2 Calculation of upstream pick-up ion spectra

Soon after their injection into the magnetized solar wind plasma flow, the pick-up ions isotropize their initial ring distribution due to effective pitch-angle scattering processes. The resulting pitch-angle isotropized distribution function, $f(t,\vec{r},v)$, is found as a solution of the following kinetic transport equation (see e.g. Lee 1983; Isenberg 1987; Bogdan et al. 1991):

$$% \frac{\partial f}{\partial t}+ \vec{U}\cdot\nabla f= \frac{... ...c{\partial f}{\partial v} \nabla\cdot\vec{U}\!+\!S(\vec{r},v)~$$ (1)

where $\vec{U}(\vec{r})$ is the solar wind velocity, v is the velocity of pick-up ions in the solar wind rest frame, $D({\bf r},v)$is the energy diffusion coefficient for pick-up ions, and $S(\vec{r},v)$ is the local production rate of freshly ionized particles.

In the following we shall assume axial symmetry with the symmetry axis parallel to the LISM wind vector $\vec{V}_{\infty}$. This in effect restricts our considerations to the vicinity of the ecliptic plane. Using spherical polar coordinates $(r,\theta,\phi)$, with the Sun as the origin and the polar axis oriented to the interstellar upwind direction (i.e. $\theta=0=\rm LISM$ upwind direction), the above transport equation (Eq. (1)) after introduction of the differential phase space density, $G=2\pi r^2 \sin\theta v^2 f$ attains the following form (see Chalov et al. 1997):

$$% \frac{\partial G}{\partial t}= \frac{\partial}{\partial r}(u_rG)- \frac{\partial}{\partial \theta}\left(\frac{u_{\theta}}{r}G\right) -\frac{\partial}{\partial v}$$ -

$$\times \!\!\left[\left(\frac{\partial D}{\partial v}+\frac{2D}{v}\frac{v... ...sin\theta}\frac{\partial u_{\theta}\sin\theta}{\partial \theta} \right)G\right]$$

$$\!\!\frac{1}{2}\frac{\partial^2}{\partial v^2}(2DG)+ 2\pi r^2 \sin\theta v^2 S(r,\theta,v)$$ + (2)

where $u_{\rm r}=u_{\rm r}(r,\theta)$ and $u_{\theta}=u_{\theta}(r,\theta)$ are the radial and tangential components of the mass-loaded solar wind velocity. Equation (2) then is solved numerically by further converting this nonlinear partial differential equation of second order into an equivalent system of linear stochastic differential equations (SDE-system). Further details can be found in papers by Chalov et al. (1995,1997) or by Fahr & Lay (2000).

The injection source $S(r,\theta,v)$ is due to neutral H atoms which become ionized at $(r,\theta)$. These neutral LISM H atoms penetrate from interstellar space into the interface region of perturbed solar wind and interstellar plasma flows and with some depletion eventually appear in the inner heliosphere being subject to ionization processes there. When these neutral atoms are ionized, they constitute the H⁺-pick-up ion production rate $q(\vec{r})$. The local heliospheric pick-up H⁺-production rate is given by the following formula (see Rucinski et al. 1993):

$$q(\vec{r})=n_{\rm H}(\vec{r})[\nu_{\rm ph}(\vec{r})+\nu_{\rm ex}(\vec{r})]$$ (3)

where $n_{\rm H}({\bf r})$ is the number density of the neutral H-atoms in the heliosphere, and $\nu_{\rm ph}(\vec{r})$ and $\nu_{\rm ex}(\vec{r})$are the relevant local photoionization and charge exchange frequencies. In Chalov et al. (1995,1997) the densities $n_{\rm H}(\vec{r})$ are calculated within the self-consistent twin-shock interface model developed by Baranov & Malama (1993) with LISM parameters fixed in accordance with settings made by these authors. The total pick-up ion production rate given in Eq. (3) is related to the injection source $S(r,\theta,v)$ in Eq. (2) by

$$q(r,\theta)=4\pi \int v^2\ S(r,\theta,v)\ {\rm d}v.$$ (4)

We assume that $S\propto \delta (v-U)$, where U is the local solar wind speed. Our assumption of a narrow initial pick-up ion distribution is a simplification (see Zank & Cairns 2000). The interplanetary magnetic field in reality will deviate from the Archimedean spiral as given by the Parker model and used here. The primary injection thus occurs into a plasma flow which has a varying magnetic field orientation, leading to different parallel component of the pick-up ions velocity. The freshly injected PUIs will be slipping with respect to the bulk flow and tend to accumulate in the regions where B is perpendicular to the flow. This situation may be represented by an injection with a broadened delta peak. Also, as discussed by Zank & Cairns (2000), the wave excitation during the pick-up in quasi-perpendicular magnetic field is less efficient than expected. However, the effect on our results would be probably restricted to the low energy part of the spectrum, and the results for the energies significantly higher than the initial spectrum width would not be changed.

The pick-up ion spectra are obtained by numerical integration of the above mentioned SDE-system of linear differential equations and reflect the effects of convection, adiabatic cooling, and momentum diffusion. We use the phase-space pick-up ion distribution function $f^{\rm PUI}$ defined by $f^{\rm PUI}\equiv {\rm d}N/{\rm d}^3x\ {\rm d}^3(\vec{v}/U_0)$ where ${\vec v}$ is the particle velocity relative to plasma and $U_0=4.5\times10^7$ cms^-1 is the reference solar wind speed. The results from SDE calculations by Chalov, Fahr & Izmodenov (Chalov et al. 1995) are expressed in terms of $\Phi^{\rm PUI}\equiv U_0 [{\rm d}N/{\rm d}^3x\ {\rm d}(E/E_0)]$ ( $E_0=mU_0^2/2\simeq 1$keV/n) which has the dimension of the differential flux (it is approximately equal to the pick-up ion differential flux integrated over the direction of particle velocity). $\Phi^{\rm PUI}$ is related to $f^{\rm PUI}$ by:

$$f^{\rm PUI}=\frac{\Phi^{\rm PUI}}{2\pi U_0 \sqrt{w}}$$ (5)

where w denotes the normalized particle kinetic energy in the plasma frame: w=(v/U₀)²=(E/E₀).

In Fahr & Lay (2000) the calculations were restricted to the upwind direction (i.e. $\theta=0$). The numerical results for $\Phi^{\rm PUI}(X,w)$ (where $X=(r/r_{\rm E})$, $r_{\rm E}=1$ AU) in the pre-shock range $1\leq X\leq X_{\rm S}(\theta=0)$( $X_{\rm S}(\theta)\equiv r_{\rm S}(\theta)/r_{\rm E}$ is the normalized distance to the termination shock) can be parametrized in the form:

$$\Phi^{\rm PUI}(X,w)=C_{\gamma}(X) w^{\gamma} {\rm e}^{-[C_K(X)(w-w_0)^K]}$$ (6)

with $C_{\gamma}(X)$, C_K(X), K, $\gamma$, and w₀ defined below. This corresponds to the following expression for the distribution function:

$$f^{\rm PUI}_1(r,\theta=0,v)=\frac{ C_{\gamma}(X) w^{\gamma} {\rm e}^{-[C_K(X)(w-w_0)^K]} }{2\pi U_0 \sqrt{w}}\cdot$$ (7)

Here the subscript "1'' denotes the preshock spectrum.

We consider two different cases, corresponding to different behaviours of the Alfvenic turbulence with the distance from the Sun: $<\tilde{B}^2>~\propto r^{\alpha}$, where $\alpha =-3$ or -2(Chalov et al. 1995). The first case ($\alpha =-3$, dissipationless wave propagation) is appropriate to solar minimum conditions (low turbulence). In this case the numerical results are best fitted by the following parameters

$$C_{\gamma}(X) 10^{4.3141}X^{-0.3363}\ [{\rm cm}^{-2}~{\rm s}^{-1}{\rm ~keV}^{-1}]$$ =

C_K(X) = 0.442 X^0.202

$$\gamma -0.1145 \ \ K=(2/3) \ \ w_0=0.8330.$$ = (8)

The other case ($\alpha =-2$) includes wave generation in interplanetary space and leads to the fit parameters:

$$C_{\gamma}(X) 10^{4.1613}X^{-1.2567}\ [{\rm cm}^{-2}~{\rm s}^{-1}{\rm ~keV}^{-1}]$$ =

C_K(X) = 0.469 X^-0.5741

$$\gamma -0.036 \ \ K=(2/3) \ \ w_0=9.7707.$$ = (9)

It should be noted that while the first case ($\alpha =-3$) corresponds to the average energy of the pick-up ions of the order of 2 keV at large distance, so that the total energy in the pick-up ions is not larger than 0.3-0.5 of the solar wind kinetic and thermal energy, the case $\alpha =-2$ is unrealistic: the average energy of the pick-up ion obtained from the resulting distribution function in this case may be as high as 100 keV. In this case too much energy is transferred from waves to the pick-up ions. Nevertheless, in the following we present some of the results obtained with $\alpha =-2$ using them as an illustration of an extreme case.

Our results are based on a simplified approach, in which the effect of wave driving by the PUIs was neglected. The studies of MHD turbulence behaviour in the outer heliosphere carried out by Zank et al. (1996) and Matthaeus et al. (1999), with the effects of mode mixing, driving by stream-stream interactions and wave excitation by interplanetary pick-up ions included in approximate form, produce the values $-10/4\ge\alpha\le -9/4$. To extend our approach to these cases would require including wave-driving in the transport equation from which the pick-up ions distribution function is derived.

Figures 1 and 2 present the evolution of the PUI distribution function with the distance in the upwind direction for these two cases together with the corresponding post-shock distribution functions (see Sect. 3).

Our aim is now to generalize these results, derived for the upwind ($\theta=0$) axis, to the general inecliptic regions with longitude angles $\theta\ge 0$. Instead of solving Eq. (2) for the non-spherically-symmetric case we shall assume that the distribution function can be derived from the solution for the upwind direction by a simple re-scaling with a direction dependent correction factor $\nu(\theta)$:

$$f^{\rm PUI}_1(r,\theta,w)=\nu(\theta)f^{\rm PUI}_1(r,\theta=0,w).$$ (10)

$\nu(\theta)$ is determined by requiring that the resulting distribution reproduces the directional dependence of the pick-up ion number density at some reference distance. This approach requires the assumption that the interplanetary turbulence levels determining the momentum diffusion coefficients $D(\vec{r},v)$ within the ecliptic only depend on the solar distance r, but not on longitude $\theta$. Momentum diffusion thus operates independent of longitude.

The correction factor $\nu(\theta)$ is given by

$$\nu(\theta)=\frac{n^{\rm PUI}(r_R,\theta)} {\int f^{\rm PUI}_1(r_{\rm R},\theta=0,v){\rm d}^3(v/U_0)}$$ (11)

where $r_{\rm R}$ is some reference distance (we take $r_{\rm R}=50$ AU), $f^{\rm PUI}_1$ is given by Eq. (7), and $n^{\rm PUI}(r,\theta)$ is the pick-up ion number density at $(r,\theta)$ which we take from the model of the heliosphere based on a gas-dynamical 5-fluid numerical solution (Fahr et al. 2000) in which the pick-up protons were considered as one of the component fluids. The same model is used to provide the function $r_{\rm S}(\theta)$ defining the shape of the solar wind termination shock.

With the help of the above formulae one can give the near ecliptic PUI upstream spectra for longitudes $\theta\ge 0$ at the shock by

$$f^{\rm PUI}_1(r_{\rm S}(\theta),\theta,v)=\nu(\theta)\frac{ C_{\g... ...ta))w^{\gamma} {\rm e}^{-[C_K(X_{\rm S}(\theta))(w-w_0)^K]}} {2\pi U_0 w^{1/2}}$$ (12)

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

Figure 1: PUI distribution function evolution with distance ($\alpha =-3$). The functions for distances 5.0, 20.0 and 80.0 AU (apex direction) are presented by dotted lines. The distribution immediately downstream from the shock is shown by the solid line. (See Sect. 3).

Figure 2: PUI distribution function evolution with distance ($\alpha =-2$). The functions for distances 5.0, 20.0 and 80.0 AU (apex direction) are presented by dotted lines. The distribution immediately downstream from the shock is shown by the solid line (see Sect. 3).

Figure 3: Preaccelerated pick-up proton spectra immediately downstream from the termination shock (model predictions: Eq. (24)). The spectra are shown for three positions on the surface of the shock: $\theta =0^\circ$ (apex), $90^\circ$ (side) and $180^\circ$ (antiapex) in the ecliptic plane. The cases of normal ($\alpha =-3$) and high turbulence ($\alpha =-2$) are presented. Also presented is the ACR spectrum at the shock derived from de-modulation of the observed spectrum (Stone et al. 1996).