3. Analytic solution for the radial acceleration
The general solution of Eq. (4) can be found using the simple argument that the energy E of the particle in the rotating reference frame is constant. If denotes the energy of the particle in the inertial rest frame, then the energy E in the uniformly rotating frame (angular velocity ) is given in the non-relativistic case by: (e.g. Landau & Lifshitz 1960). The generalisation of this equation to the relativistic case is straightforward and leads to the transformation
where is the Lorentz factor defined above.
Assume now, that in the general case a particle is injected at time and position with initial velocity . Then, from Eq. (6) it follows that the time-derivative of the radial coordinate r may be written as
where . In the case , this expression reduces to the equation given in Henriksen & Rayburn (1971).
The equation for the radial velocity , Eq. (7), can be solved analytically (Machabeli & Rogava 1994), yielding
where is the Jacobian elliptic cosine (Abramowitz & Stegun 1965, p.569ff), a Legendre elliptic integral of the first kind:
and where is defined by . By using Eq. (8), the time-derivative of r can be expressed as . Note that the Jacobian elliptic functions and are usually defined by the relations and .
Thus, in terms of Jacobian elliptic functions one gets
For the particular conditions where the injection of a test particle is described by and , the time-dependence of the radial coordinate is given by a much simpler expression: In this situation, reduces to a complete elliptic integral of the first kind and therefore, after a change of the arguments, the time-dependence of the radial coordinate becomes
If one considers non-relativistic motions, where , and the special case , Eq. (8) reduces to (cf. Abramowitz & Stegun 1965)
This expression is known to be the general solution of the equation
which describes the motion of a particle due to the centrifugal force in the non-relativistic limit. In Fig. 1, we compute the time-dependence of the radial coordinate r for different initial conditions under the (unphysical) assumption that the bead-on-the-wire motion continues until the light cylinder (with radius ) is reached. Note that in the relativistic case all particles would turn back at the light cylinder due to the reversal of the centrifugal acceleration (e.g. Machabeli & Rogava 1994).
According to our simple model, one expects that a charged test particle gains energy due to rotational motion as long as it is directed outwards. Therefore the relativistic Lorentz factor increases with distance r as the particle approaches the light cylinder. This is illustrated in Fig. 2 where we plot the evolution of the relativistic Lorentz factor as a function of r for different initial velocities and fixed (using a typical light cylinder radius of cm). Note that is not scale-invariant with respect to the injection velocity (i.e. the injection energy).
If one identifies Eq. (15) with the general expression for the centrifugal force, which reduces in the non-relativistic limit to the well-known classical expression, the centrifugal force changes its signs and becomes negative for (see Fig. 3; cf. also Machabeli & Rogava 1994). Hence, if one assumes that the bead-on-the-wire approximation holds in the vicinity of the light cylinder, the radial velocity becomes zero at the light cylinder and changes direction in any case. Accordingly, a crossing of the light cylinder within the bead-on-the-wire approximation, as mentioned for example in Gangadhara & Lesch 1997, is not physical (cf. Fig. 1). The reversal of the direction of centrifugal force according to which the centrifugal force may attract rotating matter towards the centre is well-known in strong gravitational fields (for Schwarzschild geometry: Abramowicz 1990; Abramowicz & Prasanna 1990; for Kerr geometry: Iyer & Prasanna 1993; Sonego & Massar 1996). For illustration, we compute in Fig. 3 the evolution of the effective radial acceleration as a function of the radial coordinate r for different initial velocities. Obviously, there exists a point where the effective acceleration, i.e. the centrifugal force, becomes negative.
© European Southern Observatory (ESO) 2000
Online publication: December 17, 1999