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Roche lobe overflow (stage III)


The star fills its RL and begins to transfer matter on to the companion. The mass transfer first proceeds on the thermal time-scale (see detailed discussion in van den Heuvel, 1994[205])


The common envelope stage (CE)  may be formed if the RL overflow occurs in the type C systems (in which the primary component has a well developed (degenerate) core) even for q<1; otherwise (for the type B systems), we use the condition tex2html_wrap_inline9216 for the CE stage to occur. The radius of the star at the RL filling stage is taken to be that of the equivalent RL (Eggleton 1983[46]):


Here a is the binary separation and e is the orbital eccentricity.

We distinguish different substages of RL overflow according to characteristic time-scales of the mass transfer:

Stage III. RL overflow on a thermal time-scale.  This is the most frequent case for the first mass transfer phase.

Stage IIIe. Slow (evolutionary driven) phase of mass transfer. We assume it to occur for short-period A-type binaries. However, it may occur after mass reversal during the first stage of mass exchange for B-type binaries (van den Heuvel, 1994[205]), e.g. in wide low-mass X-ray binaries.

Stage IIIs. This stage is specific to superaccreting compact companions, a shortened substage of fast mass transfer when the matter escapes from the secondary companion carrying away its orbital angular momentum. Its duration is equal to


here and below subscripts ``a'' and ``d'' refer to the accreting and donating mass star, respectively. For systems with small mass ratios, q<0.5, this time-scale corresponds to an effective q-time shortening of the thermal time for the RL overflowing star.

Stages IIIm,g. Mass transfer is controlled by additional losses of orbital angular momentum tex2html_wrap_inline9226 caused by magnetic stellar wind (MSW)  or gravitational radiation (GW).  Characteristic time of evolution is defined as tex2html_wrap_inline9228 , and in the case of MSW is (Verbunt and Zwaan, 1981[209]; Iben and Tutukov, 1987[75])


Here tex2html_wrap_inline9230 means the mass of the low-mass optical star (0.3<m<1.5), which is capable of producing an effective MSW, tex2html_wrap_inline9234 is a numerical parameter of order of unity. We used the mass-radius relation tex2html_wrap_inline9236 for main sequence stars in deriving this formula.

Gravitational radiation time is


The evolution is governed by MSW or GW if tex2html_wrap_inline9238 or tex2html_wrap_inline9240 is the shortest time-scale among all appropriate evolutionary time-scales.

Stage IIIwd. A special case when a WD overfills its RL. This stage is encountered for very short-period binaries (like 4U 1820-30) whose orbital evolution is controlled by GW or MSW losses. In order to calculate the mass loss  rate at each of the III-stages, in the vast majority of cases we use the prescription


where tex2html_wrap_inline9242 is the a priori known mass to be lost during the mass exchange phase (e.g., tex2html_wrap_inline9244 in the case of the conservative stage III, or tex2html_wrap_inline9246 in the case of III(e,m,g) or CE) and tex2html_wrap_inline9248 is the appropriate time-scale.

If one component overflows its RL, we assume the semi-major axis to change in such a way so that the donating mass star (either a normal low-mass star or a degenerate dwarf) keeps filling its RL,


next up previous contents index
Next: Wolf-Rayet and He stars Up: Principal Evolutionary States of Previous: Post main sequence stars

Mike E. Prokhorov
Sat Feb 22 18:38:13 MSK 1997