3 Results and discussion

3.1 Alfvén-wave transmission and compression ratio

If $M>\phi _{1}^{1/2}$ and $M/r^{1/2}>\phi _{2}^{1/2}$, the flow around the shock is entirely super-Alfvénic. Then all upstream Alfvén waves are convected through the shock to become downstream Alfvén waves. Some waves are transmitted retaining their propagation direction relative to the flow, but others get reflected, i.e., assume the other propagation direction. Alfvén-wave transmission through a parallel shock wave can be calculated by taking the tangential magnetic-field and velocity components to be due to waves, $\vec{B}_{t}=\delta \vec{B}$ and $\vec{u}_{t}=\delta \vec{u}$. We shall first investigate the transmission coefficients in a shock with a pre-described gas compression ratio. Thus, by using the first four jump conditions, Eqs. (8-9), together with Eqs. (15-16) and assuming a degenerate upstream cross helicity, $\delta \vec{B}_{1}=\delta \vec{B}_{1}^{\pm }$, we find after an amount of straight-forward algebra

  

$$\,\equiv \, \frac{\delta B^{h}_{2}}{\delta B^{h}_{1}}=\frac{r^{1/2}(\phi ^{1/... .../2}_{1})}{2\phi ^{1/2}_{2}}\frac{M+h\phi ^{1/2}_{1}}{M+h\phi ^{1/2}_{2}r^{1/2}}$$ (17)

$$\,\equiv \, \frac{\delta B^{-h}_{2}}{\delta B^{h}_{1}}=\frac{r^{1/2}(\phi ^{1... ...2}_{1})}{2\phi ^{1/2}_{2}}\frac{M+h\phi ^{1/2}_{1}}{M-h\phi ^{1/2}_{2}r^{1/2}},$$ (18)

where h=+[-]1 denotes waves propagating parallel [anti-parallel] to the plasma flow. These reduce to the results of Vainio & Schlickeiser (1998) when $\phi _{1}=\phi _{2}=1$. The transmission problem was first treated by McKenzie & Westphal (1969), but their results for anisotropic conditions are incorrect: they used an incorrect dispersion relation ( $\omega /k=\pm \phi ^{-1/2}V_{\rm A}$) and neglected the additional $\phi$-factor from the transverse momentum equation (see Eq. (9)).

Let us, next, calculate the gas compression ratio for a shock propagating with Mach number M into a gas with parallel plasma beta, $\beta _{\parallel }\equiv 8\pi P_{\parallel 1}/B_{0}^{2}$ and firehose factor

$$\phi _{1}=1-\frac{\beta _{\parallel }-\beta _{\perp }}{2(1+b)}\geq 0,$$ (19)

with $\beta _{\perp }\equiv 8\pi P_{\perp 1}/B_{0}^{2}$ and $b\equiv (\delta B/B_{0})^{2}$ giving the perpendicular plasma beta and the squared ratio of the wave-amplitude and ordered magnetic field B₀, respectively. Eqs. (17-18) give

$$\frac{B_{t2}}{B_{t1}}=T+R=r\frac{M^{2}-\phi _{1}}{M^{2}-\phi _{2}r},$$ (20)

which can be seen to hold for a general upstream cross helicity by transforming Eq. (9) into the de Hoffmann-Teller frame, where $\vec{u}_{t}B_{n}=u_{n}\vec{B}_{t}$ on both sides of the shock. In this frame the energy equation can be written in a form independent of tangential velocities as

$$\left[ \left( \frac{\rho }{2}u_{n}^{2}\frac{B^{2}}{B_{n}^{2}... ...2}+(\phi -1)\frac{B^{2}}{4\pi }\right) u_{n}\right] ^{2}_{1}=0$$ (21)

with B²=B_n²+B_t². Because it does not involve tangential velocities, Eq. (21) holds in any shock frame. Using Eqs. (20-21) together with Eq. (10) gives two equations,

  

$$\frac{8\pi P_{\parallel 2}}{B_{0}^{2}} \,\,=\,\, \beta _{\parallel }+2M^{2}\frac{r-1}{r}$$

$$+\,b\left\{ 2\phi _{1}-1-(2\phi _{2}-1)r^{2}\frac{(M^{2}-\phi _{1})^{2}}{(M^{2}-\phi _{2}r)^{2}}\right\}$$ (22)

$$= \, r\beta _{\parallel }+\frac{2}{5}\left\{ M^{2}\frac{r^{2}-1}{r}+2\left[ r(\phi _{1}-1)-(\phi _{2}-1)\right] \right.$$

$$+\,br\biggl [M^{2}+2(\phi _{1}-1)$$

$$\left. -\,[M^{2}+2r(\phi _{2}-1)]\frac{(M^{2}-\phi _{1})^{2}}{(M^{2}-\phi _{2}r)^{2}}\biggr ]\right\}$$ (23)

for the downstream parallel pressure, where B_1n,2n=B₀ and B²_1t=bB₀² have been used. Multiplying both sides by $r(M^{2}-\phi _{2}r)^{2}$ produces a quartic equation for r, which reduces to a cubic given by Eq. (14) of Vainio & Schlickeiser (1999) for $\phi _{1}=\phi _{2}=1$. The extra root appears in the present model, since r=1 is no longer a trivial solution, but represents a change in the pressure anisotropy accompanied by a change in the transverse magnetic field and parallel gas pressure.

One should also check for the physicality of the obtained shock solutions. We will not analyze this in detail, but note only that if the analysis of the entropy change across the shock is based on ideal gas law, we can write (Lyu & Kan 1986)

$$\Delta S\propto \ln \frac{P_{\perp 2}^{2}P_{\parallel 2}\rho _{1}^{5}}{P_{\perp 1}^{2}P_{\parallel 1}\rho _{2}^{5}}\cdot$$ (24)

Thus, for upstream $\beta _{\perp }\to 0$, the entropy change approaches infinity for a finite perpendicular pressure downstream. This may be interpreted as being due to an increase of the degrees of freedom from 1 to 3 in the gas. In the numerical analysis below, we have checked for the entropy increase as the physicality criterion for the solution.

The easiest way to analyze the compression ratio as a function of Mach number is to solve for b or $\beta _{\parallel }$ in Eqs. (22-23), regard it as a function of two variables (r and M), and to contour plot it fixing the values of other parameters. Thus, e.g.,

 

$$\beta _{\parallel } \,= \, \frac{2(4-r)}{5r}M^{2}+\frac{4[\phi _{2}-1-r(\phi _{1}-1)]}{5(r-1)}$$

$$-\,\frac{b}{5(r-1)}\Biggl \{2M^{2}r-5+2(\phi _{1}-1)(2r-5)$$

$$-\,r\bigl [2M^{2}-5r-6r(\phi _{2}-1)\bigr ]\frac{(M^{2}-\phi _{1})^{2}}{(M^{2}-\phi _{2}r)^{2}}\Biggr \},$$ (25)

which has been used in Fig. 1 to present the fast shock solution for an isotropic downstream pressure in case of large-amplitude upstream waves (b=0.1). For comparison, both low- and high-beta upstream plasmas are considered with isotropic and very anisotropic conditions. The effect of upstream anisotropy is, as expected, small for shocks with cold upstream gas, but it becomes significant as the beta becomes larger. The effect of anisotropy for a large beta is to increase the compression ratio, i.e., to give a stronger shock for a constant Mach-number. Note that in case of the largest beta, we have had to discard the solutions with smallest values of ras unphysical since they do not fullfil the entropy-increase criterion.

Figure 1: Gas compression ratio of a parallel shock with $\phi _{2}=1$propagating into a plasma with $\beta _{\parallel }=0.2$ (solid curves), 2.0 (dashed curves), and 10.0 (dot-dashed curves) with b=0.1, and $\phi _{1}=1$ (thin curves) or $\phi _{1}=\max \{0,1-\frac{1}{2}\beta _{\parallel }/(1+b)\}$ (thick curves).

The Alfvén-wave transmission coefficients for the shocks in Fig. 1 are given in Fig. 2. We choose outwards-propagating (h=-1) upstream waves consistent with their generation by counter-streaming particles. For the colder upstream gas, the transmission coefficients are affected only a little by the pressure anisotropy, but for the hotter case we see substantial effects; especially the reflection coefficient is substantially increased in presence of anisotropies. Generally, T>R, for h=-1, but for a marginally firehose-stable upstream plasma ( $\phi _{1}=0$), the downstream state tends to an equipartition between forward and backward propagating Alfvén waves ($T,R\to 2$) as $r\to 4$.

Figure 2: The transmission coefficients T and R for parallel shocks with $\phi _{2}=1$ propagating into a plasma with $\beta _{\parallel }=~0.2$(solid curves), 2.0 (dashed curves), and 10.0 (dot-dashed curves) with b=0.1, and $\phi _{1}=1$ (thin curves) or $\phi _{1}=\max \{0,1-\frac{1}{2}\beta _{\parallel }/(1+b)\}$ (thick curves). The dotted line separates the curves for R and $T\,(>\!\!R)$.

3.2 Test-particle acceleration

The most important parameter controlling the energy spectrum of cosmic rays accelerated by a shock wave is the compression ratio of the scattering centers that are responsible for the isotropization of the energetic particles. In case of Alfvén waves generated in the upstream region, we may write

$$r_{k}=\frac{u_{1}-\phi _{1}^{1/2}V_{\rm A1}}{u_{2}+H_{c2}\phi _{2}^{1/2}V_{\rm A2}}$$ (26)

where

$$H_{c2}=-\frac{T_{k}^{2}-R_{k}^{2}}{T_{k}^{2}+R_{k}^{2}}$$ (27)

is the downstream cross helicity, and T_k and R_k are the transmission coefficients of the waves at constant wavenumber k. Generally, they depend on the spectral index of the wave intensity, $I(k)\propto k^{-q}$, but for q=1 they reduce to T and R, respectively (Vainio & Schlickeiser 1999). Thus, for $q=\phi _{2}=1$,

$$r_{k}=r\frac{(T^{2}+R^{2})(M-\phi _{1}^{1/2})}{(T^{2}+R^{2})M-(T^{2}-R^{2})r^{1/2}}$$ (28)

which is plotted in Fig. 3 for the shocks analyzed in Figs. 1 and 2. The effect of anisotropy is again negligible for the shock propagating into the colder gas, but for the hotter gas, the shock is much stronger particle accelerator in case of anisotropic upstream gas. The shocks with small $\beta _{\perp }$ produce the largest values ( ${\approx} 6.5$ for the studied case of b=0.1) of the scattering-center compression ratio at relatively low r and M, which was already the result of Vainio & Schlickeiser (1998, 1999). The inclusion of anisotropic pressure, however, allows an extension of this interesting parameter range, where the spectral index of the shock-accelerated particles is between $1<\Gamma <2$, to shocks propagating into hotter gases than for the isotropic case.

Figure 3: Scattering-center compression ratio of a parallel shock with $\phi _{2}=1$ propagating into a plasma with $\beta _{\parallel }=0.2$ (solid curves), 2.0 (dashed curves), and 10.0 (dot-dashed curves) with b=0.1, and $\phi _{1}=1$ (thin curves) or $\phi _{1}=\max \{0,1-\frac{1}{2}\beta _{\parallel }/(1+b)\}$ (thick curves).

The results for the scattering-center compression ratio were also calculated for the interesting case of $\beta_{\parallel }=2$ at several values of the firehose factor (Fig. 4) to see, how large anisotropies are needed for qualitative effects on particle acceleration from pressure anisotropies. It is evident that relatively small values, below $\phi _{1}\approx 0.5$, are needed to produce large effects on particle acceleration. But, for $\phi _{1}<0.25$ the scattering-center compression ratio is above 4 for all 1<r<4.

Figure 4: Scattering-center compression ratio of a parallel shock with $\phi _{2}=1$ propagating into a plasma with b=0.1 and $\beta _{\parallel }=~2.0$, with $\beta _{\perp }=0,\, \beta _{\parallel }$ (solid curves) and $\beta _{\perp }/\beta _{\parallel }=0.75$ (dashed curve), 0.5 (dotted curve), and 0.25 (dot-dashed curve).

Perhaps the most unexpected result of our calculation is the ability of weakly compressive ( $r\gtrsim 1$) shocks to accelerate particles efficiently in warm plasmas. This result, however, depends crucially on the assumption of the shocks ability to isotropize the fluid. If r differs considerably from unity, one can probably safely assume that the small-wavelength magnetic turbulence generated in the shock transition can be responsible for the isotropization. On the other hand, the transverse magnetic field attains large values across the $\beta _{\parallel }\sim 2$ shocks with $\phi _{1}\ll 1$ regardless of the compression ratio. This can also add to the isotropization of the particles since the field changes direction across the shock at scales comparable to the ion Larmor radius. However, the shocks with small r do not contain large amounts of energy to be given for accelerated particles. When at the same time the test-particle spectral index is below 2, the pressure in accelerated particles diverges without a cutoff in the spectrum. The less there is energy available, the smaller value for the cutoff-momentum must be, and it may well turn out that the weak shocks can not be described by the test-particle picture at all. The confirmation of this requires non-linear analysis, which we shall undertake in the future.

3.3 Pressure anisotropies created by non-thermal particles

We will briefly discuss the pressure anisotropies created by the shock-related non-thermal particles assuming non-relativistic particle speeds for simplicity. For particles accelerated at the shock to speeds clearly exceeding the shock-frame scattering-center speed, V₁, particle scattering results in a distribution function, $f_{\rm st}(x,v,\mu)={\rm d}^6N/({\rm d}^3v\, {\rm d}^3 x)$, that is only weakly dependent on the pitch-angle, $\arccos \mu$. In the upstream wave frame, the small anisotropic portion of the distribution, $g_{\rm st}(\mu) = f_{\rm st}-\langle f_{\rm st} \rangle$where $\langle\cdot\rangle$ denotes angle averaging, is antisymmetric relative to $\mu=0$ if the net magnetic helicity of the waves is zero (e.g., Schlickeiser 1989), which we assume. Thus, the pressures due to these particles are

$$P_{\parallel{\rm st}} \,\,=\,\, m\int{\rm d}^3 v\, v^2(\mu-\epsilon)^2 f_{\rm st}$$ (29)

$$P_{\perp{\rm st}} \,\,=\,\, \frac m2\int{\rm d}^3 v\, v^2(1-\mu^2) f_{\rm st},$$ (30)

where $\epsilon=(u_1-V_1)/v=\phi_1^{1/2}V_{\rm A1}/v$ and the integrals are calculated in the wave frame. Using the antisymmetry of $g_{\rm st}$ and the zero net flux of particles in the shock frame, $\int{\rm d}^3v\,(v\mu + V_1) f_{\rm st}=0$, one obtains after a straight-forward calculation the contribution from these particles to the upstream pressure anisotropy as

$$P_{\parallel{\rm st}}-P_{\perp{\rm st}} = \phi_1^{1/2} (2M-\p... ...{1/2}) \frac{\rho_{\rm st}(x)}{\rho_1} \frac{B_0^2}{4\pi}\cdot$$ (31)

This increases towards the shock with the suprathermal particle density $\rho_{\rm st}$, but does not attain large values for low-Mach-number shocks. Note that this contribution is exactly zero for a marginally stable plasma ($\phi _{1}=0$).

Another contribution to the pressure anisotropy comes from the particles that have just been reflected by the shock and subsequently been picked up by the upstream waves. Let us, for simplicity, consider a cold background plasma, and assume that the shock reflects a small fraction of the incoming background-plasma ions back to the upstream region. We model those particles as a beam with a density of $\rho_{\rm b}=m n_{\rm b}$ propagating along the magnetic field away from the shock and include also the particles that return to the shock after their isotropization by the upstream waves. Specifically, in the upstream wave frame we take the distribution function for the reflected and returning particles to be

$$f_{\rm rr}(x,v,\mu)=n_{\rm b}(x)[\delta(\mu+1)+\mbox{$\frac 12$ } M/(M-\phi_1^{1/2})] F(v),$$ (32)

where $F(v)=\delta\{v-(u_1+V_1)\}/2\pi v^2$. This gives a zero net flux of particles in the shock frame and a pressure difference of

$$P_{\parallel{\rm rr}}-P_{\perp{\rm rr}}= M[4M+\mbox{$\frac 12... ...^{1/2})] \frac{\rho_{\rm b}(x)}{\rho_1}\frac{B_0^2}{4\pi}\cdot$$ (33)

If this contribution dominates the upstream pressure anisotropy, one can obtain a cubic equation for $\phi_1^{1/2}$. At high Mach numbers, the solution is approximately $\phi_1=1-4M^2\rho_{\rm b}/\rho_1$.