Appendix A: The general steady state transport in the diffusion approximation

In the case where we are concerned with the scattering of particles in relativistic bulk flows, it has been typically found useful to evaluate the scattering operator in the local Lorentz frame in which the fluid is at rest, i.e. in the so-called comoving frame K' (e.g. Webb 1985; Riffert 1986; Kirk et al. 1988). For in this frame K', a simple form of the scattering operator could be applied if one assumes, as in the present approach, the rest frame of the scattering centres to be essentially that of the background flow. Quantities, which are operated upon the scattering operator, e.g. the momentum, might then conveniently be evaluated in this comoving frame, while the time and space coordinates are still measured in the laboratory frame K characterized by its metric tensor $g_{\alpha~\beta}$. We would like to note however, that K' will in general be a non-inertial coordinate system (i.e. an accelerated frame) and therefore the related connection coefficients will not vanish. With reference to the considered particle transport, the covariant form of the Boltzmann equation is thus required, which may be achieved by replacing the ordinary (partial) space-time derivatives by their covariant derivatives. Now, by starting from the relativistic Boltzmann equation and by using a perturbation solution of the moment equations in the diffusion approximation, i.e. by assuming the deviation of the particle distribution from isotropy in the comoving frame to be small, Webb (1989) has derived the general equation describing steady state particle transport in relativistic rotating and shearing flows. Following Eq. (4.4) of Webb (1989), the special relativistic diffusive particle transport equation for the isotropic, mean scattering frame distribution (averaged over all momentum directions) $f^\prime_0(x^{\alpha}, p')~=~<f^\prime >$ may be written as

 

$$\frac{1}{p'^2} \frac{\partial}{\partial p'} \left[-\frac{p'^3~c}{... ...al p'}\right] + \nabla_{\alpha}\left(c~u^{~\alpha}~f_0'+ q^{~\alpha}\right)= 0,$$ (A.1)

with $x^{\alpha}$ the position four vector in the laboratory frame K, where the background plasma is in motion with four velocity  $u^{~\alpha}$, and p'=m' v' the (magnitude of the) particle momentum as measured in the local (comoving) fluid frame K'. Note, that in order to simplify matters, the subscript 0 and the prime superscript are omitted in the following characterisations of the isotropic part of the particle distribution function, i.e. f now denotes the isotropic part! The total particle energy and momentum in the frame K' may be written as

$$E'=p'^{~0}~c \quad {\rm and} \quad p'= \sqrt{(p'^{~0})^2-m_0^2~c^2},$$ (A.2)

respectively, with m₀ the rest mass of the particle and c the speed of light. The terms in the first line of Eq. (A.1) represent particle energy changes due to adiabatic expansion or compression of the flow (i.e. the term proportional to the fluid four divergence $\nabla_{\beta}~u^{\beta}$), due to shear energization (i.e. the term involving $\Gamma$) and due to the fact that K' is an accelerated frame (i.e. the term $\propto \dot{u}_{~\alpha}$, cf. also Webb 1985). The second line gives the effects of diffusion and convection. In Eq. (A.1), $\nabla_{\alpha}$ denotes the covariant derivative while $q^{~\alpha}$ denotes the heat flux. This heat flux contains a diffusive particle current plus a relativistic heat inertial term $\propto$ $\dot{u}_{\beta}$ and is given by

$$q^{~\alpha}=-\kappa^{~\alpha~\beta}~ \left(\nabla_{\beta}~... ...c{(p'^{~0})^2}{p'}~\frac{\partial f}{\partial p'}\right)\cdot$$ (A.3)

As shown by Webb (1989), Eq. (A.1) could be regarded as the relativistic generalization of the non-relativistic particle transport equation first derived by Earl et al. (1988). The acceleration four vector $\dot{u}_{\alpha}$ of the comoving (or scattering) frame in Eqs. (A.1) and (A.3), is defined by

$$\dot{u}_{\alpha}=u^{\beta}~\nabla_{\beta}~u_{\alpha}.$$ (A.4)

The fluid energization coefficient $\Gamma$ in Eq. (A.1) represents energy changes due to viscosity. Since the acceleration of particles draws energy from the fluid flow field, one expects on the other hand the flow to be influenced by the presence of these particles. As was shown, for example, by Earl et al. (1988) and Katz (1991), the resultant dynamical effect on the flow could be modelled by means of an (induced) viscosity coefficient. If one considers the strong scattering limit, i.e. the case where $\omega~\tau_{\rm c} \ll 1$, with $\omega$ the gyrofrequency of the particle in the scattering frame (i.e. $\omega=q~B'~c/p'^0$), $\tau_{\rm c}=1/\nu_{\rm c}$ the mean time interval between two scattering events and $\nu_{\rm c}$ the collision frequency, this fluid energization coefficient could be written as (Webb 1989, Eq. (34))

$$\Gamma =\frac{c^2}{30}~\sigma_{\alpha~\beta}~ \sigma^{\alpha~\beta},$$ (A.5)

where $\sigma_{~\alpha~\beta}$ is the (covariant) fluid shear tensor given by

 

$$\sigma_{~\alpha~\beta}=\nabla_{\alpha}u_{\beta}+\nabla_{\beta}u_{... ...eft(g_{~\alpha~\beta}+u_{\alpha}~u_{\beta} \right)~\nabla_{\delta} u^{\delta}~,$$ (A.6)

with $g_{\alpha \beta}$ the (covariant) metric tensor. Additionally, for the strong scattering limit the spatial diffusion tensor $\kappa^{\alpha \beta}$ reduces to a simple form given by

$$\kappa^{\alpha \beta}=\kappa~(g^{\alpha \beta}+u^{\alpha}~ u^{\beta})~, \quad {\rm with}\;\; \kappa= v'^2~\tau_c/3~,$$ (A.7)

the isotropic diffusion coefficient and v' the comoving particle speed.