2. The evolution of the magnetic inclination
The evolutionary equations for the field are nonlinear differential and integral equations. It is difficult to find an exact solution. To solve these equations approximately, it is helpful to inspect the order of magnitude of the quantities and the physical meaning of the equations. For a canonical pulsar, with rad and rad . The approximation is often used in the subsequent calculations. In spite of the small value of and compared with the Larmor frequency , the long time cumulative evolutionary effects, for seconds for example, will be substantial. Next let us examine the physical meaning of the equations. We note that Eq. (1) is a precession equation, and each magnetic moment vector precesses around . However, such precession is incoherent because the coordinate dependent quantity causes the phase difference in the different position. Secondly, the precession angle between the magnetic field and is very small, given approximately by
if and B are chosen to be 100 rad and Gauss, respectively. Thirdly, the incoherent precession of each magnetic moment will result in the integral magnetic field to be cancelled in the direction orthogonal to in a specific time interval required for the phase difference between the magnetic moment vectors located in the inner layer and the outer layer of magnetic region to reach . When this occurs will shift towards the rotation axis by an angle , and will precess around the new axis again for another time interval . Gradually, the magnetic axis, which is almost parallel to , will move towards the rotation axis. We can estimate as follows.
where is the average phase velocity difference between the inner and outer magnetic region boundaries, is given by
Here is the depth of the magnetic region and the Schwarzshild radius. In deriving Eq. (9), the contribution of the polar angle is taken to be and is used.
From Eqs. (7) to (9), we can obtain the decay rate of the magnetic angle as follows,
If magnetic dipole radiation is the only dissipation mechanism for a pulsar, its rotation energy loss(Sharpiro & Teukolsky 1983) is given by
Solving the differential Eqs. (10) and (11) simultaneously, we obtain the magnetic inclination evolutionary equation,
where is the intial magnetic inclination angle, () is the initial stellar angular velocity (period) and (t) (P(t)) is the angular velocity (period) at any time t. Taking typical values of the pulsar parameters, e.g. R , gauss, , M , , we obtain
where and are the internal and the dipolar magnetic fields of the neutron star in the units of Gauss, respectively. An alternative approach (Zhang 1992) to solve Eqs. (1)-(3) by perturbation expansion, in which is assumed to be small, has given a solution similar to that of Eq. (15) but with a singularity occuring at , where the assumption is clearly no longer valid. So the solution obtained here is more physical. Furthermore, we want to remark that the internal magnetic field, in general, can be larger than the dipolar field.
If the inclination angle of a pulsar does not change drastically, a good approximate expression for the stellar angular velocity at an arbitrary time is given by (see e.g. Taylor & Manchester 1977; Shapiro & Teukolsky 1983),
where is given by
and the average value of can be taken as . Then the explicit time evolution formula for the magnetic inclination angle is given by
© European Southern Observatory (ESO) 1998
Online publication: March 23, 1998