3 Determination of postshock pick-up ion spectra

We now want to derive the postshock PUI spectrum starting from the upstream spectrum at the shock given in Eq. (12). In the following we shall give expressions for spectra of those PUIs which upon arrival at $r=r_{\rm S}(\theta)$ directly pass over the local shock transition and consequently exclude those which undergo a reflection at the electric potential ramp of the shock and move upstream again (as described e.g. by Kucharek & Scholer 1995; Chalov & Fahr 1996,2000; Lee et al. 1996; Zank et al. 1996; Giacalone & Jokipii 1998; Scholer & Kucharek 1999; Le Roux et al. 2000). The relative abundance of these reflected ions as calculated by Chalov & Fahr (1996) is, however, of the order of 10 percent or less.

Here we use index "1'' for upstream quantities and index "2'' for corresponding downstream quantities. We assume that the distribution $f^{\rm PUI}$ is isotropic in velocity (i.e. due to strong pitch-angle diffusion by quasilinear wave-particle interaction). We also assume that the magnetic moment of the PUI is conserved when crossing the shock, which requires that the shock transition should be wide as compared to the PUI Larmor radius. At the PUI passage over the shock the following changes in the PUI distribution function have to be considered (we follow Fahr & Lay 2000):

1) total densities are increased by the compression ratio:

$$s(\theta)=[\rho_2/\rho_1]_{\theta}=[U_{1\perp}/U_{2\perp}]_{\theta}$$ (13)

where $U_{1\perp}$ and $U_{2\perp}$ are the upstream and downstream solar wind bulk velocity components perpendicular to the local shock surface.

2) due to the local downstream increase in the magnetic field magnitude by the local factor $\sigma(\theta)$ in connection with a conserved magnetic moment of PUIs passing over the shock there is a change in the particle energy characterized by:

$$v^2_{\perp 2}=\sigma(\theta)v^2_{\perp 1}$$ (14)

where $\vec{v}_{\perp}$ is the particle velocity perpendicular to the magnetic field. For an isotropic distribution $<v_{\perp}^2>=(2/3)v^2$. This gives the result $v_2^2=\sigma(\theta)v_1^2$,

and:

3) due to the differential flux conservation over the shock (Liouville theorem) one obtains:

$${f^{\rm PUI}_1(r_{\rm S}(\theta),\theta,v_1){\rm d}^3v_1U_{1\perp}= f^{\rm PUI}_2(r_{\rm S}(\theta),\theta,v_2){\rm d}^3v_2U_{2\perp}}$$ (15)

which leads to:

$$f^{\rm PUI}_2(r_{\rm S}(\theta),\theta,v)= f^{\rm PUI}_1(r_{\rm S... ...2}= s(\theta)J_{\theta}(v_1,v_2) f^{\rm PUI}_1(r_{\rm S}(\theta),\theta,v_1(v))$$ (16)

where $J_{\theta}(v_1,v_2)$ is the Jacobian of the transformation of preshock to postshock velocities given by: $J_{\theta}(v_1,v_2)=(4\pi v_1^2{\rm d}v_1/4\pi v_2^2{\rm d}v_2)\vert _{\theta}=\sigma(\theta)^{-3/2}$ and where again $\sigma(\theta)$ is the field compression ratio at $r=r_{\rm S}(\theta)$ which is extracted from the result of the 5-fluid counterflow simulation model published by Fahr et al. (2000) as we shall show below.

The derivation of $\sigma(\theta)$ is carried out by the following argumentation: at the position $r=r_{\rm S}(\theta)$ the upstream magnetic field can be split into components parallel and perpendicular to the shock surface, yielding:

$$\vec{B}_1(\theta)=\vec{B}_{1\parallel}+\vec{B}_{1\perp}$$ (17)

with

$$B_{1\parallel}=B_1 \cos \xi \ \ \ \ B_{1\perp}=B_1 \sin \xi$$ (18)

with $\xi$ being the angle between the upstream field $\vec{B}_1$ and the shock surface (or between the upstream solar wind velocity $\vec{U}_1$ and the shock normal $\vec{n}(\theta)$). In the ecliptic the angle $\xi$ is connected with the local curvature of the shock surface and is given by:

$${\rm tg} \xi=r_{\rm S}(\theta)/[\partial r_{\rm S}/\partial\theta]_{\theta}= r_{\rm S}/r_{S,\theta}$$ (19)

and is obtained from the form of $r_{\rm S}(\theta)$ given by Fahr et al. (2000). The downstream magnetic field now is given by:

$$\vec{B}_2=\vec{B}_{2\parallel}+\vec{B}_{2\perp}$$ (20)

with $B_{2\parallel}=s(\theta) B_1 \cos \xi$ and $B_{2\perp}=B_1 \sin \xi$.

The magnitude of the downstream magnetic field thus is given by:

$$B_2(\theta)=s(\theta)B_1(\theta)[\cos^2 \xi+\sin^2 \xi/s^2(\theta)]^{1/2}.$$ (21)

Replacing now $\cos \xi$ and $\sin \xi$ by ${\rm tg}\, \xi$ given in Eq. (19) yields:

$$\sigma(\theta)=s(\theta)(1+{\rm tg}^2\xi)^{-1/2} [1+{\rm tg}^2\xi/s^2(\theta)]^{1/2}.$$ (22)

Taking all these results together leads to:

$$% f^{\rm PUI}_2(r_{\rm S}(\theta),\theta,v)\!=\!s(\theta)\sig... ...{-3/2} f_1^{\rm PUI}(r_{\rm S}(\theta),\theta,\sigma^{-1/2}v).$$ (23)

This on basis of Eq. (12) delivers the following postshock PUI spectrum:

$$f^{\rm PUI}_2(r_{\rm S}(\theta),\theta,v) =\nu(\theta)s(\theta)\s... ...e}^{-[C_K(X_{\rm S}(\theta))\sigma(\theta)^{-K}(w-w_0)^K]}} {2\pi U_0 \sqrt{w}}$$ (24)

with $X_{\rm S}(\theta)=r_{\rm S}(\theta)/r_{\rm E}$ and w=(v/U₀)².

The resulting PUI fluxes immediately downstream of the shock are presented in Fig. 3 for three different positions at the shock surface. These are compared with extrapolated anomalous cosmic ray proton spectrum at the shock deduced from de-modulation of the observed high-energy ACR spectrum (Stone et al. 1996).