A good first indicator of the evolutionary status of a binary is its orbital period (see Chapter 16 by Tauris & Van den Heuvel; Verbunt 1993; and references therein for more extended discussion of the evolution of X-ray binaries). We show the orbital periods of X-ray emitting binaries in globular clusters in Figure 14. Most periods known are for binaries in 47Tuc. It should be noted that there is a selection effect against the discovery of long-period binaries in optical surveys. The radius R of the Roche lobe of a star with mass M in a binary with a star of mass m, is given in units of the distance a between stars as approximately
Combining this with the third law of Kepler we find
i.e. the orbital period gives the average density of a Roche-lobe filling star.
Figure 14. Orbital period distributions of X-ray-detected binaries in globular clusters. Most known orbital periods are for systems in 47Tuc, and are shown in the lower four rows. The top two rows indicate the luminous X-ray binaries and other binaries in other clusters (with symbols as for 47Tuc). The period of a cluster source in M31 is shown with an asterisk. The period range in which a main-sequence star can fill its Roche lobe is indicated; systems with shorter periods may contain degenerate stars, with longer periods (sub)giants. Periods from Table 1; 47Tuc: Edmonds et al. (2003), Freire et al. (2003), Camilo et al. (2000); other clusters: Bailyn et al. (1996), Neill et al. (2002), Kaluzny & Thompson (2002), Kaluzny et al. (1996), D'Amico et al. (2001, 2002); M31: Trudolyubov et al. (2002).
The radius of a main sequence star is roughly given by R / R M / M in the mass range of interest here. With main-sequence stars in old globular clusters limited to masses M 0.8R, we see that binaries in which mass transfer occurs, i.e. low-mass X-ray binaries and cataclysmic variables, can only have a main sequence star as the mass donor provided the orbital period is less than about 7 hr. If the orbital period is longer, the donor must be larger than a main-sequence star, i.e. a (sub)giant. From Figure 14 it then follows that, with one exception, all cataclysmic variables in globular clusters can have main-sequence donors. The one exception is AKO9, a cataclysmic variable with a slightly evolved donor in 47Tuc. Of the low-mass X-ray binaries, one may have a main-sequence donor, two must have subgiant donors; the low-luminosity low-mass X-ray binary in 47Tuc is probably a subgiant close to the main sequence. The orbital periods of most active binaries are long enough that even main-sequence stars near the turnoff mass (0.8 M) fit well within the Roche lobes; for those with the shorter periods both stars must have lower masses to be smaller than their Roche lobes. Two of the low-mass X-ray binaries have ultra-short orbital periods; at such short orbital periods the Roche filling star can be a white dwarf. With R / R 0.01(M / M)-1/3, a white dwarf fills its Roche lobe in a period Pb 48 s M / M.
The evolution of low-mass X-ray binaries and cataclysmic variables with main-sequence donors is driven by loss of angular momentum from the angular momentum of the binary Jb. The mass transfer rate is very roughly given by - / M ~ - / Jb. The loss of angular momentum from gravitational radiation alone is enough to drive mass transfer at a rate of 10-10 M yr-1; higher mass transfer rates, as witnessed by luminosities well in excess of Lx 1036 erg s-1, imply other mechanisms. The loss of angular momentum causes the orbit to shrink, and thus the orbital period to become shorter. In binaries with a (sub)giant donor, the mass transfer rate is very roughly given by the expansion rate of the donor star - / M ~ / R. Since the expansion rate of a giant becomes faster as it further ascends the giant branch, this predicts higher mass transfer, i.e. more luminous X-ray emission, for the longest periods. For the two orbital periods of low-mass X-ray binaries in globular clusters with a subgiant, expansion of the donor predicts a modest mass transfer of ~ 10-10 M yr-1. The mass transfer, combined with conservation of angular momentum, causes the orbit to expand, and the orbital period to increase. Enhanced loss of angular momentum from a stellar wind has often been invoked to explain large X-ray luminosities, in binaries with main-sequence or subgiant donors, but the actual efficiency of this loss mechanism is not known. It is worth noting that many X-ray sources show large variations in their X-ray luminosity on time scales of decades - the transients are an obvious example - indicating that the current mass transfer rate, even in apparently stable systems, may not be an accurate estimator of mass transfer rate on an evolutionary time scale.
That something is wrong with the simplest description of binary evolution is clear, however, from the orbital period distribution of the recycled radio pulsars. The expansion of a binary with a subgiant donor continues until the core of the giant is denuded of its envelope. By then the orbital period has increased by an order of magnitude. The orbital periods of the radio pulsars in 47 Tuc are less than about 2.5d, suggesting that little if any expansion has occurred during the mass transfer. On the other hand, some pulsar binaries in globular clusters, such as the pulsar binary in M4, do have periods in excess of hundred days, with fairly circular orbits, showing that expansion is strong in at least some cases.
What about the ultrashort periods? They may have white-dwarf donors; if so, their orbital period should be increasing. It has been suggested that a collision between a (sub)giant and a neutron star could lead to expulsion of the giant envelope and leave the neutron star in orbit around the core, which subsequently cools to an under-massive white dwarf. If loss of angular momentum from gravitational radiation pushes the stars closer, mass transfer begins once the white dwarf fills its Roche lobe (Verbunt 1987). Alternatively, it has been suggested that the ultrashort period systems are the outcome of an evolution which starts when a subgiant starts transferring mass to a neutron star in an orbital period less than ~ 18hr (Podsiadlowski et al. 2002). Large loss of angular momentum through a stellar wind brings the two stars closer together, and the evolution proceeds to shorter and shorter periods. The minimum period reached through such an evolutionary path is short enough to explain the 11min period of the LMXBNS in NGC6624. This binary is predicted to have a negative period derivative, as observed. There are two problems with this model, however. One is that the loss of angular momentum from the giant, required at the start of the mass transfer to convert orbital expansion into orbital shrinking, is rather high; perhaps implausibly high. Pylyser & Savonije (1988) point out that the shortest periods are only reached after a time longer than the Hubble time, because it already takes ~ 10 Gyr for a 1 M star to fill its Roche lobe in a 16hr period.
5.1.1. Some specific systems
The orbital period for the low-luminosity low-mass X-ray binary 47 Tuc X5 is too long for a Roche-lobe filling main sequence donor star with a mass less than the turnoff mass of 0.8 M. Edmonds et al. (2002b) therefore conclude that the star is smaller than its Roche lobe. We suggest an alternative possibility that the system hosts a 0.8 M subgiant donor that has recently started to transfer matter to a 1.4 M neutron star. The donor has not yet transferred much of its envelope mass: a low donor mass in an 8.666 hr orbit implies a Roche lobe for the donor that is too small to hold a subgiant. The system is very sub-luminous for a subgiant: this is expected for a donor that is losing mass.
PSR 47TucW (Chandra source 29) is a pulsar accompanied by an object whose location in the color-magnitude diagram indicates that it is too big for a white dwarf and too small for a main-sequence star. The orbital lightcurve shows clear heating by the pulsar (Edmonds et al. 2002a). If a main-sequence star is heated at constant radius, it moves up and to the left in a color-magnitude diagram, to a location below the main-sequence. If the companion to PSR 47TucW is of this nature, its position about 5 magnitudes below turnoff indicates a very low mass, of an M dwarf. This poses an interesting puzzle for the evolutionary history: if the M dwarf was in the binary from the start, it was too small to transfer mass to the neutron star and spin it up. If on the other hand the main-sequence star was captured by the pulsar tidally or via an exchange encounter, the orbit should be eccentric initially; the question is whether tidal dissipation can circularize the orbit and heat the M dwarf to its current position.
PSR NGC6397A is another pulsar accompanied by a low-mass (~ 0.25 M) companion (Ferraro et al. 2003). In this case the companion lies somewhat to the right of the turnoff, at a radius of 1.6(2) R and luminosity 2.0(4) L; notwithstanding the proximity of an energetic radio pulsar, the companion shows no sign of heating (Orosz & van Kerkwijk 2003). The position of the companion in the color-magnitude diagram is hard to explain. Orosz & van Kerkwijk invoke a stellar collision, causing a slightly evolved star near the turnoff to lose most of its envelope.
5.1.2. Black holes
The absence of known very luminous low-mass X-ray binaries with a black hole in globular clusters of our Galaxy has led to the suggestion that black holes are efficiently ejected from globular clusters through dynamical processes (Kulkarni et al. 1993; Portegies Zwart & McMillan 2000). The discovery of very luminous, soft X-ray sources in globular clusters in other galaxies shows that X-ray binaries with black holes probably exist in globular clusters.
There is no evidence that M15 contains an intermediate mass black hole; an upper limit for the mass of about 103 M can be set both from an analysis of pulsar accelerations in this cluster, and from an analysis of radial velocities of stars close to the center (Phinney 1992; Gerssen et al. 2003). A case has been made for a binary of two black holes, at least one of which must have an intermediate mass, in NGC6752 (Colpi et al. 2002). The argument for this is the presence of a white-dwarf/radio-pulsar binary in the outskirts of the cluster, which most likely was ejected from the cluster core. If the binary was ejected with the white dwarf companion to the pulsar already formed, the very small eccentricity of its orbit implies that the orbit of the other binary involved in the scattering was much larger. To still produce an ejection velocity for the pulsar binary high enough for it to reach the outer cluster region then requires at least one black hole with a mass ~ 100 M in the scattering binary (Colpi et al. 2002). To solidify the case for a binary black hole it would have to be demonstrated that the pulsar indeed belongs to NGC6752 (as is probable), and that the pulsar binary was ejected before the formation of the white dwarf (which is not obvious). The optical identification of the white dwarf companion to this pulsar shows that the white dwarf is young compared to the age of the globular cluster; this strengthens the case for a scenario in which a binary consisting of a main-sequence star and a neutron star was ejected from the cluster core, and subsequent evolution of the main-sequence star led to circularization of the orbit (Bassa et al. 2003).