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We shall assume that the rotator is surrounded by an ideally conductive plasma with a density and a sound velocity at a sufficiently far distance from the rotator. The rotator moves relative to the environment with a velocity . Under the action of gravitational attraction, the surrounding matter should fall on to the rotator. A rotator without a magnetic field would capture a stationary flow of matter, , which can be estimated by Bondy-Hole-Lyttleton formulae (Bondi and Hole, 1944; Bondi, 1952; McCrea, 1953) in the form
where is a dimensionless factor of the order of unity. When one of the velocities, or , far exceeds the other, the accretion rate is determined by the dominating velocity, and can be written in a convenient form as
where is the gravitation capture radius:
In the real astrophysical situation, the parameters of the surrounding matter at distances can be taken as conditions at infinity.
As already noted, the matter surrounding a NS or a WD is almost always in the form of a high-temperature plasma with a high conductivity. Such accreting plasma must interact efficiently with the magnetic field of the compact star (Amnuel' and Guseinov, 1968). Hence, the interaction between the compact star and its surroundings cannot be treated as purely gravitational and therefore the accretion is not a purely gasdynamic process. In general, such interaction should be described by the magnetohydrodynamical equations. This makes the already complicated picture of interaction of the compact star with the surrounding medium even more complex.
The following classification of magnetic rotators is based on the essential characteristics of the interaction of the plasma surrounding them with their electromagnetic field. This approach was first proposed by Schwartzman (1970a) who distinguished three stages of interaction of magnetic rotators: the ejection stage, the propeller stage, which was later rediscovered by Illarionov and Sunyaev (1975) and named as such, and the accretion stage. Using this approach, Schwartzman (1971) was able to predict the phenomenon of accreting X-ray pulsars in binary systems. New interaction regimes discovered later have led to a general classification of magnetic rotators (Lipunov, 1982a, 1984;[98, 104] Kornilov and Lipunov, 1983a).
It should be noted that the interaction of the magnetic rotator with the surrounding plasma is not yet understood in detail. However, even the first approximation reveals a multitude of interaction models. To simplify the analysis, we assume the electromagnetic part of the interaction to be independent of the accreting flux parameters, and vice versa.
Henceforth, we shall assume in almost all cases that the intrinsic magnetic field of a rotator is a dipole field:
This is not just a convenient mathematical simplification. We will show that the magnetoplasma interaction takes place at large distances from the surface of the magnetic rotator, where the dipole moment makes the main contribution. Moreover, the collapse of a star into a NS is known to ``cleanse'' the magnetic field. Indeed, the conservation of magnetic flux leads to a decrease of the ratio of the quadrupole magnetic moment q to the dipole moment in direct proportion to the radius of the collapsing star, . It should be emphasized, however, that the contribution of the quadrupole component to the field strength at the surface remains unchanged.
The light cylinder radius is the first important characteristric of the rotating magnetic field:
where c is the speed of light.
A specific property of the field of the rotating magnetic dipole in vacuum is the stationarity of the field inside the light cylinder and formation of magnetodipole radiation beyond the light cylinder. The luminosity of the magnetic dipole radiation is equal to (Landau and Lifschitz, 1971):
This emission exerts a corresponding braking torque
leading to a spindown of the rotator.