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evolution equation

The evolution equation presented above indicates that an accreting compact star must endeavor to attain an equilibrium state in which the resultant torque vanishes (Davidson and Ostriker, 1973[38]; Lipunov and Shakura, 1976[109]). This hypothesis is confirmed by observations of X-ray pulsars.

By equating the right-hand side of equation (4.10.2) to zero, we obtain the equilibrium period:

where
s yr
, and

s yr
,
,
days.

Alternatively:

Let us turn to the case of disk accretion. The above model of the spin-up and spin-down torques possesses an unexpected property. The equilibrium period obtained by setting the torque to zero is connected with the critical period through a dimensionless factor:

The parameters
and
must be such that
. Since
, the equilibrium period in the case of disk accretion is close to the
critical period,
, separating accretion stage *A* and the propeller stage ** P**.
In the case of the supercritical accretion the equilibrium
period is determined by formula (Lipunov, 1982b[99])

Sat Feb 22 18:38:13 MSK 1997