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Astron. Astrophys. 346, 831-842 (1999)

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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 [FORMULA] are assumed to be cold and to expand with explosion energy [FORMULA] into a homogeneous ambient medium which has a hydrogen number density [FORMULA] 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, [FORMULA], [FORMULA] and [FORMULA]. 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


where [FORMULA] 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 [FORMULA], we obtain the clump density after the reverse shock has passed at time [FORMULA] as


for an adiabatic index [FORMULA].

Correspondingly, the post-reverse shock temperature of the clump is related to that of the surrounding uniform ejecta, [FORMULA], as


which generalizes by the factor [FORMULA] 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 [FORMULA] and [FORMULA], respectively. [FORMULA] is given by


where [FORMULA] is the reverse-shock velocity in the rest frame of the unshocked ejecta (McKee & Truelove 1995). In Eq. (3), [FORMULA] 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 [FORMULA] is related to [FORMULA] as


for [FORMULA] (Sgro 1975). In particular, [FORMULA] if [FORMULA] (no clumpiness), and [FORMULA] appearing in Eq. (3) is less than unity for [FORMULA]

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 [FORMULA]. 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 [FORMULA] is treated as described below.

Since we trace the clump material, it is preferable to introduce Lagrangian particle abundances, defined by


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 [FORMULA], the number abundance [FORMULA] of a nuclear species k with charge Z and i orbital electrons (i.e. [FORMULA] times ionized, and [FORMULA] follows


where [FORMULA] and [FORMULA] are the nuclear [FORMULA]-decay, ionization and recombination rates, respectively, of the ion. For 44Ti, [FORMULA], and thus [FORMULA] where [FORMULA] 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, [FORMULA], is given by [FORMULA] 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, [FORMULA], which defines the Boltzmann distribution, and of the matter density [FORMULA] (or the ion number density, [FORMULA]), which enters in the absolute electron number density [FORMULA].

The decrease by expansion of the temperature [FORMULA] from [FORMULA] of Eq. (3) would roughly be proportional to [FORMULA] (for [FORMULA] and [FORMULA]. Two considerations are due, however. One concerns the possibility that both [FORMULA] and the ion temperature, [FORMULA], 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 [FORMULA] and [FORMULA] would occur quickly, in which case [FORMULA] already at [FORMULA]. 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 [FORMULA], which is a function primarily of [FORMULA], [FORMULA] and the number densities of ions and electrons. Assuming that [FORMULA] at [FORMULA], and that the total kinetic energy


is constant during a short time interval [FORMULA], we solve Eq. (8) simultaneously with the abundance network equations to determine the degree of ionization of a dominant species, and thus [FORMULA].

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 [FORMULA] and [FORMULA] are the change of [FORMULA] during [FORMULA] and the corresponding net energy consumption. The equilibrium temperature T is then calculated from Eq. (9).

Finally, [FORMULA] and [FORMULA] are obtained from the current values by multiplying [FORMULA]

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Online publication: June 17, 1999