## 3. The modelThe 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 ## 3.1. A model for young supernova remnantsIf sufficiently high temperatures happen to be reached behind the
reverse shock, 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 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 where is the uniform background density of the ejecta which expand homogeneously. ## 3.2. Thermodynamic conditions just behind the reverse shockWe 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 for an adiabatic index . Correspondingly, the post-reverse shock temperature of the clump is related to that of the surrounding uniform ejecta, , as 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 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 for (Sgro 1975). In particular, if (no clumpiness), and appearing in Eq. (3) is less than unity for ## 3.3. Post reverse-shock evolutionWe 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 where d ## 3.3.1. Ionic abundancesAt a given time , the number
abundance of a nuclear species
where and
are the nuclear
-decay, ionization and recombination
rates, respectively, of the ion. For Since we assume that material which contains The reaction rates entering in Eq. (7) are discussed in Appendix A. ## 3.3.2. Ion and electron temperaturesThe 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
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), 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 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 where and
are the change of
during
and the corresponding net energy
consumption. The equilibrium temperature Finally, and are obtained from the current values by multiplying © European Southern Observatory (ESO) 1999 Online publication: June 17, 1999 |