Issue 
A&A
Volume 505, Number 1, October I 2009



Page(s)  329  337  
Section  The Sun  
DOI  https://doi.org/10.1051/00046361/200810755  
Published online  03 August 2009 
Injection to the pickup ion regime from high energies and induced ion powerlaws
H.J. Fahr^{1}  I. V. Chashei^{2}  D. Verscharen^{1,3}
1  Argelander Institute for Astronomy, University of Bonn, Auf dem Hügel 71, 53121 Bonn, Germany
2  Lebedev Physical Institute, Leninskii pr. 53, 117924 Moscow,
Russia
3  Max Planck Institute for Solar System Research, MaxPlanckStr. 2, 37191 KatlenburgLindau, Germany
Received 6 August 2008 / Accepted 29 July 2009
Abstract
Though pickup ions (PUIs) are a wellknown phenomenon in the inner heliosphere,
their phasespace distribution nevertheless is a theoretically unsettled
problem. Especially the question of how PUIs form their suprathermal
tails, extending to far above their injection energies, still now is
unsatisfactorily answered. Though Fermi2 velocity diffusion theories have
revealed that such tails are populated, they nevertheless show that
resulting population densities are much less than seen in observations
showing powerlaws with a velocity index of ``5''.
We first investigate here, whether or not observationally suggested powerlaws can be the result
of a quasiequilibrium state between suprathermal ions and
magnetohydrodynamic turbulences in energy exchange with each other. We
demonstrate that such an equilibrium cannot be established, since it would
require too high PUI pressures enforcing a shockfree deceleration
of the solar wind. We furthermore show that Fermi2 type energy diffusion in
the outer heliosphere is too inefficient to determine the shape of the
distribution function there. As we can show, however, powerlaws beyond the
injection threshold can be established, if the injection takes place at
higher energies of the order of 100 keV. As we demonstrate here, such an
injection is connected with modulated anomalous cosmic ray (ACR) particles at the
lower end of their spectrum when they again start being convected outwards
with the solar wind. Therefore, we refer to these particles as ACRPUIs. In our quantitative calculation of the
PUI spectrum resulting under such conditions we in fact find again powerlaws,
however with a velocitypower index of ``4'' and fairly
distanceindependent spectral intensities. As it seems these facts are
observationally well supported by VOYAGER measurements in the lowest
energy channels.
Key words: plasmas  solar wind  cosmic rays
1 Introduction
Suprathermal ions, picked up by the supersonic solar wind flow as ionized neutral atoms, have become known as pickup ions (PUIs) and are produced all over the inner heliosphere with a typical upwinddownwind asymmetry with respect to the inflow direction of the neutral ISM inflow vector (Fahr & Rucinski 1999; Rucinski et al. 1993). In the case of PUI protons, their production is due to photoionization and charge exchange of interstellar Hatoms (see Fahr & Rucinski 1999; Rucinski & Fahr 1991; Bzowski et al. 2008; Rucinski et al. 2003). Their spatial distribution seems well understood, while the PUIphasespace transport is a much less settled subject. Especially it exists an ongoing debate of how efficiently PUIs just after the pickup process are accelerated to higher energies due to nonlinear waveparticle interactions (see e.g. Bogdan et al. 1991; Chalov et al. 2004; Isenberg 1987; Fichtner 2001; Fichtner et al. 1996; Chalov & Fahr 1996,1998) and whether at all energy diffusion plays a relevant role in this transport.Some hint is given by the solar wind proton temperature behavior with distance. The observed nonadiabatic temperature behavior namely proves that a specific solar wind proton heating must operate in the outer heliosphere which can only be due to energy absorption from PUI generated turbulence, since convected turbulence amplitudes quickly die out with distance (see Smith et al. 2001; Fahr & Chashei 2002a).
Freshly injected PUIs represent keVenergetic protons in the supersonic solar wind frame and may be called here: ``primary pickup ions'' (or: PUIs). The velocity distribution of these newly produced PUIs is toroidal and unstable (see Winske & Leroy 1984; Lee & Ip 1987; Fahr & Ziemkiewicz 1988; Winske et al. 1985). With the free energy of this unstable distribution PUIs drive Alfvénic wave power. The latter enforces pitchangle isotropization of the initial velocity distribution (see Chalov & Fahr 1998,1999a). Due to wavewave coupling, the wave energy generated by PUIs at the injection wavelength is diffusively transported in wavevector space both to smaller wavelengths where it can be absorbed by solar wind protons and to larger wavelengths where it is reabsorbed by all PUIs. This effect is seen as the main reason of solar wind proton heating occurring in the outer heliosphere (Chashei et al. 2003; Smith et al. 2001; Stawicki 2004; Fahr & Chashei 2002b). Only a small fraction of about 5% of the PUIgenerated wave energy reappears in the observed proton temperatures. Freshly injected PUIs excite turbulences that can organize a powerlaw distribution. From this distribution, both the solar wind ions and the PUIs themselves can absorb energy as shown by Chashei et al. (2003). Also the approach by Isenberg et al. (2003) where energy diffusion of PUIs is not taken into account shows that only a low degree (25%) of the PUI driven wave energy is absorbed by solar wind protons in form of thermal energy. This raises the question where the major portion of the wave energy produced during the primary pickup process goes to. To clarify the energy redistributions, kinetic and spectral details of the relevant processes have to be investigated. A detailed numerical study of the PUI velocity distribution and the spectral Alfvénic/Magnetosonic wave power evolution has meanwhile been carried out (Chalov et al. 2004,2006b) and presents a simultaneous solution of a coupled system of equations consistently describing the isotropic velocity distribution function of PUIs and the spectral wave power intensity.
As one can see from this study, the largest portion of the selfgenerated wave energy is reabsorbed by PUIs themselves as a result of the cyclotron resonant interaction and leads to PUIacceleration. It could perhaps be hoped that this energization of pickup protons due to Fermi2 stochastic acceleration processes eventually leads to the ubiquitous powerlaw PUItails pointed out by Fisk & Gloeckler (2006,2007). To the opposite, however, as reflected in the results presented by Chalov et al. (2004,2006b) it is evident that this is not the case: even at larger distances close to the termination shock (100 AU) the PUI distributions show a rapid cutoff at energies higher than the injection energy. The question thus is raised here why powerlaws have been seen at all. An explanation that we are favoring here is a new injection source to the PUI regime from high energies connected with modulated anomalous cosmic ray (ACR) particles. These protons are primary ACR particles that occur with a spectrum down to the typical energy of the usually assumed PUIs. At this part of the spectrum, both particle species cannot be distinguished. Therefore, we refer to them as ACRPUIs.
In Sect. 2, we investigate the physical possibility of powerlaw ions in the outer heliosphere as they are recently proposed by several authors and we find that they cannot occur with a powerindex of 5. As we show in Sect. 3, the proposed processes are not effective enough to produce the desired ion tails which means that another mechanism has to lead to the observed spectrum. In Sect. 4, we show how a highenergy source can be derived by taking a modulated ACR spectrum upstream of the solar wind termination shock. This injection mechanism is discussed in Sect. 5 where we show that these highenergy ions can lead to powerlaw ion tails, however with a powerindex of 4. The results are discussed and compared with observations in Sect. 6.
2 Can powerlaw ion distributions be in equilibrium with hydromagnetic turbulence?
Challenged by recent results concerning ion spectra at large distances (Fisk & Gloeckler 2006; Decker et al. 2005; Király 2005; McDonald et al. 2003), we look into the problem of PUIphasespace transport under these new given auspices. First we discuss the argument given by Fisk & Gloeckler (2006,2007) that PUIs under resonant interaction with ambient compressive turbulences enter a quasiequilibrium state with saturated powerlaw distributions of a somehow sacrosanct spectral velocity index of .
The related, wellknown Kolmogorov formalism is based on a
``dimensional'' reasoning: The problem concerning the energy
distribution in eddies of a typical wave number k is considered
using two different dimensional quantities: namely the spectral
energy density E_{k} and the wave number k. The spectral
energy flux is then defined by
and has to be constant for stationary cases. is the typical life time of eddies with scale and is given by
with the typical vortex velocity v_{k} given by
v_{k}=(E)^{1/2}.  (3) 
Requiring a constant energy flux S given by Eq. (1), with given by Eq. (2) then automatically leads to the Kolmogorov spectrum in the form E_{k} .
In their thermodynamical approach to particle spectra in
equilibrium with waves, Fisk & Gloeckler (2006,2007), however, only
consider one singledimensional quantity, i.e. the kinetic energy
T, because in their case the energy distribution
has the dimension
.
The flux combination
(4) 
is then expected by them to be constant and, thus, the energy gain is given by
(5) 
and hence the above requirements result in the requirement that .
This shows that the analogy of the FiskGloeckler approach with
the Kolmogorov formalism is not complete, since in their theory
the quantity
is not specified. Only if we play the same
game with the Kolmogorov turbulence using instead of
Eq. (2) the assumption
,
where u is
some independent external speed (especially independent of k),
we will find from Eq. (1) also the result
(6) 
analogous to the result obtained by FiskGloeckler.
The problem may be briefly inspected here whether or not some spatial/temporal disturbances in the solar wind plasma can be considered as waves. They should be considered as waves, if the convection time is much greater than the passage period of these waves in the wind reference frame where r denotes the radial distance to the sun and U the solar wind bulk velocity. The Alfvén speed is designated as with the corresponding wavevector k_{0} for the turbulence. For an inequality of the kind , this then leads to the definition of the principal wave turbulence correlation scale given by . A more exact definition of the turbulence correlation scale is given in the paper of Chashei et al. (2003). This value is also in accordance with the value used by Fahr (2007) for his estimation of the upper possible velocity border. Some measurements indicate smaller values for the turbulence correlation length than calculated by us in this paper (see e.g. Matthaeus et al. 2005). However, the following conclusions in our paper drawn on the effects of turbulent heating of suprathermal protons become even less promising for an MHDequilibrium to be established, if smaller correlation lengths prevail. One can expect that at larger distances, , the value for increases proportionally to r, since a similar dependence is also valid for the outer scale of turbulence k_{0}^{1}(see Chashei et al. 2003). Here we estimate the maximum energy for protons resonating with these largest scales at distances of about 100 AU and find with , where , and are the maximal ion speed, the associated Lorentz factor and the ion gyrofrequency, respectively. Protons with these speeds have energies of about independent of distance. Thus, ions with can be expected to be scattered by waves, whereas ions with can only be scattered by large scale velocity structures in the solar wind.
Since in any case, however, powerlaws seem to be a fact well supported by observations within a certain range of solar distances and ion energies, we shall nevertheless take this finding as serious here as deserved and determine in the following the absolute spectral intensity of this PUIpowerlaw distribution and its consequences.
If the PUI distribution is given in the form
(7) 
with the velocitypower index , then the PUI density is given by
(8) 
where is a local normalization value, and v_{0} and are lower and upper velocity limits of the quasistationary PUIpowerlaw.
Using
,
one obtains for the powerlaw PUI pressure
as the second moment of the distribution function the following
result:
Evidently, the definition of requires the determination of all three local values , v_{0}(r), and which we aim at below.
2.1 Determination of the inner and outer velocity border
Using the definition of the lower velocity border as done by
Fahr (2007) one finds
With the above result one obtains the PUI distribution in the form
The local PUI density can be derived from the interplanetary Hatom density
,
which depends on the radial distance
r and the inclination angle
with respect to the upwind
axis, and the effective (charge
exchange + photoionization)induced injection rate
.
Here
,
,
U,
denote the solar wind proton density, the charge exchange
cross section, the solar wind bulk velocity and the
photoionization frequency. The Hatom density at distances
in the upwind hemisphere is satisfactorily well
given by the following expression (see Fahr 1971):
(12) 
and thus leads to the following PUI density (also see Fahr & Rucinski 1999)
From the above one derives the following radial space derivative
which later on in this paper will be needed:
To calculate the PUI pressure, one more quantity in addition is
needed, namely the upper velocity border
.
To determine
we follow the idea presented by Fisk & Gloeckler (2006,2007)
assuming that PUIpowerlaw distributions result from a specific
quasiequilibrium state selfestablishing such that the wave field
transfers per unit of time as much energy to PUIs by energy
diffusion, as energy is expended in the solar wind frame for the
work done by the pressure gradient of comoving PUIs against the
magnetosonic fluctuations. The important restriction to energy
diffusion by nonlinear interaction with the compressive
fluctuations is that the typical diffusion period
should be much larger than the convection period given by
(see Chalov et al. 2003). This leads to the requirement
(15) 
with as the mean free path for particles parallel to the magnetic field. The uppermost velocity is the limit at which this condition is just fulfilled:
(16) 
We assume the interaction of the particles with a slab Alfvénic turbulence field. The mean free path is given by
(17) 
with the pitchangle diffusion coefficient
(18) 
from Chalov et al. (2003) for cyclotron resonant waveparticle interaction with unpolarized, onedimensional, and isotropic turbulence, which leads to the velocityindependent expression
(19) 
for the mean free path, with the reference value
(20) 
for the diffusion coefficient. Therefore, the upper velocity border is given by
(21) 
which evaluates to
(22) 
by taking and (Chalov & Fahr 1999b). The ratio
is needed in the later calculation and should be compared with the result obtained by Fahr (2007) deriving the upper velocity border from the study of the uppermost resonance possibilities of ions with the largest prevailing correlation lengths existing in the solar wind velocity structures, yielding the result
(24) 
which gives smaller values than those derived for conditions when balanced pressure equilibrium in the sense of Fisk & Gloeckler (2007) is adopted. Here below we shall demonstrate that this value for , which is the direct consequence of the assumptions by Fisk, definitely leads to unreasonable consequences when we investigate the associated PUI pressure.
2.2 The upstream PUI pressure
We calculate the PUI pressure resulting from powerlaw distributed
PUIs upstream of the shock and find with
Eqs. (9)(11) and (23)
(25) 
where we have introduced . We obtain the pressure gradient by differentiation:
=  
(26) 
which leads with Eq. (14) to
=  
(27) 
If now we derive the effective upstream Mach number, neglecting
thereby solar wind electron and proton pressures compared to the
PUI pressure, i.e. assuming
,
and following the definitions by
Fahr & Rucinski (1999), we obtain
=  
(28) 
With and the PUI abundance (see Fahr & Rucinski 1999), we obtain
(29) 
For a shock position at and with the lower reference distance , this expression yields
(30) 
which means that the effective upstream Mach number is lower than 1.
Hence, powerlaw PUI pressures in balance with the wave fields would not allow for the occurrence of a termination shock of the solar wind (i.e. needing upstream Mach numbers ).
The nonexistence of the above ``turbulenceparticle''equilibrium
(TPE) can also be concluded along a different line of
argumentations together with the calculation of the downstream PUI
pressure connected with powerlaw distributed PUIs. Using the
results presented in Fahr & Lay (2000), one obtains on the basis of
Liouville's theorem and conservation of magnetic moment at the
passage from upstream to downstream over the shock the
PUIdistribution function downstream of the shock (indicated by
index 2) by the following relation:
(31) 
depending on the upstream distribution function , where s denotes the compression ratio at the shock. From that relation one obtains for the downstream PUI pressure the following expression
=  
=  (32) 
which states that the downstream PUI pressure is enhanced with respect to the upstream PUI pressure by the factor s^{2}. Reminding that the latter pressure for TPEconditions is given by
(33) 
one finds that the downstream PUI pressure should amount to
(34) 
which normalized with the upstream solar wind kinetic energy density would require that
(35) 
with denoting the PUI abundance at the shock. This would mean that the downstream thermal energy of the PUIs is much higher than the kinetic energy of the upstream solar wind which is forbidden by physical reasons. This again leaves to conclude that PUIs cannot exist in pressure equilibrium with the compressional magnetosonic turbulence that was assumed by Fisk & Gloeckler (2007) and cannot be responsible for the stochastic particle acceleration up to regions near the solar wind termination shock.
Requiring that the downstream thermal energy of the PUIs stays
below the
upstream kinetic energy of the PUIs would require instead an upper border
of the PUI power spectrum defined by
=  
=  (36) 
yielding
(37) 
Evaluating this formula with s=2.5,
,
and
leads to the result
(38) 
and means that and , i.e. much lower than required for pressure equilibrium conditions.
3 The relative effectiveness of energy diffusion and convective changes at ion phasespace transport
In the following we want to clarify the role of energy diffusion
in determining the shape of the PUIdistribution function. We
start from the transport equation adequate to describe the
phasespace behavior of the PUIs by a distribution function
f(t,r,v) (see e.g. Isenberg 1987; Chalov & Fahr 1996)
where is the solar wind bulk speed, D_{vv} is the velocity diffusion coefficient, and, and are functions describing PUIinjection sources and phasespace losses. Terms on the lefthand side of the above Eq. (39) under steady state conditions induce changes with a typical convection time given by
where r is of the order of the heliocentric distance.
Considering quasilinear velocity diffusion of particles due to
Fermi2 type interactions with the Alfvénic or magnetosonic
turbulence, both of which are leading to analogous expressions
(see Toptygin 1985; Le Roux & Ptuskin 1998; Chalov & Fahr 2000). For our estimates
here, one can use the diffusion coefficient derived by
Schlickeiser (1989) which for estimate purposes can be
represented in the following form:
where is the resonant wave number, the proton cyclotron frequency, the Alfvén speed, k_{0} the turbulence outer scale, (or ) the power exponent of the 1Dturbulence spectrum, and is the fractional turbulence level
where is the induction of local interplanetary magnetic field. Typical velocity diffusion times characterizing the action of the first term of the righthand side of Eq. (39) in changing the distribution function fcan be defined by
Combining the relations given by Eqs. (40), (42), (41), and (43) yields as a typical ratio of the characteristic times
(44) 
At heliocentric distances , one can assume (Chashei et al. 2003) , , , for IroshnikovKraichnan turbulence (i.e. a powerlaw for the spectral energy density ) or for Kolmogorov turbulence (i.e. ), which are the two mostly found and discussed spectral energy distributions in solar wind turbulence (Horbury et al. 2005; Bale et al. 2005). Then near 1 AU, we find a ratio showing that convection and diffusion processes here are of comparable importance for the particles with , while convection effects are of increasing importance for particles with higher velocities, i.e. with v>U. This result is in good agreement with other considerations (Horbury & Balogh 2001; Jokipii & Kota 1989; Zank et al. 1996; Matthaeus & Goldstein 1986; Zank & Matthaeus 1992). Isenberg (2005), however, concludes that the effect of energy diffusion is small for PUIs. His similar treatment is parametrized by the percentage of turbulence occurring in slab Alfvénic fluctuations. Nevertheless, these two approaches can be conciliated if this percentage is assumed to amount about 20%.
At larger heliocentric distances,
,
the frozenin
Parker magnetic field near the ecliptic is nearly azimuthal and
decreases with (1/r). Correspondingly, the resonant wavevector
behaves like
,
which leads to
.
Besides that, as argued by
Chashei et al. (2003), the following rdependences can be expected:
and
.
Consequently, the radial dependence of
is exclusively
determined by the rdependence of
meaning
that
in case of Kolmogorov turbulence. The above Eq. (45) shows that at larger solar distances energy diffusion of particles cannot determine the shape of the suprathermal tail of the resulting PUIdistribution function.
In fact, assuming for reasons of a better clarification, just to the contrast here, a dominance of the energy diffusion, i.e. neglecting all the terms in Eq. (39) except for the diffusion term, would deliver as a result of the transport equation the distribution . This distribution thus should evidently be much flatter than the obviously observed distribution (Fisk & Gloeckler 2006) indicating that energy diffusion in fact plays an inferior role.
This result also turned out from more quantitative calculations of waveparticle interactions in the outer heliosphere by Chalov et al. (2006a) or Fahr & Chashei (2007) in which a consistent treatment of PUI transport in phasespace and wave turbulence transport in kspace was presented. From these calculations it becomes evident that in fact in the outer heliosphere PUIs drive turbulence powers, but are not efficiently enough profiting from energy diffusion to produce powerlaw tails.
The only way to fill up ion tails at energies larger than 1 keV resulting in some powerlaw distribution, as it seems to us, is to inject ions into the PUI regime at higher energies of about 50 keV and then let them cool to smaller energies at their coconvection with the solar wind. This would be a process similar to the one of PUIs originating from freshly ionized neutral atoms which are injected at about 1 keV (see Fahr 2007; Siewert & Fahr 2008). In the following, we study such a highenergy injection mechanism to the PUI energy regime resulting from ACR ions that are adiabatically cooled to lower energies and, therefore, can be called ACRPUIs.
4 Injection to the PUI regime from the highenergy side
Here we first want to study the modulated part of the ACRs with
origin near the solar wind termination shock and investigate
whether they can perhaps serve as a possible injection seed from
high energies into the PUI regime. We start from the analytic
expression for the modulated ACR spectral intensity given for the
case of a spherically symmetric solar modulation. The differential
ACR intensity for this case is given by the following expression
(Stawicki et al. 2000):
j(r,p)  =  p^{2}f(r,p)  
=  
(46) 
where S(r_{0},p_{0}) denotes the ACR source function with source coordinates r_{0} and p_{0}, is the modified Bessel function of the first kind, U denotes the solar wind bulk velocity, and h and have the following definitions:
(47) 
(48) 
with the following additional quantities
(49) 
(50) 
(51) 
and the following assumptions of the r and pdependences:
(52) 
with an arbitrary reference momentum for the bulk velocity and the spatial diffusion coefficient .
As shown by Stawicki et al. (2000), the above expression can be
simplified for low values of ACR particle momenta p and then
yields the following expression for the ACR distribution:
(53) 
The modulation of the ACR spectrum due to spatial diffusion is contained in y_{0}. This shows that, hence, the ansatz by Stawicki et al. (2000) is an analytical way to express the modulated distribution function of the ACRs in the heliosphere. In the frame of its accuracy, this approach yields a valuable way to take the modulated ACR spectrum as the source of our cooling mechanism. The spectrum of the ACR particles at the solar wind termination shock serves as the ACR source function and, thus with good reasons, one adopts the function developed by Drury (1983) for an ideally planar shock which is given by
(54) 
where the spectral index q is connected with the shock compression ratio s by q=3s/(s1). , are appropriate values of the Fermi1 injection momentum and of the upper cutoff momentum, respectively. This then leads to
f(r,p)  
(55) 
This expression represents a constant distribution function.
Numerical simulations by Fichtner & Sreenivasan (1999) show a linear
dependence of the ACR flux on the energy (). This means
that the distribution function in phasespace is constant which
can be shown by considerations about normalization of the
differential flux and the distribution function:
Therefore, the f(r,p) of the ACR particles is given by a
constant value which follows from modulation theory:
(57) 
From the above expression we develop the total streaming
(see Gleeson & Axford 1967; Fahr 1990; Fahr & Verscharen 2009) for the ACRPUIs
in the range between the injection momentum
and an
upper border
.
ACR particles with
higher momenta propagate only according to the transport equation
for cosmic rays and do not participate in the diffusive injection.
With these considerations we find
as the total particle streaming at each place r. Since we expect that , the lower border can be neglected in Eq. (58).
Now we consider the local spatial divergence of this ACRPUI
streaming at some lower momentum border and take this to be the
highenergy injection source Q to the PUI regime. With this
choice, we obtain
as the divergence of the ACR particle flux, constituting a PUI source.
5 PUI BoltzmannVlasov equation with ACRinduced injection
In the ``solar'' rest frame (SF), the representative
BoltzmannVlasov equation (BVE) is given by
(60) 
under the prevailing conditions in the outer heliosphere, i.e. negligible energy diffusion, for the stationary case (Fahr 2007). The corresponding BVE equation in the ``solar wind'' rest frame (WF) has the form (Fahr 2007)
(61) 
where the second term on the lefthand side describes the velocityspace divergence of the phasespace flow connected with the magnetically induced deceleration (i.e. magnetic cooling), which is indicated by the subscript m. The coordinate t=t(r) denotes the proper time in the comoving reference frame (WF). Furthermore, is the local ion injection rate to the PUI regime.
For low energies (1 keV), this rate is due to locally freshly ionized neutral Hatoms and is given by with being the local PUI production rate (Fahr & Siewert 2008).
For high energies (50 keV for the ACRPUIs), the relevant injection, in contrast to the normal subkeV PUIs, in our view is given by the upper expression derived from the modulated ACR spectrum and given by Eq. (59).
Furthermore,
is determined by the
magnetically induced velocity decrease of particles with a
velocity v, when they are convected outwards with the solar wind
bulk flow at a mean velocity U to larger distances where the
coconvected interplanetary magnetic field B appears reduced in
magnitude (Fahr 2007). At larger distances AU near
the ecliptic, the magnetic field decreases like (1/r) (i.e. in
case of the nearly azimuthal, distant Parker field). Under these
conditions, one finds the following vdependent magnetic
velocityspace drift
(62) 
and its associated radial gradient
(63) 
in the form given by Fahr & Siewert (2008).
Ions which are picked up at r_{v} with a velocity U will, without other processes being involved, have ``magnetically'' cooled down to a velocity v at r if the relation r_{v}(v)=rv/U is fulfilled. The injection of freshly created PUIs at r_{v}(v) with an initial velocity v=U will be responsible for ions with velocity v at r.
Taking all these constraints together, one finally finds, when
reminding that the time and distance coordinates are related to
each other by
,
that for velocities
the solution for
in the WF is given by
(see Fahr 2007; Siewert & Fahr 2008)
This distribution function
can be evaluated
for larger solar distances
in the
upwind hemisphere for nearecliptic positions assuming that at
such solar distances the upwind Hatom density can be considered
as essentially constant, meaning that
.
This in fact is an acceptable approximation for solar
distances AU and velocities
,
and
then leads to
(65) 
i.e. to the astonishing fact that under pure magnetic cooling, the resulting PUIdistribution function for velocities is a powerlaw with the interesting powerindex predicted and confirmed by Fisk & Gloeckler (2006,2007); however, in their case expected as result of an assumed quasiequilibrium state established between magnetoacoustically driven ion energy diffusion and magnetoacoustic turbulence generation.
The above result is valid only for ions with . One possibility that we now start to see here is that for ions with velocities , i.e. much higher than the original PUIinjection threshold, one has to consider in addition some highenergy injection rate due to modulated ACRs as we have derived above.
For those ACRinduced ions (i.e. for the ACRPUIs) with the
relevant source function
(66) 
we find analogously to Eq. (64) for ACRPUIs
(67) 
which finally leads to an rindependent distribution function for the range in the form
(68) 
Altogether, thus, we obtain the total PUI distribution in the
following form:
=  
=  (69) 
where H(x) is the wellknown step function with and .
In any case, the above result shows that a velocitypower index of ``5'' is obtained for the velocity range and a powerindex of ``4'' is obtained for the velocity range where .
Rescaling velocity in units of U (i.e. X=v/U) leads to the
total distribution function
=  
=  (70) 
We introduce now two dimensionless characterizing quantities.
First, we define
(71) 
where the charge exchange cross section (Fahr 1971) and the hydrogen density (Izmodenov et al. 2003) are used.
With
(72) 
one can finally find
Measurements by the SWICS instrument on the ULYSSES space probe show a PUIphasespace density of about at two times the solar wind velocity at a distance of 5.26 AU (Gloeckler 2003). Equation (73) leads to a value for the PUIphasespace density of about , which is in good accordance to the observations. The difference could be a consequence of the assumption of a constant , especially at small distances from the sun.
Observations by the LECP instruments on VOYAGER 1 and 2
(Lanzerotti et al. 2001) show an rindependent proton intensity in
the energy range 0.571.78 MeV of about
particles per (cm^{2} s sr MeV), which corresponds to
particles per (cm^{2} s sr erg) in Gaussian units.
This value can be converted to a phasespace density
(cf. Eq. (56)):
(74) 
If we assume that these ions observed by Lanzerotti et al. (2001) are ACRPUI particles with the proposed behavior, this leads to a value for obtained from the measurements of
(75) 
We take this observational value to avoid uncertainties at the determination of the ACR intensity and the injection border . The calculated spectra are shown in Fig. 1.
6 Comparison of our results with data
Now we want to compare the above results with VOYAGER and ULYSSES
spectral ion data. For that purpose, we first transform the above
velocity distribution function into a distribution of spectral
energy flux j(E) given in units of (ions/(cm^{2} s sr MeV)).
This then leads to
j(E)  
(76) 
where and are the velocity and the energypower indices, respectively. As one can easily see, leads to , whereas would lead to . Looking into the data obtained by VOYAGER 1 during the year 2004 (see Decker et al. 2005,2006) before the shock crossing occurred, one can find  even though the data are very timevariable in this interval  that for ions with higher energy keV) the energypower index has been observed with , whereas interestingly enough for ions with lower energy (i.e. a smaller powerindex of seems to be indicated. That supports our above derived theoretical prediction that the velocitypower index of ACRinduced PUIs in the outer heliosphere is , rather than as for the ions with higher energy (MeV).
Also during the most recent VOYAGER 2 crossing of the shock (see Decker et al. 2008) it became evident that the energypower indices registered both before and after the shock crossing show values of which also nicely confirms a velocity index of .
Figure 1: Phasespace density spectra for PUIs and ACRPUIs. The PUI distribution is dependent on the distance from sun, whereas the ACRPUIs are not. PUIs cannot gain velocities above U, i.e. X=1. The solid line is led down to the ACRPUI spectrum at the cutoff at X=1 artificially. 

Open with DEXTER 
In addition, it is important to recognize that the particle detectors of the VOYAGER spacecraft did not see radial changes of the spectral flux intensity j(E) at the lower energy channels (0.351.5 MeV) during the period from years 1995 through 2005 (Krimigis et al. 2003; Lanzerotti et al. 2001; Decker & Krimigis 2003). This is also a support for the newly derived theoretical expression given in Eq. (73) and showing that a distanceindependent intensity can be expected at energies keV. Only the part of the spectrum that is a result of cooled PUIs shows a dependence on the distance due to the different PUI abundances. The ACRPUIs that are injected from ACRs and cooled from high energies, however, are proven to have no radial variation.
Our model explains occurring powerlaw tails with an index 4. There are, however, observations around 5 AU undoubtedly showing a powerindex 5 for suprathermal ion tails (e.g. Gloeckler 2003). As we can show, the favored explanation by Fisk & Gloeckler (2006,2007) cannot hold in the described way. Maybe other processes (such as anomalous Fermi2 type waveparticle energy diffusion), which are not treated in our approach, can lead to a powerindex of 5 at energies just above 5 keV. But this interaction does not explain the occurrence of extended powerlaw tails at larger distances (Chalov et al. 2004).
It is perhaps still a little bit an open question, where, i.e. at
what lower energy, one should cut off the modulated ACR spectrum
to calculate the injection to the PUI regime. Even though this
does not count very much in quantitative terms, since the lower
energy part of the modulated ACR spectrum has in fact a constant
distribution function (see Eq. (56)), it may
nevertheless represent an intellectually interesting question, to
decide up to what energies convection and Fermi2 energy diffusion
are dominant, and from what energies upwards Fermi1 acceleration
and spatial diffusion dominate. There is one clear hint given to
answer this question, namely by the momentum dependence of the
spatial diffusion coefficient
generally given
in the form
(77) 
where, dependent on the momentum range, the exponent is expected to be in the range (see Le Roux & Potgieter 1992; Jokipii 1996). This shows that spatial diffusion becomes less and less efficient, the lower the ion momentum p is. Below some critical value , ions lose their degree of kinetic freedom to spatially diffuse relative to the solar wind background flow and, thus, they are simply convected outwards with the solar wind bulk flow then.
The determination of the absolute height of the ACRPUI spectrum is a slightly problematic endeavor. In our calculation, we normalize the absolute spectrum with the aid of observational data. The ACR intensity and the limit for the momentum, up to which the injection is effective due to the total streaming (see Eq. (58)), are quite uncertain. Another open question results from a lack of data. Especially, the transition from the PUI to the ACRPUI part has not been covered sufficiently by particle detectors in the outer parts of the heliosphere. Maybe future missions can provide a closer look to this part of the spectrum.
Acknowledgements
H.J. F. and D. V. are grateful to the Deutsche Forschungsgemeinschaft for financial support within the frame of the DFG project Fa 97/312. One of us, I. V. C., is grateful for financial support in the frame of DFG project 436 RUS113/110/04.
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All Figures
Figure 1: Phasespace density spectra for PUIs and ACRPUIs. The PUI distribution is dependent on the distance from sun, whereas the ACRPUIs are not. PUIs cannot gain velocities above U, i.e. X=1. The solid line is led down to the ACRPUI spectrum at the cutoff at X=1 artificially. 

Open with DEXTER  
In the text 
Copyright ESO 2009
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