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:
of
its mass in the form of a stellar wind; some of this material will be
captured gravitationally by the companion (stellar wind accretion).
0.7 of the star break-up velocity) B-emission stars
often in highly eccentric (e 
0.3) binary systems.
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.
where
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).
and
) matter falls in a
small region located near the polar cap(s) of the compact object causing
the radiation at the Earth to be modulated with the star rotation period
P. If 
matter can get through the magnetosphere without be
forced to follow the magnetic field lines and accretion occurs at all
star latitudes.
and cannot advance any further
because, 
, the drag exerted by the magnetic field is
super-Keplerian. Some or all the material might be ejected beyond the
accretion radius via the propeller mechanism (centrifugal
inhibition of accretion). If the wind material accumulates at or nearby
the magnetosphere more rapidly than the ejection rate, a buildup of
material outside the magnetosphere may occur (Maraschi, Traversini &
Treves 1983).
, beyond which the magnetic
field lines cannot propagate, is larger than , so that the spin
period must be
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.
and/or 
.
