Appendix C: Green's solutions for the steady state diffusive particle transport equation

For general applications one may search for (two-dimensional) Green's solutions of the steady state transport equation (see also WJM 94), i.e. solutions of Eq. (B.28) with source term

$$Q=q_0~\delta(r-r_{\rm s})~\delta(p'-p_{\rm s}')~\delta(z-z_{\rm s}),$$ (C.1)

or equivalently (i.e. utilizing the properties of the delta function) with source term

$$Q=\frac{q_0}{p_{\rm s}'}~\delta(r-r_{\rm s})~\delta(\Phi-\Phi_{\rm s})~\delta(z-z_{\rm s}),$$ (C.2)

describing monoenergetic injection of particles with momentum $p'=p_{\rm s}'$ at position $r=r_{\rm s}$, $z=z_{\rm s}$. For consistency, the constant q₀ in Eqs. (C.1) and (C.2) has to be defined such, that the relevant expression satisfies the requirement that $\delta(\vec{r}-\vec{r}_{\rm s})~ \delta(\vec{p}~'-\vec{p}_{\rm s}~')$ vanishes unless $r=r_{\rm s}$, $\phi=\phi_{\rm s}$, $p'=p_{\rm s}'$ and integrate to unity (or $N_{\rm s}$ if $N_{\rm s}$ particles are injected) over all space and momentum directions. Using cylindrical coordinates this requires

$$q_0=\frac{N_{\rm s}}{8~\pi^2~p_{\rm s}'^2~r_{\rm s}}\cdot$$ (C.3)

Solutions of the transport Eq. (B.28) may then be found by applying Fourier techniques, i.e. by using the double Fourier transform defined by

 

$$F(r;\mu,\nu)=\int_{-\infty}^{\infty} {\rm d}z \int_{-\infty}^{\infty} {\rm d}\Phi\; \exp[i~(\nu~\Phi+\mu~z)]\;f(r,z,\Phi),$$ (C.4)

where the inverse Fourier transform is given by

 

$$f(r, z, \Phi)=\frac{1}{4\pi^2} \int_{-\infty}^{\infty}{\rm d}\mu\;\int_{-\infty}^{\infty} {\rm d}\nu\;\exp[-i~(\nu~\Phi+\mu~z)] F(r;\mu,\nu).$$ (C.5)

Denoting the Fourier transform of the source term $-Q/\kappa$ by $\tilde{Q}$, we have

$$\tilde{Q} -\int_{-\infty}^{\infty}{\rm d}z \int_{-\infty}^{\infty} {\rm d}\... ...~p_{\rm s}'} \delta(r-r_{\rm s})~\delta(\Phi-\Phi_{\rm s})~ \delta(z-z_{\rm s})$$ =

$$-\frac{q_0}{\kappa_{\rm s}~p_{\rm s}'}~ \exp[i~(\nu~\Phi_{\rm s}+\mu~z_{\rm s})]~\delta(r-r_{\rm s}).$$ = (C.6)

Taking the Fourier transform of the transport Eq. (B.28) one arrives at

 

$$\frac{\partial^2 F}{\partial r^2}+\left(\frac{1+\beta}{r}+[3+\alp... ...~v_z}{\kappa}~i~\mu + (1+\gamma^2\frac{v_z^2}{c^2})~\mu^2\right] F =\tilde{Q}~.$$ (C.7)

Let $F_{\rm G}$ be the (two-dimensional) Green's solution of this equation satisfying homogeneous, i.e. zero Dirichlet boundary conditions, then Fourier inversion [i.e. Eq. (C.5)] yields the required Green's function solution $f_{\rm G}(r,z,p';r_{\rm s},z_{\rm s},p_{\rm s}')$. Generally, the considered Fourier techniques yield the Green's function for infinite domains. The proper Green's function for bounded domains (i.e. for finite z) may be obtained, for example, by the method of images (e.g. Morse & Feshbach 1953, pp. 812-816) in the case where the boundaries are restricted to straight lines in two dimensions or planes in three dimensions. The steady state version of the Green's formula (cf. WJM 94, Eqs. (3.24) and (7.12)) may then be applied in order to arrive at the general solution of the considered transport equation.

Yet, even for the simplification v_z=0 and $\gamma=\rm const.$ (i.e. using a simple galactic rotation law $\Omega \propto 1/r$), the Green's function solution for the steady state transport Eq. (B.28) is not straightforward to evaluate (see WJM 94). However, as we are primarily interested in an analysis of the azimuthal effects of particle acceleration in rotating jet flows, we may be content with a z-independent solution of the transport equation, i.e. with an investigation of the so-called one-dimensional Green's function which preserves much of the physics involved (cf. Morse & Feshbach 1953, pp. 842-847). This (one-dimensional) Green's function may be found by integrating the two-dimensional ring source Q (which depends on the space coordinates $r_{\rm s}, z_{\rm s}$) from $z_{\rm s}=-\infty$ to $z_{\rm s}=\infty$; i.e. the one-dimensional Green's function represents the Green's solution for a steady state monoenergetic injection of particles from a infinite cylindrical surface parallel to the jet axis at radius $r_{\rm s}$. Hence, by utilizing the fourier integral theorem it could be directly shown, that the one-dimensional Green's function $f_{{\rm G},1D}$ corresponds to a z-independent solution of the transport Eq. (B.28) with source term $Q(r,p')=q_0\;\delta(r-r_{\rm s})\;\delta(p'-p_{\rm s}')$, i.e. the (one-dimensional) Green's function we are seeking for in the following analysis, represents the solution of the modified transport equation

  

C.1 Rigid rotation profiles

In the case of solid body (uniform) rotation shearing in the background flow is absent since the fluid moves without internal distortions. For $\Omega = \Omega_0= \rm const.$, Eq. (C.8) reduces to the purely spatial transport equation

 

$$\frac{\partial^2 f}{\partial r^2}+\left[\frac{1+\beta}{r}+ (3+\al... ...^2}{(1-\tilde{\Omega}_0^2~r^2/c^2)} \right]~ \frac{\partial f}{\partial r}=Q_0,$$ (C.9)

where the constant $\tilde{\Omega}_0$ is defined by

$$\tilde{\Omega}_0=\frac{\Omega_0}{\sqrt{1-v_z^2/c^2}}~,$$ (C.10)

while $Q_0= - q_0~\delta(r-r_{\rm s})~\delta(\Phi-\Phi_{\rm s})/ (\kappa_{\rm s} ~p_{\rm s}')$, with q₀ given by Eq. (C.3), and where the diffusion coefficient is assumed to be of the form

$$\kappa=\kappa_{\rm o}~\left(\frac{p'}{p_{\rm o}'}\right)^{\alpha} \left(\frac{r}{r_{\rm o}}\right)^{\beta}\cdot$$ (C.11)

with $\kappa_{\rm o}, p_{\rm o}'$ and $\alpha, \beta$ as constants, cf. Eqs. (A.7) and (B.23).
For rigid rotation Eq. (B.18) yields

$$H(r,p')= p'^0~c~~(1-\Omega_0^2~r^2/c^2-v_z^2/c^2)^{1/2} = \frac{p'^0~c}{\gamma},$$ (C.12)

where $\gamma$ denotes the Lorentz factor of the flow. As $H=H_{\rm s}(r_{\rm s},p_{\rm s}')$ is a constant of motion (cf. Noether's theorem, see also WJM 94), the particle momentum p' in the comoving frame could be simply expressed as a function of the radial coordinate

$$p'(r)=m_0~c~\sqrt{\frac{H_{\rm s}^2}{m_0^2~c^4~(1- \Omega_0^2~r^2/c^2-v_z^2/c^2)}-1}~,$$ (C.13)

with m₀ the rest mass of the particle.

Basically, the solution space of the homogeneous part of Eq. (C.9) could be described by a set of two independent solutions, e.g. by the functions y₁(r) and y₂(r) with Wronskian $W(r)\equiv W(y_1(r),y_2(r))=y_1~\mbox{d}y_2/\mbox{d}r - y_2~\mbox{d}y_1/\mbox{d}r$, where for an appropriate choice $y_2(r)\equiv 1$. The relevant analytical expressions might be directly written down for some special cases of interest:
a.) for a constant diffusion coefficient, i.e. $\alpha =\beta =0$, two independent solutions are given by $y_2(r)\equiv 1$ and

$$y_1(r)=\sqrt{1-\tilde{\Omega}_0^2~r^2/c^2}~ \left(\frac{4}{3}-\fr... ...ght) -\ln\frac{c~(1+\sqrt{1-\tilde{\Omega}_0^2~r^2/c^2})} {\tilde{\Omega}_0~r}~$$ (C.14)

where for the Wronskian one simply finds

$$W(r)=-r^{-1}~ \left(1-\frac{\tilde{\Omega}_0^2~r^2}{c^2}\right)^{3/2};$$ (C.15)

b.) in the case, where $\beta$ is negative, the solution y₁(r) could be expressed in terms of the incomplete Beta function (cf. Abramowitz & Stegun 1965, p. 263), i.e. one finally may arrive at the system $y_2(r)\equiv 1$ and

$$y_1(r)=\frac{1}{2}\left(\frac{\tilde{\Omega}_0}{c}\right)^\beta \... ...}, \frac{5+\alpha}{2}\right) \qquad{\rm for}\;\;\alpha>-5,~{\rm and}\;\beta<0 .$$ (C.16)

Note, that now the solutions y₁, y₂ have been defined such that the appropriate Wronskian reduces to Eq. (C.15) for $\alpha =\beta =0$, i.e. we have

$$W(r)=-r^{-(1+\beta)}~ \left(1-\frac{\tilde{\Omega}_0^2~r^2}{c^2}\right)^{(3+\alpha)/2}\cdot$$ (C.17)

The general (one-dimensional Green's) solution of the inhomogeneous differential equation Eq. (C.9) with monoenergetic source term Q₀ defined above could then be written as (e.g. Morse & Feshbach 1953, p. 530)

 

$$f(r,p')= y_1(r)~ \left[k_1 -\int^r \frac{Q_0~y_2(r)}{W(r)}\mbox{d}r \right] +y_2(r)~\left[k_2 +\int^r \frac{Q_0~y_1(r)}{W(r)} \mbox{d}r \right],$$ (C.18)

where k₁, k₂ are integration constants specified by the boundary conditions.

In the disk-jet scenario the accretion disk is usually assumed to supply the mass for injection into the jet, thus for simplicity one may consider a rather hollow jet structure (cf. Marcowith et al. 1995; Fendt 1997a; Subramanian et al. 1999) where the plasma motion in the azimuthal direction is restricted to a region $r_{\rm in} \leq r \leq r_{\rm out} < r_{\rm L}$ where $r_{\rm in}$ denotes the jet inner radius, $r_{\rm out}$ the relevant outer radius and $r_{\rm L}$ the light cylinder radius. Particles are supposed to be injected at position $r_{\rm s}$ with initial momentum $p_{\rm s}'$, where $r_{\rm in} < r_{\rm s} < r_{\rm out}$. By chosing homogeneous boundary conditions $f(r = r_{\rm in})=0$ and $f(r = r_{\rm out})=0$, the integration constants in Eq. (C.18) are determined by

$$k_1=-k_2~\frac{1}{y_1(r_{\rm in})} \quad\; {\rm and}$$

$$k_2=\frac{\tilde{q}~~[y_1(r_{\rm out})-y_1(r_{\rm s})]} {1-y_1(r_{\rm out})/y_1(r_{\rm in})}~,$$ (C.19)

where

$$\tilde{q}= - \frac{q_0}{\kappa_{\rm s}~p_{\rm s}'~W(r_{\rm s})}~\delta(\Phi-\Phi_{\rm s})~,$$ (C.20)

with q₀ given by Eq. (C.3), i.e. $q_0=N_{\rm s}/(8~\pi~p_{\rm s}'^2~r_{\rm s})$.
Therefore the (one-dimensional) Green's solution may be written as

 

$$f(r,p';r_{\rm s},p_{\rm s}')= y_1(r)~\left[k_1~\theta(r-r_{\rm s})+k_1~ \theta(r_{\rm s}-r) -\tilde{q}~\theta(r-r_{\rm s})\right]$$

$$\qquad +\left[k_2~\theta(r-r_{\rm s})+k_2~\theta(r_{\rm s} -r)+ \tilde{q}~y_1(r_{\rm s})~\theta(r-r_{\rm s})\right]~,$$ (C.21)

where $\theta(x)$ denotes the Heaviside step function. The delta function in Eqs. (C.20) and (C.21) indicates that the particle momentum in the comoving frame is directly related to the relevant radial position by Eq. (C.13). In order to gain insight into the efficiency of the acceleration process one may introduce a spatial weighting function N(r) defined by

 

$$f(r,p';r_{\rm s},p_{\rm s}')= N(r)~\delta(\Phi-\Phi_{\rm s}) = N(r)~H_{\rm s}~\delta(H-H_{\rm s}).$$ (C.22)

C.2 Keplerian rotation profiles

In the case of Keplerian rotating background flow with $\Omega(r)=k~r^{-3/2}$, $k=\sqrt{G~M}$ generally both, shear and centrifugal acceleration, will occur. By applying a simple Fourier transformation (cf. Eqs. (C.4) and (C.5)), i.e.

$$F=\int_{-\infty}^{\infty}\mbox{d}\Phi~ \exp[i~\nu~\Phi]~f~,$$ (C.23)

where the inverse Fourier transform is given by

$$f=\frac{1}{2~\pi} \int_{-\infty}^{\infty} \mbox{d}\nu~ \exp[-i~\nu~\Phi]~F~,$$ (C.24)

the transport equation (C.8) could be written as

 

$$\frac{\partial^2 F}{\partial r^2}+ \frac{1}{r}~\left(a_1 + a_2 \f... ...\frac{\tilde{\gamma}(r)^4}{r^3}\left(i~\nu~a_3+a_4~\nu^2\right) F =\tilde{Q}_0,$$ (C.25)

where $\tilde{\gamma}(r)$ is defined by $\tilde{\gamma}(r)= \gamma(r)~\sqrt{1-v_z^2/c^2}$, with $\gamma$ the Lorentz factor of the flow, and where $\tilde{Q}_0$ denotes the Fourier transform of the source term $Q_0=-q_0 ~\delta(r-r_{\rm s})~\delta(\Phi-\Phi_{\rm s})~/(\kappa_{\rm s} ~p_{\rm s}')$, i.e.

$$\tilde{Q}_0=-\frac{q_0~\exp[i~\nu~\Phi_{\rm s}]}{\kappa_{\rm s}~p_{\rm s}'} ~\delta(r-r_{\rm s})~.$$ (C.26)

The abbreviations a₁, a₂, a₃, a₄ in Eq. (C.25) are defined by

  

$$a_1=(1+\beta)~,$$ (C.27)

$$a_2=(3+\alpha)~\frac{G~M}{(c^2-v_z^2)}~,$$ (C.28)

$$a_3=\frac{9}{20}~(3+\alpha)~\frac{G~M}{(c^2-v_z^2)}~,$$ (C.29)

$$a_4=\frac{9}{20}~\frac{G~M}{(c^2-v_z^2)}\cdot$$ (C.30)

An analytical evaluation of Eq. (C.25) is rather complicated. However, a simple set of solutions may be found in the case of r being large such that the rotational velocity becomes non-relativistic and the approximation $\tilde{\gamma}(r)= (1-G~M/[(c^2-v_z^2)~r])^{-1/2} \simeq 1$ holds. For, the Fourier transformed transport equation then simplifies to

 

$$\frac{\partial^2 F}{\partial r^2}+ \frac{1}{r}~\left(a_1 + \frac{... ... F}{\partial r} -\frac{1}{r^3}\left(i~\nu~a_3+a_4~\nu^2\right) F =\tilde{Q}_0~,$$ (C.31)

which, using the substitution y=a₂/r, $a_2 \neq 0$ (i.e. $\alpha \neq -3$), leads to

$$y~\frac{\partial^2 F}{\partial y^2} + (2-a_1-y)~\frac{\partial F}{\partial y} -\frac{i~\nu~a_3 + a_4~\nu^2}{a_2}~F=0,$$ (C.32)

for the homogeneous part of Eq. (C.31). Equation (C.32) is known in the literature as Kummer's equation (e.g. Abramowitz & Stegun 1965, p. 504). For the general case where $a_2 \neq 0$ and $(2-a_1) \neq -n$, $n \in N_0$, the complete solution of this equation, i.e. of the homogeneous part of Eq. (C.31), may be written as

$$F_H(r,\nu) = c_1~f_1(r,\nu) + c_2~f_2(r,\nu)$$ (C.33)

where the functions f₁, f₂ are given by

 

$$f_1(r,\nu) = M\left(\frac{i~\nu~a_3 + a_4~\nu^2}{a_2}, 2-a_1, \frac{a_2}{r}\right)$$ (C.34)

$$f_2(r,\nu) = U\left(\frac{i~\nu~a_3 + a_4~\nu^2}{a_2}, 2-a_1, \frac{a_2}{r}\right)\cdot$$ (C.35)

Here, M(a,b,y) and U(a,b,y) denote the confluent hypergeometric functions (cf. Abramowitz & Stegun 1965, pp. 504f; Buchholz 1953, pp. 1-9), with M(a,b,y) being characterized by the series representation

$$M(a,b,y)=1+\frac{a}{1!~b}~y + \frac{a~(a+1)}{2!~b~(b+1)}~y^2 + \frac{a~(a+1)~(a+2)}{3!~b~(b+1)~(b+2)}~y^3 + \ldots$$ (C.36)

while U(a,b,y) is given by the series

$$U(a,b,y)=\frac{\pi}{\sin \pi b} \left(\frac{M(a,b,y)} {\Gamma(1+a-b)~\Gamma(b)}- y^{1-b}~\frac{M(1+a-b,2-b,y)}{\Gamma(a)~\Gamma(2-b)} \right),$$ (C.37)

with $\Gamma(x)$ the Gamma function. M(a,b,z) has a simple pole at b=-n for $a \neq -m$ or for a=-m if m>n, and is undefined for b=-n, a=-m and $m \leq n$. U(a,b,z), however, is defined even for $b \rightarrow \pm n$. For the relevant Wronskian one has (e.g. Abramowitz & Stegun 1965, p. 505)

$$W(y)\equiv W\left(M(a,b,y),U(a,b,y)\right)=-\frac{\Gamma(b) ~y^{-b}~{\rm e}^y}{\Gamma(a)}\cdot$$ (C.38)

Thus, for the Wronskian $W(r)\equiv W(f_1,f_2)$ we find

$$W(r,\nu)=\frac{a_2}{r^2}~ \Gamma(2-a_1)~\left(\frac{a_2}{r}\right)^{-(2-a_1)} \frac{{\rm e}^{a_2/r}}{\Gamma(a(\nu))}~,$$ (C.39)

where

$$a(\nu)=\frac{i~\nu~a_3 + a_4~\nu^2}{a_2}\cdot$$ (C.40)

For the general solution of the inhomogeneous fourier equation (C.25) one finds (cf. Eq. (C.18))

 

$$F(r,\nu) f_1(r,\nu)~\left[k_1~\theta(r-r_{\rm s})+ k_1~ \theta(r_{\rm s}-r) -\tilde{q}(\nu)~ f_2(r_{\rm s},\nu)~ \theta(r-r_{\rm s})\right]$$ =

$$+ f_2(r,\nu)~\left[k_2~\theta(r-r_{\rm s})+ k_2~ \theta(r_{\rm s}-r) +\tilde{q}(\nu)~f_1(r_{\rm s},\nu)~ \theta(r-r_{\rm s})\right]~,$$ (C.41)

where $\tilde{q}(\nu)$ has been defined by

$$\tilde{q}(\nu)= - q_0~ \frac{\exp[i~\nu~\Phi_{\rm s}]}{\kappa_{\rm s} ~p_{\rm s}'~W(r_{\rm s},\nu)}~,$$ (C.42)

with q₀ given by Eq. (C.3) and where k₁, k₂ are integration constants specified by the boundary conditions. For homogeneous Dirichlet conditions at the boundaries $r_{\rm in}$ and $r_{\rm out}$, i.e. $F(r_{\rm in},\nu)=F(r_{\rm out},\nu)=0$ with $r_{\rm in} < r < r_{\rm out}$ and $r_{\rm in} < r_{\rm s} < r_{\rm out}$ the integration constants are fixed (for each $\nu$) and given by

$$k_2(\nu) -\tilde{q}(\nu)~f_1(r_{\rm in},\nu)~ \frac{f_2(r_{\rm out},\nu)~f... ...{\rm out},\nu)~f_1(r_{\rm in},\nu)- f_1(r_{\rm out},\nu)~f_2(r_{\rm in},\nu)}~,$$ = (C.43)

$$k_1(\nu) \frac{f_2(r_{\rm in},\nu)}{f_1(r_{\rm in},\nu)}\;k_2~.$$ = (C.44)

By Fourier inversion, we now may obtain the required (one-dimensional) Green's function

$$f(r,\Phi; r_{\rm s},\Phi_{\rm s})=\frac{1}{2~\pi}~\int_{-\infty}^{\infty}\mbox{d}\nu~ \exp[-i~\nu~\Phi]~F(r,\nu)~,$$ (C.45)

with $F(r,\nu)$ given by Eq. (C.41). As may be obvious from the foregoing investigation this fourier inversion is not easy to evaluate, not even using numerical methods. However, in order to cope with the integration, we may consider the following substitution ( $\alpha \neq -3$)

$$\omega=\frac{3}{\sqrt{20~(3+\alpha)}}~ \left(\nu+\frac{3+\alpha}{2}~i\right)~,$$ (C.46)

for which one finds (cf. Eq. (C.40) and Eqs. (C.27)-(C.30))

$$a(\nu)=\omega^2 + \sigma^2~,$$ (C.47)

where $\sigma$ is defined by $\sigma=3~\sqrt{3+\alpha}/\sqrt{80}$. Performing the substitution in Eq. (C.45), one arrives at an integral with the path of integration A now in the complex plane, i.e. parallel to the real axis at a distance $3~(3+\alpha)~i/[2~\sqrt{20~(3+\alpha)}~]$, extending from -R to R with $R \rightarrow \infty$. We may close the path by choosing a rectangular contour C which consists of the stated parallel A, the real axis, and the outer lines B₁, B₂ parallel to the imaginary axis at R and -R, respectively. For the relevant ranges of $\alpha$ ( $\alpha \neq -3$; $2-a_1 \neq -n$) of interest, the integrand has no poles within the region bounded by C and thus, by virtue of Cauchy's integral theorem, the value of the contour integral around C sums to zero. For $R \rightarrow \infty$ the integrals over B₁ and over B₂ vanish as might be verified by using asymptotic expansion formulas for the integrand. Noting that by means of Euler's equation ${\rm e}^{i~x}=\cos x + i~\sin x$ we have

$$\int_{-\infty}^{\infty} {\rm e}^{-i~x}~f(x^2)~\mbox{d}x= 2 \int_0^{\infty} f(x^2)~\cos x ~\mbox{d}x,$$ (C.48)

and collecting all relevant expressions together, one finally may arrive at the integral ( $2-a_1 \neq -n; \alpha \neq -3$)

 

$$f(r,p';r_{\rm s},p_{\rm s}') g(r_{\rm s},p_{\rm s}',\alpha,\beta)\; \exp\left[-\frac{3+\alpha}... ...alpha)}}{3}~\omega~(\Phi-\Phi_{\rm s})\right) \times \Gamma(\omega^2+\sigma^2)~$$ =

$$\times \left[f_1(r, \omega^2)~f_2(r_{\rm out},\omega^2)~ \frac{h(... ...N(r_{\rm in}, r_{\rm out};\omega^2)}\right] \qquad {\rm for}\quad r > r_{\rm s}$$ (C.49)

$$g(r_{\rm s},p_{\rm s}',\alpha,\beta)\; \exp\left[-\frac{3+\alpha}... ...alpha)}}{3}~\omega~(\Phi-\Phi_{\rm s})\right) \times \Gamma(\omega^2+\sigma^2)~$$ =

$$\times \left[f_1(r,\omega^2)~f_2(r_{\rm in},\omega^2)~ \frac{h(r_... ...N(r_{\rm in}, r_{\rm out};\omega^2)}\right] \qquad {\rm for}\quad r < r_{\rm s}$$ (C.50)

where we have introduced the following abbreviations

$$g(r_{\rm s},p_{\rm s}',\alpha,\beta) = \frac{2~q_0~r_{\rm s}\sqrt... ...ma(2-a_1)} \left(\frac{a_2}{r_{\rm s}}\right)^{1-a_1} {\rm e}^{-a_2/r_{\rm s}},$$ (C.51)

$$f_1(r,\omega^2)=M(\omega^2+\sigma^2, 2-a_1,a_2/r)~,$$ (C.52)

$$f_2(r,\omega^2)=U(\omega^2+\sigma^2, 2-a_1,a_2/r)~,$$ (C.53)

$$h(r_{\rm in}, r_{\rm s};\omega^2)= f_1(r_{\rm in},\omega^2)~f_2(r_{\rm s},\omega^2) - f_2(r_{\rm in},\omega^2)~f_1(r_{\rm s},\omega^2)~,$$ (C.54)

$$h_N(r_{\rm in}, r_{\rm out};\omega^2)= f_2(r_{\rm out},\omega^2)~f_1(r_{\rm in},\omega^2) - f_1(r_{\rm out},\omega^2)~f_2(r_{\rm in},\omega^2),$$ (C.55)

with q₀ given by Eq. (C.3). Equation (C.49) may be evaluated using numerical methods (cf. Wolfram 1996). $f_1(r,\omega^2)$ and $f_1(r,\omega^2)$ are bounded for $\omega = 0$ while the integrand behaves well enough when $\omega \rightarrow \infty$ to allow for a numerical evaluation.

C.3 Flat rotation profiles

In the case of flat (galactic-type) rotation with $\Omega(r)= \Omega_0~r_0/r$, the co-operation of centrifugal and shear effects could be analysed more directly. Applying Fourier transformation (cf. Eq. (C.23)) the transport Eq. (C.8) simplifies to

$$\frac{\partial^2 F}{\partial r^2} +\frac{a}{r}~\frac{\partial F}{\partial r} +\frac{b(\nu)}{r^2}~F=\tilde{Q}_0,$$ (C.56)

where a and $b(\nu)$ are defined by

$$a = (1+\beta)+(3+\alpha)~\gamma^2~ \eta^2~,$$ (C.57)

$$b(\nu) = \left[(3+\alpha)~i~\nu + \nu^2 \right] ~\frac{\gamma^4~\eta^2}{5}~(1-v_z^2/c^2)$$ (C.58)

with $\gamma=1/\sqrt{1-\eta^2-v_z^2/c^2}$ the Lorentz factor of the flow and $\eta \equiv v_{\theta}/c=\Omega_0~ r_0/c$ its rotational velocity, and where $\tilde{Q}_0$ is given by Eq. (C.26). Solutions for the homogeneous part of Eq. (C.56) could then be directly written down, i.e. one has

 

$$F_1(r,\nu)= r^{\frac{1}{2}\left(1-a-\sqrt{(a-1)^2+4~b(\nu)}\right)},$$ (C.59)

$$F_1(r,\nu)= r^{\frac{1}{2}\left(1-a+\sqrt{(a-1)^2+4~b(\nu)}\right)},$$ (C.60)

with Wronskian $W(r,\nu)$ fixed by

$$W(r,\nu)=\frac{\sqrt{(a-1)^2 + 4~b(\nu)}}{r^a}\cdot$$ (C.61)

The general solution of the inhomogeneous fourier Eq. (C.56) in the case of homogeneous Dirichlet conditions at the boundaries $r_{\rm in}$ and $r_{\rm out}$, may then be determined following the steps given in the previous subsection, e.g. see Eqs. (C.41)-(C.45). In order to calculate the inverse Fourier transform, it proves useful to apply the following substitutions

$$\omega = \nu +\frac{1}{2}~(3+\alpha)~i$$ (C.62)

$$\sigma = \frac{3+\alpha}{2}~ \sqrt{1+\frac{5}{\gamma^2~\eta^2~(1-v_z^2/c^2)} \left(\frac{\beta}{3+\alpha}+\gamma^2~\eta^2\right)^2} ,$$ (C.63)

for which one has

$$\frac{1}{2}\sqrt{(a-1)^2 + 4~b(\nu)}=\frac{\gamma^2~\eta}{\sqrt{5}} \sqrt{1-v_z^2/c^2}~ \sqrt{\omega^2 +\sigma^2}.$$ (C.64)

The complete solution may then be found by proceeding as presented in the previous subsection.