2 Model background

By starting from the relativistic Boltzmann equation and by using both the differential moment equations and a simple BKG (Bhatnagar, Gross & Krook) time relaxation for the scattering term, the relativistic particle transport equation in the diffusion approximation was derived by Webb (1989, 1992; cf. also Appendix A). In the underlying physical picture, it is assumed that scattering of high energy particles occurs by small-scale magnetic field irregularities carried in a collisionless, systematically moving background flow. In each scattering event the particle momentum is randomized in direction, but its magnitude p' is assumed to be conserved in the (local) comoving flow frame, where the electric field vanishes. Since the rest frame of the scattering centres is regarded to be essentially that of the background flow, particles neither gain energy nor momentum merely by virtue of the scattering if there is no shear or rotation present and the flow is not diverging. However, in the case of a shear in the background flow, the particle momentum relative to the flow changes for a particle travelling across the shear. As the particle momentum in the local flow frame is preserved in the subsequent scattering event, a net increase in particle momentum may occur (cf. Jokipii & Morfill 1990). Thus, if rotation and shear is present, high energy particles, which do not corotate with the flow, will sample the shear flow and may be accelerated by the centrifugal and shear effects (cf. Webb et al. 1994).

For the present application, we consider a rather idealized (hollow) cylindrical jet model where the plasma moves along the z-axis at constant (relativistic) v_z, while its velocity component in the plane perpendicular to the jet axis is purely azimuthal and characterized by the angular frequency $\Omega$. Using cylindrical coordinates for the position four vector, i.e. $x^{\alpha}=(ct,r,\phi,z)$, the metric tensor becomes coordinate-dependent (i.e. $(g_{\alpha \beta})={\rm diag}\{-1,1,r^2,1\}$) and for the chosen holonomic basis the considered flow four velocity may be written in shortened notation as

 

$$u^{\alpha}= \gamma_f~(1,0,\Omega/c,v_z/c)~,$$ (1)

$$u_{\alpha}= \gamma_f~(-1,0,\Omega~r^2/c,v_z/c),$$ (2)

where the normalization

$$\gamma_f=1/\sqrt{1-\Omega^2 r^2/c^2-v_z^2/c^2}$$ (3)

denotes the Lorentz factor of the flow and where the angular frequency may be a function of the radial coordinate, i.e. $\Omega=\Omega(r)$. As suggested by Webb et al. (1994), the resulting transport equation may be cast in a more suitable form if one replaces the comoving variable p' by the variable $\Phi=\ln(H)$, where H is given by

$$H=p'^0~c~\exp\left(-\int^r \mbox{d}r'\frac{\gamma_f^2~\Omega^2~r'}{c^2} \right)\cdot$$ (4)

In the case of highly relativistic particles (with $p'^0 \simeq p'$) and using an (isotropic) diffusion coefficient of the form

$$\kappa=\kappa_0\; p'^{\alpha}~r^{\beta},$$ (5)

we finally arrive at the steady state transport equation (cf. Appendix B) for the (isotropic) phase space distribution function f(r,z,p')

 

$$\frac{\partial^2 f}{\partial r^2}+\left(\frac{1+\beta}{r}+[3+\alp... ...\gamma_f^2~v_z^2/c^2)~\frac{\partial^2 f}{\partial z^2} =-\frac{Q}{\kappa}\cdot$$ (6)

In general, the solution of Eq. (6) can be quite complicated. However, as we are especially interested in the azimuthal effects of particle acceleration in a rotating flow further along the jet axis, it seems sufficient for a first approach to search for a z-independent solution of the transport equation, i.e. we may be content with an investigation of the one-dimensional Green's function, which preserves much of the physics involved and which corresponds to the assumption of a continuous injection along the jet (cf. Appendix C). In this case the corresponding source term becomes

$$Q=\frac{q_0}{p_{\rm s}'}\; \delta(r-r_{\rm s})~\delta(\Phi-\Phi_{\rm s})~,$$ (7)

describing mono-energetic injection of particles with momentum $p'=p_{\rm s}'$ from a cylindrical surface at $r=r_{\rm s}$. Here, $\delta$ denotes the Dirac delta distribution and the constant q₀ is defined as $q_0=N_{\rm s}/(8~\pi^2~p_{\rm s}'^2~r_{\rm s})$ with the total number $N_{\rm s}$ of injected particles. In order to solve the z-independent transport equation, we then apply Fourier techniques and consider the Green's solutions satisfying homogeneous, i.e. zero Dirichlet boundary conditions.