4. An example: mass transfer in red giant - neutron star binaries
Consider a binary system composed of a neutron star and a less evolved star. The less evolved star burns fuel, expanding and chemically evolving. If the binary orbital period is sufficiently short, the evolving star will eventually fill its Roche lobe and transfer mass to its companion. We now consider this problem, in the case where mass transfer starts while the donor is on the red giant branch (The so-called case B. See e.g.: Iben & Tutukov 1985 .).
The global properties of an isolated star are functions of the stellar mass and time from zero age main sequence. Alternately, red giant structure can be parameterised by total mass and core mass. Detailed models show that the dependence on total mass is weak, so the radius and luminosity are nearly functions of the core mass, only. We use the fit formulae by Webbink (1975 ), who writes:
Here , and the parameters , and are the result of fits to the red giant models (See Table 3.). Eq. (67) assumes that the red giant's luminosity comes solely from shell hydrogen burning by the CNO cycle, which produces energy at a rate (Webbink et al. 1983 ).
Table 3. Parameters fitted to a series of red giant models, both for Pop I (, ) and Pop II (, ). Data are transcribed from Webbink et al. (1983 ), and are applicable over the ranges and for Pop I and Pop II, respectively. Pop I figures due to Webbink (1975 ); Pop II from Sweigart & Gross (1978 )
The red giant eventually fills its Roche lobe, transferring mass to the other star. If mass transfer is stable, according to the criteria in Sect. 3, then one can manipulate the time derivatives of r and to solve for the rate of mass loss from the giant. The stellar and Roche radii vary as:
The first term in Eq. (69) takes into account changes in the Roche radius not due to mass transfer, such as tidal locking of a diffuse star or orbital decay by gravitational wave radiation. For the models considered here, the Roche lobe evolves only due to mass transfer. Eqs. (69), (70), and (71) then reduce to the two equations:
is a consequence of the star being a red giant.
The equations for the evolution of the core mass, red giant mass, neutron star mass, orbital period, and semimajor axis (obtainable from Eqs. (67), (73), (74), (11), (31), and (30)), form a complete system of first order differential equations, governing the evolution of the red giant and the binary. The core mass grows, as burned hydrogen is added from above. This causes the star's radius to increase, which forces mass transfer, increasing a and P, at .
4.1. The code
The program used to generate the numbers presented here follows the prescription outlined by Webbink et al. 1983 , in the treatment of initial values, numeric integration of the evolution, and prescription for termination of mass transfer. The solar mass, radius, and luminosity are taken from Stix (1991 ).
Initial values were provided for the masses of the neutron star, the red giant, and its core at the start of the contact phase (red giant filling its Roche lobe): , , and , respectively, as well as for the parameters (, , ...) of the mass transfer model. A tidally-locked system was assumed and spin angular momentum of the stars neglected. The program then solved for the orbital period P and separation, a, using relations (2), (63), and (65).
Integration of Eqs. (10), (6), (67), and (73) was performed numerically by a fourth-order Runge-Kutta scheme (cf.: Press et al. 1986 ) with time steps limited by , , and . The first two criteria are those used by WRS; the last was added to this code, to insure that care is taken when the envelope mass, , becomes small toward the end of the integration.
Detailed numeric calculations (Taam 1983 ) show that a red giant cannot support its envelope if . The code described here follows that of WRS and terminates mass transfer at this point.
An overdetermined system of , , , and P was integrated numerically by the code, allowing for consistancy checks of the program. Tests performed at the end of the evolution included tests of Kepler's law (Eq. (2)), angular momentum evolution (Eq. (29) or (42)), and a check of the semi-detatched requirement . Plots of various quantities versus red giant core mass show the evolutionary histories of binary systems, in Figs. 8, 9, and 10. Population I stars were used in all calculations (see Table 3 ; ).
Re-emission would presumably be in the form of propeller ejecta (Ghosh & Lamb (1978 ) or a bipolar outflow, such as the jets seen in the galactic superluminal sources. The evoution of a binary, subject to this constraint, is shown in Fig. 11. Comparing Fig. 11 with Figs. 8 and 9, one sees that the low core mass systems, with their corresponding low mass transfer rates, mimic conservative systems. The faster evolving systems behave more like systems with pure isotropic re-emission.
Almost all ring-forming systems are unstable to thermal and/or dynamical timescale runaway of mass transfer, and are not displayed.
© European Southern Observatory (ESO) 1997
Online publication: April 6, 1998