## 3. Linear stability analysis of the mass transferMass transfer will proceed on a timescale which depends critically on the changes in the radius of the donor star and that of its Roche lobe in response to the mass loss. The mass transfer might proceed on the timescale at which the mass transfer was initially driven (e.g.: nuclear, or orbital evolutionary), or at one of two much higher rates: dynamical and thermal. If a star is perturbed by removal of mass, it will fall out of hydrostatic and thermal equilibria, which will be re-established on sound crossing (dynamical) and heat diffusion (Kelvin-Helmholtz, or thermal) timescales, respectively. As part of the process of returning to equilibrium, the star will either grow or shrink, first on the dynamical, and then on the (slower) thermal timescale. At the same time, the Roche lobe also grows or shrinks around the star in response to the mass loss. If after a transfer of a small amount of mass, the star's Roche lobe continues to enclose the star, then the mass transfer is stable, and proceeds on the original driving timescale. Otherwise, it is unstable and proceeds on the fastest unstable timescale. In stability analysis, one starts with the equilibrium situation and examines the small perturbations about it. In this case, the question is whether or not a star is contained by its Roche lobe. Thus, one studies the behaviour of the quantity which is the (dimensionless) variation in the difference in radius between the star and its Roche lobe, in response to change in that star's mass. Here is the difference between the stellar radius and the volume-equivalent Roche radius . The star responds to this loss of mass on two widely separated different timescales, so this analysis must be performed on both of these timescales. The linear stability analysis then amounts to a comparison of the exponents in power-law fits of radius to mass, : where and Each of the exponents is evaluated in a manner consistent with the physical process involved. For , chemical abundance and entropy profiles are assumed constant and mass is removed from the outside of the star. For , mass is still removed from the outside of the star, but the star is assumed to be in the thermal equilibrium state for the given chemical profile. In calculating , derivatives are to be taken along the assumed evolutionary path of the binary system. In the following subsections these exponents are described a bit further and computed in the case of mass loss from a binary containing a neutron star and a Roche lobe-filling red giant. Such systems are thought to be the progenitors of the wide orbit, millisecond pulsar, helium white dwarf binaries. They are interesting, both by themselves, and as a way of explaining the fossil data found in white dwarf - neutron star binaries. The problem has been treated by various authors, including Webbink et al. (1983 ), who evolved such systems in the case of conservative mass transfer from the red giant to the neutron star. ## 3.1. Adiabatic exponent:The adiabatic response of a star to mass loss has long been understood (see, for example, Webbink (1985 ) or Hjellming & Webbink (1987 ) for an overview), and on a simplistic level, is as follows. Stars with radiative envelopes (upper-main sequence stars) contract in response to mass loss, and stars with convective envelopes (lower-main sequence and Hayashi track stars) expand in response to mass loss. The physics is as follows. A star with a radiative envelope has a positive entropy gradient near its surface. The density of the envelope material, if measured at a constant pressure, decreases as one samples the envelope at ever-increasing radii. Thus, upon loss of the outer portion of the envelope, the underlying material brought out of pressure equilibrium expands, without quite filling the region from which material was removed. The star contracts on its dynamical timescale, in response to mass loss. A star with a convective envelope has a nearly constant entropy profile, so the preceding analysis does not apply. Instead, the adiabatic response of a star with a convective envelope is determined by the scalings among mass, radius, density, and pressure of the isentropic material. For most interesting cases, the star is both energetically bound, and expands in response to mass loss. Given the above physical arguements, the standard explanation of mass-transfer stability is as follows. A radiative star contracts with mass loss and a convective star expands. If a convective star loses mass by Roche lobe overflow, it will expand with possible instability if the Roche lobe does not expand fast enough. If a Roche lobe-filling radiative star loses mass, it will shrink inside its lobe (detach) and the mass transfer will be stable. This analysis "is of only a meagre and unsatisfactory kind" (Kelvin 1894 ), as it treats stellar structure in only the most simplistic way: convective vs. radiative envelope. One can quantify the response of a convective star by adopting some
analytic model for its structure, the simplest being an isentropic
polytrope. This is a model in which the pressure and the constituent gas has an adiabatic exponent related to the polytropic index through . Other slightly more realistic cases include those with , applicable to radiative stars; composite polytropes, with different polytropic indices for core and envelope; and centrally condensed polytropes, which are polytropes with a point mass at the center. These are all considered in a paper by Hjellming & Webbink (1987 ). We use the condensed polytropes, as they are simple, fairly realistic models of red giant stars, which tend in the limit of low envelope mass to (pointlike) proto-white dwarfs which are the secondaries in the low mass-binary pulsar systems. What follows is a brief treatmant of standard and condensed polytropes, as applicable to the adiabatic response to mass loss. Scaling arguements give for standard
polytropes. The pressure is an energy density and consequently scales
as ; density scales as .
The polytropic relation (Eq. (49)) immediately gives the scaling
between In particular, when
().
A far better approximation to red giant structure, and only slightly more complex, is made by condensed polytropes, which model the helium core as a central point mass (see, e.g.: Hjellming & Webbink (1987 )). Admittedly, this is a poor approximation, as concerns the core. However, the star's radius is much greater than that of the core, so this is a good first-order treatment. Furthermore, differences between this approximation and the actual structure occur primarily deep inside the star, while the star responds to mass loss primarily near the surface, where the fractional change in pressure is high. Overlap between the two effects is negligible. Analysis of the condensed polytropes requires integrating the
equation of stellar structure for isentropic matter (Lane-Emden
equation), to get a function , and
differentiating The Lane-Emden equation is (cf.: Clayton (1968 )): Here, is the scaled radial coordinate and
is the density, scaled to its central value
The polytropic radius is set by the position of the first root of
and is therefore at .
Similarly, . Eq. (51) shows that the density
() may be rescaled, without affecting the
length scale, so Alternately, the polytrope admits a length scale, which depends only on specific entropy, so the polytrope's radius is independent of the mass. Generalising to condensed polytropes, Eq. (52) suggests the extension: For this model, the stellar radius, stellar mass, and core mass are for some and . The core mass fraction is a monotonic function in , and increases from 0 to 1 as increases from 0 to . As the core mass fraction increases towards 1, the polytrope's radius decreases from to 0, so the more condensed stars are also the smaller ones. The adiabatic exponent should be evaluated at constant core mass, as opposed to mass fraction. Thus, for condensed polytropes, where is chosen to solve Eq. (57). This solution matches that given by Eq. (50) when there is no core. Furthermore, is an increasing function of the core mass fraction, which diverges, as tends towards unity. These are general features of the condensed polytropes, and hold for polytropic indices . The procedure for calculating is described in detail in Hjellming & Webbink (1987 ). Results for a variety of core mass fractions of polytropes are given, both in Table 2 and graphically, in Figs. 1 and 3.
The function can be reasonably well fit by the function: (Hjellming & Webbink 1987 ), and to better than a percent by either of the functions (in order of increasing accuracy): as shown graphically in Fig. 2.
## 3.2. Thermal equilibrium exponent:Analytic and numeric modeling of red giants have shown that the luminosity and radius depend almost entirely on the mass of the star's helium core (Refsdal & Weigert 1970 ; cf.: Verbunt (1993 )). Since the core mass changes on the nuclear timescale, and we are interested in changes in the radius on the (much shorter) thermal timescale, the radius may be taken as fixed, giving . ## 3.3. Roche radius exponent:Since the exponent must be computed
according to the evolution of the binary with mass transfer, it is
sensitive to mass transfer mode, as are ,
We rewrite , in a form which depends explicitly on previously calculated quantities: The derivatives and appear in Sect. (2), both for the unified model, and for the ring; as well as in a tabulated form in Appendix A. All that remains is , which will be calculated using Eggleton's (1983 ) formula for the volume-equivalent Roche radius: To get an idea of how changing the mode of mass transfer effects
stability, has been plotted vs. One notices several things in these graphs. Fig. 4 shows the
extreme variation in with changes in mode of
mass transfer. Each of the three curves 'ring', 'wind', and 'iso-r'
differs greatly from the conservative case. At least as important is
the extent to which they differ from one another, based only on the
way in which these three modes account for the variation of angular
momentum with mass loss. When angular momentum is lost at an enhanced
rate, as when it is lost to a ring or a wind from the less massive
star (direct wind, at low-
Second, Fig. 4 shows that ring formation leads to rather high , even for modest . The formation of a ring will usually lead to instability on the dynamical timescale (). The slower thermal timescale instability will occur only if the giant is rather evolved and has a high coremass fraction (with its high ). Delivery of mass to a ring would imply either a substantially larger, simultaneous flow of mass through the stabler of the two Jeans's channels, or else orbital decay, leading to dynamical timescale mass transfer. Fig. 6 shows the variation of due only
to the differences between the isotropic wind and isotropic
re-emission, in a family of curves with . At
, stability is passed from the mostly isotropic
re-emission modes at low
For , mass transfer is stabilised by trading
isotropic re-emission for wind, so families of
curves lie below their respective curves in
this region of the A family of curves with one wind of fixed strength and one wind of
varying strength (as shown in Fig. 5, with
, and variable ), will
also intersect at some value of Notice in each case, intersections of various unified wind model
curves occur at . One
can understand this, in the framework of the arguments of Sect. (A).
Depending on which side of one lies, either
or has the majority of the
angular momentum. When , angular momentum will
be lost predominantly by the isotropic wind and
will be high. When , angular momentum losses
will be modest and low. The situation is
reversed for isotropic re-emission. The two curves form a figure-X on
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