3. Results of nucleosynthesis calculations
In the first example, we start with a relativistic flow (polytropic index ) with the accretion rate , where, is the Eddington accretion rate. We use the mass of the central black hole to be throughout. We choose a very high viscosity and the corresponding parameter (Shakura & Sunyaev 1973) is 0.2 in the sub-Keplerian regime. The cooling is not as efficient as in a Keplerian disk: , where, and are the heat generation and heat loss rates respectively. The specific angular momentum at the inner edge is (in units of ). The flow deviates from a Keplerian disk at 4.15 Schwarzschild radii. It is to be noted that includes all possible types of cooling, such as bremsstrahlung, Comptonization as well as cooling due to neutrino emissions. We assume that the flow is magnetized so that only ions have larger viscosity. Due to poor supply of the soft photons from Keplerian disks, the Comptonization in the boundary layer is not complete: we assume a standard value (Chakrabarti, & Titarchuk 1995) in this regime: , i.e., ions (in te radiation-hydrodynamic solution) are one-tenth as hot as obtained from the hydrodynamic solutions. [For high accretion rate, , and ions and electrons both cool to a few KeV ( oK)]. The typical density and temperature near the marginally stable orbit are gm cm-3 and oK respectively where the thermonuclear depletion rates for the , and reactions are given by gm-1 s-1, gm-1 s-1 and gm-1 s-1 respectively. Here, is the element abundance on the left, is the reaction cross-section, v is the Maxwellian average velocity of the reactants. At these rates, the time scales of these reactions are given by, s, s and s respectively indicating that the deuterium burning is the fastest of the reactions. In fact, it would take about a second to burn initial deuterium with . The does not form at all because the dissociates to D much faster.
The above depletion rates have been computed assuming Planckian photon distribution corresponding to ion temperature . The wavelength at which the brightness is highest at is shown in Fig. 1 in the dashed curve (in units of cm). Also shown is the average wavelength of the photon (solid curve) obtained from the spectrum . The average has been performed over the region 2 to 50 keV of the photon energy in which the hard component is usually observed
where, and are computed from 2 and 50 keV respectively. The average becomes a function of the energy spectral index (), which in turn depends on the ion and electron temperatures of the medium. We follow Chakrabarti & Titarchuk (1995) to compute these relations. We note that is lower compared to for all ion temperatures we are interested in. Thus, the disintegration rate with Planckian distribution that we employed in this paper is clearly a lower limit. Our assertion of the formation of a neutron disk should be strengthened when Comptonization is included.
Fig. 2 shows the result of the numerical simulation for the disk model mentioned above. Logarithmic abundance of neutron is plotted against the logarithmic distance from the black hole. First simulation produced the dash-dotted curve for the neutron distribution, forming a miniature neutron torus. As fresh matter is added to the existing neutron disk, neutron abundance is increased as neutrons do not fall in rapidly. Thus the simulation is repeated several times in order to achieve a converging steady pattern of the neutron disk. Although fresh neutrons are deposited, the stability of the distribution is achieved through neutron decay and neutron capture reactions. Results after every ten iterations are plotted. The equilibrium neutron torus remains around the black hole indefinitely. The neutron abundance is clearly very significant (more than five per cent!).
We study yet another case where the accretion rate is smaller () and the viscosity is so small () and the disk so hot that the sub-Keplerian flow deviates from a Keplerian disk farther away at . The polytropic index is that of a mono-atomic (ionized) hot gas . The Compton cooling factor is as above since it is independent of the accretion rates as long as the rate is low (Sunyaev & Titarchuk 1980; Chakrabarti & Titarchuk 1995). The cooling is assumed to be very inefficient because of lower density: . The specific angular momentum at the inner edge of the disk is . In Fig. 3, we show the logarithmic abundances of proton (p), helium () and neutron (n) as functions of the logarithmic distance from the black hole. Note that dissociates completely at a distance of around where the density and temperatures are gm cm-3 and oK. Maximum temperature attained in this case is oK. Both the neutrons and protons are enhanced for , the boundary layer of the black hole. This neutron disk also remains stable despite neutron decay, since new matter moves in to maintain equilibrium. The abundance is insignificant.
© European Southern Observatory (ESO) 1999
Online publication: March 10, 1999