## 4. An example: mass transfer in red giant - neutron star binariesConsider 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 ).
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 So long as the star remains in contact with its Roche lobe, 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 ## 4.1. The codeThe 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 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
Another interesting case of mass transfer is isotropic re-emission at the minimum level necessary to ensure Eddington limited accretion: 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 |