## 3. Analytic solution for the radial accelerationThe general solution of Eq. (4) can be found using the simple
argument that the energy 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 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 Using Eq. (7), the Lorentz factor may be written as a function of
the radial coordinate 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
Using the definition of the Lorentz factor, the equation for the accelerated motion, Eq. (4), may also be written as (cf. Chedia et al. 1996; Kahniashvili et al. 1997). By inserting the above relations, the solution for the radial acceleration could be expressed in terms of the Jacobian elliptic functions: 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
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
© European Southern Observatory (ESO) 2000 Online publication: December 17, 1999 |