A&A 439, 1-22 (2005)
R. Kallenbach1 - M. Hilchenbach2 - S. V. Chalov3 - J. A. le Roux4 - K. Bamert 5
1 - International Space Science Institute, Hallerstrasse 6, 3012 Bern, Switzerland
2 - Max Planck Institute for Solar System Research, Postfach 20, 37191 Katlenburg-Lindau, Germany
3 - Institute for Problems in Mechanics of the Russian Academy of Sciences, Prospect Vernadskogo 101-1, 117526 Moscow, Russia
4 - Institute of Geophysics and Planetary Physics, University of California, Riverside, CA 92521, USA
5 - Institut für Experimentelle und Angewandte Physik, University of Kiel, Leibnizstrasse 19, Kiel 24098, Germany
Received 14 February 2005 / Accepted 6 April 2005
This article presents an integrated analytical model on the injection efficiencies of the different ion species of the Anomalous component of the Cosmic Rays (ACRs) at the solar wind termination shock. We find that the injection into diffusive (first-order Fermi) acceleration is dominated by parallel ion diffusion and not by perpendicular diffusion unless the angle between the shock normal and the heliospheric magnetic field is almost exactly 90 ( ). In steady state the threshold speed for injection into first-order Fermi acceleration at a not exactly perpendicular solar wind termination shock - with the Parker shock angle - adjusts itself self-consistently. Increased anisotropic ACR flux amplifies Alfvénic turbulence which in turn suppresses parallel diffusion. It therefore increases the injection threshold and decreases the ACR flux until equilibrium is reached. For this equilibrium situation, we estimate the injection efficiencies of different species of suprathermal ions at the termination shock. We consider the following pre-acceleration processes: 1) momentum diffusion in compressional (ion-acoustic and magnetosonic) turbulence in the upstream supersonic solar wind and adiabatic cooling during convection to the termination shock; 2) reflection, transmission, and acceleration in the electric potential of the termination shock; and 3) momentum diffusion (stochastic or second-order Fermi acceleration) in the subsonic solar wind downstream of the termination shock in the inner heliosheath region. Our model results are compared to data from instruments on board the SOHO, ACE, Ulysses, and Voyager spacecraft.
Key words: solar wind - plasmas - acceleration of particles - turbulence - shock waves - solar system: general
Anomalous Cosmic Rays (ACRs) are thought to originate from suprathermal pick-up ions derived from the interstellar gas (Fisk et al. 1974), ionized in the heliosphere and convected to the solar wind termination shock where they are accelerated (Pesses et al. 1981). The energy threshold for the injection of suprathermal ions into diffusive acceleration at the solar wind termination shock has been estimated to be of order 100 (Zank et al. 2001) to 1000 keV/amu (Jokipii 1992). Ions in the energy range above this threshold can be observed in-situ near the termination shock using the LECP instrument on board the Voyager 1 spacecraft (Krimigis et al. 2003; McDonald et al. 2003; Fisk 2003). Suprathermal ions in the energy range from 1 to 100 or 1000 keV/amu i.e. in the energy range below the commonly presumed injection threshold may be studied by "remote'' imaging of the energetic neutral atoms (ENAs) that originate from the heliospheric interface region and reach Earth's orbit on undistorted trajectories. In the energy range 58-88 keV/amu ENAs have been observed with the SOHO/CELIAS/HSTOF sensor near Earth's orbit (Hilchenbach et al. 1998). Upper limits on the ENA flux in the energy range 10-50 keV/amu have been found with the IMAGE/HENA instrument near Earth's orbit (E. Roelof, private communication). However, ENAs may only provide observational data to verify models on the injection of pick-up ions at the termination shock if the ENA flux from the outer heliosphere is not obscured by ENAs created in the inner and middle heliosphere.
In this article, we give theoretical estimates for the suprathermal particle flux in the heliosphere inside and outside the termination shock. We describe the processes of momentum diffusion (stochastic or second-order Fermi acceleration) in compressional and Alfvénic turbulence in the supersonic solar wind, the interaction of suprathermal ions with the electric field of the termination shock, and momentum diffusion in Alfvénic turbulence of the heliosheath plasma downstream of the termination shock. The power spectral density of Alfvénic fluctuations in the heliosheath is yet unknown, but we try to estimate it from theory on turbulence transmission through shocks (McKenzie & Westphal 1969; Vainio & Schlickeiser 1998) and from the recently observationally verified efficiency of Alfvén wave amplification by suprathermal and energetic particles (Bamert et al. 2004). In a self-consistent manner, these Alfvénic fluctuations determine the injection threshold at the termination shock. We do not model the stochastic acceleration of suprathermal ions in compressional turbulence downstream of the termination shock because of the high uncertainty of its fluctuation levels.
The study of this article extends the work by Chalov et al. (2003) in so far as it includes 1) a more detailed assessment of momentum diffusion in ion-acoustic solar wind turbulence; 2) fully analytical approximations which are easy to handle; 3) an estimate of the self-consistent injection threshold at the solar wind termination shock; and 4) in particular a more detailed comparison to spacecraft data. We compare our model results with suprathermal pick-up ion flux data from ACE/SWICS at 1 AU and Ulysses/SWICS at heliocentric distances out to 5 AU (Gloeckler 2003), with ENA flux data in the energy range 58-88 keV/amu from SOHO/CELIAS/HSTOF (Hilchenbach et al. 1998), and with ion flux data at energies larger than 100 keV/amu from Voyager 1. In particular, we refer to a recent evaluation of the spectra of suprathermal pick-up ions transmitted through the quasi-perpendicular bow shock of Jupiter (Gloeckler et al. 2004). The study presented in this article should give guidelines to estimate the contributions from the regions inside and outside the termination shock to the ENA flux observed at 1 AU. This is of importance for future space missions such as the Interstellar Boundary Explorer (IBEX).
The majority of suprathermal ions in the middle heliosphere is thought to be derived from pick-up ions of interstellar origin. Pick-up ions in the supersonic solar wind are selectively accelerated in co-rotating interaction regions (Balogh et al. 1999) and at interplanetary coronal mass ejections (Bamert et al. 2002). Even in the quiet solar wind, undisturbed by interplanetary shocks, ubiquitous suprathermal tails in the pick-up ion distributions are observed (Gloeckler 2003), (Fig. 1). These pick-up ions, pre-accelerated in the supersonic solar wind and convected to the outer heliosphere, may provide a seed population for injection into diffusive acceleration at the termination shock.
|Figure 1: Left: quiet-time proton spectra in the CME-dominated slow solar wind and in polar coronal holes during solar activity maximum (Gloeckler 2003). The average suprathermal proton flux in the slow solar wind is about a factor 5 higher than during the quiet times and even higher in the CIR-dominated slow solar wind during solar activity minimum. Therefore, the shown suprathermal proton flux is a lower limit for the flux in the inner and middle heliosphere. Flux of suprathermal H+ and He+ upstream and downstream of Jupiter's quasi-perpendicular bow shock. Right: comparison of suprathermal tails of H+at 1 AU and at about 5 AU. Adapted from Gloeckler (2003).|
|Open with DEXTER|
The acceleration process that creates the suprathermal tails of pick-up ions remains uncertain, however. They may be created through acceleration of pick-up ions which are multiply reflected at quasi-perpendicular shocks of co-rotating interaction regions (le Roux et al. 2000) and their successors in the outer heliosphere, or through selective acceleration at shocks driven by coronal mass ejections (Kallenbach 2002). If these shocks are responsible for the pre-acceleration of pick-up ions the process may cease during solar activity minimum at a heliocentric distance of about 15 AU, where the shocks of merged interaction regions cease to exist (Gazis et al. 1999). The process may cease even closer to the Sun during solar activity maximum because the shocks of interplanetary coronal mass ejections - which supply the majority of energetic particles during activity maximum - weaken rapidly with heliocentric distance.
The suprathermal tails may also arise from stochastic acceleration in turbulent waves. Dwyer et al. (2004) suggest that suprathermal ions predominantly originate in the compressed region between the forward and reverse shock of co-rotating interaction regions. The momentum diffusion parameter of ions trapped in compressional fluctuations of the solar wind plasma has been derived by Bykov & Toptygin (1993). Compressional fluctuations may be classified as ion-acoustic turbulence (Gurnett et al. 1979a; Mangeney et al. 1999) and magnetosonic turbulence. Schwadron et al. (1996) have studied the correlation between suprathermal ion flux and the strength of magnetic field fluctuations near co-rotating interaction regions in order to verify the model on statistical acceleration in magnetosonic wave turbulence (Fisk 1976). Stochastic acceleration also occurs in non-compressional Alfvénic solar wind turbulence. The strength of these types of turbulence relative to the energy density of the bulk plasma is predicted to persist or even to increase in the solar wind out to the region of the termination shock (Zank et al. 1996).
Any of the acceleration processes of suprathermal ions has to
compete with the adiabatic cooling in the radially expanding solar
wind. The dynamics of suprathermal ions convected with the solar
wind plasma is described by the pitch-angle averaged Fokker-Planck
equation (Chalov et al. 2003):
We assume small-scale Alfvénic turbulence scaling as a power
with spectral index
above a minimum wavenumber
is about 0.03 AU at a heliocentric distance
(Goldstein et al. 1995). To estimate the approximate efficiency for
momentum diffusion in this Alfvénic turbulence we apply standard
quasi-linear theory (Hasselmann & Wibberenz 1968). For further
use in this article we introduce the parameter describing the
total relative Alfvénic magnetic field fluctuations
The parallel scattering mean free path
follows from cyclotron-resonant interaction
the Alfvénic fluctuations, where
is the angular speed of the gyro-motion of an
ion with atomic mass-to-charge ratio
We also give
the parallel spatial diffusion parameter
In the heliocentric frame, where ions have speed v, this momentum diffusion parameter is valid for or . For suprathermal ions with , such as the observed ions in the suprathermal tails of the solar wind (Gloeckler 1999), the above momentum diffusion parameter can be applied. For more refined models on the diffusion parameters in Alfvénic fluctuations, we refer to Schlickeiser (2002).
The rms-amplitude of the average Alfvénic fluctuations in the solar wind scales as (Zank et al. 1996). The angular speed of the gyro-motion scales with B0, which in turn scales with in the outer heliosphere. At 1 AU the proton angular speed is about 0.5 s-1. The Alfvén speed is assumed to be about 45 km s-1 at 1 AU and does not vary much with heliocentric distance. The correlation length increases less than or about proportional to (Zank et al. 1996), and most power spectra are Kolmogorov-like with spectral index s = 5/3.
Therefore, the parameter scales approximately with . The relative magnetic turbulence power in the undisturbed slow solar wind ranges from 0.01 (Bamert et al. 2004) to 0.03 (Forsyth et al. 1996). However, only a fraction of 0.15 of the turbulence represents parallel Alfvén waves (Bieber et al. 1996) so that we take 10-3 as typical value.
The momentum diffusion parameter
becomes dimensionless in units of the convection time scale
at 1 AU by multiplying it by
and dividing it by
It then reads
The interaction of suprathermal ions with magnetosonic fluctuations differs from the interaction with Alfvénic fluctuations because magnetosonic waves are compressional and have
an electric field amplitude along the ambient magnetic field. The
dimensionless diffusion parameter arising from large-scale
magnetosonic fluctuations, which have a correlation length of
AU, is (Chalov et al. 2003):
Another type of compressional fluctuations in the solar wind are ion-acoustic fluctuations. Ion-acoustic waves mainly propagate along the ambient magnetic field and have an electric field amplitude along the ambient magnetic field.
We need to distinguish two types of ion-acoustic turbulence regions: 1) regions, where the size of the turbulence region along the magnetic field is large compared to the parallel scattering mean free path and 2) regions, where . There is a basic difference in the momentum diffusion in these two types of regions. If the bulk plasma fluid transfers momentum contained in the velocity fluctuations to the suprathermal ions. According to Bykov & Toptygin (1993) the detailed scattering mechanism that determines is not important, and the momentum diffusion parameter only depends on the total amplitude of the velocity fluctuations, but not on its spectral distribution. In the solar wind and near collisionless shocks, we can use the mean free path for pitch-angle scattering in Alfvénic turbulence as upper limit for the parallel mean free path to verify the condition .
In the case the ion only gains energy in the fluctuating electric field of the compressional turbulence. We will treat the acceleration due to the Landau resonance between ions and ion-acoustic waves. The power spectral density of the electric field fluctuations then enters the momentum diffusion parameter.
For the situation that suprathermal ions are trapped in the region
of ion-acoustic turbulence, i.e.,
the momentum diffusion in ion-acoustic
fluctuations also scales with u2 as in the case of magnetosonic
fluctuations (le Roux 2004):
The dispersion relation of ion-acoustic waves is
The power spectral densities in these ion-acoustic turbulence
regions have been found to be as large as
V2 m-2 Hz-1 at 1 AU and at a
frequency of about 3 kHz; at smaller heliocentric distance the
wave intensity has been found to be larger, further out it is
smaller. We estimate the rms-amplitude of the speed fluctuations
from the power
From Eq. (8) the momentum diffusion parameter is
The above values only apply for the strongest ion-acoustic waves near interplanetary shocks. At solar activity maximum, these shocks are mainly CME-driven shocks. If a few CMEs per day with a typical size of the turbulence region 10-3 AU pass the heliosphere, the time-averaged power spectral density of the ion-acoustic turbulence may be about 3 10-2 of that observed downstream of CME-driven shocks. This still would yield .
Ion-acoustic activity may also be present in the ambient solar
wind. The over-all power spectral density of ion-acoustic
fluctuations in the heliosphere may drop inversely with
heliocentric distance, however, this is not well known, in
particular it is unknown for the outer heliosphere. As baseline we
take for the average momentum diffusion coefficient in
ion-acoustic turbulence with
In the quiet slow solar wind, ion-acoustic turbulence may also be present in the form of weak double layers (Salem et al. 2003). Although these structures are subject to further investigation, we model the potential efficiency of stochastic ion acceleration in their electric field fluctuations. Their typical net potential is reported to be about 1 mV, and they are observed as spikes that last about 1 ms and typically occur at a rate of 1 s-1 at 1 AU. This gives a rough estimate for their spatial extension (Mangeney et al. 1999), which in fact is smaller than the value for the maximum wavelength mentioned above. For such small values of the condition is unlikely to apply. Therefore, another approach than that of le Roux (2004) needs to be taken. The ion is not energized by transfer of momentum from the bulk fluid i.e. by the velocity fluctuations but by the electrostatic field fluctuations in a narrow bandwidth given by the Landau resonance.
In quasi-linear theory the momentum diffusion parameter in
electrostatic field fluctuations is calculated as (Treumann & Baumjohann 1997)
The power spectral density
of the electrostatic fluctuations of the ion-acoustic
waves is up to
V2 m-2 Hz-1and
over the spikes (Lacombe et al. 2002). Typical mean power spectral
densities in the slow solar wind during solar activity minimum are
V2 m-2 Hz-1 (Issautier et al. 1999). In our evaluation we allow for a general power law
with spectral index s. Then, the diffusion parameter is
Using the above diffusion coefficients, we first solve the homogeneous part of the above equation i.e. with Q=0 and S=0and then add a source of freshly ionized pick-up ions or a source of pick-up ions pre-accelerated at quasi-perpendicular shocks during multiple reflections. We first assume due to turbulence regions which are larger than the mean free path for pitch-angle scattering. Then, we add to the problem with resulting from e.g. weak double layers which are smaller than the mean free path for pitch-angle scattering.
If the distribution function is factorized as
then we obtain a
differential equation for
The above integral over the source function is limited to the
0 < u' < 2. This means that for u > 2 the injection
function is a constant
It follows then from the above differential equation
The speed characterizes the importance of stochastic acceleration by small-scale electrostatic fluctuations such as the weak double layers. Large results in large , while results in and . The complete solution is a power law for , where typically . Therefore, in the range u > 2 i.e. outside the pick-up ion shell, where the suprathermal tails can be observed, the theory predicts power laws, as for .
If the distribution function is factorized as
then we need to solve for
We conclude that stochastic acceleration in small-scale electrostatic fluctuations (ion-acoustic waves), with described by a momentum diffusion parameter , is mainly effective at low speeds. The spectral index at larger speeds is determined by the diffusion parameter . However, the momentum diffusion described by can boost the injection efficiency to some extent. We note, that for the injection of solar wind ions, the ion-acoustic waves have the "advantage'' that their phase speed can be quite low for frequencies around the proton plasma frequency . Consequently, the resonance between ion-acoustic waves and the tails of the bulk solar wind is more efficient than the resonance with Alfvén waves.
As baseline for our work we take the phase space densities
We also estimate the typical spatial scale with which the phase space densities vary in the heliosphere. For suprathermal tails with spectral index , the ratio of the adiabatic cooling rate to the inverse convection timescale is . Therefore, spatial variations occur on a scale . Variations of at least a factor 5 are observed in the suprathermal ion flux (Gloeckler 2003), for instance when comparing quiet-time flux with the average flux. This implies that the "acceleration site'' of the quiet-time suprathermal tails may be as distant as about , which is about 0.5 AU near Earth for spectral indices . This needs to be kept in mind when correlating measured turbulence levels and suprathermal ion flux. The tails during quiet times may be remnants from acceleration in regions with higher turbulence such as the downstream regions of interplanetary shocks or regions of stream-stream interactions.
All the processes of stochastic acceleration in plasma waves described above prefer pick-up ions over bulk solar wind ions. They require that the ions are faster than the wave phase speed in the solar wind frame. While this is usually the case for pick-up ions, the vast majority of solar wind ions is slower than the wave phase speeds, even in the case of ion-acoustic waves. Therefore, an additional process such as direct acceleration at the shock may be required for the bulk solar wind ions. On the other hand, direct acceleration of ions at interplanetary shocks is again more efficient for pick-up ions because part of the pick-up ions has the same velocity as the shock, which, for instance, is necessary for shock surfing to operate (Lee et al. 1996).
Two processes are commonly discussed in the context of ion acceleration near the termination shock: (i) diffusive (first-order) Fermi acceleration and (ii) shock drift acceleration in the motional electric field of the shock. In fact, during diffusive acceleration at the termination shock ions seem to gain most energy in the motional electric field as well (Jokipii 1992). In the following we derive an expression for the injection threshold speed into first-order Fermi acceleration from quasi-linear theory. Then we apply quasi-linear theory to estimate the approximate strength of Alfvén wave amplification by ACRs and the resulting increase of the injection threshold speed. This leads to a self-consistent adjustment of the ACR intensity. Additionally, the effect of the electric shock potential on low-energy suprathermal ions is quantified.
The standard theory of diffusive shock acceleration (first-order
Fermi process) predicts a power law for the energetic ion distribution,
Diffusive acceleration at a shock operates if the ion is
diffusively returned to the shock from the downstream plasma. The
mean speed of this diffusive motion must be larger than the
convection speed with which the ion is transported in downstream
direction. In the steady state of first-order Fermi acceleration
this is the upstream convection speed
the upstream and the downstream plasma. The diffusion speed is the
spatial diffusion parameter
divided by the gradient scale
of the distribution function of super-Alfvénic ions. The gradient scale cannot be smaller than the gyro-radius
(Jokipii 1992; Zank et al. 2001). Taking the gyro-radius
as gradient scale gives an upper limit for the "diffusion speed'' and thus a lower limit for the injection
According to standard kinetic
theory we get the injection condition
In the following we express lengths in units of (1 AU) and speeds in units of the downstream plasma speed
The gyro-radius in units of
The parallel mean free path is
The effective mean free path perpendicular to the termination
shock is approximately
From Eq. (33) we estimate the injection threshold. For the expression for the injection threshold yields the value . This very small value suggests that the "injection problem'' in this form may only be a problem that applies for a strictly perpendicular shock. For a not exactly perpendicular termination shock, the injection threshold is simply given by the condition that the ions must be super-Alfvénic in the upstream and downstream plasma. The simple condition that the ions should be super-Alfvénic apparently advantages the pick-up ions for injection into diffusive acceleration.
However, the "injection problem'' may arise for the following reasons: 1) the quasi-linear theory does not apply for only slightly super-Alfvénic particles. In this article, we do not address this problem which has to be tackled by numerical simulations. 2) Once ACRs are accelerated, they generate Alfvén waves themselves and increase the factor . In that sense the ACR production is "self-limiting''. High ACR flux means high Alfvén wave amplification and thus a large factor and an increasing injection threshold , which causes the ACR production to be less efficient. In fact, the injection threshold speed may easily be in the highly super-Alfvénic range, where quasi-linear theory is usually a good approach. We pursue these thoughts in the following. First, we show that the acceleration times of ACRs are sufficiently short to allow for acceleration within the convection time scale of the heliosheath plasma and within the ionization time of the singly charged pick-up ions to become doubly charged.
The typical time for an ion to be accelerated by the first-order
Fermi process from speed v0 to v1 is (e.g. Zank et al. 2001,
and references therein)
The maximum time for ACRs to remain singly charged (Ellison et al. 1999) is . The convection time scale in the heliosheath is approximately the same. This means that the upper speed limit in ACR acceleration is quite high. We need, however, to explore the factor in more detail. It turns out that the factor can be quite large.
The above acceleration times are upper limits because the ions also gain energy in the motional electric field during each shock transition (Jokipii 1992).
of Alfvén wave generation upstream of
the termination shock is derived from the efficiency of Alfvén wave generation through pick-up ions and through anisotropic energetic particles. Here, we report on recent observations near
interplanetary travelling shocks (Bamert et al. 2004) and try to
extrapolate these results to the situation of the termination
shock. The increase of wave power is given by a factor
|Figure 2: a) Voyager 1 and Voyager 2 ACR proton energy spectra in the outer heliosphere at about 85 AU during the intensity enhancement in 2002 (2002/2092002/364), along with the solar minimum spectra observed by Voyager 1 in 1998/11999/182 ( dashed lines) and the predicted shock source spectra for a weak shock ( dotted lines). From McDonald et al. (2003). b) Phase space density of protons upstream of the main interplanetary shock driven by the Bastille Day CME at 1 AU. The dash-dotted line denotes the phase space density that the protons would assume if they were propagated from the shock to the upstream region without amplifying Alfvén waves of the ambient solar wind. The dashed line denotes the model prediction by Lee (1983). c) Power spectral density of magnetic field fluctuations measured at the same time and location as the proton spectrum in b). The increase of the wave power - with respect to the ambient solar wind level ( dash-dotted line) - due to amplification by the energetic protons shown in panel b) is by about a factor 30 at . Panels b) and c) from Bamert et al. (2004).|
|Open with DEXTER|
The unmodulated ACR phase space density at 1 MeV (see Fig. 2)
may still be a factor 100 higher right at the termination shock.
This can be expected from the high-energy part of the spectra in
Fig. 2a. As
scales with and
does not vary with ,
expect a wave amplification near the termination shock of perhaps
for a quasi-parallel
shock. The Alfvén wave generation scales as (Lee 1983), where
is the angle of the ambient magnetic
field to the shock normal. This yields typically
termination shock. Introducing the resonance condition
to the heliosheath speed scale
the amplification factor becomes
With the expected rather high amplification of Alfvén waves -
even at a quasi-perpendicular shock - the parallel mean free path
is strongly reduced and the perpendicular mean free path is
increased accordingly. In Eq. (33) we need to
include the full expression for the effective mean free path. We
calculate the injection threshold
Fermi acceleration to operate. We use
as the ratio between the proton ACR phase space density with spectral index
and the ACR phase space density
s3 km-6 which causes an amplification of about
for Alfvén waves resonating with 1 MeV protons. The ratio
also expresses so-to-say the ratio between the
modelled ACR phase space density and the phase space density
expected from the observation of the unmodulated high-energy tail of the ACRs. We perform some algebra:
Equation (38) shows that the injection threshold is lower for ions with high mass-to-charge ratio. This is in general agreement with the observed composition of the ACRs (Cummings et al. 2002). Equation (38) qualitatively predicts that the injection threshold rises when the ACR intensity rises. This is due to the fact that increasing ACR intensity causes increasing Alfvénic turbulence and, thus, a reduction of the diffusion speed and consequently a higher injection threshold. Therefore, some level will result in a self-consistent manner.
To estimate the level on which the ACR intensity adjusts itself
self-consistently we additionally need to know from which source
the ACRs are "fed''. We treat the most simple case that power-law
distributions are injected into first-order Fermi acceleration.
is derived from the phase space density
of Eq. (29) at high speeds
and from the expression in Eq. (38) for the injection threshold as a function of
|Figure 3: Left: flux of ACRs relative to the expected maximum unmodulated ACR flux at the termination shock, corresponding to s3 km-6, as a function of the injected phase space density , where is the parameter of the abscissa. The range of extrapolated phase space densities from observations at 1 AU and 5 AU is shown in grey shade. Right: injection threshold energy of protons corresponding to the relative ACR proton flux levels shown in the left panel for spectral indices and .|
|Open with DEXTER|
The injection efficiency is commonly defined as the ratio of the
pressure of the ACRs of species S at the termination shock with
respect to their dynamical pressure in the upstream solar wind:
We need to insert the full function
of Eq. (29) which is based on an injected power law
convected to the termination shock, where, for instance,
s3 km-6 and
to allow for a more general expression for the injected population at the termination shock. We get
|Figure 4: Injection efficiencies of H+, He+, and O+ ACR ions as a function of the injected phase space density of the proton suprathermal tails characterized as in Fig. 3.|
|Open with DEXTER|
Any shock has a cross-shock electric potential and an associated electric field. Non-parallel shocks have additionally a motional electric field in the upstream plasma viewed from the shock frame. The motional electric field is parallel to the shock front. The electric fields will accelerate, decelerate and/or reflect suprathermal ions. A recent analysis of data from the SWICS instrument on board Ulysses (Gloeckler et al. 2004) indicates that the distribution functions of suprathermal ions are considerably modified when they pass Jupiter's bow shock (Fig. 5). A possible explanation of this phenomenon is the interaction of the ions with the shock potential and the convective (motional) electric field in the foreshock region.
|Figure 5: Flux of suprathermal H+ and He+ upstream and downstream of Jupiter's quasi-perpendicular bow shock. The suprathermal tails of H+ are enhanced by a factor 10, while the He+ tails are enhanced by a factor 4-5. Data from Gloeckler et al. (2004).|
|Open with DEXTER|
We do not present a detailed discussion of the direct pre-acceleration of pick-up ions at the quasi-perpendicular termination shock during multiple reflections of pick-up ions (le Roux et al. 2000). Rather, we give a rough estimate of the modification of a nearly isotropic power-law distribution function of suprathermal ions already present in the upstream supersonic solar wind and convected through the termination shock. In order to do so we calculate a transmission coefficient for a given ion speed u. This transmission coefficient is based on the distinction of ion pitch-angles that allow for transmission and those pitch-angles which lead to a reflection of the ion. Ions which are reflected will encounter the termination shock again, but eventually with a different pitch-angle and somewhat higher speed due to the acceleration in the motional electric field near the shock. As the total number of pick-up ions is conserved during the passage through the shock, the downstream spectrum is to first order given by the normalized product of the transmission function times the upstream spectrum. We neglect the modification of the spectral index of the suprathermal ions by the passage through the termination shock. The data of Fig. 5 indicate that the spectral index of protons does not change very much.
We begin with the motion of test particles in the shock potential
described in the shock frame:
We continue with the solution to the quadratic part of Eq. (45), which is valid for the majority of ions:
Three populations approach the termination shock from the upstream solar wind: 1) the bulk solar wind ions idealized as a pencil beam ; 2) the freshly ionized pick-up ions in a shell distribution ; and 3) the suprathermal tails for . The suprathermal tails at the termination shock presumably reach down to almost because the speeds of the waves causing these tails are much smaller than the speed of the supersonic bulk solar wind. Population 2) is presumably negligible at the termination shock. For the protons at the Jovian bow shock it is already hard to identify population 2) below the suprathermal tails (Fig. 5), and the flux of the tails scales as while the flux of the freshly ionized pick-up ions scales as . In a very idealized picture the cross-shock potential is characterized by which stops the bulk protons to zero speed and conserves the number of suprathermal ions at . Of course, in reality is less than unity because the downstream plasma does not have zero speed.
We define a normalized transmission function
for the suprathermal tails which yields the downstream distribution function when multiplied with
the upstream distribution function. The upstream distribution
function is assumed to be a power-law above the minimum speed
with the same spectral index
as upstream. The data of Fig. 5 suggest that this is a valid
approach. Therefore, the normalized transmission function
for the suprathermal tails at speeds
The data of Fig. 5, however, suggest . This presumably is due to the fact that some of the bulk protons are also injected into the suprathermal tails. We include this "injection efficiency'' into the ratio . The data of the Jovian bow shock give . This ratio is probably smaller at the termination shock because the ratio of the bulk proton flux to the flux in the suprathermal tails is probably lower, possibly by a factor 5/85.
Subsequent stochastic acceleration in the subsonic heliosheath plasma further increases the phase space density of the suprathermal ions transmitted through the termination shock. Ions with energies above the injection threshold diffuse back to the upstream solar wind plasma. We derive a model for the suprathermal ion population created by stochastic acceleration in the heliosheath plasma. We compare the theoretical flux with ENA flux data measured with CELIAS/HSTOF at 1 AU. This section very closely follows the work by Kallenbach et al. (2004).
The transport Eq. (18) also applies for the heliosheath, although the parameters are different. The convection speed is only 45 km s-1 according to Voyager 1 data (Krimigis et al. 2003), if it really has entered the heliosheath. This is a factor less compared to the region far upstream from the termination shock i.e. at a few AU. Of course, this is a controversial number which may have to be modified in the future, but we take it as baseline here. The Alfvén speed in the heliosheath is probably about the same as in the upstream solar wind. The pick-up ion source term diminishes to a lower speed scale because the ions are picked up by a much slower plasma. The Alfvénic turbulence power is increased by a factor ranging from to or even higher, depending on the efficiency of upstream Alfvén wave generation. Both, the increase of the convection time scale as well as the increase of the relative turbulent power increase the importance of stochastic acceleration in the heliosheath compared to the situation of the supersonic solar wind. The momentum diffusion parameter is increased by a factor of if the transport Eq. (18) is rewritten in speed units of , . We do not treat the effect of statistical acceleration in large-scale magnetosonic waves and in ion-acoustic waves because we do not know the speed fluctuations in the heliosheath. Therefore, the efficiency of momentum diffusion in Alfvénic turbulence gives a lower limit of the minimum contribution of ENAs originating in the heliosheath to the ENAs observed by HSTOF (Hilchenbach et al. 1998).
The source of the population accelerated by Alfvén waves is assumed to be the suprathermal pick-up ion distribution which is convected to the heliosheath from the upstream solar wind region. The population is described by a power law with spectral index . As already mentioned, we take as baseline for slow solar wind conditions the numbers s3 km-6 for H+ and s3 km-6 for He+, which are close to observations (Gloeckler 2003). We define s3 km-6, s3 km-6, 106 s3 km-6, and 105 s3 km-6. The pick-up ions freshly ionized in the heliosheath plasma are neglected.
To keep the problem treatable, we assume that scales with
in the heliosheath.
As the heliopause is perhaps at
the scaling with
is similar to a -dependence over the range of heliocentric distance in the heliosheath. The transport equation is then:
Without stochastic acceleration, the function
Including the term describing the stochastic acceleration the function
We try to solve the inhomogeneous equation again by the "variation
of the constant''. Multiplying
by another function
and entering the product
into the transport equation leads to
For speeds , it can be shown that despite the exponential term in the differential Eq. (53) for . This term is compensated by the exponential term of . For , the function approximately fulfills the transport equation of the heliosheath.
Therefore, the sum of the functions
fulfills fairly well both the transport
equation for the heliosheath and the boundary condition at the
termination shock. For speeds
the total distribution function is
finally expressed as
The differential flux
of ENAs near Earth's orbit in units of cm-2 s-1 keV-1 amu sr-1 is calculated from the phase space density
in s3 km-6 of
heliospheric suprathermal pick-up ions (Gruntman et al. 2001) as
Equation (55) provides a simple analytical formula to estimate the contribution of suprathermal particles inside and outside the termination shock to the energetic neutral atom production. The first term of , , is the convected power law matching SWICS observations in the slow solar wind between 1 AU and 5 AU (Gloeckler 1999) and the low-energy part of the Voyager 1 spectra at 85 AU (Krimigis et al. 2003). The second term describes the power-law distribution of suprathermal ions transmitted through the termination shock and neutralized in the heliosheath. The term includes the acceleration of the ions in the motional electric field, but not stochastic acceleration in the heliosheath. The third term describes the stochastic acceleration by Alfvén waves in the heliosheath. Note that again the self-consistent damping of Alfvén waves by energetic ions in the heliosheath can be introduced to first order by replacing by . If the contribution to the ENA flux from the heliosheath is reduced by a factor of about 15 compared to the case of .
|Figure 6: Model functions of Eq. (55) for four situations: typical suprathermal tails of the slow solar wind fed into the heliosheath in the apex direction ( upper left) and into the heliotail ( upper right), and typical suprathermal tails of the fast solar wind fed into the heliosheath in the apex direction ( lower left) and into the heliotail ( lower right). The numbers 1, 2, 3, 4, and 5 denote the contributions to the ENA flux from the region outside the termination shock without stochastic acceleration in the heliosheath (1), from the region inside and outside the termination shock without stochastic acceleration in the heliosheath (2), from stochastic acceleration with turbulence parameters (see text) (3), (4), and (5). The parameters and , which reproduce the injection efficiency of the bulk solar wind protons at the Jovian bow shock, have been used. Data of the IMAGE/HENA instrument are only upper limits for the flux (Ed Roelof, private communication). HSTOF data distinguish spectra of ENAs detected in the apex direction and in the heliotail direction. Highest flux has been detected during solar activity minimum.|
|Open with DEXTER|
If the injection threshold at the termination shock is above about 100 keV/amu, the ENA flux detected by HSTOF near Earth's orbit must come from the population described by Eq. (55). Figure 6 shows the model functions and spacecraft data for four stationary situations: suprathermal tails of the slow or fast solar wind fed into either the upwind heliosheath or into the heliotail, respectively. Charge exchange cross sections and survival probabilities are incorporated numerically from (Gruntman et al. 2001). We note that the ENA flux detected by HSTOF near Earth's orbit is compatible with the stationary situation when the typical suprathermal tails of the slow solar wind are fed into the heliosheath for both the apex and the anti-apex direction. Only moderate Alfvén wave amplification by ACRs and/or pick-up ions upstream from the termination shock is necessary to reconcile the final ENA flux near Earth's orbit. The field of view of the HSTOF sensor is always near the ecliptic plane, so that HSTOF may preferentially detect ENAs from heliosheath regions that are fed by slow solar wind. In fact, the ENA flux detected by HSTOF was considerably higher during solar activity minimum, when fairly stable slow solar wind streams but also co-rotating interaction regions (CIRs) were present in the ecliptic plane. Both, the slow solar wind and CIRs generate strong suprathermal tails. Modest acceleration of pick-up ions in the electric field of the termination shock and in the turbulence of the heliosheath would both explain HSTOF data.
However, enhanced suprathermal particle flux in the inner and middle heliosphere would do so as well (Kóta et al. 2001), which makes it difficult to decide which fraction of ENAs detected by HSTOF originate in the heliosheath, and which fraction in the supersonic solar wind. The spectra marked by label 1 in the upper panels of Fig. 6 denote the ENAs originating from suprathermal ions neutralized in the supersonic slow solar wind. Their flux is not much below the flux detected by HSTOF.
Data of the IMAGE/HENA instrument - detecting neutrals at even lower energies than HSTOF i.e. possibly further below the injection threshold - are compatible with the situation when fast solar wind is fed into the heliosheath. The field of view of the HENA instrument very often includes regions of the heliosheath that are fed by the fast solar wind. The seeming discrepancy between the HENA data and the HSTOF data - HSTOF has detected ENAs, while HENA has not - may be caused by the different view directions of the instruments. Furthermore, HENA has only operated during solar activity maximum, while HSTOF data are dominated by times during solar activity minimum.
Regarding the observed abundances of Anomalous Cosmic Rays i.e. the observed relative enrichment of He over H by a factor 10 and of O over He by another factor 10, we note that Eq. (54) predicts an enhancement of the abundance in the suprathermal tails that scales with . For singly charged ions this means an ordering by atomic mass A2/3. This factor comes from the product of with . This product is proportional to . If we multiply the injected abundances of He and O by these factors we arrive at the injection efficiencies plotted in Fig. 7. These injection efficiencies match observations (Cummings et al. 2002).
|Figure 7: Injection efficiencies of H+, He+, and O+ ACR ions as a function of the injected phase space density of the proton suprathermal tails characterized as in Fig. 3. Same as Fig. 4 except that injected abundances of He and O are modified by the processes of acceleration in the electric potential of the termination shock and by stochastic acceleration in Alfvénic fluctuations in the heliosheath.|
|Open with DEXTER|
There are, in our view, three possibilities to explain the Voyager 1 observations near 85 AU at 34 N of long-lasting energetic particle enhancements below 1 MeV/amu.
Possibly, the parallel mean free path is reduced so much that the first-order Fermi process is rather dominated by perpendicular diffusion. Perpendicular diffusion has increasing acceleration time in high levels of Alfvénic fluctuations (Eq. (34)), which also limits the efficiency of ACR production.
Part of this work was supported by INTAS grant WP-0270 and the Swiss National Science Foundation.