We have considered a model of the evolution of the pick-up ion distribution including both the pre-shock and the post-shock regions of the heliosphere. The pre-acceleration of the pick-up ions by the turbulence in the pre-shock solar wind was taken into account. The results suggest that the energetic ion spectrum near the termination shock may, for the energies below 50-100 keV, be determined by the shock-modified pre-accelerated pick-up ion spectrum rather than the power-law spectrum expected from the shock acceleration.

The ion flux at the shock obtained in the model is at low energy (up to 60-70 keV) significantly higher than the extrapolated ACR power law spectrum. In result, also the ENA flux from the pick-up ions is high and agrees with the observations by CELIAS/HSTOF (the alternative interpretation, assuming that the ENA come from the shock-accelerated ACR particles, may have problems with explaining the observed ENA flux level: see Czechowski et al. 2001). The directional dependence of the ENA flux is also in agreement with observations, with the flux intensity peaks from the (approximately) anti-apex direction. This may be an indication that the ENA flux comes from the post-shock region, although there is a possible alternative (Kota et al. 2001) connected with the CIR accelerated ions interacting by charge-exchange with the helium atoms in the region where the helium density is increased due to gravitational focusing (the helium cone). Although the errors are large, the data suggest the anti-apex to apex flux ratio lower than calculated. This may imply that the model of the heliosphere used in the calculations (Fahr et al. 2000; Czechowski et al. 2001) should be modified. Another possibility is that a part of the ENA flux comes from another source, with no peaks from the anti-apex direction.

$\begin{figure} \par\includegraphics[width=8.5cm,clip]{h2973f13.eps}\end{figure}$

Figure 13: CELIAS/HSTOF data compared with the PUI ENA hydrogen energy spectrum (flux from the forward direction, $\alpha =-3$ ). Only the first three low energy data points (marked by filled triangles) are expected to represent the neutral flux (see the text). The data points above 90 keV (open triangles) depend significantly on the quiet time proton flux. The background (not shown) is of the order of1- $2 \times 10^{-4}~({\rm cm^2\ s\ sr\ keV})^{-1}$ .

The results are consistent with the "normal turbulence'' case, with the amplitude of the magnetic field perturbations evolving adiabatically ( $\tilde{B}^2\propto r^{-3}$ case). The other version of the model (persisting source of the perturbations, $\tilde{B}^2\propto r^{-2}$ ) does not fit the ENA data, but it is not realistic enough in any case (too large total energy in the PUI spectrum).

The calculated ENA energy spectrum is steeper than the spectrum deduced from CELIAS/HSTOF data. The experimental spectrum is, however, sensitive to assumptions about the proton energy spectrum and the transmission function. Also, the PUI model disregards the effects of diffusive shock acceleration, which can in fact become important in the considered energy range and make the proton and the ENA spectrum less steep.

The energy range of the CELIAS/HSTOF observations (58-88 keV) coincides with the region where the PUI and PUI ENA spectrum decrease very fast (for the normal turbulence case) and other contributions may appear. The forthcoming data from INCA/Cassini (Krimigis et al. 2000) and HENA/Image (Mitchell et al. 2000; first results: Roelof 2000) which should include also the lower energy region where the model predicts high ENA flux, are therefore of particular interest.

6 Conclusions