Appendix B: Cosmic ray diffusion and adiabatic cooling

An important question is whether the particles produced in the shock discussed in Sect. 3.3.1 can indeed diffuse out of the shock region, in which case they will freely escape and propagate through the Galaxy essentially with the energy obtained in the shock, or whether they are trapped inside the shocked gas until it expands adiabatically after the shock has passed and activity has ceased. In the latter case, the particles will lose a significant amount of energy to adiabatic expansion.

Following the discussion in Sect. 3.3.1, we distinguish two cases: dissipation in the forward and in the reverse shock.

The escape time of the particles out of the shock can be estimated as

$$\tau \sim \frac{{R_{\rm shock}}^2}{\kappa}$$ (B.1)

where $R_{\rm shock}$ is the typical size scale of the shock and $\kappa$the relevant diffusion coefficient.

While the shocked ISM must still be magnetically connected with the unshocked ISM, the jet plasma will be situated on field lines advected out from the central engine, which are likely not connected with the ISM. In the forward shock, the relevant diffusion coefficient should then be taken as $\kappa_{\rm forward} \sim \kappa_{\parallel}$, the diffusion coefficient parallel to the mean magnetic field, while for the reverse shock one has to consider diffusion across the magnetic boundary of the contact discontinuity between shocked jet plasma and shocked ISM in addition to diffusion to the contact discontinuity and away from it.

A lower limit on the diffusion time out of the shock is thus given by the value for the forward shock, since the particles which have diffused out of the reverse shock must, in addition, also propagate through the forward shock.

B.1 Forward shock

Using the simple approximate expression for $\kappa_{\parallel}$(e.g., Kennel & Petschek 1966), and assuming a Kolmogorov turbulence spectrum for the magnetic field originating on scales of order the shock size $R_{\rm shock}=l_{\rm jet}~\theta=10^{15}~{\rm cm}~l_{16}~\theta_{0.1}$and containing a fraction $\epsilon \equiv {B_{\rm turb}^2}/{B_{\rm tot}^2}$ of the total magnetic energy, the parallel diffusion coefficient for a particle with energy $\gamma~m~c^2 \sim \Gamma_{\rm jet}~m~c^2$can be written as

$$\kappa_{\parallel} \sim \frac{2}{\pi}~ v~ r_{\rm G} \frac{\nu_{\rm gyro}}{\nu_{\rm scatter}} \sim \frac{2~ r_{\rm G}~ c}{\pi}\frac{B^2}{3~kB_{k}^2}$$

$$\sim \frac{2 ~ r_{\rm G}~ c}{3\pi~\epsilon}~\left(\frac{R_{\rm shock}}... ...~\epsilon}\left(\frac{\gamma~ m~ c^2}{e~B}~l_{\rm jet}^2~\theta^2\right)^{1/3}.$$ (B.2)

Here, $\nu_{\rm scatter}$ is the scattering frequency for particles with gyro radius $r_{\rm G}$ off magnetic turbulence with energy density $k B_{\rm k}^2/8\pi$ and wave numbers of order $k \sim r_{\rm G}^{-1}$. Scattering can also be induced by collective resonant interactions of particles with the field, in which case the parameter $\epsilon$ denotes the efficiency of this process.

Writing the shock area as $A_{\rm shock} = \pi R_{\rm shock}^2 = \pi \times 10^{30}~ {\rm cm^{2}}\ \theta_{0.1}^2~ l_{16}^2$ gives an approximate hot spot pressure $p_{\rm shock}$ of

$$p_{\rm shock} \sim \frac{L_{\rm kin}}{A_{\rm shock}\ c} = 3 \... ...ergs\ cm^{-3}}~ L_{38.5}~ l_{16}^{-2}~ \theta_{\rm 0.1}^{-2}.$$ (B.3)

If the magnetic field is at a fraction $\varphi$ of the equipartition field, $\varphi \equiv B/B_{\rm eq}$, we have $B \equiv \varphi~B_{\rm eq} \sim 0.5~{\rm G}~ \varphi~ L_{38.5}^{1/2}/l_{16}~ \theta_{0.1}$.

The comoving (i.e., measured in the frame of the shocked plasma) limit to the proton escape time $\tau_{\parallel}'$ is then

 

$$\tau_{\parallel}' \mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$ }}}\hbox{$>$ }}}\frac{R_{\rm shock}^2}{\kappa_{\parallel}} \sim l_{\rm jet}~\theta~\frac{3\pi~\epsilon~\varphi^{1/3}}{2~c}~ \left(\frac{e}{\gamma~m~c^2}\right)^{1/3}~ \left(\frac{L_{\rm jet}}{\pi~c}\right)^{1/6}$$

$$\sim 4\times10^{7}~{\rm s}~\frac{l_{16}~\theta_{0.1}~\epsilon~\varphi^{1/3}~ L_{38.5}^{1/6}} {\Gamma_{\rm jet}^{1/3}}\cdot$$ (B.4)

If we write the Lorentz factor of the shocked ISM gas as $\Gamma'$, then time dilation gives $\tau_{\parallel} = \Gamma'~\tau_{\parallel}'$ in the observer's frame.

If the turbulent velocity inside the region of interest is comparable to the expansion velocity, and if large scale turbulence is present (which was the underlying assumption in out estimate of $\epsilon$ above), then turbulent transport could aid particle escape: in a simple mixing length approach, the diffusion coefficient can be approximated by $\kappa_{\rm turb} \sim 1/3~v_{\rm turb}~l_{\rm turb}$, where $v_{\rm turb}$ is the characteristic turbulent velocity and $l_{\rm turb}$ the scale length of the largest scale turbulence. The turbulent transport time is then

$$\tau_{\rm turb} \sim \frac{3 R_{\rm shock}^2}{v_{\rm turb}~l_... ...ck}}{v_{\rm turb}}\frac{0.1~R_{\rm shock}}{l_{\rm turb}}\cdot$$ (B.5)

If the shock region is expanding roughly ballistically as it propagates, the adiabatic loss time scale measured in the observer's frame is

$$\tau_{\rm ad} = \frac{\gamma}{\dot{\gamma}} = 4\frac{p}{\dot{... ...es 10^{5}~{\rm s}\ l_{16}~\left(\frac{c}{\dot{l}}\right)\cdot$$ (B.6)

If the forward shock is relativistic, we need to take time dilation into account when comparing $\tau _{\rm ad}$ and $\tau_{\parallel}$. It follows that the adiabatic loss time scale $\tau _{\rm ad}$ is longer than the escape time only if

$$\epsilon~\varphi~ L_{38.5}^{1/6}~ \theta_{0.1}~\frac{\dot{l}_{\rm jet}}{c}~\Gamma_{5}^{2/3} < 4 \times 10^{-3}.$$ (B.7)

Since both $\epsilon$ and $\varphi$ might be significantly smaller than 10^-5, it is not implausible that $\tau_{\parallel} \mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$ }}}\hbox{$<$ }}}\tau_{\rm ad}$. In the case of strong turbulent transport, this condition simplifies to

$$v_{\rm turb}~l_{\rm turb} > 3 R_{\rm shock} \dot{R}_{\rm shock}.$$ (B.8)

If the external pressure is parameterized as $p_{\rm ISM} \equiv 10^{-11}~{\rm ergs\ cm^{-3}}~ p_{-11}$, the maximum reduction in particle energy possible through adiabatic losses is roughly

$$\left(\frac{p_{\rm ISM}}{p_{\rm shock}}\right)^{1/4} \sim 10^{-2} \left(\frac{p_{-11}~ l_{16}^2~ \theta_{0.1}^2}{L_{38.5}}\right)^{1/4}$$ (B.9)

$$\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$ }}}\hbox{$<$ }}} \frac{\langle\gamma - 1\rangle_{\rm min}}{\langle\gamma - 1\rangle_{\rm shock}} < \left(\frac{p_{\rm ISM}}{p_{\rm s}}\right)^{2/5}$$

$$\hspace{48pt} \sim 6\times 10^{-4} \left(\frac{p_{-11}~ l_{16}^2~ \theta_{0.1}^2}{L_{38.5}}\right)^{2/5},$$

where the left hand side of the inequality corresponds to a relativistic equation of state in the hot spot and the right hand side to a non-relativistic equation of state. This, of course, means that a large fraction of the energy injected originally into CRs could be lost adiabatically, reducing the amount of energy available (estimated in Sect. 2.1) by up to a factor of $\sim$1000.

B.2 Reverse shock

In jets where dissipation occurs mainly in the reverse shock, the jet geometry is similar to AGN jets (i.e., the jets are effectively reflected by the ISM, thus inflating a cocoon with spent jet fuel).

Particles accelerated in the shock will eventually be advected out of the shock region and into the cocoon (see Fig. 2). The timescale for this process is

$$\tau_{\rm advect} \sim \frac{R_{\rm shock}}{v_{\rm advect}} \... ...s}~l_{16}~\theta_{0.1}~\left(\frac{c}{v_{\rm advect}}\right),$$ (B.10)

where $v_{\rm advect} \mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$ }}}\hbox{$<$ }}}c/3$, since the limiting downstream velocity in a strong relativistic shock is c/3.

The diffusion time towards the contact discontinuity is given by Eq. (B.4). In order to enter the forward shock, the particules have to propagate across the magnetic boundary at the contact discontinuity, which introduces an additional cross-field diffusion term. The contact discontinuity will have a typical thickness of the order of the Larmor radius $r_{\rm G}$ of the particles, thus the particles will have to traverse a region of size $r_{\rm G}$ perpendicular to the field in order to cross the contact discontinuity (this approximation is valid as long as the parallel diffusion time over one coherence length of the field is longer than the perpendicular diffusion time across one Larmor radius, otherwise the perpendicular diffusion time across one coherence length should be used). The lower limit to the diffusion time across the field is then $\tau_{\perp} \mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$ }}}\hbox{$>$ }}}\frac{r_{\rm G}^2}{\kappa_{\perp}}$.

The perpendicular diffusion coefficient $\kappa_{\perp}$ can be approximated as $\kappa_{\parallel}~(k B_{\rm k}^2/B^2)^2$(e.g., Parker 1965), i.e.,

 

$$\tau_{\perp} \sim \tau_{\parallel}~ \left(\frac{r_{\rm G}}{R_{\rm shock}}\right)^2~... ...(\frac{r_{\rm G}}{R_{\rm shock}}\right)^{2}~\left(\frac{B^2}{kB_{k}^2}\right)^2$$

$$\sim \frac{3\pi}{2~c~\epsilon}~R_{\rm s}^{2/3}~r_{\rm G}^{1/3} \sim 16... ...\gamma^{1/3}~l_{16}~\theta_{0.1}} {\epsilon~L_{38.5}^{1/6}~\varphi^{1/3}} \cdot$$ (B.11)

The total diffusion time out of the reverse shock should then be of the order of $\tau \sim \tau_{\parallel} + \tau_{\perp}$, which has a minimum when $\left(\epsilon~\varphi^{1/3}~L_{38.5}^{1/6}/\Gamma_{\rm jet}^{1/3}\right)_{\rm min} \sim 3\times 10^{-3}$, with $\tau_{\rm min} \sim 2\times 10^{5}~l_{16}~\theta_{0.1}$. Thus, even in the optimal case, the diffusion time out of the reverse shock is longer than the advection time unless $v_{\rm advect} \mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$ }}}\hbox{$<$ }}}0.1~c$.

Thus, it is rather likely that the bulk of the particles are advected out of the shock and into the cocoon. The pressure driven expansion of a cocoon can be approximated by a simple spherically symmetric model: Dimensional analysis suggests that the size of the cocoon $R_{\rm c}$follows a simple scaling (Castor et al. 1975)

$$R_{\rm c} \sim \left(\frac{L~t^3}{\rho_{\rm ISM}}\right)^{1/5}\cdot$$ (B.12)

Using this scaling to obtain order of magnitude estimates of the conditions inside the cocoon, we can write the cocoon pressure $p_{\rm c}$ as

 

$$p_{\rm c} \sim \frac{1}{2}~\rho_{\rm ISM} \dot{R}_{\rm c}^2$$

$$\sim 2\times 10^{-5}~{\rm ergs~ cm^{-3}}~R_{16}^{-4/3}~L_{38.5}^{2/3}~n_{\rm ISM}^{1/3},$$ (B.13)

where $R_{16} = R_{\rm c}/10^{16}~{\rm cm} \sim l_{\rm 16}$ is the cocoon radius. The magnetic field is

$$B_{\rm c} \sim 3\times 10^{-2}~{\rm G}~\varphi~R_{16}^{-2/3}~L_{38.5}^{1/3}~n_{\rm ISM}^{1/6}.$$ (B.14)

Comparison of Eq. (B.13) with Eq. (B.3) shows that a (relativistic) particle advected out of the shock into the cocoon will suffer adiabatic losses of the order

$$\frac{\langle \gamma - 1 \rangle_{\rm c}}{\langle \gamma - 1 ... ...~R_{16}^{2/3}~n_{\rm ISM}^{1/3}}{L_{38.5}^{1/3}}\right)^{1/4}$$ (B.15)

which is only weakly dependent on the source parameters.

Once the particles are inside the cocoon, the escape time is again given by $\tau \sim \tau_{\parallel} + \tau_{\perp}$ with the expressions for $\tau_{\parallel}$ and $\tau_{\perp}$ from Eqs. (B.4) and (B.11), though with different values:

$$\tau_{\parallel} \sim 7\times 10^{8}~{\rm s}~R_{16}~\left[\e... ...{1/9}~\Gamma^{-1/3}~ L_{38.5}^{1/9}~n_{\rm ISM}^{1/18}\right]$$ (B.16)

and

$$\tau_{\perp} \sim 4.5\times 10^{3}~{\rm s}~R_{16}~\left[\eps... ...\Gamma^{-1/3}~ L_{38.5}^{1/9}~n_{\rm ISM}^{1/18}\right]^{-1}.$$ (B.17)

The adiabatic loss timescale is simply

$$\tau_{\rm ad} = 4\frac{p_{\rm c}}{\dot{p}_{\rm c}} \sim \fra... ...0^{5}~{\rm s}~R_{16}^{5/3}~n_{\rm ISM}^{1/3}~L_{38.5}^{-1/3}.$$ (B.18)

Since this is proportional to R^5/3, while $T_{\parallel} \propto R^{10/9}$ and $\tau_{\perp} \propto R^{8/9}$, the particles are more likely to escape out of large, old cocoons than out of small, young ones.

B.3 Summary

The diffusion of CRs out of radio lobes is clearly an important problem not only in the context of CR emission from microquasars, but also for extragalactic radio sources and their contribution to the CR flux in clusters of galaxies. Since the current level of understanding is still very rudimentary, further study of this aspect is necessary. Given the large uncertainties in these estimates, and given that we tried to provide conservative estimates whenever possible, we feel that there is a good chance that the mechanisms outlined in this paper will work under realistic circumstances and that a measurable CR contribution from microquasars can be expected.