3 Applications

3.1 Rigid rotation - no shear

In order to gain insight into the particle transport and acceleration process, let us first consider the influence of rigid rotation profiles. One may associate such rotation profiles with dynamo action in the inner accretion disk around a spinning black hole, which creates a jet magnetosphere filled with disk plasma and rotating with the angular frequency of its foot points concentrated near the innermost stable orbit (e.g. Camenzind 1996). The acceleration of particles due to rigid rotation could represent an efficient process for the production of high energy particles. This was demonstrated by Rieger & Mannheim (2000) for the centrifugal acceleration of charged test particles at the base of a rigidly rotating jet magnetosphere for conditions assumed to prevail in AGN-type objects. In such cases an upper limit for the maximal attainable Lorentz factor can be established, which is given by the breakdown of the bead-on-the-wire approximation in the vicinity of the light cylinder. There are several similarities to the application presented here: If we consider the transport of relativistic particles in a rigidly rotating background flow (i.e. $\Omega = \Omega_0= \rm const.$) by utilising the above model, shearing is absent (i.e. $\mbox{d}\Omega/\mbox{d}r =0$) and thus particle energization only occurs as a consequence of the interaction with the accelerating background flow, i.e. due to the presence of centrifugal effects. It can then be readily shown that for the case considered here, Eq. (4) is analogous to the Hamiltonian for a bead on a rigidly rotating wire (cf. Webb et al. 1994, also Rieger & Mannheim 2001a), i.e.

$$% H=\frac{p'^0~c}{\gamma_f},$$ (8)

while the transport equation becomes purely spatial. The variable H introduced above, could thus be regarded as describing the balance between the centrifugal force and the inertia of the particle in the comoving frame. By means of Noether's theorem, H could be shown to be a constant of motion, and hence Eq. (C.12) indicates that the ratio of particle to flow Lorentz factor is fixed. The comoving particle momentum p' can then be simply expressed as a function of the radial coordinate r, i.e. p' = p'(r) (cf. Fig. 1). We may easily solve the resultant (spatial) transport equation analytically, assuming homogeneous boundary conditions at the jet inner radius  $r_{\rm in}$ and its relevant outer radius $r_{\rm out}$. In the present case, one may choose the size of the inner radius $r_{\rm in}$ to be of the order of the radius of the innermost stable orbit around a rotating black hole, while the outer radius $r_{\rm out}$ is at any rate physically constrained to be smaller than the light cylinder radius (note that for relativistic MHD winds, the Alfvén point may be close to the light cylinder). Results concerning the particle transport are presented in Figs. 1 and 2. They indicate that the energetic particles injected at $r_{\rm s}$ will gain energy by being transported in the rigidly rotating background flow, i.e. due to the impact of centrifugal acceleration the particle momentum increases with position as expected. The precise evolution in the immediate vicinity of the light cylinder should however be treated with some caution due to the limitations imposed by the applied diffusion approach. Figure 2 reveals a decrease in efficiency if the time $\tau_{\rm c}$ between two collisions increases with momentum (in case of a very weak spatial dependence), i.e. if scattering occurs less rapidly for the higher than for the lower energy particles.

Figure 1: Particle momentum p' as a function of the radial coordinate in the case of rigid rotation, for particles injected at $r_{\rm s}=0.1~ r_{\rm L}$ with initial Lorentz factor $\gamma _{\rm s}=10$ (solid) and 15 (dashed). Here, $r_{\rm L}$ is defined by $r_{\rm L}=c~ (1-v_z^2/c^2)^{0.5}/\Omega_0$

3.2 Keplerian rotation - shear dominance

Let us now consider the acceleration of particles due to a rotating background flow with keplerian rotation profile $\Omega(r)=k~r^{-3/2}$, where $k=\sqrt{G~M}$. Again, such flow profiles might be associated with jets or disk winds originating from the accretion disk around the black hole and dragging the Keplerian disk rotation with them. Hence, for reasons of self-similarity, keplerian rotation may be intuitively regarded as one of the most characteristical descriptions with respect to the velocity profile of intrinsically rotating flows. For example, Lery & Frank (2000) recently investigated the structure and stability of astrophysical jets including Keplerian rotation in the outermost part of the outflow and rigid rotation close to its axis (cf. also Hanasz et al. 2000). They also studied the application to non-relativistic outflows from young stellar objects. One may thus eventually consider a simple model where particles, accelerated in a rigidly rotating flow, are subsequently injected into a Keplerian rotating flow. Generally, if we consider Keplerian rotation, both shear and centrifugal effects are present. For non-relativistic rotation, analytical solutions of the Fourier transformed transport equation are given in the Appendix C.2. in terms of the confluent hypergeometrical functions. Such solutions should be appropriate for the outer jet regions. As the relative strength of the contribution by shear to that of centrifugal energization evolves with r, shear effects will eventually dominate over centrifugal acceleration (see discussion). This is particularly illustrated in Fig. 3, where we have plotted the logarithm of the (normalized) particle distribution function f above $p'/p_{\rm s}' \geq 3$ for $\beta =0$ (i.e. allowing no spatial dependence of the diffusion coefficient) and different momentum dependence of $\kappa$.

Figure 2: Spatial distribution $N(r)/N(r_{\rm s})$ for rigid rotation using a different energy dependence of the diffusion coefficients, i.e. $\alpha =\beta =0$ (solid line), $\alpha =-2$, $\beta =-0.01$ (dotted line) and $\alpha =2$, $\beta =-0.01$ (dashed line). Boundary conditions $r_{\rm in}=0.05~r_{\rm L}$, $r_{\rm s}=0.1~ r_{\rm L}$ and $r_{\rm out}=0.999~r_{\rm L}$ have been used for the calculations.

Figure 3: The momentum-dependence of the (normalized) distribution function f for Keplerian rotation using $\beta =0$, calculated for $\alpha =0.5, 1.0, 1.5, 2.0, 3.0$ at fixed $r=40~r_{\rm ms}$. Boundary and injection conditions have been specified as $r_{\rm in}=10~r_{\rm ms}$, $r_{\rm out}=1000~r_{\rm ms}$, $r_{\rm s}=20 ~r_{\rm ms}$, where $r_{\rm ms}$ is given by $r_{\rm ms}=G~M/(c^2-v_z^2)$.

Most interestingly, well-developed power law momentum spectra are recovered, which suggest a spectral slope linearly related to the momentum index of the diffusion coefficient, so that we have $f \propto p'^{-(3+\alpha)}$. This excellently confirms the results previously derived by Berezhko & Krymskii (1981). For a collisionless plasma with a simple shear flow $U(y)~ \vec{e}_x$, and by using a simple BGK term, they found the shear acceleration to give rise to a power-law momentum spectrum for the steady state comoving particle distribution $n(r,p') \propto p'^{~2}~ f \propto p'^{-(1+\alpha)}$, if the collision time $\tau_c$ depends on momentum as $\tau_{\rm c} \propto p'^{~\alpha}$ and $\alpha >0$. However, if the momentum index $\alpha$ is smaller than zero, i.e. $\alpha <0$, an exponential spectrum may developed.

3.3 Flat rotation - interplay between shear and centrifugal effects

In the case of flat rotation (i.e. $\Omega=\Omega_0~r_0/r=v_{\phi f}/r$) one may easily investigate a more complex interplay between shear and centrifugal effects. For in this case, the relative strength of shear to centrifugal effects is independent of the radial coordinate and the general solution of the fourier-transformed equation could be cast in simple analytical terms allowing basic inverse Fourier integration to be done, using homogeneous boundary conditions at $r_{\rm in}$ and $r_{\rm out}$. For the present application, we have considered typical jet flows with relativistic v_z/c=0.95 and different azimuthal velocities $v_ {\phi f}$ (i.e. for a range of $\gamma_f \sim (3-4)$). The corresponding results are plotted in Fig. 4 using a constant diffusion coefficient (i.e. $\alpha =\beta =0$). Again, the calculated distribution functions reveal a powerlaw-type momentum dependence, where for low azimuthal velocities very steep momentum spectra are recovered, i.e. the momentum exponents are found to be near by -8.8 (for $\beta_{\phi}=0.075$), -6.9 (for $\beta_{\phi}= 0.10$), -5.1 (for $\beta_{\phi}=0.15$) and -4.4 (for $\beta_{\phi}= 0.20$). The observed flattening of the spectra with increasing azimuthal velocities is indicative of the increasing impact of centrifugal effects.

Figure 4: The momentum-dependence of the (normalized) distribution function f(r,p') for flat rotation, calculated for $r_{\rm in}/r_{\rm out}=0.02$, $r_{\rm s}/r_{\rm out}=0.04$ at position $r/r_{\rm out}=0.2$. Chosen azimuthal velocities are $v_{\phi f}/c=0.075$ (short-dashed), 0.10 (dotted),0.15 (solid), 0.20 (long-dashed).