Astron. Astrophys. 346, 831-842 (1999) 4. ResultsWe present here our results, which we obtained with the model described above. They depend on the following parameters: , and two values for q, given as and , which define the lower and upper boundaries of the mass shell where the ^{44}Ti-containing clumps are assumed to exist. As we mentioned earlier, the values of the first three can be constrained in the case of Cas A from the observational analyses of the blastwave radius and velocity . Below we consider mainly the parameter space which is consistent with these constraints. While we treat as a free parameter, a value of the order of 10 seems to be reasonable as a first guess, based on the density contrast which develops in hydrodynamic instabilities in the envelope of the progenitor star during the supernova explosion (Fryxell et al. 1991). 4.1. Thermodynamic evolution of post-shock clumpsFig. 1 shows the evolution of the ^{56}Fe number density and of the electron temperature in the ^{56}Fe-dominated clumps at four different locations in the ejecta, given by the corresponding mass coordinate q, after the reverse shock has passed the clumps. The post-shock expansion decreases the density as well as the equilibrium temperature, while first increases in its attempt to reach equilibrium between electrons and ions. The results are plotted for the chosen set of parameter values quoted therewith.
4.2. Effective decay rates of ^{44}Ti and its radioactivity as an observableLet us now consider ^{44}Ti in a clump that is located anywhere in the mass coordinate of the ejecta. If ^{44}Ti is near the surface (large q), it suffers from the reverse shock early. Since the electron temperature is still relatively low (see Fig. 1), there is no possibility to reach a high degree of ionization (thus an appreciable reduction of the decay rate). On the other hand, ^{44}Ti embedded in clumps located in the innermost region may become highly ionized as the temperature is generally highest, resulting in the smallest ^{44}Ti decay rate. Since is closer to t, the "age" of the remnant, however, the net effect of the reduced decay rate appears weaker. Therefore, the maximum effect of the retarded -decay is obtained if ^{44}Ti is at intermediate values of q. This situation is depicted in the panels of Fig. 2.
The top panel of Fig. 2 shows the average numbers of electrons that are bound to ^{44}Ti and to ^{56}Fe at and 300 years after the explosion. The increase of toward the low side of q indicates that the time span between and t was not long enough for ionization. At the high end of q-values, the effects of post-shock recombination are visible. When combined with from Fig. 1, Fe) determines the total number of ionization electrons, , in a clump: Fe)] with . The retardation of the ^{44}Ti decay by ionization is illustrated in the middle panel of Fig. 2. The effective decay rate in the post-shock period, , is defined as the time-average of [given by Eq. (18) in Appendix A] between the time and a time . It can be expressed as where is the ^{44}Ti abundance at mass coordinate q and time (age) t, and is the corresponding value at the shock impact time , which has been reduced from its initial value according to . In order to estimate the observable consequences of the reduced -decay rates, one has to recall that the measurable -ray activity per (normalized) unit mass of the remnant is the product of the current -decay rate and the current abundance, . This introduces a nonlinear relation between the observable activity given in the bottom panel of Fig. 2 and the effective decay rates displayed above, particularly in the absence of quick recombination. In Fig. 3, we compare the results obtained with the input physics as described in Sect. 3 to the cases where either the non-adiabaticity of the equilibration process of electrons and ions owing to the energy consumption by the ionization process is neglected (upper panel ), or where instantaneous equilibration (i.e., always) is assumed (lower panel ). It can be seen that in both cases the trends of the "full" model shown in Fig. 2, where both the non-adiabaticity and the gradual equilibration process are taken into account, are somewhat enhanced. This can be understood by the fact that the omission of either of the two effects leads to higher electron temperatures and thus to a higher degree of ionization in the clumps.
4.3. ^{44}Ti radioactivity in Cas AThe ^{44}Ti radioactivity of the whole supernova remnant as observable at an age t, relative to the activity level that would result in the absence of any retardation of the -decay rate by ionization, is defined by the ratio Here, is the total number of ^{44}Ti nuclei in the ^{56}Fe clumps at . For a remnant like Cas A, which is in the transition phase from the ejecta-dominated stage to the Sedov-Taylor phase, the reverse shock has passed through most of the ejecta. Therefore, the factor must be expected to be larger than unity when the remnant is observed at this stage of its evolution. This implies that the initial abundance inferred from the current -ray activity due to the ^{44}Ti decay is lower by a factor than the amount of produced ^{44}Ti that is estimated on the basis of the laboratory decay rate . For certain combinations of the characterizing parameters of the supernova remnant model, , and , the delayed decay of ionized ^{44}Ti can have a sizable effect on the -ray activity of Cas A which is observable at the present age of about 320 years. In Tables 1-3 the ratio denotes the currently observable activity according to our model, normalized to the expected activity if ionization effects are not taken into account. The results depend on the assumed density enhancement in the clumps relative to the density of the homogeneous ejecta, and on the location of the clumps within a shell in the expanding ejecta bounded by the lower and upper mass coordinates and , respectively. In addition, in Tables 1-3 numbers are given for the blastwave radius and the blastwave velocity as predicted by the McKee & Truelove (1995) model at the current age of the Cas A supernova remnant. Table 1. Illustration of the variability of the ^{44}Ti radioactivity ratios as defined by Eq. (12) with different values of the model parameters. The tabulated results are at the time (age) of 320 y after the explosion for an explosion energy of erg and various combinations of values for the ejecta mass , the hydrogen number density in the ambient medium, , the over-density factor of the clumps, , and the lower and the upper boundaries and of the mass shell where the clumps are assumed to exist. and are the radius and velocity of the blastwave at that time as computed from the McKee & Truelove (1995) model for the given sets of and values Table 2. Same as Table 1 but for erg Table 3. Same as Table 1 but for erg The ratio exhibits the following tendencies. For fixed explosion energy and ejecta mass, increases with the density of the circumstellar gas of the supernova remnant because the expansion and dilution of the ejecta are slowed down for higher (i.e., and are smaller at the same time). An increase of the ratio can also be seen when the explosion energy becomes larger but all other parameters are kept constant. The opposite trend is visible if the explosion energy and ambient density are fixed, in which case the ratio decreases with larger ejecta mass. Although the blastwave velocity at the present time is higher, it was lower early after the explosion and therefore the blastwave radius is smaller for the larger ejecta mass. It is not easy to give simple explanations for these trends. The ratio , which scales with the current abundance of ^{44}Ti and its effective decay rate, reflects the whole time evolution of the remnant and contains contributions from all parts of the shocked ejecta with their different ionization histories. The nonlinear dependence of on the remnant parameters is a result of the interplay between a number of effects. For example, the post-shock density and temperature in the ejecta, in particular in the clumps, are important for the degree of ionization. The time when the reverse shock hits the clumps determines the time left for the equilibration between electrons and ions and for the duration of the delay of the ^{44}Ti decay. This delay lasts from the moment of nearly complete ionization until recombination takes place again. In order to obtain a large effect on the ^{44}Ti radioactivity that is measured in Cas A presently, high post-shock density and temperature are favorable for efficient ionization of ^{44}Ti. On the other hand, the present-day -ray emission of Cas A means that the temperature must not have decreased too slowly and the density not too rapidly. If that happened, ^{44}Ti would be hindered from recombination and the observed -ray emission could not be explained. The maximum effect from the delayed decay of ionized ^{44}Ti is found if most of the ^{44}Ti-carrying clumps have been mixed roughly half-way into the ejecta, i.e., if they are assumed to be located in the q-interval between 0.4 and 0.6 (compare Fig. 2), and if the density enhancement in the clumps is around 10 for the higher explosion energies of and (Tables 2 and 3, respectively). In case of the lowest considered value of the explosion energy, (Table 1), a clump over-density of about 5 seems preferable because of the higher density of the more slowly expanding ejecta. Higher than optimum -values do not ensure rapid and strong ionization, whereas smaller than optimum -values do not allow for quick recombination so that the decay rate stays low. If ^{44}Ti is located very far out in the remnant (large q), the temperature behind the reverse shock is too low for efficient ionization (see Fig. 1). If ^{44}Ti sits deep inside the ejecta (small q), the clumps have been reached by the reverse shock not sufficiently long ago so as to show a significant effect from ^{44}Ti ionization. We found maximum values of between roughly 1.5 and 2.5 for a large number of different combinations of remnant parameters and assumed clump locations and density enhancement factors. This means that the present-day ^{44}Ti radioactivity of Cas A could be 50% up to more than a factor of 2 higher for a certain amount of ^{44}Ti, if the decay half-life of ^{44}Ti was stretched by ionization effects. Or, reversely, the current -ray emission of Cas A due to ^{44}Ti decay might be explained with a significantly lower production of this nucleus during the supernova explosion. Fig. 4 shows that the relative effect from the retardation of the ^{44}Ti decay will increase with the age of the remnant.
For the different considered combinations of parameter values, Cas A has reached slightly different stages of the remnant evolution around the transition time from the ejecta-dominated to the Sedov-Taylor phase. The tabulated blastwave radius and velocity can be compared with observations which give values for the circumstellar density of about 20 hydrogen atoms per cm^{3}, an estimated ejecta mass of about and a blastwave radius of 2-3 pc at a distance of 3.4 kpc (see references given in Sect. 2). © European Southern Observatory (ESO) 1999 Online publication: June 17, 1999 |