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

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2. Mass transfer and the evolution of orbital parameters

In this section, expressions for the variation of orbital parameters with loss of mass from one of the stars are derived. In what follows, the two stars will be referred to as [FORMULA] and [FORMULA], with the latter the mass losing star.

A binary composed of two stars with radii of gyration much less than the semimajor axis a will have an angular momentum:


where the period P is related to a and the total mass [FORMULA] through Kepler's law:


Here, the reduced mass [FORMULA], the mass ratio [FORMULA], and e is the eccentricity. Tidal forces circularise the orbits of semidetatched binaries on timescales of [FORMULA], 1 the nuclear timescale on which the binaries evolve (Verbunt & Phinney 1995 ). Furthermore, one expects stable mass transfer by RLOF to circularise orbits. Consequently, an eccentricity [FORMULA] will be assumed for the remainder of this work.

Eqs. (1) and (2) combine to give expressions for the semimajor axis and orbital period, in terms of the masses and angular momentum:


with logarithmic derivatives:


There are various modes (or, borrowing from the terminology of nuclear physics, channels) of mass transfer associated with the different paths taken by, and destinations of mass lost from [FORMULA].

The details of mass transfer must be considered, in order to calculate the orbital evolution appropriate to each mode. Several of these modes are described here, roughly following Sect. 2.3 of the review article by van den Heuvel (1994 ).

The mode which is most often considered is accretion, in which matter from [FORMULA] is deposited onto [FORMULA]. The accreted component (slow mode) conserves total mass [FORMULA] and orbital angular momentum L. In the case considered here, of a Roche lobe filling donor star, matter is lost from the vicinity of the donor star, through the inner Lagrangian point, to the vicinity of the accretor, about which it arrives with a high specific angular momentum. If the mass transfer rate is sufficiently high, an accretion disk will form (cf: Frank, King, and Raine 1985 ). The disk forms due to viscosity of the proffered fluid and transports angular momentum away from and mass towards the accretor. As angular momentum is transported outwards, the disk is expands to larger and larger circumstellar radii, until significant tides develop between the disk and the mass donor. These tides transfer angular momentum from the disk back to the binary and inhibit further disk growth (Lin & Papaloizou 1979 ). The mass transfer process is conservative if all mass lost from the donor ([FORMULA]) is accreted in this way.

The second mode considered is Jeans's mode, which van den Heuvel calls, after Huang (1963 ), the fast mode; the third is isotropic re-emission. Jeans's mode is a spherically symmetric outflow from the donor star in the form of a fast wind. The best known example of Jeans's mode: the orbital evolution associated with Type II supernovae in binaries (Blaauw 1961 ) differs markedly from what is examined here. In that case, mass loss is instantaneous. A dynamically adiabatic outflow ([FORMULA]) is considered here.

An interesting variant of Jeans's mode is isotropic re-emission. This is a flow in which matter is transported from the donor star to the vicinity of the accretor, where it is then ejected as a fast, isotropic wind. The distinction between Jeans's mode and isotropic re-emission is important in considerations of angular momentum loss from the binary. Mass lost by the Jeans mode carries the specific angular momentum of the mass loser; isotropically re-emitted matter has the specific angular momentum of the accretor.

Wind hydrodynamics are ignored.

The fourth case considered is the intermediate mode, or mass loss to a ring. No mechanism for mass loss is hypothesised for this mode. The idea is simply that the ejecta has the high specific angular momentum characteristic of a circumbinary ring.

The differences among each of these modes is in the variation of the quantities q, [FORMULA], and most importantly [FORMULA], with mass loss. For example, given a [FORMULA] binary, mass lost into a circumbinary ring ([FORMULA]) removes angular momentum at least 25 times faster than mass lost by isotropic re-emission. This difference will have a great effect on the evolution of the orbit, as well as on the stability of the mass transfer.

In the following sections, formulae for the orbital evolution of a binary under different modes of mass loss are presented and integrated to give expressions for the change in orbital parameters with mass transfer. The different modes are summarised in Table (1).

Due to the formality of the following section and the cumbersome nature of the formulae derived, one should try to develop some kind of intuitition, and understanding of the results. The two sets of results (i.e.: one for a combination of winds, the other for ring formation) are examined in the limits of extreme mass ratios, compared to one another, and their differences reconciled in the first Appendix.


Table 1. Names, brief description of, and dimensionless specific angular momentum of the modes of mass loss explored in this paper. Parameters [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA], and A are defined in Eqs. (7), (8), (35), (37), and (14), respectively. The factor A in the fast mode is typically unity

2.1. Unified wind model

Consider first a model of mass loss which includes isotropic wind, isotropic re-emission, and accretion, in varying strengths. Conservative mass transfer, as well as pure forms of the two isotropic winds described above, can then be recovered as limiting cases by adjusting two parameters of this model:


Here, [FORMULA] is an infinitesimal mass of wind from star 2 (isotropic wind), and [FORMULA] an infinitesimal mass of wind from star 1 (isotropic re-emission). Both amounts are expressed in terms of the differential mass [FORMULA], lost from star 2. A sign convention is chosen such that [FORMULA] is positive if the star's mass increases, and [FORMULA] is positive if removing matter (from [FORMULA]). Given the above, one can write formulae for the variation of all the masses in the problem:


Neglecting accretion onto the stars from the ISM and mass currents originating directly from the accretor, all transferred mass comes from the mass donating star ([FORMULA]) and [FORMULA], [FORMULA], and the accreted fraction [FORMULA] all lie between 0 and 1, with the condition of mass conservation [FORMULA], imposed by Eq. (9). For the remainder of this section, the subscript on [FORMULA] will be eschewed.

If mass is lost isotropically from a nonrotating star, it carries no angular momentum in that star's rest frame. In the center of mass frame, orbital angular momentum L will be removed at a rate [FORMULA], where [FORMULA] is the mass loss rate and h the specific angular momentum of the orbit.

If a wind-emitting star is rotating, then it loses spin angular momentum S at a rate [FORMULA], where [FORMULA] is that star's rotation rate and [FORMULA] the average of the square of the perpendicular radius at which the wind decouples from the star. Strongly magnetised stars with ionised winds will have a wind decoupling radius similar to their Alfvén radius, and spin angular momentum may be removed at a substantially enhanced rate.

The above consideration becomes important in the case of a tidally locked star, such as a Roche lobe filling red giant. In this case, if magnetic braking removes spin angular momentum from the star at some enhanced rate, that star will begin to spin asynchronously to the orbit and the companion will establish tides to enforce corotation. These tides are of the same form as those between the oceans and the Moon, which are forcing the length of the day to tend towards that of the month.

Strong spin-orbit coupling changes the evolution of orbital angular momentum in two ways. First, for a given orbital frequency, there is an extra store of angular momentum, due to the inertia of the star. Thus, for a given torque, L evolves more slowly, by a factor [FORMULA]. The second is from the enhanced rate of loss of total angular momentum, due to the loss of spin angular momentum from the star. Extreme values of [FORMULA] allow the timescale for loss of spin angular momentum to be much less than that for orbital angular momentum. Thus, the second effect will either compensate for or dominate over the first, and there will be an overall increase in the torque due to this wind.

This enhancement will be treated formally, by taking the angular momentum loss due to the fast mode to occur at a rate A times greater than what would be obtained neglecting the effects of the finite sized companion.

Keeping the above discussion in mind, the angular momentum lost from the system, due to winds is as follows:


which can be simplified by substituting in expressions for the [FORMULA]:


So far, the equations have been completely general; [FORMULA], [FORMULA], and A may be any functions of the orbital elements and stellar properties. Restricting these functions to certain forms leads to simple integrable models of orbital evolution. If constant fractions of the transferred mass pass through each channel ([FORMULA], [FORMULA] constant), then the masses are expressable as simple functions of the mass ratio q:


Furthermore, if the enhancement factor A is also constant 2, then the angular momentum is an integrable function of q, as well:


where the exponents 3 are given by:


Finally, substitution of Eqs. (22), (23), and (24) into Eqs. (3) and (4) gives expressions for the evolution of the semimajor axis and orbital period in terms of the changing mass ratio q:


The derivatives of these functions may be evaluated either by logarithmic differentiation of the above expressions (Eqs. (28) and (29)), or by substitution of Eqs. (11), (12), (13), and (19) into Eqs. (5) and (6). Either way, the results are:


The reader should be convinced that the above equations are correct. First, and by fiat, they combine to give Kepler's law. Second, they reduce to the familiar conservative results:


in the [FORMULA] limit. Finally, and again by construction, the formulae are composable. Ratios such as [FORMULA] are all of the functional form [FORMULA], so if one forms e.g.: [FORMULA], the result is immediately independent of the arbitrarily chosen intermediate point [FORMULA].

Note that the results for isotropic re-emission obtained by Bhattacharya & van den Heuvel (1991 , Eq. (A.6)), and again by van den Heuvel (1994 , Eq. (40)) are not composable, in the sense described above. The correct expression for [FORMULA] in the case of isotropic re-emission may be obtained by setting [FORMULA]:


Also note that Eq. (A.7) of Bhattacharya & van den Heuvel (1991 ) should have its exponential in the numerator, as opposed to the denominator. In van den Heuvel (1994 ), Eq. (37) should read [FORMULA]. In Eq. (38), an equals sign should replace the plus. (These corrections were also found by Tauris (1996 )).

Thus, if there is mass loss from one star, with constant fractions of the mass going into isotropic winds from the donor and its companion, one can express the variation in the binary parameters, [FORMULA], a, and P, in terms of their initial values, the initial and final values of the ratio of masses, and these mass fractions.

2.2. Formation of a coplanar ring

Now consider a model in which mass is transferred by accretion and ring formation, as described above. For concreteness, follow a standard prescription (cf: van den Heuvel (1994 )) and take the ring's radius, [FORMULA] to be a constant multiple [FORMULA] of the binary semimajor axis. This effectively sets the angular momentum of the ring material, since for a light ring,


Formulae describing orbital evolution can be obtained acording to the prescription of the previous section. If a fraction [FORMULA] of the mass lost from [FORMULA] is used in the formation of a ring, then Eqs. (9) and (19) should be replaced as follows:


Following the procedure used in Sect. (2.1), and with [FORMULA] ([FORMULA]):



Unless the ring is sufficiently wide ([FORMULA]), it will orbit in a rather uneven potential, with time-dependant tidal forces which are comparable to the central force. In such a potential, it would likely fragment, and could fall back upon the binary. Consequently, stability probably requires the ring to be at a radius of at least a few times a. A 'bare-minimum' for the ring radius is the radius of gyration of the outermost Lagrange point of the binary. This is between a and [FORMULA], depending on the mass ratio of the binary (Pennington 1985 ). The ring should not sample the potential at this radius. For most of what follows, we will work with a slightly wider ring: [FORMULA] and [FORMULA].

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© European Southern Observatory (ESO) 1997

Online publication: April 6, 1998