantially larger than the value of the immediate surroundings.
This is due to the radiative losses becoming so high that cooling
no longer takes place isobarically. In the limit that the cooling
timescale,
,
is much less than the appropriate dynamical
timescale,
,
the cold gas would cool isochorically. Here
is the timescale for the hot post-shock gas to respond to the
rapid depressurization of this material, and is of the order of the
timescale for
collapse of the reverse shock,
,
where l is the length scale of the shocked ejecta, and
(
)
is the
velocity of the contact discontinuity (reverse shock).
The thermal parameters of the cool region at t = 0.5, 1.0, and 1.4 yr
are plotted as crosses in Fig. 3.
Beyond t=1.4 yr, as the cloud cools further, its thermal energy as a
fraction of its total energy (thermal plus kinetic) approaches the
round-off error of our calculation (10-4), and we
cannot follow its evolution past this time. There is also the well-known
problem of the "eating away'' of the edges of hot material. This affects
all numerical schemes, and results from the numerics smearing the
temperature gradient and placing cells at intermediate temperatures, which
then undergo high cooling (often the line-cooling bump at
K).
In this fashion cool regions can rapidly grow as hot regions are
"eaten away''.
Model | ![]() |
![]() |
![]() |
Cool |
(
![]() |
![]() |
(
![]() |
Regions | |
A | 106 | 0 |
![]() |
Y |
B | 104 | 0 |
![]() |
N |
C | 104 | 3000 |
![]() |
Y |
However, it is clear that the gas will continue to cool, and eventually
reach
K, as the following argument demonstrates.
With
the gas cools isochorically so
.
At t=1.4 yr, the cloud
is at a temperature
K and an ionization
parameter
.
It therefore moves approximately along
a line of
slope -1 in Fig. 3, and encounters the thermal equilibrium
curve at
and
K. This point is
thermally stable and the rapid decrease in the temperature is brought
to a halt. The temperature of the cool region,
,
then remains
relatively constant, and ajusts only for changes in the value of the
ionization parameter,
,
which occur on a sound-crossing
timescale as the surrounding post-shock material gradually repressurizes
the cool gas to its level. The sound
crossing time within the shocked ejecta is short compared to the dynamical
time of the remnant, so
,
where
is
the ionization parameter of the surrounding shock material (
in the shocked ambient gas). There is therefore no immediate
danger of
increasing to the point that the cooled gas begins to reheat towards the
Compton temperature. Thus
approaches
and the
cooled region moves along the thermal equilibrium curve to the left.
The density of the region,
,
increases during this process until
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Figure 5:
The evolution of the remnant of a type-II supernova for
a constant density medium with
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Observations of the HILs have revealed that the optical depth is less than
but of order unity, and that
.
The observed range of the ionization parameter of the HIL region is
(Kwan & Krolick 1981), which is
somewhat higher than that inferred for the LIL region. The systematic
blue-shift of the HILs with respect to the LILs is interpreted as the
HILs forming in a region with bulk outflow (possibly inflow) and that
the red-shifted emission is obscured. Line widths, which reflect local
gas velocities, are typically several
.
Hence our
model results are in harmony with the temperature,
ionization parameter, column density, and velocity dispersion of the
observed HIL BELR clouds.
![]() |
Figure 6:
As Fig. 5 (Model A) but for a constant
density medium with
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We now check to see if our results are in agreement with the two
conditions which must be satisfied for the shocked gas to cool down to
K, namely
and
.
For n=12,
,
E = 1051 ergs,
,
,
A = 0.19 and
,
we obtain
from Eq. (A.5)
= 7 yr. Assuming solar abundance for
the ejecta,
,
and
yr. Hence
as required. The fact
that the cooled region first forms at
yr indicates that
we are underestimating the cooling rate at lower temperatures where
line-cooling is dominant. For these parameters, we further obtain
from Eq. (1)
,
and from
Eq. (A.13)
,
and thus Model A
also satisfies the requirement that
.
In reality the ejecta will be enriched in metals, which will reduce
the above estimate of
.
In Fig. 6 we show the results for a simulation
with
(matching the earlier simulation of Model A shown in
Fig. 5) but with a lower ambient density
(
=
)
and AGN flux (e.g.
with the remnant 1 pc distant).
We call this Model B. As a result of the lower ambient density, it has a
longer evolution time (
yrs) and the remnant expands further
than in Model A. Relative to Model A, the lower flux from the central engine
also increases the cooling timescale of the shocked gas
(
yrs). Although the ionziation parameter of the shocked gas
at
is less than
[
,
as for
Model A], because
we do not expect any of
the shocked material to cool to
K.
Once the reverse shock reaches the core radius
at t = 32 yr, the solution swiftly diverges from a
self-similar form (despite the cooling of the post-shock material, many
aspects such as the radius and velocity of the shocks are not too far
removed from such an evolutionary form prior to this point). At this time the
coolest shocked gas is at
K. The subsequent
acceleration of the reverse shock towards the
centre of the remnant rapidly de-pressurizes the shocked region, increasing
the ionization parameter of the shocked gas, and ultimately leading to its
reheating towards the Compton temperature. Thus, by simply adjusting the
density of the ambient medium, whilst keeping all other parameters constant,
we have shown that a cool region may not form if the effective ram pressure
of the ambient medium is not sufficiently high.
We consider whether there is also a mechanism which prevents the formation
of cool regions in SNRs once the density of the ambient material increases
past some limit. This question can be addressed in the
following way.
As
increases,
must similarly increase to
maintain a given value for
.
For a given
this means a
reduction in
(which might be consistent with an increase
in
). A higher value of
leads to a faster evolution of the
remnant, which given that the cooling timescale of the shocked gas also
shortens, does not prevent the possibility of
cool regions forming. However, physically it does mean that the lifetime
of the cool regions is also very short, since the evolved time
between their
formation and their destruction, which presumably occurs shortly after
the interaction of the ejecta core with the reverse shock, is
short too. Hence at some value of
,
the contribution to the
BELR emission will be on such short timescales that it will become an
unimportant component of the overall emission.
We note that it is unlikely that the ejecta density actually has such a
sharp cut in its gradient. However, once the ejecta passing through the
reverse shock starts deviating from an r-12 profile the volume
of the shocked region will
start expanding with an inevitable rise in the ionization parameter. This
would occur at an earlier time relative to the sharp cut-off case and
would tend to reduce the contribution of mass to the BELR by this model.
Assuming a constant value of
and
,
this argument also introduces a lower bound to
the value of
at which a cool region can form.
Conversely, for a given
,
an increase in
and constant
means a corresponding increase in
.
In a similar
way to the above argument, this introduces an upper bound to the
value of
at which a cool region can form. These conditions together
place upper bounds on the values of
and
inferred from
observations.
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Figure 7:
As Fig. 6 (Model B) but in this
simulation (Model C) the ambient medium has a flow velocity towards the
remnant of
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The distance to the central engine influences the value of
through the ionzing flux (Eq. (A.8))
Therefore there is an upper limit to
after which not even
SNe in the densest environments will be able to produce cool regions.
If the ambient medium is in motion relative to the
remnant, the parameter space over which cool regions can form is extended
(Model B (Fig. 6) vs. Model C
(Fig. 7)).
The second constraint is that the ambient medium needs to be dense enough for the cooling timescale of the shocked gas to be less than the age of the remnant at the time when the reverse shock reaches ejecta which do not have a strongly radially dependent density. If this is not the case no cool regions can form, as illustrated by Model B (Fig. 6).
We now compare our results to some of the observational correlations
that have been determined. Korista et al. (1995) noted that the blue and
red wings respond faster to continuum changes than the core. If one
assumes that the expansion velocity of the supernova remnant is the
dominant velocity component, this observation can be explained with
our model in the following way. Remnants which are surrounded by the
highest ambient densities evolve most rapidly,
and therefore form cool regions
with the highest expansion velocities. If the ambient density increases
towards the central engine, there will be a tendency
to form cool regions with, on average, higher velocities. Being
close to the central engine, these regions respond more
rapidly to any continuum changes, as the observed correlation requires.
Even if
is not the dominant component, the above
argument implies that our assumed model nevertheless
provides a statistical bias in this direction.
There now appears to be conclusive evidence that the higher ionization lines respond faster than lower ionization lines to changes in the continuum variability (see references in Fromerth & Melia 2001). If the LIL are formed in an accretion disc close to the central engine, this implies that the thermal timescale in the disc is longer than the corresponding timescale for the HIL. Alternatively the delay could be explained if the bulk of the LIL heating is due to back-scattered X-rays from the gas responsible for the HILs and the general ambient medium (see Collin-Souffrin et al. 1988). Therefore, our model is consistent with these observations.
Our calculations show that it is the shocked ejecta which cool to form
the HILs, and that once the ejecta core reaches the reverse shock, the
shocked region rapidly depressurizes. Since this presumably leads to
the destruction of any cool regions, it seems sensible to suppose that
for the remnant parameters we have chosen, an upper limit to the HIL
mass would be
per remnant. With further appropriate assumptions
we can then estimate the required supernova rate needed to sustain the
mass in the gas responsible for the HIL. With the assumption that
each remnant can eventually cool
of gas, and that the
lifetime of the cool gas in the remnant is of order 10 yr, to
sustain
of gas in the HIL would require a supernova rate of
2 yr-1. Although our estimates are very uncertain,
this rate is acceptable:
certainly for high luminosity QSOs, supernova rates
10 yr-1 are conceivable (Terlevich et al. 1992).
A less steep density distribution of
the ejecta envelope (if n < 12 more mass would be available per SN),
and the suppression of dust formation in intermediate-mass
AGB stars in the BELR region (which may reduce the minimum
zero-age main sequence mass required for supernovae; Hartquist et al.
1998) are two
additional possibilities which could further reduce our estimate of the
required supernova rate. Additionally, it is possible that conditions
exist for the swept-up ambient material to also cool to temperatures
appropriate for the HIL, as the cooling time estimated by Eq. (A.7)
is temperature independent. The fact that the reverse shocked material
cools first in our simulations is simply due to enhanced cooling by
line emission starting earlier due to its lower initial temperature.
In Model A where
we therefore might expect the
swept-up ambient material to form HIL gas as well. At
,
the forward shock is estimated to be at a radius of 0.03 pc and to have
swept up
of ambient gas. This gas could therefore be a
significant contributer to the total HIL emission.
In this first paper we have been solely interested in the question of whether cool regions could form from supernova shocks given that they are bathed in the hard radiation flux of the central engine. For simplicity we restricted our modelling to the simplest 1D approach, and assumed solar metallicites in our calculations. The fact that the ejecta (and also the swept up medium) may be responsible for the HIL emission warrants a careful consideration in future models. We will also perform calculations on 2D axisymmetric hydrodynamic grids.
Another future goal is the inclusion of the effect of the QSO radiation field
on the dynamics of the remnant. At high temperatures, the effective
cross-section of gas is the Thomson scattering cross-section,
.
Once the gas begins to cool, the effective cross-section,
,
increases, enhancing the radiative driving. As noted by Williams
et al. (1999),
increases to roughly
2 at 106 K, 40 at 105 K,
and to
104 when the gas has fully cooled (see also
Arav & Li 1994; Arav et al. 1994).
For gas cooler than
K,
it is reasonable to take the radiative acceleration as
(Röser 1979; Dyson et al.
1981). We expect that
substantial radiative driving will critically alter the
dynamics of the interaction between the supernova ejecta and the ambient
medium (e.g. Falle et al. 1981; Williams 2000).
Although we have chosen to model the interaction of supernova ejecta with the ambient medium, it is possible that a wind from a group of early-type stars may also provide the necessary conditions for the formation of cool regions. This interaction may be more relevant in the nuclei of Seyfert galaxies, since supernova explosions will evacuate all but the most tightly bound gas in them (Perry & Dyson 1985). Finally it is clear from our models that whilst the supernova-QSO wind interaction is conceptually simple, the BELR is likely to be a very complicated region in practice.
Acknowledgements
JMP would like to thank PPARC for the funding of a PDRA position. We would like to thank R. J. R. Williams and R. Coker for many helpful conversations and T. Woods for the use of his cooling and heating tables. This research has made use of Nasa's Astrophysics Data System Abstract Service. We would also like to thank an anonymous referee whose suggestions improved this paper.
We now derive an expression for the evolution of the ionization parameter
of post-shock gas cooled back to equilibrium with the radiation field,
.
From Perry & Dyson (1985) we have