Evolution of a bi nary system.

The evolution of a binary system of two main-sequence stars can significantly affect the evolution of both component stars if the orbital separation is sufficiently small. If the orbital period is less than about 10 days, tidal interactions will have circularized the orbit during the pre- and early main-sequence phase. [60, 167, 168] Both stars start in the main sequence with the mass of the primary , and the mass of the secondary , defined such that . The binary system is described by the orbital separation , and the mass ratio of the components . The gravitational potential of the binary system is described by the Roche model where each star dominates the gravitational potential inside regions called Roche lobes. The two Roche lobes meet at the inner Lagrange point along the line joining the two stars. Figure 5 shows equipotential surfaces in the orbital plane for a binary with . If either star fills its Roche lobe, matter will stream from the Roche lobe filling star through the inner Lagrange point to the other star in a process known as Roche lobe overflow (RLOF). This mass transfer affects both the evolution of the components of the binary as well as the binary properties such as orbital period and eccentricity.

Figure 5: Cross section of equipotential surfaces in the orbital plane of a binary with . The values of the potential surfaces are 5.0, 3.9075, 3.8, 3.559, 3.2, 3.0, and 2.8. The units have been normalized to the orbital separation, so .

Roche lobe overflow can be triggered by the evolution of the binary properties or by evolution of the component stars. On the one hand, the orbital separation of the binary can change so that the Roche lobe can shrink to within the surface of one of the stars. On the other hand, stellar evolution may eventually cause one of the stars to expand to fill its Roche lobe. When both stars in the binary are main-sequence stars, the latter process is more common. Since the more massive star will evolve first, it will be the first to expand and fill its Roche lobe. At this stage, the mass exchange can be conservative (no mass is lost from the binary) or non-conservative (mass is lost). Depending on the details of the mass exchange and the evolutionary stage of the mass-losing star there are several outcomes that will lead to the formation of a relativistic binary. The primary star can lose its envelope, revealing its degenerate core as either a helium, carbon-oxygen, or oxygen-neon white dwarf; it can explode as a supernova, leaving behind a neutron star or a black hole; or it can simply lose mass to the secondary so that they change roles. Barring disruption of the binary, its evolution will then continue. In most outcomes, the secondary is now the more massive of the two stars and it may evolve off the main sequence to fill its Roche lobe. The secondary can then initiate mass transfer or mass loss with the result that the secondary also can become a white dwarf, neutron star, or black hole.
The relativistic binaries that result from this process fall into a number of observable categories. A WD-MS or WD-WD binary may eventually become a cataclysmic variable once the white dwarf begins to accrete material from its companion. If the companion is a main-sequence star, RLOF can be triggered by the evolution of the companion. If the companion is another white dwarf, then RLOF is triggered by the gradual shrinking of the orbit through the emission of gravitational radiation. WD-WD cataclysmic variables are also known as AM CVn stars. If the total mass of the WD-WD binary is above the Chandrasekhar mass, the system may be a progenitor to a type I supernova.

The orbit of a NS-MS or NS-WD binary will shrink due to the emission of gravitational radiation. At the onset of RLOF, the binary will become either a low-mass X-ray binary (if the donor star is a WD or MS with ), or a high-mass X-ray binary (if the donor is a more massive main-sequence star). These objects may further evolve to become millisecond pulsars if the NS is spun up during the X-ray binary phase [34, 134]. A NS-NS binary will remain virtually invisible unless one of the neutron stars is observable as a pulsar. A BH-MS or BH-WD binary may also become a low- or high-mass X-ray binary. If the neutron star is observable as a pulsar, a BH-NS binary will appear as a binary pulsar. BH-BH binaries will be invisible unless they accrete matter from the interstellar medium. A comprehensive table of close binary types that can be observed in electromagnetic radiation can be found in Hilditch [74].

The type of binary that emerges depends upon the orbital separation and the masses of the component stars. During the evolution of a star, the radius will slowly increase by a factor of about two as the star progresses from zero age main sequence to terminal age main sequence. The radius will then increase by about another factor of 50 as the star transitions to the red giant phase, and an additional factor of 10 during the transition to the red supergiant phase. These last two increases in size occur very quickly compared with the slow increase during the main-sequence evolution. Depending upon the orbital separation, the onset of RLOF can occur any time during the evolution of the star. Mass transfer can be divided into three cases related to the timing of the onset of RLOF.

Case A:
If the orbital separation is small enough (usually a few days), the star can fill its Roche lobe during its slow expansion through the main-sequence phase while still burning hydrogen in its core.
Case B:
If the orbital period is less than about 100 days, but longer than a few days, the star will fill its Roche lobe during the rapid expansion to a red giant with a helium core. If the helium core ignites during this phase and the transfer is interrupted, the mass transfer is case BB.
Case C:
If the orbital period is above 100 days, the star can evolve to the red supergiant phase before it fills its Roche lobe. In this case, the star may have a CO or ONe core.
The typical evolution of the radius for a low metallicity star is shown in Figure 6. Case A mass transfer occurs during the slow growth, case B during the first rapid expansion, and case C during the final expansion phase. The nature of the remnant depends upon the state of the primary during the onset of RLOF and the orbital properties of the resultant binary depend upon the details of the mass transfer.

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