The first suggestion that X-ray binaries were powered by accretion onto a compact object was made by Shklovsky in 1967, five years after the discovery of Sco X-1, and three years after Salpeter's original recognition of the possible importance of accretion as an astrophysical energy source (in quasars). To date a large number of collapsed objects has been discovered and made accessible to a detailed study through observations in the X-ray band. Accretion of matter in the strong gravitational field of these objects is in most cases responsible for the efficient production of radiative energy. Up to 10-42% (depending on the compact object) of the rest-mass of the accreting flow can, in principle, be converted into radiation, the bulk energy of which falls in the X-ray band. X-ray binaries consist of a neutron star or a stellar mass black hole accreting matter from a non-collapsed companion in a close binary orbit. Accreting white dwarfs in binary systems, i.e. Cataclysmic Variables (CVs), have long been known to be X-ray emitters.
There are four main reasons for which mass transfer takes place at some stage in the evolution of close binary:
If accretion takes place with transfer mass rate onto a neutron star (or white dwarf) with mass and radius R, the resulting steady-state release of gravitational potential energy (in the Newtonian theory), also called ``accretion-luminosity'', will be:
(where G is the gravitational constant) or
where , , and .
In the case of spherical symmetry and steady state, the accretion luminosity can be written in terms of the limiting ``Eddington'' luminosity . Above this value a star of given mass M starts blowing away its outer layers by radiation. is defined as the luminosity at which the radiation force on ionised hydrogen plasma balances the gravitational force exerted by the star
where r is the distance to the stellar center, is the proton mass and is the Thompson scattering cross section ( =6.6 10 cm ). The Eddington luminosity sets an upper limit to the accretion luminosity of an accreting compact object, since for ;SPMgt; , further accretion of matter will be inhibited by radiation. The maximum accretion rate possible is
(assuming that all the accreting matter is converted into X-ray radiation)
The majority of the observed X-ray pulsators have companions of early spectral type (O or B). Such system can be very luminous and many of the first galactic X-ray sources to be discovered (e.g. Cen X-3, SMC X-1 and Vela X-1) fall into this category. Early-type stars are known to lose mass in the form of a stellar wind both intense, with mass loss rates as high as 10 -10 , and highly supersonic, with velocity
at a distance r from the center of the early-type star. and are the mass and radius of the primary star and is the escape velocity at its surface. For typical parameters is generally a few thousand km/s which greatly exceeds the sound speed . As a consequence the accreting gas is far to be in hydrostatic equilibrium.
A compact object (which I henceforth assume to be a NS) with mass moving with velocity through a medium with sound speed will gravitationally capture matter from a roughly cylindrical region with axis along the relative wind direction which represents the volume where wind particle kinetic energy is less than the gravitational potential one (e.g. Hoyle & Lyttleton 1939; Bondi & Hoyle 1944; Bondi 1952; Davidson & Ostriker 1973; Lamb, Pethick & Pines 1973; for a review see also Henrichs 1983, Frank, King & Raine 1992, and King 1995). The radius of the cylinder, called the accretion radius or gravitational capture radius, is given by:
( is the NS orbital velocity). Note that it is usually possible to neglect the term because in most cases (X-ray heating may, however, cause to be comparable to ). The net amount of gas captured and accreted by the NS can be obtained by combining the above relationship with Kepler's third law and the continuity equation (assuming spherically symmetric and steady mass loss)
where a is the orbital separation and the wind density near the accretion radius; finally one obtains
Note that the above results are obtained in the approximation of nonrelativistic monoatomic gas, for which the polytropic index is 5/3 implying that the solution is everywhere subsonic and that fluid falls essentially freely onto the NS surface (for a rigorous treatment see Bondi 1952). Moreover , for typical wind parameters, is usually in the range 10 -10 . Hence, in order to have one needs , which can only be found in main-sequence stars more massive than about 20-25 and in blue supergiants of mass 15-20 .
In this scenario the net amount of specific angular momentum carried by the gas stream and captured by the accreting object is mainly due to the asymmetry of the accretion. It is possible to demonstrate that a density and velocity gradient is always present on the cylindric surfaces because of a different amount of particles is captured at the far-side and at the near-side surface (Shapiro & Lightman 1976). Taking into account only the radial density gradient in the spherically symmetric stationary expansion of the stellar wind, the captured angular momentum J with respect to the NS is approximatively given by
where is the orbital angular velocity and J is written per unit mass.
Stellar winds from low mass and/or late type stars are not usually strong (except perhaps when the star reaches the asymptotic red-giant branch) and mass transfer occurs mainly through Roche-lobe overflow (or X-ray irradiation). In the Roche approximation the gravitational field generated by the two stars, and revolving about their center of mass in a circular orbit, is approximated by that of two point masses. It is also assumed that stars corotate with the binary system. Under these conditions, it is possible to derive the potential (gravitational plus centrifugal) and its corresponding equi-potential surfaces. Close to each object, the potential is dominated by the gravitational potential of the star, thus the surfaces are almost spherical. As one moves farther from a stellar center, two effects start to became important: (a) the tidal effect, which causes an elongation in the direction of the companion and (b) flattening due to the centrifugal force. Consequently the surfaces are distorted in a way that their largest dimension is along the line of centers.
The most important equi-potential surface, from the point of view of binary star evolution, is the critical surface with a figure-of-eight cross section which passes through the inner Lagrangian point . The two cusped volumes which are enclosed by this critical surface are called the Roche lobes of the respective stars.
The importance of the Roche geometry lies in the fact that stars which fill their Roche lobe start to transfer mass through (actually in a region nearby) to their companion. In the nomenclature of ``interacting'' binary systems the case where one of the two stars fills its critical Roche surface is called semi-detached and include both NS X-ray binaries and CVs (for a review see Shore, Livio & van den Heuvel 1994). Mass transfer can be triggered either by the expansion of a star (due to its own evolution) to the point that it fills its Roche lobe, or by contraction due to systematic angular momentum losses. The rate of mass transfer depends largely on (a) the change in the Roche lobe radius with respect to the mass of the lobe filling star, (b) the change of the radius of the star with respect to its mass (for a fixed composition profile and specific entropy) and (c) the change of the radius of a star in thermal equilibrium with respect to its mass (for a fixed composition profile).
Depending on the above conditions three mass transfer regimes can be identified which take place on dynamical timescale (b;SPMlt;a), thermal timescale (c;SPMlt;a;SPMlt;b) and orbital evolution or nuclear timescale (a;SPMlt;b and c). In the first case, the mass losing component is unable to contract as fast as its Roche lobe as it is losing mass, therefore, mass transfer is enhanced. In the second case, the star is unable to remain in thermal equilibrium as it loses mass (because that would lead to a too high mass loss rate). Its departure from thermal equilibrium will allow the star to keep contact with the Roche lobe and transfer mass on a thermal timescale. In the third case, the mass transfer process is not self-fed, but rather it occurs as a result of the expansion of the lobe-filling star (as a consequence of its evolution) or because of the fact that angular momentum losses from the system decrease the size of the Roche lobe. For a companion star with mass 1.5-2.0 , mass transfer rates below can be obtained, in the range 10 -10 /yr. So, Roche lobe overflow can only produce steady long-lived X-ray sources with .
In this scenario, the specific angular momentum J with respect to the orbiting NS is given by (Petterson 1978)
where is the Roche lobe radius and it has been assumed that all the matter from the companion star leaves the critical lobe near .
Apart from the ``steady'' forms of mass loss by Roche-lobe overflow in a binary, or by a ``steady'' wind in the case of massive star, there is a third well-known type of mass loss among ordinary stars: the irregular outburst of equatorial mass loss observed in rapidly rotating B-star, so-called B-emission stars. Such B-type stars show, at irregular time intervals, outbursts of equatorial mass ejection which produce a rotating ring of gas around the star, giving rise to a sudden appearance of hydrogen emission lines in the optical spectrum. Mass ejection is usually intrinsic to the B-star (i.e. is independent of the presence or absence of a companion star). The star can appear as a normal main-sequence B-star for many years and suddenly go through a B-emission phase, becoming a Be star for periods ranging from a few weeks to many years (Slettebak & Snow 1987), while others are Be-stars almost permanently.
If such Be-star has a NS companion, this can suddenly became a bright X-ray transient when the B-star goes through a B-emission phase and ejects a ring or disc of matter from its equator, part of which is captured at a distance by the NS. The accretion mechanism of Be stars works more efficiently when the system eccentricity is low as the NS passes closer to the Be star and in a denser region around it. Large orbital eccentricities probably arise because tides have not yet circularised the binary after the supernova explosion that gives rise to the NS. The accretion mechanism is not far from the case of stellar wind accretion and the same description can be assumed to work as a first approximation. The relations () and () can be used for the accretion radius and the specific angular momentum transferred, respectively.
There is evidence that the winds of Be stars consist of two distinctly different regions: a high-velocity (10 km s ), low-density polar region, and a low-velocity (10 km s ), high-density equatorial region (e.g. Dachs et al. 1986; Waters 1986), sometimes referred to as the ``disc model''. The disc model can account for many features observed in Be stars. The high density equatorial regions produce the hydrogen emission lines and the IR excess. The wind density in the equatorial regions drops with distance as with n between 2 and 4. X-ray luminosities in Be/X-ray binaries during outburst can be understood if the NS is embedded in the slow, dense, equatorial wind of the Be star where the mass accretion mechanism is more efficient.
Accreting matter forms a disc when its specific angular momentum J is too large to hit the accreting object directly. This requires that the circularisation radius
(where the matter would orbit if it lost energy but not angular momentum) is larger than the effective size of the accretor (R). By means of equations () and () is possible to obtain the size of the expected radius at which a disc begins to form.
Is important to emphasise that in X-ray binaries where the accretor is a NS or WD, the condition () almost always holds if accretion is via Roche lobe overflow. In fact, for typical orbital parameters we obtain
which is much larger than the radius of a NS ( 10 cm) and slightly larger than the radius of a WD ( 10 cm). The outcome is less clear if wind accretion is at work, as J and are much lower (eq. (b) and ). In this case the typical size of is
For this value comfortably exceeds the radius of a NS (a WD is ruled out). Usually it is not easy to decide if disc formation occurs in stellar wind accretion. In the generally accepted scenario is that in the case of wind accretion a small accretion disc can form occasionally both in the prograde and retrograde direction, such that the net sign of J changes in an erratic manner.
In the ``standard'' accretion disc theory matter can orbit at and accretion can only take place through a sequence of (almost Keplerian) circular orbits with gradually decreasing J (for a review see Frank, King & Raine 1992 and references therein; King 1995). Most of the original angular momentum is carried outwards beyond and it is typically returned to the secondary star's orbit through tides. Viscosity is the agent responsible for both energy dissipation and angular momentum transport, providing a torque between the shearing orbits. The physical origin of the viscosity is still unclear (perhaps it is due to chaotic magnetic fields in the disc and other effects).
In the thin disc approximation, the self-gravity of the disc is neglected and the radial and vertical velocities of the flow are assumed to be much smaller than the Keplerian angular velocity of the disc material:
This implies that a stationary thin disc can be described as a one-dimensional structure, in which all quantities are replaced by their averages in the vertical direction. The binding energy of a gas element of mass in the Kepler orbit which arrives on the surface of the compact object is . Since the gas elements start to accrete at large distance from the star with negligible binding energy, the total luminosity released in a steady state disc is
where is the accretion luminosity in () and (). The other half of is released very close to the star surface.
The required Kepler angular velocity () cannot be maintained at the inner edge of the disc if it is to join smoothly to a non-magnetic accreting star spinning at below the break-up velocity . The region over which gas moving at Keplerian velocities in the disc is decelerated to match the star angular velocity is called the boundary layer (BL). In such region, if there exists a point where implying that the shear vanishes at this point. Thus viscosity is no longer able to transport angular momentum into or out of the star, and the accreting object simply gains the angular momentum of the accreting matter.
If the star spins more slowly than the break-up value, the BL must release a large amount of energy as the accreting matter comes to rest at the stellar surface. Some of this is used to spin up the star, but there remains an amount
to be dissipated (here is the luminosity emitted in the BL). For this is one-half of the total accretion luminosity. Not all the has to appear as radiation, but it is clear that for accretion on to a slowly rotating star the boundary layer can emit a luminosity comparable to that of the disc. For a rapidly rotating star is always negative and viscosity transports angular momentum outwards at all radii. Correspondingly reduces drastically.
The picture of the boundary layer accretion described above can only be relevant if the disc extends right down to the surface of the accreting star. Quite often this is not the case. The presence of highly coherent pulsations, the detection of cyclotron line in the X-ray spectra of NSs in binary systems and the high level of linear and circular polarisation observed in optical spectra of an increasing number of CVs as well, point unambiguously to the existence of strong magnetic field around compact objects ( 10 G for WD and 10 G for NS). The presence of the magnetic field plays an increasingly important role in the dynamics of the gas flow as it approaches to the surface of the collapsed star leading to the disruption of the disc.
Gas which is at least partially ionised and falling towards a magnetised star will at some point have its motion affected by the magnetic field. The magnetosphere is usually defined as that volume (not in general spherical) within which the field strongly effects the dynamical properties of the infalling flow, such as the trajectory, energy and angular momentum (for a review see Vasyliunas 1979, Lamb 1979, Henrichs 1983). For spherically symmetric infall, the radius of the magnetosphere is usually determined from the balance of magnetic pressure and ram pressure of the infalling gas (Davidson & Ostriker 1973):
where B is the magnetic field strength and is the velocity of the infalling matter. Assuming a magnetic dipole field ( , where is the magnetic moment of the star), and an infall velocity comparable to the free-fall velocity , by using the continuity equation () a magnetospheric radius of
is derived, where , and .
For stream accretion balance results
where is the cross section of the stream and v is the velocity of the stream . Correspondingly
Beside and , typical accretion length scales, there is a third characteristic length scale, namely the corotation radius which is the distance at which NS rotation velocity matches the Keplerian one. This happens where the centrifugal force just balances the local gravity, i.e.
where P is the spin period ( is independent from the accretor size). is then the minimum requirement for a binary X-ray source to show a significant magnetic behaviour. It is thus possible to distinguish several likely physical regimes depending on the relative size of , , and as well as the magnetic field strength (Illarionov & Sunyaev 1975; Stella, White & Rosner 1986).
In addition, for accretion onto a NS surface to occur, the condition must also be satisfied. Setting implies
Thus only for spin period longer than this, i.e., for slowly rotating NS, the mechanism of magnetic inhibition of accretion occurs before the centrifugal barrier acts.