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

4. Results

We 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 ⁴⁴Ti-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 clumps

Fig. 1 shows the evolution of the ⁵⁶Fe number density and of the electron temperature in the ⁵⁶Fe-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.

Fig. 1. Examples for the evolution with time t (in years) of the 56Fe number density (top ), and of the electron temperature (bottom ) in the 56Fe-dominated clumps at four different locations indicated by the mass coordinate q. The dashed lines represent the asymptotic behavior of the equilibrium temperatures (as described in Sect. 3.3.2) at and 0.6. The left edge of each solid curve corresponds to . For example, the reverse shock took 50 y to reach . The results are given for the following set of parameter values: erg, , cm-3 and

4.2. Effective decay rates of ⁴⁴Ti and its radioactivity as an observable

Let us now consider ⁴⁴Ti in a clump that is located anywhere in the mass coordinate of the ejecta. If ⁴⁴Ti 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, ⁴⁴Ti embedded in clumps located in the innermost region may become highly ionized as the temperature is generally highest, resulting in the smallest ⁴⁴Ti 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 ⁴⁴Ti is at intermediate values of q. This situation is depicted in the panels of Fig. 2.

Fig. 2. Top panel: The average numbers of electrons that are bound to 44Ti and to 56Fe at mass coordinate q and at time t (in y) after the explosion. Middle: Effective decay rates of the shocked 44Ti during the time span between and t [Eq. (11)] in units of the laboratory rate . Bottom: The corresponding 44Ti radioactivity observable at time t ("age") relative to the case that assumes no reduction of the -decay rate. The parameter values used here are the same as those for Fig. 1

The top panel of Fig. 2 shows the average numbers of electrons that are bound to ⁴⁴Ti and to ⁵⁶Fe 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 ⁴⁴Ti 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 ⁴⁴Ti 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.

Fig. 3. 44Ti radioactivities at times t (in y) in the clumps located at radial positions q, relative to the radioactivity of the model with most detailed input physics, when the non-adiabaticity of the thermal evolution of the clumps is ignored (upper panel ) and when instantaneous equilibration between and (toward T) is assumed (lower panel )

4.3. ⁴⁴Ti radioactivity in Cas A

The ⁴⁴Ti 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 ⁴⁴Ti nuclei in the ⁵⁶Fe 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 ⁴⁴Ti decay is lower by a factor than the amount of produced ⁴⁴Ti 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 ⁴⁴Ti 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 ⁴⁴Ti 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 ⁴⁴Ti 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 ⁴⁴Ti 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 ⁴⁴Ti radioactivity that is measured in Cas A presently, high post-shock density and temperature are favorable for efficient ionization of ⁴⁴Ti. 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, ⁴⁴Ti would be hindered from recombination and the observed -ray emission could not be explained. The maximum effect from the delayed decay of ionized ⁴⁴Ti is found if most of the ⁴⁴Ti-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 ⁴⁴Ti 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 ⁴⁴Ti 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 ⁴⁴Ti 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 ⁴⁴Ti radioactivity of Cas A could be 50% up to more than a factor of 2 higher for a certain amount of ⁴⁴Ti, if the decay half-life of ⁴⁴Ti was stretched by ionization effects. Or, reversely, the current -ray emission of Cas A due to ⁴⁴Ti 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 ⁴⁴Ti decay will increase with the age of the remnant.

Fig. 4. The decrease with time of the amount of 44Ti in the supernova remnant, , by -decays, starting from the initial amount (right scale in logarithm), and the corresponding, mostly increasing activity ratio (left scale ). Values of 3 (upper panel ) and 5 (lower ) are used for , while erg, cm-3 and are adopted in both cases. The different lines show results obtained for different assumptions of the radial locations of the clumps: = [0.0, 0.5] (solid line ), [0.4, 0.6] (dash-dotted line ), and [0.2, 0.8] (dashed line ). The exponential decrease of the 44Ti abundance with the laboratory rate is plotted by the dotted line

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 cm³, 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
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