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Astron. Astrophys. 327, 620-635 (1997)

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3. Linear stability analysis of the mass transfer

Mass 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 [FORMULA] is the difference between the stellar radius [FORMULA] and the volume-equivalent Roche radius [FORMULA]. 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, [FORMULA]:


where [FORMULA] and m refer to the mass-losing, secondary star. Thus, [FORMULA] and [FORMULA]. Stability requires that after mass loss ([FORMULA]) the star is still contained by its Roche lobe. Assuming [FORMULA] prior to mass loss, the stability condition then becomes [FORMULA], or [FORMULA]. If this is not satisfied, then mass transfer runs to the fastest, unstable timescale.

Each of the exponents is evaluated in a manner consistent with the physical process involved. For [FORMULA], chemical abundance and entropy profiles are assumed constant and mass is removed from the outside of the star. For [FORMULA], 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 [FORMULA], 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: [FORMULA]

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 P and density [FORMULA] vary as


and the constituent gas has an adiabatic exponent related to the polytropic index through [FORMULA]. Other slightly more realistic cases include those with [FORMULA], 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 [FORMULA] for standard polytropes. The pressure is an energy density and consequently scales as [FORMULA] ; density scales as [FORMULA]. The polytropic relation (Eq. (49)) immediately gives the scaling between R and M. Since the material is isentropic, the variation of radius with mass loss is the same as that given by the radius-mass relation of stars along this sequence. Consequently, for polytropic stars of index n,


In particular, [FORMULA] when [FORMULA] ([FORMULA]). 4 The above approximation is fine towards the base of the red giant branch, where the helium core is only a small fraction of the star's mass. It becomes increasingly poor as the core makes up increasingly larger fractions of the star's mass, which happens when the star climbs the red giant branch or loses its envelope to RLOF. Mathematically speaking, the scaling law that lead to the formula for [FORMULA] is broken by the presence of another dimensionless variable, the core mass fraction.

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 [FORMULA], and differentiating R at constant specific entropy S (adiabatic requirement) and core mass [FORMULA] (no nuclear evolution over one sound crossing time). In general, the Lane-Emden equation is non-linear, and calculations must be performed numerically. The cases of [FORMULA] and [FORMULA] are linear and analytic. The case of [FORMULA] is presented below, as a nontrivial, analytic example, both for understanding, and because it can be used as a check of numeric calculations of [FORMULA].

The [FORMULA] Lane-Emden equation is (cf.: Clayton (1968 )):


Here, [FORMULA] is the scaled radial coordinate and [FORMULA] is the density, scaled to its central value 5. The substitution [FORMULA] allows solution by inspection:


The polytropic radius is set by the position of the first root of [FORMULA] and is therefore at [FORMULA]. Similarly, [FORMULA]. Eq. (51) shows that the density ([FORMULA]) may be rescaled, without affecting the length scale, so R is independent of M, and [FORMULA].

Alternately, the [FORMULA] polytrope admits a length scale, [FORMULA] 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 [FORMULA] and [FORMULA]. The core mass fraction


is a monotonic function in [FORMULA], and increases from 0 to 1 as [FORMULA] increases from 0 to [FORMULA]. As the core mass fraction increases towards 1, the polytrope's radius decreases from [FORMULA] to 0, so the more condensed stars are also the smaller ones.

The adiabatic [FORMULA] exponent [FORMULA] should be evaluated at constant core mass, as opposed to mass fraction. Thus, for condensed [FORMULA] polytropes,


where [FORMULA] is chosen to solve Eq. (57). This solution matches that given by Eq. (50) when there is no core. Furthermore, [FORMULA] is an increasing function of the core mass fraction, which diverges, as [FORMULA] tends towards unity. These are general features of the condensed polytropes, and hold for polytropic indices [FORMULA].

The procedure for calculating [FORMULA] is described in detail in Hjellming & Webbink (1987 ). Results for a variety of core mass fractions of [FORMULA] polytropes are given, both in Table 2 and graphically, in Figs. 1 and 3.


Table 2. Adiabatic [FORMULA] relation vs. core mass fraction [FORMULA], as in Hjellming & Webbink (1987 ), Table 3. Columns two and four are the core-mass fraction and mass-radius exponent, respectively. The parameter E in columns one and three is an alternate description of the degree of condensation of the polytrope, used by Hjellming and Webbink. The data presented here are in regions where the residuals of the fit formula Eq. (61) are in excess of 0.001

[FIGURE] Fig. 1. Plot of [FORMULA] versus core mass fraction, of an isentropic, [FORMULA] red giant star. As the fraction of mass in the core grows, the star becomes less like a standard polytrope. Important to note is the crossing of the [FORMULA] and [FORMULA] curves near [FORMULA]

The function [FORMULA] 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.

[FIGURE] Fig. 2. Differences between various fit formulae and the function [FORMULA]. The labeled curves are as follows: [FORMULA], [FORMULA], and [FORMULA]
[FIGURE] Fig. 3. Plots of [FORMULA] versus q, assuming a fiducial and constant [FORMULA]. The three solid-line curves correspond, in ascending order, to core masses of 0.14, 0.30, and [FORMULA]. Note that this figure is on the same scale as Figs. (4), etc., so that they may be overlaid

3.2. Thermal equilibrium exponent: [FORMULA]

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 [FORMULA].

3.3. Roche radius exponent: [FORMULA]

Since the exponent [FORMULA] must be computed according to the evolution of the binary with mass transfer, it is sensitive to mass transfer mode, as are [FORMULA], a, and P. The results for the various modes of nonconservative mass transfer are both interesting, and sometimes counterintuitive. It therefore makes sense to discuss them systematically and at some length.

We rewrite [FORMULA], in a form which depends explicitly on previously calculated quantities:


The derivatives [FORMULA] and [FORMULA] 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 [FORMULA], 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, [FORMULA] has been plotted vs. q, for various models, in Figs. 4, 5, 6, and 7.

One notices several things in these graphs. Fig. 4 shows the extreme variation in [FORMULA] 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-q ; isotropic re-emission at high-q), the orbit quickly shrinks in response to mass loss and [FORMULA] is high. By contrast, in the case of the isotropic wind in the high-q limit, angular momentum is retained despite mass loss and the orbit stays wide, so [FORMULA] is lower than it is in the conservative case and the mass transfer is stabilised.

[FIGURE] Fig. 4. [FORMULA] with all mass transfer through a single channel. For each curve, all mass from the donor star is transferred according to the indicated mode: conservative (cons); isotropic wind from donor star (wind); isotropic re-emission of matter, from vicinity of 'accreting' star (iso-r); and ring formation, with [FORMULA]. Since the unified model (winds+accretion) always has [FORMULA], the [FORMULA] curves labeled cons, wind, and iso-r also form an envelope around all curves in the unified model
[FIGURE] Fig. 5. This illustrates the effect of an increasing fraction of the mass lost from the donor star ([FORMULA]) into a fast wind from that star. Solid lines correspond to [FORMULA] from top to bottom on the graph's right side; [FORMULA] corresponds to conservative mass transfer. Note that at [FORMULA], [FORMULA] is independent of q

Second, Fig. 4 shows that ring formation leads to rather high [FORMULA], even for modest [FORMULA]. The formation of a ring will usually lead to instability on the dynamical timescale ([FORMULA]). The slower thermal timescale instability will occur only if the giant is rather evolved and has a high coremass fraction (with its high [FORMULA]).

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 [FORMULA] due only to the differences between the isotropic wind and isotropic re-emission, in a family of curves with [FORMULA]. At [FORMULA], stability is passed from the mostly isotropic re-emission modes at low q to mostly wind modes at high q. The [FORMULA] crossing of all these curves is an artifact of the model. Mass loss depends on the parameters in the combination [FORMULA]. Angular momentum loss has a dependence [FORMULA]. The two parameters [FORMULA] and [FORMULA] are then equivalent at [FORMULA].

[FIGURE] Fig. 6. The change in [FORMULA] for all mass lost to isotropic winds, with varying fractions in isotropic wind and isotropically re-emiiited wind. From top to bottom on the right side, the solid curves are for [FORMULA] of [FORMULA], [FORMULA], [FORMULA], [FORMULA], and [FORMULA]. Note that all curves cross at [FORMULA]. Any set of curves [FORMULA] will intersect at [FORMULA], as at this point, [FORMULA] and [FORMULA] have the same coefficients in [FORMULA]

For [FORMULA], mass transfer is stabilised by trading isotropic re-emission for wind, so families of [FORMULA] curves lie below their respective [FORMULA] curves in this region of the q - [FORMULA] graph. This becomes interesting when one examines the [FORMULA] curves, at high [FORMULA]. For total wind strengths of less than about 0.85 and [FORMULA], [FORMULA]. Thus, for modest levels of accretion (at least [FORMULA]), with the remainder of mass transfer in winds, red giant - neutron star mass transfer is stable on the thermal timescale, so long as [FORMULA]. If the donor red giant has a modest mass core, so that [FORMULA], then the process will be stable on he dynamical timescale, as well.

A family of curves with one wind of fixed strength and one wind of varying strength (as shown in Fig. 5, with [FORMULA], and variable [FORMULA]), will also intersect at some value of q. It is easy to understand why this happens. For concreteness, take a constant- [FORMULA] family of curves. Based on the linearity of the equations in [FORMULA] and [FORMULA], [FORMULA], with some f and g. If one evaluates [FORMULA] at a root of [FORMULA], then the result is independent of [FORMULA].

Notice in each case, intersections of various unified wind model [FORMULA] curves occur at [FORMULA]. One can understand this, in the framework of the arguments of Sect. (A). Depending on which side of [FORMULA] one lies, either [FORMULA] or [FORMULA] has the majority of the angular momentum. When [FORMULA], angular momentum will be lost predominantly by the isotropic wind and [FORMULA] will be high. When [FORMULA], angular momentum losses will be modest and [FORMULA] low. The situation is reversed for isotropic re-emission. The two curves form a figure-X on the q - [FORMULA] diagram and must cross when neither q, nor [FORMULA] is too great; that is, near [FORMULA].

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© European Southern Observatory (ESO) 1997

Online publication: April 6, 1998