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
 |
(3) |
Combining this with the third law of Kepler we find
 |
(4) |
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.
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.
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).