Astron. Astrophys. 346, 831-842 (1999)
3. The model
The remnant model employed in this work uses the analytic
description by McKee & Truelove (1995) for the hydrodynamic
evolution. The ejecta with mass are
assumed to be cold and to expand with explosion energy
into a homogeneous ambient medium
which has a hydrogen number density
and is composed of 10 hydrogen atoms per helium atom, corresponding to
a helium mass fraction of 28.6%. Inside the homogeneous ejecta, we
assume the existence of denser clumps that contain a small fraction of
the ejecta mass but most of the produced 44Ti. The
treatment of the interaction of the reverse shock with these clumps
and of the corresponding effects on the 44Ti decay are
described below. Ionization of 44Ti is most efficiently
induced by electrons. If and how long a high degree of ionization is
sustained depends on the reverse-shock characteristics and the
postshock evolution of the matter containing 44Ti.
3.1. A model for young supernova remnants
If sufficiently high temperatures happen to be reached behind the
reverse shock, 44Ti may become fully ionized, in which case
its decay is prevented until cooling due to the expansion of the gas
leads to (partial) recombination of the innermost electrons.
To follow the radioactivity of the supernova remnant requires
knowledge of the evolution of the shocked ejecta. The analytical model
of McKee & Truelove (1995) yields scaled relations for the radius,
velocity and postshock temperature of the reverse shock as functions
of time t. Also, the model provides the density of the
unshocked (homogeneous) ejecta at time t, which allows one to
calculate the postshock density from the density jump at the reverse
shock. In order to describe the density history of each mass shell
after the reverse shock has passed through it, we assume that the
shocked matter moves (approximately) with the same velocity as the
contact discontinuity. Knowing the motion of the latter, therefore, we
can estimate the dilution of the shocked gas by the expansion of the
supernova ejecta.
The time-dependent analytical solutions are given by the three
parameters, ,
and
. A combination of values of these
parameters is considered to yield a suitable description of a certain
supernova remnant, if the model gives a present-day radius and
velocity of the blastwave that is compatible with observational
data.
We extended the analytic remnant model of McKee & Truelove
(1995) by assuming that the iron-group elements together with the
explosively nucleosynthesized 44Ti are concentrated in
overdense clumps of gas. These clumps are assumed to move outward
through the dominant mass of homogeneous supernova ejecta with high
velocity. Since the clumps should contain only a minor fraction of the
total mass of the ejecta, the dynamics of the young supernova remnant
as described by the McKee & Truelove (1995) model is assumed not
to be altered by their presence. The radial position r of the
iron clumps within the ejecta can be measured by the relative mass
coordinate q which is defined as
![[EQUATION]](img32.gif)
where is the uniform background
density of the ejecta which expand homogeneously.
3.2. Thermodynamic conditions just behind the reverse shock
We now consider the properties of a clump of metal-rich matter
within the homogeneous ejecta. Denoting the density enhancement factor
of the assumed clump relative to the surrounding ejecta by
, we obtain the clump density after
the reverse shock has passed at time
as
![[EQUATION]](img36.gif)
for an adiabatic index .
Correspondingly, the post-reverse shock temperature of the clump is
related to that of the surrounding uniform ejecta,
, as
![[EQUATION]](img39.gif)
which generalizes by the factor
the result of Sgro (1975) to the case considered here, where the
chemical composition of the clump is different from that of the
surrounding ejecta as described by the corresponding mean molecular
weights and
, respectively.
is given by
![[EQUATION]](img43.gif)
where is the reverse-shock
velocity in the rest frame of the unshocked ejecta (McKee &
Truelove 1995). In Eq. (3), is the
ratio of the pressure values in the regions behind the reflected shock
and in front of it. The reflected shock is formed when the reverse
shock hits a dense clump in the ejecta (Sgro 1975; Miyata 1996). The
ratio is related to
as
![[EQUATION]](img45.gif)
for (Sgro 1975). In particular,
if
(no clumpiness), and appearing in
Eq. (3) is less than unity for
3.3. Post reverse-shock evolution
We now follow the fate of the homogeneous ejecta and of the matter
in clumps after they have experienced the impact of the the reverse
shock at time . As we have mentioned
earlier, we assume that the shock itself would continue to proceed
inward through the cold ejecta as prescribed by the McKee &
Truelove (1995) model. The thermodynamic and chemical evolution of the
clumps at times is treated as
described below.
Since we trace the clump material, it is preferable to introduce
Lagrangian particle abundances, defined by
![[EQUATION]](img51.gif)
where dN is the number of particles at the mass coordinate
q in the interval dq.
3.3.1. Ionic abundances
At a given time , the number
abundance of a nuclear species
k with charge Z and i orbital electrons (i.e.
times ionized, and
follows
![[EQUATION]](img56.gif)
where and
are the nuclear
-decay, ionization and recombination
rates, respectively, of the ion. For 44Ti,
, and thus
where
is the 44Ti decay rate in
the cold ejecta, i.e. its laboratory value.
Since we assume that material which contains 44Ti
consists dominantly of one single stable nuclide (56Fe in
practice), the number of ionization electrons,
, is given by
for that species. In practice, we
further assume that both 56Fe and 44Ti are
ionized to some degree in the reverse shock itself, which we cannot
resolve. Typically, we start our network calculations with ionic
systems of 10 bound electrons, This choice is made since our major
concern is to see the reduction of the orbital-electron capture rates
in the domains of high shock temperatures.
The reaction rates entering in Eq. (7) are discussed in Appendix
A.
3.3.2. Ion and electron temperatures
The network requires the knowledge of the thermodynamic evolution
of the shocked matter, and in particular of the electron temperature,
, which defines the Boltzmann
distribution, and of the matter density
(or the ion number density,
), which enters in the absolute
electron number density .
The decrease by expansion of the temperature
from
of Eq. (3) would roughly be
proportional to (for
and
. Two considerations are due,
however. One concerns the possibility that both
and the ion temperature,
, may not necessarily be the same as
T. Secondly, some non-adiabatic effects may appear as the
result of ionization and recombination.
It is sometimes asserted that the collisionless equilibration
between and
would occur quickly, in which case
already at
. Save this possibility, the
equipartition of the shock energy to ions and electrons may not be
achieved at once. The equilibration process through the Coulomb
interaction is described by a non-linear equation (Spitzer 1962),
![[EQUATION]](img75.gif)
in terms of the equilibration time
, which is a function primarily of
,
and the number densities of ions and electrons. Assuming that
at
, and that the total kinetic
energy
![[EQUATION]](img78.gif)
is constant during a short time interval
, we solve Eq. (8) simultaneously
with the abundance network equations to determine the degree of
ionization of a dominant species, and thus
.
In order to include the non-adiabatic effect caused by ionization
and recombination, we assume that the energy required for unbinding
additional electrons (net energy loss) is taken from electrons only.
Namely, after performing the network calculation at each time step and
solving Eq. (8), we make a replacement
![[EQUATION]](img80.gif)
where and
are the change of
during
and the corresponding net energy
consumption. The equilibrium temperature T is then calculated
from Eq. (9).
Finally, and
are obtained from the current values
by multiplying
© European Southern Observatory (ESO) 1999
Online publication: June 17, 1999
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