4 Pick-up protons beyond the termination shock

In the region downstream from the termination shock the plasma flow is slower ($\sim$100 kms^-1) and the temperature higher ($\sim$150-300 eV) than in the region upstream. The protons picked up by the flow will then be mixed with the thermal population.

To estimate the post-shock spatial and energetic distributions of the pick-up ions originating in the upstream region we apply the method used previously in the study of the low-energy ACR distribution (see Czechowski et al. 1995,1999c,2001; an application to PUI was presented in Czechowski et al. 1999a). The pitch-angle-averaged distribution function is obtained as a solution of Parker's transport equation. The PUI distribution function immediately after the shock obtained in the previous section is used as a boundary condition for the numerical solution downstream.

The solution downstream will thus be shaped by the effects of convection by the plasma flow (described by the velocity field $\vec{U}(\vec{x})$), losses predominantly due to charge-exchange with the neutral background atoms (loss rate $\beta$), spatial diffusion described by the diffusion tensor $\kappa$ and adiabatic energy change due to nonzero divergence of plasma flow $\nabla\cdot\vec{U}$. The plasma flow and its divergence (and also the background neutral density needed to determine $\beta$) are taken from Kausch's model (Kausch 1998; Fahr et al. 2000). The charge-exchange cross sections are taken from Barnett et al. (1990). We make the scalar diffusion approximation, which in the region between the termination shock and the heliopause may be adequate in lowest order. The diffusion coefficient is assumed to change at the heliopause. Inside the heliopause we assume $\kappa=\kappa_1=(1/3)\kappa_{\parallel}$ with $\kappa_{\parallel}$ the parallel diffusion coefficient derived from the modulation models (we take the formula of Le Roux et al. 1996) while outside the heliopause the diffusion coefficient is taken to be much larger than inside: $\kappa_2=10^2 \ \kappa_1$.

Figure 4: PUI density distribution for 2.5 keV protons outside the termination shock. The density profiles are shown for $\theta =0^\circ$, $104^\circ$, $133^\circ$, $162^\circ$ and $180^{\rm o}$.

Figure 5: PUI density distribution for 100 keV protons outside the termination shock. Note the proton density enhancements due to adiabatic acceleration caused by compression of the flow in the heliotail region.

Parker's equation for the pitch angle averaged distribution function f

$$\frac{\partial f}{\partial t}=\nabla\cdot\kappa\cdot\nabla f ... ...\frac{\partial f}{\partial \log p}\nabla\cdot \vec{U} -\beta f$$ (25)

(the drift speed term is omitted, because it is small in the low energy range we are interested in). The equation was derived (Parker 1965) for the case of particles moving much faster than the background plasma. Downstream of the shock the plasma flows at the speed of less than 100 kms^-1. We take the low energy cutoff at 1 keV corresponding to the particle speed of $\approx$450 kms^-1.

Figure 6: Evolution of the PUI energy spectrum with heliocentric distance downstream from the shock (apex direction). The curves labelled 1 to 5 correspond to the distances of 89 AU ($\sim$shock), 159 AU, 271 AU, 446 AU and 990 AU, respectively. The solid (dotted) lines illustrate the effect of different choices of the lower boundary value in energy (at which the constant slope of the spectrum was assumed): 0.63 keV (1 keV).

Figure 7: As Fig. 6, but in the anti-apex direction. The curves labelled 1 to 5 correspond to the distances of 187 AU ($\sim$ shock), 249 AU, 345 AU, 505 AU and 990 AU, respectively. Note the crossing of the curves, related to the PUI density enhancements in the heliotail (Fig. 5).

We assume the solution to be axially symmetric with respect to the LISM flow apex-antiapex line. As the arguments leading to the post-shock form of the distribution are applicable only in the ecliptic, we must interpret the result as applicable also only near to the ecliptic plane. The boundary conditions used in our approach are defined as follows. At the termination shock surface we set the distribution function $f(\vec{x},v)$ to be equal to the post-shock form of the upstream solution. At large distance the solution is required to match the asymptotic form (corresponding to constant plasma speed $\vec{U}(\vec{x})=\vec{V}_\infty$). At the boundaries in energy space we prescribe the slope of the energy spectrum, which we set to the value of the slope of the downstream shock spectrum.

The choice of the boundary conditions in energy, which must be done at each point in space, is (except for the requirement of consistency with the energy spectrum assumed at the shock) an arbitrary procedure, unless the behaviour of the ion distribution at the limits in energy is known (Czechowski et al. 1999c,2001). To estimate the resulting uncertainty we have checked different cases, varying the energy limits and assuming different slopes. We found that the choice of the boundary conditions in energy affects the solution in the region close to the energy limits, but the effect in the remaining region is small.

The calculations were done for two cases, corresponding to two cases of the upstream solution: $\alpha =-3$ and $\alpha =-2$ (the latter case is the extreme one used only for comparison, see Sect. 1). Because the proton flux behaves differently for these cases, we set the upper energy limit to 100 keV for the first case and to 1000 keV for the second. The low energy limit we set to 1 keV (see above).

The spatial pick-up protons distributions downstream from the shock for two values of energy are presented in Figs. 4 and 5 in the form of constant direction profiles. The angle $\theta$ is counted from the LISM apex direction. The density profiles start at the termination shock (which has the radius dependent on $\theta$ i.e. is non-spherical) and show a sharp change in slope at the heliopause. In Fig. 4 one can see a trough in the low energy (2.5 keV) proton density near $\theta=180^\circ$, which is responsible for the dip in the low energy (<3 keV) ENA flux from the antiapex direction (see Fig. 9). Figure 5 presents the typical distribution in the high PUI energy range (100 keV). Because the PUI energy spectrum is steep at high energy the effect of the adiabatic acceleration in the heliotail (in the region where $\nabla\cdot \vec{U}<0$: Czechowski et al. 2001) is quite pronounced (a "bump'' in the $\theta=162^\circ$ and $180^\circ$ profiles in Fig. 5).

The energy spectra are presented in Figs. 6 and 7 for different heliocentric distances. The higher depletion of low energy particles with the distance is caused by charge-exchange processes (Czechowski et al. 1999c).