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

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2. Type I X-ray burst model

As a type I X-ray burst model for the rp-process, we adopt a plane parallel approximation by Fujimoto et al. (1981). This model is reasonable enough to investigate the nuclear process during the shell flash if we assume that physical quantities are averaged over the accumulated layers, hydrostatic equilibrium is maintained, and the configuration is spherically symmetric. In fact, Hashimoto et al. 1983, and HSH have performed the calculation of the nucleosynthesis under these assumptions. Let us summarize the formulation of the model for the following discussion.

A hydrostatic equilibrium equation to determine the structure of the accreting neutron star is written as follows:

[EQUATION]

Here, P is the pressure and [FORMULA] is the rest mass contained interior to the radius r. For a plane parallel configuration which can be legitimate as far as the ratio of the radius to the pressure scale height is as large as [FORMULA], the column mass density [FORMULA] and the surface gravity [FORMULA] are introduced. Using the total gravitational mass [FORMULA], the accreted proper mass [FORMULA] and the radius R at the surface, they are defined as follows:

[EQUATION]

[EQUATION]

where [FORMULA] is the general relativistic correction factor of Schwarzschild metric. We have [FORMULA] = 1.3 and log [FORMULA] = 14.4 for a model with [FORMULA] and [FORMULA] km. The amount of accreted matter [FORMULA] can be estimated from [FORMULA] to [FORMULA] (Fujimoto et al. 1987).

Eq. (1) is integrated and reduced to be a constant pressure if we adopt a plane parallel approximation:

[EQUATION]

where [FORMULA] cm s-2 and [FORMULA] g cm-2 are considered to be parameters which are assumed to be constant during the burst. For example, around the ignition, we have [FORMULA] for [FORMULA] with the helium mass fraction [FORMULA] and that of CNO elements [FORMULA] (Fujimoto et al. 1987).

The energy equation is written as

[EQUATION]

where T is the temperature, [FORMULA] is the specific heat at the constant pressure, [FORMULA] is the nuclear energy generation rate, and [FORMULA] is the neutrino energy loss rate by [FORMULA]-decays associated with the nuclear reactions. The neutrino loss due to the direct interactions between electrons and neutrinos can be neglected during the flash. Radiative energy loss rate [FORMULA] is approximated by

[EQUATION]

where [FORMULA] K, a is the radiation density constant and [FORMULA] is the mean molecular weight per electron which in the early phase of the flash would be approximated to be [FORMULA] with the hydrogen mass fraction X, when hydrogen is dominant. For opacity [FORMULA], the Compton scattering opacity is adopted (e.g. Ebisuzaki et al. 1983).

The nuclear reaction network has been coupled to the thermodynamical equations through [FORMULA]. The rate equations of abundance are written as follows:

[EQUATION]

Here the first and second terms account for the destruction and production, respectively, of the 0-th abundance [FORMULA] and [FORMULA] expresses the rate of reaction or decay [FORMULA] [FORMULA] [FORMULA], where [FORMULA] denote the species of particles concerned: nucleus, neutron, proton, electron, positron, neutrino, antineutrino, and photon.

Once a set of parameters [FORMULA] or [FORMULA] is specified, using the initial values of temperature and abundances, we can get density and other thermodynamical quantities for the next time step from the equation of state. Here the total pressure P consisted of the contributions from partially relativistic and/or degenerate electrons and positrons in thermal equilibrium, ions, radiation, and the Coulomb interactions (Slattery et al. 1982, Yakovlev & Shalybkov 1989):

[EQUATION]

We note that in the calculation of [FORMULA], we have taken into account the effects of non-ideal gas as P. The maximum temperature [FORMULA] can be estimated from [FORMULA]:

[EQUATION]

Here, the region is assumed to be radiation dominated around the peak of the flash. HSH have examined the shell flashes for [FORMULA] and [FORMULA]; During the shell flash, the peak temperature in units of [FORMULA] K ranges from [FORMULA] and the corresponding density ranges from [FORMULA] g cm-3.

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© European Southern Observatory (ESO) 1999

Online publication: February 22, 1999
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