Astron. Astrophys. 342, 464-473 (1999)

4. Computational results

In Fig. 1, we compare the nuclear energy generation rates under the condition of = 1.5 and = 10⁶ g cm^-3 for REACLIB (case C) and HA85 (see also Wormer et al. 1994). We cannot discern the appreciable differences between them: the effects of uncertainties of nuclear data are not clear. This is not the case for X-ray bursts as described below.

Fig. 1. Comparison of the energy generation rate under the constant temperature = 1.5 and density = 106 g cm-3. Solid line is the results with the use of new rates (REACLIB) and dotted line is those with old rates (HA85).

According to the spherically symmetric models, to ignite the shell flashes we have (Hanawa & Fujimoto 1982). However, the parameters cannot be specified well from the one-zone model, because the ignition conditions depend sensitively on , helium abundances, and . Therefore, we have selected the following parameter sets of the pressure and the surface gravity: (log P, log ) = (22.75, 14.5), (22.9, 14.75), (23, 14.25), (23, 14.5), and (23, 14.75), respectively; these are the cases with an appreciable amount of hydrogen left unburnt after the flash (see HSH) which are relevant to the present investigation. It is noted that in these ranges of parameters, we see that for km.

Let us study the effects of uncertainties of the nuclear data on the shell flash using the typical parameter (log P, log ) = (23, 14.75) with the initial composition HCNO. Before the stage of the flash, the HCNO cycle is regulated by intervening -decays; the energy generation rate is governed by the stable hydrogen burning of . Then, the shell flash begins to occur as is seen from Fig. 2. Note that the scale of the lower abscissa corresponds to cases A and B, and that of the upper one corresponds to case C. The nuclear process over a shell flash can be classified in three categories. For , HCNO cycle operates rather slowly. Beyond , break out from HCNO cycle occurs very rapidly through the reactions of and which trigger the explosive combined hydrogen and helium burning . The ignition of the flash leads to the first sharp peak in the energy generation rate as seen in Fig. 2 which shows the formation of the iron peak elements. The second peak corresponds to the formation of ⁵⁶Ni. Transition from ⁵⁶Ni to ⁶⁴Ge makes the third peak; the peaks in cases A and B is wider compared with case C due to the effects of the different Q-values for ⁶⁴Ge. A small difference between cases A and B is seen for s because the transition of the abundance peak from ⁶⁴Ge to ⁶⁸Se affects the decrease in the energy generation. When , the rp-process proceeds appreciably beyond . The nucleosynthesis depends on the Q-value of the waiting nuclei; The nuclei shown in Table 2 play an important role to determine the rp-process path. In particular, the Q-value of ⁶⁸Se is not yet known; proton drip line along some key nuclei is uncertain. Schatz et al. (1998) assumed the Q-value of keV and described the uncertainty in Q-values of (p,) reactions.

Fig. 2. Comparison of the energy generation rate during the flash for the models of log P = 23, log = 14.75. Solid line: case A, dotted line: case B (scale of the lower abscissa), and dashed line: case C (scale of the upper one).

From Fig. 3, once the breakout from the HCNO cycle occurs, the nuclear flash leads to the peak temperature . The typical changes in the temperature and the density can be seen in Fig. 4. However, comparison of Figs. 5 and 6 reveals that new reaction rates change the time variation of compositions significantly; break out from the HCNO cycle is appreciable for case C before the depletion of . Consequently, before the shell flash begins the rp-process proceeds up to the formation of and through successive (p,) and (,p) reactions changing the nucleosynthesis path by -decays; this leads to steep rise in temperature during the flash phase due to abundant seed heavy nuclei as seen in Figs. 3 and 6. Then, compared with case B or HSH, more nuclear energy has been released when the flash begins; higher peak temperature is attained as illustrated in Fig. 4: the locus extends to lower density for case C. The effects of the Q-values are clear as inferred from Table 2. For case A, the waiting point results from the decays of ⁶⁴Ge and ⁶⁸Se. For case B, decay of ⁶⁸Se corresponds to the final waiting point. For case C, decay of ⁷² Kr might lead to a new waiting point as suggested by Mathews (1991). The large differences in the Q-values for affect the degree of the decrease in the tails as seen in Figs. 2 and 3. When the temperature decreases down to K, only weak interactions are active. As is seen from Tables 3 and 4, the final products at this temperature depend significantly on the nuclear data, which could affect the modeling of type I X-ray bursts in especially 10 minutes intervals.

Fig. 3. Time variation of the temperature during the flash for the same models as in Fig. 2.

Fig. 4. Same as Fig. 2 but for the changes of the temperature and the density during the flash.

Fig. 5. Changes of mass fractions during the flash. Case B of log P = 23, log = 14.75.

Fig. 6. Same as Fig. 5 but for case C.

Table 4. Final mass fractions after the shell flashes for calculated models with the HCNO initial compositions. All models are computed by case C.

We should note that since the thermal history is crucial to X-ray bursts as is shown in Figs. 2, 3, and 4, the condition of the constant temperature and density or an artificial assumption of "adiabatic expansion" (e.g. Rembges et al. 1997) is inadequate to investigate the nuclear process during the shell flashes.

We can examine how the shell flashes are affected by the initial compositions as shown in Table 3. Since the solar seeds have more heavy elements of , the elapsed time to the peak temperature is shorter than that for HCNO by an order of magnitude. However, the values of the peak temperature and the amounts of the final products are similar for both initial compositions though hydrogen is rather consumed for the initial Solar abundances by the seed nuclei. Therefore, as far as the simple one zone model is concerned, it is reasonable to adopt the initial compositions as HCNO.

The radiative luminosity is given as , where and R are fixed in our sequence of calculations. Then, light curve of our model is characterized in terms of as shown in Fig. 7. We should note that profiles of the light curves are affected significantly by the different nuclear data. On the other hand, the Eddington luminosity in the local frame is given by

Fig. 7. Light curves with the same parameters as in Fig. 2.

With the use of the corresponding Eddington luminosity , we could obtain

It appears that if (see Tables 2 and 4). However, it should be noted that the condition of K is satisfied around the bottom of the burning shell in the actual situation: in evaluating , R would be the radius of the photosphere and will be different layer by layer inside the neutron star atmosphere. Then, remains below as long as both the spherically symmetric configuration (Hashimoto et al. 1993) and hydrostatic equilibrium are assumed (see also Ebisuzaki et al. 1983).

In Table 4 we presented the final abundances with the use of the nuclear data of case C for several sets of parameters which had been performed by HSH. We can see that the final products depend on the peak temperature and the radiative energy loss rate; contrary to the results of HSH, if log 14.5 and 1.5, the main final products are not ⁶⁸Ge but ⁷⁴Se and ⁷⁴ Kr. This indicates that the waiting point is beyond ⁶⁸Se and the flow of abundances will be beyond Kr isotopes. However it must be noted that the final products and/or the amount of the fuel left after the flash depend on both the Q-value of seen in Table 3 and the radiative energy loss rate which may be too simplified for our one zone model.

© European Southern Observatory (ESO) 1999

Online publication: February 22, 1999
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