Kinetic-Hydrodynamic Models of the SolarWind

8/6/2019 Kinetic-Hydrodynamic Models of the SolarWind

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Space Sci Rev (2009) 142: 2372

DOI 10.1007/s11214-008-9409-1

Kinetic-Hydrodynamic Models of the Solar Wind

Interaction with the Partially Ionized Supersonic Flowof the Local Interstellar Gas: Predictions

and Interpretations of the Experimental Data

Vladimir B. Baranov

Received: 10 January 2008 / Accepted: 10 June 2008 / Published online: 30 July 2008

Springer Science+Business Media B.V. 2008

Abstract At present there is no doubt that the local interstellar medium (LISM) is mainly

partially ionized hydrogen gas moving with a supersonic flow relative to the solar system.

The bulk velocity of this flow is approximately equal 26 km/s. Although the interactionof the solar wind with the charged component (below plasma component) of the LISM can

be described in the framework of hydrodynamic approach, the interaction of H atoms with

the plasma component can be correctly described only in the framework of kinetic theory

because the mean free path of H atoms in the main process of the resonance charge ex-

change is comparable with a characteristic length of the problem considered. Results of self-

consistent, kinetic-hydrodynamic models are considered in this review paper. First, such the

model was constructed by Baranov and Malama (J. Geophys. Res. 98(A9):15,15715,163,

1993). Up to now it is mainly developed by Moscow group taking into account new experi-

mental data obtained onboard spacecraft studying outer regions of the solar system (Voyager

1 and 2, Pioneer 10 and 11, Hubble Space Telescope, Ulysses, SOHO and so on). Predictions

and interpretations of experimental data obtained on the basis of these models are presented.

Kinetic models for describing H atom motion were later suggested by Fahr et al. (Astron. As-

trophys 298:587600, 1995) and Lipatov et al. (J. Geophys. Res. 103(A9):20,63120,642,

1998). However they were not self-consistent and did not incorporate sources to the plasmacomponent. A self-consistent kinetic-hydrodynamic model suggested by Heerikhuisen et al.

(J. Geophys. Res. 111:A06110, 2006, Astrophys. J. 655:L53L56, 2007) was not tested on

the results by Baranov and Malama (J. Geophys. Res. 111:A06110, 1993) although it was

suggested much later. Besides authors did not describe in details their Monte Carlo method

for a solution of the H atom Boltzmann equation and did not inform about an accuracy of

this method. Therefore the results of Heerikhuisen et al. (J. Geophys. Res. 111:A06110,

2006) are in open to question and will not be considered in this review paper. That is why

below we will mainly consider a progress of the Moscow group on heliospheric modelling

endeavours in the kinetic-hydrodynamic approach. Criticism of the models that treat inter-

stellar hydrogen in the heliosphere as several fluids is given. It is shown that the multi-fluid

V.B. Baranov ()

Institute for Problems in Mechanics, Russian Academy of Sciences, Prospect Vernadskogo, 101, k. 1,

119526 Moscow, Russia

e-mail: [email protected]
mailto:[email protected]:[email protected]


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24 V.B. Baranov

models give rise to unreal results especially for distributions of neutral component parame-

ters. Magnetohydrodynamic (MHD) modelling of the solar wind interaction with the LISM

gas is also reviewed.

Keywords Solar wind Interstellar medium Heliopause Termination shock Bowschock Pickup ions Charge exchange

1 Introduction

Constructing a quantitative theoretical model for the prediction and explanation of exper-

imental data is an important goal in various branches of scientific knowledge. However,

such a model is useful if it has a reliable and physically correct theoretical basis. Other-

wise, an interpretation of space experiments on the basis of theoretical models could lead to

misleading conclusions. Numerical instabilities are often interpreted by theorists as physi-

cal phenomena in astrophysical objects, could be examples of this statement. To construct

global models of physical phenomenon occurring in astrophysics or space physics the fluid

dynamic approach is the most frequently used. A theoretical science which constructs mod-

els of physical phenomena in space by means of the fluid dynamic methods is named the

cosmic gas dynamics. There are many examples of pioneering models that made valuable

contribution to astrophysics and space science emphasizing an important role of cosmic gas

dynamics:

(a) The solar wind as a physical phenomenon has been predicted by Parker ( 1958) on the

basis of the one-dimensional and one-fluid hydrodynamic equations. Later the solar windwas discovered by space experiments on the boards of the first spacecrafts (see Gringauz et

al. 1960; Neugebauer and Sneider 1962).

(b) The analytical hydrodynamic model of the interplanetary plasma interaction with the

Earths magnetosphere taking into account a bow shock formation has first been suggested

by Zhigulev and Romishevsky (1959) in the Newton approximation of thin layer and pla-

nar magnetic dipole. The formation of the bow shock is experimentally confirmed by all

spacecraft.

(c) The hydrodynamic model of the solar wind interaction with cometary ionospheres

taking into account mass loading effect was suggested by Biermann et al. (1967). The

exact numerical solution of their model was obtained almost 20 years later by Baranov andLebedev (1986) before comet Halley missions. Many theoretical predictions were confirmed

by Vega 1/2, Giotto and Suisei spacecraft in March 1986.

(d) The axial symmetric hydrodynamic model of the solar wind interaction with the su-

personic flow of the interstellar gas (two-shock model) was firstly suggested by Baranov et

al. (1970) in the Newton approximation of thin layer. The development of this model gave

rise to predictions of several physical phenomenon discovered later by spacecraft investigat-

ing outer regions of the solar system (see below).

(e) Jeans (1928) has developed the model of galaxy formation based on the gravitational

instability. Jeans instability criterion

= 2 /k > a0

/0G j ,

where and k are the wave length and wave number, a0 and 0 are the constant sound

velocity and mass density respectively, G is the gravitational constant, on the basis of hy-

drodynamic equations is obtained.


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Kinetic-Hydrodynamic Models of the Solar Wind Interaction 25

(f) A spiral structure of galaxies can be explained by Lins waves of density. The wave

nature of spiral patterns was firstly suggested by Lin and Shu (1964) on the basis of an exact

solution of the hydrodynamic equations for the cold planar galactic disc.

Pioneer models, considered in points (a)(f), were later developed to be more adequate

for explaining real physical phenomena obtained on the basis of experimental data. In doingso it was often necessary to go out from a framework of classical cosmic gas dynamics. In

particular, in the problem considered in point (d) the motion of interstellar H atoms, interact-

ing with protons of the plasma component via processes of the resonance charge exchange,

cannot be correctly described in the framework of a hydrodynamic approach due to Knudsen

number Kn = l/L 1 (l and L are mean free path of H atoms and a characteristic length ofthe problem, respectively). Therefore, a model of the solar wind interaction with partially

ionized hydrogen gas of the LISM must be constructed on the basis of a self-consistent solu-

tion of the hydrodynamic equations for plasma component and kinetic Boltzmann equation

for H atoms (kinetic-hydrodynamic model). Such a model was first constructed by Baranov

and Malama (1993).

At present time there is no doubt that the Sun is moving through a warm (T 6500 K)and partly ionized local interstellar gas with the velocity V 26 km/s. Main componentsof the interstellar gas are electrons and protons (below plasma component) and H-atoms

(below neutral component). An effect of the interaction between neutral and plasma com-

ponents due to processes of the resonance charge exchange is a determining factor for the

heliosphere structure.

It is impossible to review all papers in the considered problem published after the lastreview by author (Baranov 1990). This paper does not pretend to be a complete review of

all theoretical studies done in the field. We could refer, for example, to the complete reviews

by Zank (1997), Izmodenov (2000), Lallement (2001), Baranov and Izmodenov (2006).

A development of the well-grounded kinetic-hydrodynamic model by Baranov and Malama

(1993) will be mainly considered in this review paper. It is a continuation of the review by

Baranov (1990) and can be considered as a logical description of the theoretical problem

connected with an interaction of the solar wind and the local interstellar medium. Physi-

cal parameters in the LISM and at the Earths orbit, which can be considered as boundary

conditions for a theoretical model, in Sect. 2 are discussed. In Sect. 3 physical ideas for

constructing a model of the solar wind (SW) interaction with the supersonic flow of the

local interstellar medium (LISM) relative to the solar system are considered. Mathematical

formulation of the model by Baranov and Malama (1993) and its basic results in Sect. 4

are given. Development of the kinetic-hydrodynamic model to take into account of H atom

ionization by the electron impact, galactic and anomalous component of cosmic rays, inter-

stellar magnetic field, 11-years solar activity cycles, and nonequilibrium of pickup and solar

wind protons are considered in Sect. 5. Calculations of the heliosphere extent in the tail

region are also given in this section. Comparisons of the kinetic-hydrodynamic approachresults with the results of multi-fluid models are shown in Sect. 6. Review of magneto-

hydrodynamic (MHD) effects on the heliospheric structure and on the interaction of the

interplanetary strong discontinuities with the termination shock in Sects. 8 and 9 is given.

Critical analysis of heliopause stability models is in Sect. 9. Theoretical predictions and their

connection with experimental data are analyzed in Sect. 10.


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26 V.B. Baranov

2 Experimental Foundation for Constructing a Model of the Solar Wind Interaction

with the Local Interstellar Medium (LISM)

The structure of the outer heliosphere and heliospheric boundary are determined by the

interaction of the solar wind (SW) with the interstellar neighborhood of the Sun (local in-

terstellar medium or LISM). Choice of an adequate model of this interaction depends on the

undisturbed SW and LISM parameters.

2.1 Solar Wind Parameters

The solar wind parameters are regularly detected by instruments on the board of spacecraft

beginning with 60th years. Direct investigations of the distant solar wind became possible

due to the launch of Voyager 1 and 2, Pioneer 10 and 11 in the middle of the 70-th. However,

all direct measurements were made only in the vicinity of the ecliptic plane until the middle

of the 90-th when the Ulysses spacecraft began to study the solar wind parameters out of theecliptic plane. As it will be seen below, many physical processes in the solar wind are deter-

mined by H-atoms which penetrate into the solar system from the LISM. At the Earths orbit

(at 1 AU) the flux of interstellar hydrogen atoms is quite small due to losses by processes

of photoionization and charge exchange with protons. Therefore, the solar wind parameters

at 1 AU can be considered as undisturbed. Measurements of the pickup protons, born due

to the charge exchange of interstellar H-atoms with the solar wind protons, also show that

they have no dynamical influence on the original solar wind flow near the Earths orbit.

Therefore, the Earths orbit can be taken as an inner boundary in the model of the solar wind

interaction with the LISM. At this boundary the average solar wind bulk velocity, proton

(or electron) number density and proton temperature are equal, respectively, VE 400600km/s, neE npE 1020 cm3, TE (14) 105K (Mach number ME 10). The elec-tron temperature is any higher than the temperature of protons. The experimental data, ob-

tained on the Ulysses spacecraft, show that at the polar regions of the Sun the solar wind

velocity is larger and the number density is smaller as compared with the same parameters

at the ecliptic plane. This difference is much smaller in the solar maximum. Besides, the so-

lar wind parameters at 1 AU depend, generally speaking, on a time which is determined by

the solar activity. However, these parameters assumed to be constant in the most theoretical

models (spherical symmetric and stationary solar wind).

There is also the interplanetary magnetic field (IMF). The analysis of Voyager and

Pioneer 11 data show (see, for example, Burlaga and Ness 1993; Burlaga et al. 1998;Ness and Burlaga 2001) that in the vicinity of the ecliptic plane the radial variation of the

IMF is consistent with the theoretical prediction by Parker (1963), i.e. the macrostructure

of the IMF is described by the Archimedean spiral connected with the solar magnetic field

frozen in the radial solar wind. The analysis of the magnetic field measurements on the

Ulysses spacecraft show (see, for example, Forsyth et al. 1996) that the IMF out of the

ecliptic plane are also not in contradiction with the Parkers model which predicts that the

radial component of the magnetic field Br 1/r 2 and azimuthal component B 1/r (r isthe heliocentric distance). An effect of the IMF on the solar wind flow is determined by

the Alfven Mach number MA =

(4)1/2V /B , where , V and B are the solar wind mass

density, velocity and strength of the IMF, respectively. At the Earths orbit we have approx-

imately Br B 1= 105G, i.e MAE 1. Taking into account the change of the IMFin the Parkers model and the hypersonic character of the solar wind (V const VE and 1/r 2) we have MA 1 for the supersonic region of the solar wind, i.e. the effect of theIMF on the radial flow is negligible in this region although it can give rise to small azimuthal

component of the solar wind velocity ( 20 km/s).


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2.2 Interstellar Gas Parameters

For understanding the processes connected with the LISMs parameters the data of the

ground-based astronomical observations by Lallement and Bertin (1992) turned out to be

very important. They have showed that the Sun is embedded in a partially ionized gas of thelocal interstellar cloud (LIC) moving relative to the solar system with a velocity V 26km/s. This velocity is supersonic at the LIC temperature being T 7000 K, which wasinterpreted at an analysis of absorption lines obtained on the HST spacecraft (see, for ex-

ample, in Lallement 1996), i.e. this motion (interstellar wind) is supersonic relative to the

thermal sound velocity a =

RT ( is the ratio of specific heats, R is the gas con-stant). The local interstellar temperature and velocity can be also inferred from direct mea-

surements of interstellar helium atoms by the Ulysses/GAS instrument (Witte et al. 1996;

Witte 2004) because these atoms penetrate the solar system without collisions with other par-

ticles due to large mean free passes connected with charge exchange and elastic collisions.

Observations of the H atom and proton number densities in the LISM give usually rise totheir estimations with an accuracy of a factor 2 and an order or more of magnitude, respec-

tively. In particular, the observed magnitude of the electron number density was estimated

in the range from ne 0.003 cm3, deduced from LISM observations of the ionizationstate of magnesium (Frisch et al. 1990) up to about 0.1 cm3 according to analysis of theLISM ionization degree by the integrated fluxes of the celestial EU V radiation (Reynolds

1990). There is no clear observational upper limit to the local magnitude of np ne. Forexample, an interpretation of the neutral magnesium absorption, detected by Goddard High-

Resolution Spectrograph (GHRS) onboard of the Hubble Space Telescope (HST), could

give rise to a very large proton number density np

0.3 cm

3 (Lallement 1996; Lalle-

ment et al. 1992, 1994) whereas an estimation of sodium ionization gives rise to a value

np 0.05 cm3 (Lallement and Ferlet 1997). The electron (proton) number density inthe interstellar medium is often taken to be ne 0.04 cm3 according to the pulsar dis-persion measurements. However, these data are not very reliable because they are strongly

depended on typical scales of averaging and are different for different pulsars (see, for exam-

ple, Manchester and Taylor 1977). For example, the electron number density for the pulsar

PSR 1642-03 gives rise to an estimation ne 0.210.24 cm3 instead of 0.04 cm3.The most reliable information concerning the number density of H-atoms in the LISM

(nH) is given by measurements of the scattered solar radiation at the wavelength 12.16 nm.

The magnitude ofnH 0.08 cm3 was estimated on the basis of the pioneer measurementsof this radiation and their interpretations (Bertaux and Blamont 1971; Thomas and Krassa

1971; Blum and Fahr 1970; Fahr 1974). However, these estimations did not take into account

an effect of the resonance charge exchange in the interface region separating the supersonic

solar wind and the supersonic flow of the LISM plasma component. The important role of the

interface region, introduced by Baranov et al. (1970, 1979), was first estimated by theorists

(Wallis 1975; Baranov et al. 1979; Ripken and Fahr 1983; Fahr and Ripken 1984). As for

to observers, up to 1985 (see, for instance, Bertaux et al. 1985) they used the so-called hot

model (in details, see Izmodenov 2006) for interpreting the backscattered solar radiation

measured onboard many space vehicles (Prognoz, Mars, Venera and others). In this

model there is no the interface region and, therefore, the main process of the LISM hydrogen

atom losses, namely, the process of the charge exchange between the LISM hydrogen atoms

and the LISM protons was not take into account.

A parametric investigation taking into account the formation of the heliospheric interface

region was carried out by Izmodenov et al. (2003b) within the framework of the kinetic-

hydrodynamic model. Authors used (a) the proton and H-atom number densities ranging


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28 V.B. Baranov

Table 1 Local Interstellar Parameters

Parameter Direct measurements/estimations

Sun/LIC relative velocity 26.3 0.4 km s1 (direct He atoms1)

25.7 km s1

(Doppler-shiftedabsorption lines2)

Local interstellar temperature 6300 340 K (direct He atoms1)6700 K (absorption lines2)

LIC H atoms number density 0.2 0.05 cm3 (estimate based onpickup ion observations3)

LIC proton number density 0.030.1 cm3 (estimate based onpickup ion observations3)

Local Interstellar magnetic field Magnitude: 24 GDirection: unknown

Pressure of low-energetic 0.2 eV cm3cosmic rays

1Witte (2004), Moebius et al. (2004);

2Lallement (1996);

3Gloeckler (1996), Gloeckler et al. (1997), Geiss et al. (2006).

from 0.03 cm3 to 0.1 cm3 for ne np and from 0.16 cm3 to 0.2 cm3 for nH, whichapproximately correspond to the data of observations for the LISM (see Lallement 1996), (b)

all modern experimental data obtained on the basis of the scattered solar radiation in Lyman-

alpha (see, for example, the data obtained by Quemerais et al. (1999) onboard SOHO),

(c) measurements of the pickup protons (Gloeckler 1996; Gloeckler et al. 1997; Geiss et

al. 2006) and helium atoms (Witte 2004; Moebius et al. 2004) onboard Ulysses and (d)

estimations of the local interstellar helium ionization (Wolff et al. 1999). Interstellar helium

ions and solar wind alpha particles, which were neglected by Baranov and Malama (1993,

1995), were also taken into account by authors. The results of the parametric investigationsby Izmodenov et al. (2003b) give rise to the most probable estimation of the hydrogen atom

and proton number densities in the LISM, namely, nH = 0.185 0.01 cm3 and np ne = 0.05 0.015 cm3, which is in agreement with the data of observations (Lallement1996).

At present neither magnitude nor direction of the magnetic field in the vicinity of the

solar system are measured. Rough estimations using an average galactic magnetic field give

a magnitude which is the order of 24 G. Some estimations of the interstellar magnetic

field direction are given from measurements of the scattered solar Lyman-alpha radiation

onboard SOHO spacecraft and results of the MHD modelling (Lallement et al. 2005, and

Izmodenov et al. 2005a, respectively). Therefore, the LISM magnetic field can be considered

as a free parameter in theoretical models. The LISM parameters (index ) which can beused as boundary conditions in the problem considered are given in Table 1 (Izmodenov and

Baranov 2006).

As we see from this Section the LISM is the partially ionized hydrogen gas with the

supersonic velocity relative to the solar system (M = V/a > 1). Therefore, the hy-


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drodynamic models by Parker (1961) with M 1 and by Baranov et al. (1970), whereM 1 but the LISM was assumed to be the fully ionized gas, are not real.

3 General Physical Ideas of the Solar Wind Interaction with the LISM

Parameters, which are given in Table 1, show that the flow of the LISM relative to the solar

system is supersonic (V > a, where a is the thermal sound velocity). The interaction ofthe plasma component of the LISM with the solar wind plasma can be described within the

framework of the continuum model (hydrodynamic approximation) because the effective

Knudsen number Kn = l/L 1 for charged particles, where l is their mean free path and Lis the characteristic length in the problem under consideration (see, for example, Baranov,

2000, 2006a). As it was first shown by Baranov et al. (1970) the interaction of the solar

wind with the supersonic flow of the LISM plasma component gives rise to the formation

of the interaction region, which between the bow shock (BS) and the solar wind terminationshock (TS) is located (see Fig. 1). The plasma of the solar origin and the LISM plasma

component are separated by a tangential discontinuity or heliopause (HP at Fig. 1). The

region between BS and TS is often called the heliospheric interface or heliosheath (inner

heliosheath between TS and HP and outer heliosheath between BS and HP).

Main neutral component of the LISM flow is hydrogen atoms (cosmic abundance of

helium is equal nHe/nH 0.1 ). The effect of the hydrogen atoms on the solar wind interac-tion with the LISM flow is connected with main process of the resonance charge exchange

(H+H+ = H++H ) because the collisional cross section of this process is much more thanthat for the elastic collisions. However, the Knudsen number for HH+ collisions accompa-nying by the charge exchange is Kn 1, i.e. the interaction of the solar wind with the LISMFig. 1 Qualitative pattern of the

solar wind interaction with the

LISM. Here, BS and TS are the

bow and termination shocks,

respectively, HP is the heliopause

separating the solar wind and the

plasma component of the LISM.

Trajectories of different H atom

populations are shown by dashed

lines


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30 V.B. Baranov

hydrogen atoms can not be described within the framework of the hydrodynamic approxi-

mation. Kinetic approximation must be used for this interaction because the hydrodynamic

approximation is not correct in this case. The LISM hydrogen atoms can penetrate into the

solar system across strong discontinuity surfaces BS, HP and TS. Their charge exchange

with protons of the supersonic solar wind (region 1), the inner heliosheath (region 2), theouter heliosheath (region 3) and the supersonic LISM flow (region 4) gives rise to the forma-

tion of newborn H-atoms, which have the proton parameters of these regions, and newborn

protons with the parameters of neutral H-atoms. The newborn protons are then picked-up

by plasma component, altering thus the momentum and the energy of these flows. As a re-

sult of the resonance charge exchange process, three main populations of hydrogen atoms

are formed.

The trajectories of the HSW hydrogen atoms born in regions 1 and 2 (atoms of populations

1 and 2, respectively) are shown in Fig. 1. These atoms with the parameters of the protons

from regions 1 and 2 penetrate into the LISM and alter the parameters of the undisturbed

flow ahead of the bow shock BS due to the charge exchange with the LISM protons. The

population 3 is formed in the region 3 due to the charge exchange of the LISM hydrogen

atoms with the LISM protons heated and decelerated in the bow shock BS. The population 3

of H atoms has parameters of the outer heliosheath protons. And, finally, population 4 is the

LISM hydrogen atoms which penetrate into the solar system without charge exchange (pri-

mary H-atoms). The charge exchange of the LISM hydrogen atoms in the outer heliosheath

and the formation of the population 3 of H-atoms leads to effective filtering for penetrat-

ing primary LISM hydrogen atoms into the solar system as it was shown for the first time

on a qualitative level by Wallis (1975) and quantitatively by Baranov et al. (1979).

It should be noted at the end of this Section that Williams et al. ( 1997) have suggestedthat elastic collisions between hydrogen atoms and between H atoms and protons (HH and

HH+ collisions, respectively) are important in the problem considered. However, carefulconsideration of the momentum transfer cross sections both elastic collisions and charge

exchange made by Izmodenov et al. (2000) has showed that elastic collisions are negligible

as compared with the charge exchange.

4 Kinetic-Hydrodynamic Model of the Solar Wind Interaction with the Partially

Ionized Hydrogen Flow of the Local Interstellar Medium

The self-consistent model of the solar wind interaction with the supersonic flow of the par-

tially ionized hydrogen LISM gas is considered in this Section in kinetic-hydrodynamic

approximation. In this approximation the interaction of the solar wind with the plasma com-

ponent of the LISM is described by hydrodynamic (Euler) equations with source terms

determining a change of the momentum and energy due to proton collisions with the H

atoms accompanying by the processes of the resonance charge exchange. Source terms

are calculated on the basis of a solution of the Boltzmann equation for H atom distribution

function by Monte Carlo method. This self-consistent model of the solar wind interaction

with the partially ionized flow of the LISM was, first, constructed by Baranov and Malama

(1993) in a stationary and axisymmetric approximation. We will consider here this model in

detail because it: (i) is most well-founded; (ii) is often used for interpretation of experimen-

tal data and (iii) is developed up to now to take into account new physical phenomena. Early

models, which were constructed before 1993, are in details described in Baranov (2006b)

and Izmodenov (2006).


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4.1 Mathematical Formulation of the Problem

From the qualitative pattern of the interaction between the solar wind and the partially ion-

ized supersonic interstellar gas flow, described in the previous section, it follows that a com-

plicated mathematical problem of the quantitative description of the considered physicalphenomenon arises. As we have seen from Sect. 3, its intricacy lies in the fact that the inter-

action of the interstellar plasma component with the solar wind can be described within the

framework of the hydrodynamic equations, whereas the motion of the interstellar H atoms

interacting with the solar wind and LISM protons due to processes of the resonance charge

exchange can only be described in the framework of the kinetic theory.

For stationary problem the equations of mass, momentum and energy conservation have

the following form (one-fluid approximation for the plasma component):

V

=0, (1)

VV+ 1p = F1[fH(r, wH, ), , V,p)], (2)

V

+ p

+ V

2

2

= F2[fH(r, wH, ), , V,p)] (3)

p = ( 1) (4)

where p, , V and are pressure, mass density, bulk velocity and internal energy of the

plasma component, respectively; F1 and F2 are source terms, describing the change of

momentum and energy of plasma component due to collisions between H atoms and protons,which are accompanied by the resonance charge exchange (mass density is not change in

this process of collisions), and fH(r, wH) is the H atom distribution function depending on

radius-vector r and individual velocity wH of H atom.

The trajectories of H atoms are calculated by the complicated Monte Carlo scheme with

splitting of the trajectories (Malama 1991) in the field of the plasma component parame-

ters. Such an approach allows the calculation of the source terms (F1 and F2) within the

framework of kinetic description of H atoms (the processes of multiple charge exchange

are also taken into account naturally by the Monte Carlo method). The H atom distribution

function in the undisturbed LISM (fH) is assumed to be a shifted (on bulk velocity V)Maxwellian distribution with the temperature T and the number density nH. Effect ofthe solar gravitational force Fg and the force of the solar radiation pressure Fr on the H

atom motion are also taken into account by Baranov and Malama ( 1993). The use of the

Monte Carlo method, as was proposed by Malama (1991) for the solution of this problem,

is identical to the numerical solution of the Boltzmanns equation

wH fHr

+ Fr + FgmH

fHwH

= fp(r, wH)|wH wH|ex fH(r, wH)dwH

fH(r, wH) |wH wp|ex fp(r, wp)dwp (5)for distribution function of H atoms. Equation (5) is a linear equation relative to the function

fH because the hydrodynamic equations (1)(4) for the ideal gas are correct if to take that

the proton distribution function fp(r, wp) is the local Maxwellian one with hydrodynamic

values (r), V(r) and T (r). Here, wp is an individual velocity of protons.


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32 V.B. Baranov

If the distribution function fH is known, then the source terms F1 and F2 in (2) and

(3) can be calculated exactly according to the Monte Carlo procedure of Malama (1991),

namely

F1 =1

np

dwH

dwpex | wH wp | (wH wp)fH(r, wH)fp(r, wp), (6)

F2 = mH

dwH

dwpex | wH wp | (w2H/2w2p/2)fH(r, wH)fp (r, wp), (7)

nH =

dwHfH(r, wH), np =

dwpfp(r, wp), (8)

where ex is the effective cross section of resonance charge exchange collisions.

To solve numerically the hydrodynamic part of the problem, i.e. to solve the equa-

tions (1)(4), Baranov and Malama (1993) used the discontinuity-fitting second order

technique, which is based on the scheme of Godunovs method (Godunov et al. 1976;Falle 1991). In this case the Rankine-Hugoniot relations on the shocks BS and TS (see

Fig. 1), the condition of static pressure equality and the no-flow condition for plasma com-

ponent through the tangential discontinuity (heliopause HP) are automatically satisfied. The

magnitudes of hydrodynamic parameters at the Earths orbit and in the undisturbed LISM

flow were used as boundary conditions in accordance with results presented in Sect. 3. To

solve the problem as a whole Baranov and Malama (1993) used an iterative method sug-

gested by Baranov et al. (1991). The iterative method consists of several steps. At first, the

trajectories of H atoms by the Monte Carlo method using the distribution of plasma parame-

ters obtained without source terms for the fully ionized hydrogen plasma were calculated

(see, for example, zero iteration made by Baranov et al. 1991). Then the momentum and

energy sources F1 and F2 in (1)(4) on this step are calculated using (6)(8). In the first

iteration the hydrodynamical equations (1)(4) with these sources are solved using the hy-

drodynamic boundary conditions as formulated above. Then, the new distribution of plasma

parameters was used for the next Monte Carlo iteration for H atoms. The hydrodynamic

problem was solved again with the new source terms of this second iteration and so on.

Baranov and Malama (1993) continued this process of iterations until the results of two

subsequent iterations were practically coinciding.

4.2 Main Numerical Results of the Self-Consistent Model

The iteration method described in Sect. 4.1 was realized by Baranov and Malama (1993) for

following boundary conditions at the Earths orbit and in the undisturbed LISM

npE = 7 cm3, VE = 450 km/s, ME = 10, np = 0.07 cm3,V = 25 km/s, nH = 0.14 cm3, M = 2.

The problem has an axial symmetry in the coordinate system where the Sun is in its

origin, Oz-axis and the vector of the LISM velocity are in the opposite direction and the

solar wind is spherically symmetric. In this case all parameters depend on the heliocentricdistance r and the polar angle .counted of the Oz axis.

To calculate trajectories of H atoms the following values of the charge exchange cross

section ex and the ratio of solar radiation pressure Fr and the force of the gravitational

force Fg were used

ex = (A1 A2 ln u)2 cm2, = Fr /Fg = 0.75,


11/50


Fig. 2 Geometrical pattern of the interface. Here, BS and TS are bow and termination shocks, HP is the

heliopause. The results by Baranov and Malama (1993) at nH = 0 (left) and at nH = 0.14 cm3 (right)are shown. Resonance charge exchange processes give rise to disappearing reflected shock (RS), tangential

discontinuity (TD), Mach disc (MD) and triple point A in the tail region. (From Izmodenov and Alexashov

2003)

where u is the velocity (in cm/s) of protons relative to the H atoms, A1 = 1.64 107,A2 = 6.95 109 (Maher and Tinsley 1977).

The geometrical pattern of the flow considered in Fig. 2 is demonstrated. We see that

the results obtained without taking into account the resonance charge exchange processes

(nH =

0 in left picture) give rise to a formation of the complicated structure of the tail

region consisting of reflected shock (RS), tangential discontinuity (TD), and termination

shock (TS) turning into the Mach disk (MD) at triple point A. This complicated picture

disappears at nH = 0 (right picture) forming only the bow shock (BS), heliopause (HP)and smooth termination shock (TS). We also see from Fig. 2 that for nH = 0.14 cm3 theinterface region (or heliosheath) is shifted toward the Sun and the heliocentric distance of

the TS in the upwind direction ( = 0) is about 2.5 time less than in downwind direction( = ), i.e. about 96 AU and 250 AU, respectively. The results by Baranov and Malama(1993) have also shown that charge exchange between the solar wind protons and the LISMs

hydrogen atoms gives rise to the subsonic flow in the inner heliosheath (sonic line disappears

here although it is conserved in the outer heliosheath).It is interesting to note here, that Voyager-1 and Voyager-2 have crossed the termination

shock, respectively, in December 2004 at the heliocentric distance about 94 AU (Burlaga et

al. 2005; Fisk2005; Decker et al. 2005; Stone et al. 2005) and in August 2007 at the distance

about 84 AU (see presentations on the AGU Conference, December 2007), i.e. this distance

was theoretically predicted more than 10 years ago by Baranov and Malama (1993) in the

kinetic-hydrodynamic model and more than 23 years ago by Baranov et al. (1981) (see, also,

Baranov 1990) for more simple hydrodynamic model, where the following source terms

was used (Holzer 1972)

F1 = c(VH V), c = nHex U, (9)U = [(VH V)2 + 128k(T + TH)/9 mH]1/2,F2 = c[(V2H V2)/2+ 3k(TH T )/2mH]. (10)

Here VH, nH and TH are the bulk velocity, number density and temperature of primary H

atoms, k is the Boltzmann constant. It was also assumed (Baranov et al. 1981) that parame-


12/50

34 V.B. Baranov

Fig. 3 Hydrogen wall at the

different angles . (From

Baranov and Izmodenov 2006)

ters of H atoms are described by the following equations

VH = V = const, TH = T = const, (11) nHVH =cnp, (np = /mH). (12)

The approximation (9)(12) was later used for 3D magnetohydrodynamic (MHD) sim-

ulation of the solar wind interaction with the LISM by Linde et al. ( 1998) although it is a

compromise between the well-grounded kinetic-gasdynamic model, considered above, and

the simplified hydrodynamic model taking into account only a sink of primary LISM hy-drogen atoms due to the resonance charge exchange (see the continuity equation (12)).

An unexpected result was obtained by Baranov et al. (1991) on the first step of the itera-

tion (calculation of the H atom parameters by the Monte Carlo method in the hydrodynamic

field of parameters obtained without source terms), namely, the non-monotonic distribu-

tion of the H atom number density with the decrease of the heliocentric distance. A maxi-

mum of this distribution is in the vicinity of the heliopause. This effect was confirmed by the

complete solution of the self-consistent problem, considered above (Baranov and Malama

1993), and it was named hydrogen wall. Curves 1 and 3 in Fig. 3 demonstrate this effect

along the axis of symmetry (

=0 and

=, respectively) and curve 2along the direction

which is normal to this axis ( = /2).We see the hydrogen wall which is determined as sharp gradients of the H atom number

density in the vicinity of the maximum. Their locations depend on the LISMs fractional

ionization of hydrogen (Baranov and Malama 1995). The hydrogen wall is most clearly

expressed near the stagnation point on the heliopause. This effect is smaller at > 0 and it

is absent in the downwind direction ( = ) as one can see from Fig. 3. A formation of thehydrogen wall is due to the creation of secondary H atoms with decreased velocity in the

vicinity of the HP corresponding to the decreased velocity of compressed (in the bow shock)

interstellar protons.

The theoretical prediction of the hydrogen wall formation was experimentally con-firmed by Linsky and Wood (1996) and Linsky (1996). Details of their discovering will be

considered below. It is only interesting to note here, that the concept of a hydrogen wall

on qualitative level was earlier used by Quemerais et al. (1993) to explain the intensity of

Lyman alpha glow as a function of the heliocentric distance which has been observed by

Voyager 1 and 2. In so doing authors placed the hydrogen wall at an arbitrary heliocentric

distance.


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5 Development of the Kinetic-Hydrodynamic Model

New experimental data obtained onboard the spacecraft require refining and developing the

self-consistent model by Baranov and Malama (1993, 1995) considered in Sect. 4. The main

difficulty in constructing a complete model consists in the multi-component nature of the

LISM and the solar wind (galactic and anomalous components of cosmic rays, pickup ions,

magnetic fields and so on) and in non-steady-state conditions in interplanetary plasma con-

nected with solar cycles, solar activities, plasma fluctuations and so on. It should be noted

here that sometimes taking account of certain processes, which at first glance have a small

effect on the results provided by the model, can play an important role in interpreting ex-

perimental data. For example, taking into account the solar wind alpha-particles and ionized

helium in the LISM leads to some variations in the termination shock, heliopause and bow

shock locations (Izmodenov et al. 2003a, 2003b, 2004), which can amount to about 2 AU,

13 AU and 40 AU (of about 2%, 8% and 10%), respectively. These values can be impor-

tant for interpreting the measurements onboard the spacecraft Voyager 1 and 2 which moveaway from the Sun at a velocity of about 3.5 AU/yr. In this section we will consider a devel-

opment of the model presented in previous section to take into account certain of physical

phenomena mentioned above.

5.1 Effect of the H Atom Ionization by the Electron Impact

The LISM hydrogen atoms penetrating into the solar wind can be ionized by solar wind

hot electrons (mainly in the inner heliosheath, where solar wind plasma is heated by the

TS). This physical phenomenon was taken into account by Baranov and Malama (1996).

The source term q

=0 (its form see below in Sect. 5.6) on the right hand side (RHS) of

the continuity equation must be taken into account in this case at calculations of H atom

trajectories by Monte Carlo method. It was shown that the effect of H atom ionization by

electron impact is negligible for the geometrical pattern of the flow considered (the locations

of the TS, HP and BS are almost no changed). However, the main effect is connected with

appearing the strong increase of the electron number density in the inner heliosheath (from

the termination shock TS to the heliopause HP). We hope that this theoretical prediction can

be important for interpretation of the kHz radiation detected by Voyager (see, for example

Gurnett et al. 1993; Gurnett and Kurth 1995) and will be confirmed by Voyager 1 and 2 in

nearest future.

5.2 Effect of Galactic and Anomalous Components of Cosmic Rays

The galactic and anomalous components of cosmic rays, whose spectra are appreciable dif-

ferent, can be considered as high-energy (often relativistic) populations with a negligible

mass density as compared with that of the plasma, but a considerable (not negligible) en-

ergy density. The origin of the galactic cosmic ray (GCR) acceleration is outside of the solar

system, while the origin of the anomalous cosmic ray (ACR) acceleration is inside of the

solar system. On the hydrodynamic level, the cosmic ray effect on the flow of the plasma

component is described by the gradient of the cosmic ray pressure pcr and by the energytransport V

pcr . In this case the momentum and energy equations will have the form

(compare with the (2) and (3)

VV+ 1(p + pcr ) = F1[fH(r, wH, ), , V,p)], (13)

V

+ p

+ V

2

2

= F2[fH(r, wH, ), , V,p)] V pcr . (14)


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36 V.B. Baranov

The cosmic ray pressure is determined by the formula

pcr = 43

0

fcr (r, | p |, t ) | p |4 d | p |,

where fcr is the isotropic distribution function of the cosmic rays and |p| is the magnitude ofthe particle momentum. Integration of the equation for fcr (the form of this equation can be

found, for example, in Alexashov et al. 2004a) over |p| gives rise to the following equationfor pcr (in stationary case)

[pcr cr (V+Vd)pcr ] + (cr 1)V pcr +Q= 0. (15)

Here, is the coefficient of the cosmic ray diffusion, V is the bulk velocity of the plasma

component, Vd is the drift velocity in the heliospheric or interstellar magnetic field averagedover the distribution function fcr , cr is the politropic index of cosmic rays, and Q is the

energy injection rate describing energy gains of ACRs from hot protons. We have Q= 0 forGCR because the origin of their acceleration is outside of the heliosphere.

The system of (1), (4)(8) and (13)(15) at Q = 0, was numerically solved by Myasnikovet al. (2000a, 2000b). It was shown that the effect of the GCR on the flow considered is

negligible as compared with the effect of the resonance charge exchange although this effect

is not negligible at nH = 0, i.e. at the real conditions of the partially ionized LISM, theGCR do not influence on the distribution of the plasma component and H atom parameters

as well as on the geometrical pattern of the flow (see Fig. 2 right).

To take into account the effect of the ACR the following expression for the source term

of (15) was used by Chalov and Fahr (1996, 1997)

Q=p V . (16)

Here is the coefficient determining the intensity of charged particle injection subjected to

the acceleration up to the anomalous component of cosmic rays in the TS and p is the static

pressure of the plasma component determined from the system of hydrodynamic equations.

Usually, the coefficient is a free parameter, whose order of magnitude is determined by

plasma properties.The dynamic effect of the anomalous component of cosmic rays on the heliospheric

interface structure was studied by Alexashov et al. (2004a). The system of (1), (4)(8),

(13)(16) was numerically solved by iteration method considered in Sect. 4.1. The influence

of the ACR diffusion coefficient was studied at a constant coefficient of the injection ,

since at present coefficients of the diffusion in the outer heliosphere and, in particular, in

the heliosheath are poorly known. The main effect of the ACR is connected with a structure

of the flow in the vicinity of the heliospheric termination shock TS, namely, it reduces to

smooth deceleration of the solar wind in the so-called precursor followed by the subshock.

As it is seen from Fig. 4 the TS intensity decreases and, as a result, it is located at a greater

heliospheric distance than in the case in which the ACR effect does not take into account

(Baranov and Malama 1993). Both the TS intensity and the value of its displacement depend

on the value of the diffusion coefficient. A decrease of the TS intensity is accompanied by a

decrease of the plasma component temperature in the inner heliosheath, which is important

for planning in USA measurements of the H atom fluxes of population 2 onboard the IBEX

(Interstellar Boundary Explorer) spacecraft in 2008.


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Fig. 4 Effect of ACR on the flow in the vicinity of the termination shock (TS). Upwind positions of the

termination shock for three different values of the diffusion coefficient: = 3.75 1020; 3.75 1021 and3.75 1022 cm s1 (curves 1, 2 and 3, respectively). Curve 4 shows the TS position in the case when ACRsis absent. The precursor is pronounced at small and medium values of . (From Alexasov et al. 2004a)

A maximum TS displacement ( 4 AU) is realized for intermediate value of the dif-fusion coefficient (Alexashov et al. 2004a). The heliospheric TS precursor is most clearly

expressed for low values of the diffusion coefficient and vanishes at its large values. This

can be attributed to the fact that in the former case the diffusion scale length is much smaller

than the distance to the TS and the ACR pressure behind the TS being comparable with the

static pressure of the plasma component. In the latter case the ACR pressure is negligible as

compared with the plasma component static pressure, so that the effect of the ACR on the

results by Baranov and Malama (1993, 1995, 1996) can be neglected.

It is important to note here that a clearly expressed angular asymmetry in the ACR energy

exists due to the difference in the amounts of the energy injected into the ACR in the forward(relative to the oncoming interstellar gas flow) and the tail regions of the TS This difference

is due to the fact that the static pressure of the plasma component is lower in the tail than in

the forward region (Alexashov et al. 2004a).

5.3 Effect of the Interstellar Magnetic Field

The effect of the interstellar magnetic field on the flows of the plasma component and the

hydrogen atoms was first considered by Alexashov et al. (2000) within the framework of the

model by Baranov and Malama (1993) presented in Sect. 4. To consider the axi-symmetric

problem the vector of the interstellar magnetic field induction B was assumed by authorsto be parallel to the velocity vector V. It was shown that the effect of the interstellar mag-netic field on the positions of the TS, BS and HP are significantly smaller when compared to

the model without interstellar H atoms (Baranov and Zaitsev 1995). The numerical solution

was performed with various Alfven Mach number MA = V

4/B in the undis-turbed LISM. It was found that the bow shock straightens out with decreasing MA (with


16/50

38 V.B. Baranov

increasing the magnetic field). The BS approaches the Sun near the axis of the symmetry,

but recedes from it on the flanks. By contrast, the nose of the HP recedes from the Sun due

to the tension of magnetic field lines, while the HP in its wings approaches the Sun under

magnetic pressure. As a result, the region of the compressed interstellar plasma component

around the heliopause (or pileup region) decreases by almost 30%, as the magnetic fieldincreases from zero to 3.5 106 G. It was also shown that H atom filtration and heliosphericdistributions of primary and secondary interstellar atoms (populations 4 and 3) are virtually

unchanged over the entire range of the interstellar magnetic field. The magnetic field has

the strongest effect on the distribution of the population 2 hydrogen atom density, which

increases by a factor of almost 1.5 as the interstellar magnetic field increases from zero to

3.5 106 G.The interstellar magnetic field effect on the heliospheric interface structure was studied

by Izmodenov et al. (2005a) and Izmodenov and Alexashov (2006) for the general three-

dimensional (3D) case. The following magnetohydrodynamic (MHD) equations for plasma

component were used

V = q, (17)

(V )V= 1p + 1

4( B)B+ F1, (18)

(VB)= 0, B= 0, (19)

V

1p

+ V

2

2+ 1

4B [VB]

= F2. (20)

Here q is the source term connected with the change of the plasma component mass den-

sity due to processes of the photoionization and ionization of H atoms by the electron impact.

The system of (17)(20) together with the (5), where the processes of the photoionization

and ionization of H atoms by the electron impact were also taken into account, and (6)(8)

was numerically solved by these authors at the boundary conditions

V = 26.4 km/s, np = 0.06 cm3, nH = 0.18 cm3,M = V/a = 2, B = 2.5 G s,VE = 432 km/s, npE = 7.39 cm3, ME = 10.

It was also assumed by Izmodenov et al. (2005a) that the vector of the interstellar magnetic

field is inclined to the vector of the interstellar gas velocity at angle 45 and the distribution

function of the H atoms in the LISM is Maxwellian. For the values of parameters, listed

above, MA = V/aA = 1.18 and Mf s = V/a+ = 1.01, where aA and a+ are thealfvenic speed and the speed of the fast magnetosonic wave in the LISM, respectively.

The strong discontinuities (bow and heliospheric termination shocks and tangential dis-

continuity or heliopause) are plotted at Fig. 5 in the xOz plane determined by the vectors

V

and B

. For the purpose of comparison the TS, HP and BS are shown for the case

B = 0 (dashed curves). Figure 5 clearly indicates an asymmetry of the flow relative to theOz axis. Taking the magnetic field into account causes the TS to approach to the Sun while

the heliocentric distance to the BS increases in the whole region of the flow. In the upwind

direction the TS and HP are closer to the Sun by 10 AU and 20 AU, respectively. The he-liocentric stand-off distance of the HP depends on the ratio of the magnetic field pressure to

the magnetic tension. In the region, where the magnetic tension is larger than the magnetic


17/50


Fig. 5 The effect of the interstellar magnetic field on the geometrical pattern of the flow in the case when

the angle between the vector B and V is equal 45. For comparison the positions of the BS, HP and TSat B = 0 are shown (dashed lines). (From Izmodenov et al. 2005a)

field pressure, the heliopause recedes from the Sun (see, also, Baranov and Zaitsev 1995;

Alexashov et al. 2000). We can see this effect in Fig. 5 comparing the HP locations for

x > 0 and x < 0 The magnetic tension is more than the magnetic field pressure at the polar

angle about > 80 for x > 0. Calculations by Izmodenov et al. (2005a) showed that the

stagnation point on the heliopause is about 10 above the Oz axis (x > 0). In the vicinity of

this point the density of the interstellar plasma component reaches a maximum, while its ve-locity vector has a considerable Ox component Vx . Since the parameters of population 3 H

atoms are consistent with those of the plasma component in the outer heliosheath, they also

have a velocity component along the Ox axis. As might be expected, in this case a maximum

intensity of the hydrogen wall is reached precisely near the stagnation point ( 100).The vector VH of the H atom bulk velocity was calculated as the moment of the H atom

distribution function fH. Its component along the Ox axis is not zero at even small helio-

centric distances. The calculations showed that for the magnitude and direction of the vector

B, accepted by Izmodenov et al. (2005a), the angle between VH within the heliopause andthe LISM velocity V

is changing between 3 and 5. The same direction of the H atom

bulk velocity was recently detected by Lallement et al. (2005) from the measurements of

the scattered solar Lyman-alpha radiation onboard of the SOHO (Solar and Heliospheric

Observatory) spacecraft which could be explained by the effect of the interstellar magnetic

field.

An influence of different angles between the LISMs bulk velocity and magnetic field

on the heliosphere structure was also considered by Izmodenov and Alexashov (2006) in


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40 V.B. Baranov

the kinetic-fluid approach. It was shown that the interstellar magnetic field can be a reason

of difference heliocentric distances to the termination shock detected by Voyager 1 and

Voyager 2.

5.4 Effect of the 11-Year Solar Activity Cycles

The solar wind parameters have been measured onboard spacecraft during about four so-

lar cycles (about 45 years). The measurements showed (see, for example, Gazis 1996;

Richardson 1997) that the solar wind momentum flux changes by a factor of about two

upon transition from the solar activity maximum to its minimum. This experimental result

gave rise to the qualitative scenario, suggested by Richardson (1997), in which the bow

shock BS disappears due to the motion of the TS and HP to the Sun during decreasing the

solar wind momentum flux. However, the exact hydrodynamic calculations by Baranov and

Zaitsev (1998) without taking into account H atoms showed that the response of the TS po-

sition to the changes of the solar wind parameters is within 12% from the mean location,

the response of the HP is much smaller and the response of the BS is negligible, i.e. the sce-

nario by Richardson (1997) cannot be physically realized. It has been shown theoretically

that such variations of the solar wind momentum flux strongly influence on the heliosheath

structure (e.g. Karmesin et al. 1995; Baranov and Zaitsev 1998; Wang and Belcher 1999;

Zaitsev and Izmodenov 2001; Zank and Mueller 2003). However, most global models of

solar cycle effects either ignored the interstellar H atom component or took into account this

component in the framework of the hydrodynamic approximation, which is not correct due

to Kn = l/L 1 (see Sect. 3). The non-stationary, self-consistent model of the heliosphericinterface structure, taking into account the interstellar H atoms in the kinetic approximation,

was developed by Izmodenov and Malama (2004a, 2004b) and Izmodenov et al. (2005b).This axisymmetric model is described below.

The model by Izmodenov and Malama (2004a, 2004b) and Izmodenov et al. (2005b) is

non-stationary version of the kinetic-hydrodynamic model by Baranov and Malama (1993)

described in Sect. 4. In addition to the Baranov and Malama (1993) consideration, the ion-

ized helium component as well as the processes of the photoionization and the ionization by

the electron impact were taken into account. For studying the solar cycle effect a periodic

solution of the time-dependent version of the hydrodynamic equations for the plasma com-

ponent (1)(4) together with the time-dependent kinetic equation (5) for H atoms was nu-

merically obtained. For solving the non-stationary kinetic equation a time-dependent Monte

Carlo method was developed. Periodic values of the hydrodynamic parameters at the Earthsorbit were taken as the boundary conditions. In particular, the results were obtained for

ideal solar cycle, in which the solar wind number density oscillates harmonically, while

the bulk velocity and temperature remain constant, i.e.

npE = npE0((1+ n sin t), V E = const= VE0, TE = const= TE0. (21)For the solar wind disturbances determined by (21) the ratio of the maximum to minimum

momentum flux is equal to = (1 + n)/(1 n). Following Baranov and Zaitsev (1998)the calculations were performed for n = 1/3, so that = 2. As it was pointed out by Zankand Mueller (2003) the solar cycle effects on the heliospheric interface remain the samewhen the variation of the solar wind dynamic pressure is caused by the solar wind velocity

variation. The effects of the realistic solar cycle were preliminary studied by Izmodenov

et al. (2003a). The results presented below were obtained by Izmodenov et al. (2005b) for

following constant parameters

npE = 8 cm3, VE = 445 kms1,


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Fig. 6 Variations of the TS, HP and BS locations with the solar cycles. (From Izmodenov et al. 2005a)

which are averaged magnitudes over a few solar cycles. The following magnitudes of the

LISM parameters were also used

V = 26.4 km/s, T = 6500 K, np = 0.06 cm3, nH = 0.18 cm3. (22)

These particular values of the interstellar gas velocity and temperature were chosen on the

basis of the interstellar He atom observations by GAS/Ulysses (Witte et al. 1996; Witte 2004;

Gloeckler et al. 2004). The choice of nH and np is based on an analysis of the Ulyssespickup ion measurements (see, e.g., Izmodenov et al., 2003a, 2003b).

The 11 year-periodic variations of the termination shock, heliopause and bow shock lo-

cations are shown in Fig. 6 in the nose region of the heliosphere and along the axis of

symmetry. We see that the TS, HP and BS oscillation amplitudes are equal to about 7.5 AU,2 AU and less than 0.7 AU, respectively. As it was shown by Izmodenov et al. ( 2005b) the

oscillation amplitude increases from the nose to the tail region and can be equal to 25 AU.

The phase of the TS downwind fluctuations is shifted by 3.5 years as compared with thephase of the upwind fluctuations. The TS response to the momentum flux variation at the

Earths orbit takes place with a delay of about two years as it is seen from Fig. 6. The plasma

parameters also perform oscillations with an 11 year period throughout the entire interface

region. However, their wavelength in the solar wind (within the heliopause) is larger than

the heliocentric distance to the HP in the nose region of the heliosphere. This means that

time snap-shots of the plasma parameter (density, velocity and temperature) disturbances

are not qualitatively different from stationary solutions. The situation is essentially different

in the outer heliosheath, where the HP periodic motion produces a number of additional

weak shocks and waves of the rarefaction as well as in the model by Baranov and Zaitsev

(1998) who did not take into account H atoms. The amplitudes of these waves decrease

while they propagate away from the Sun due to the increase of their surface areas and due

to the interaction between the shocks and rarefaction waves.


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42 V.B. Baranov

Fig. 7 Interstellar plasma number density, velocity, pressure and temperature as functions of the heliocentric

distance for two different moments of time: t1 = 1 year (curves 1) and t2 = 6 year (curves 2). Stationarysolution (curves 3) and averaged over 11 years time-dependent solution (dots) are shown. (From Izmodenovet al. 2005a)

Figure 7 presents distributions of plasma density, velocity, pressure and temperature as

functions of the heliocentric distance at two different moments of the time (curves 1 for

t1 = 1 year and curves 2 for t2 = 6 year). From Fig. 7 we see a wave structure with thecharacteristic wavelength 40 AU in the upwind direction of the outer heliosheath (leftcolumn). This structure is practically absent in the post-shocked plasma of the downwind

direction (right column). Figure 7 shows also a comparison of the 11-year averaged distri-

butions of the interstellar plasma component parameters (dots) with those obtained from a

stationary solution (curves 3). It is interesting to note here that the heliocentric distances of

the TS, HP and BS averaged on the solar cycles as well as the distributions of the plasma

component parameters are similar to those obtained within the framework of the stationary

model by Baranov and Malama (1993) with the solar cycle averaged boundary conditions

(21), (22). Figure 8 demonstrates the distributions of H atom parameters of populations 1

4 obtained by the stationary model (dots) with the time-dependent solution averaged over


21/50


Fig. 8 Number densities (top row), bulk velocities (middle row) and kinetic temperature (bottom row)of populations 4 and 3 (left column) and populations 1 and 2 (right column) of H atoms are shown in the

upwind direction as functions of the heliocentric distance. Dots, which represent the stationary solution, are

practically coincident with solid curves, which represent time-dependent solution. (From Izmodenov et al.

2005a)


22/50

44 V.B. Baranov

11 years. These distributions are practically coinciding. Although only distributions in the

upwind direction are presented in Fig. 8 this conclusion remains valid for all of the compu-

tational domains.

5.5 Heliosphere Extent in the Tail Region

The main purpose for modelling the tail region of the heliospheric interface is a search of

answers to two fundamental questions: (i) where is the edge of the solar wind plasma and

(ii) how long the solar wind exerts some influence on the surrounding interstellar gas flow?

To supply an answer to the first question it is necessary to define the solar wind plasma

boundary. This boundary in the nose region is the heliopause separating the solar wind

and the plasma component of the LISM. However, such a definition would be incorrect

in the tail region since the HP is an unclosed surface here. The tail part of the heliopause

could extend up to infinity in the absence of mixing between the solar wind plasma and

the interstellar gas flow at large heliocentric distances. In order to solve this problem andto answer on both questions mentioned above the tail region structure of the heliospheric

interface was calculated by Izmodenov and Alexashov (2003), Alexashov and Izmodenov

(2003) and Alexashov et al. (2004b) in detail at large heliocentric distances (up to 30,000

AU). It should be noted here that the heliospheric tail region was calculated by Baranov and

Malama (1993, 1995) only to the distances of about 700 AU (see Fig. 2).

The results of the calculations performed by Alexashov et al. (2004b) are presented in

Fig. 9. Immediately after crossing the heliospheric shock TS the solar wind plasma acquires

a subsonic velocity of about 100 km/s and a temperature of about 1.5 106 K. Then thesolar wind velocity continues to decrease due to loading by new protons, born as a resultof the resonance charge exchange processes, and gradually approaches to the magnitude of

the undisturbed LISM flow velocity (V 25 km/s). The charge exchange processes giverise to effective cooling of the solar wind in the tail region because the interstellar H atom

temperature is much lower than the proton temperature downwind the TS. As a result of

Fig. 9 Mach number contours

for the solar wind and interstellar

plasmas in the tail region of the

heliospheric interface. (From

Alexashov et al. 2004b)


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this cooling the solar wind Mach number increases with increasing the heliocentric distance

as it is seen from Fig. 9. The solar wind becomes again supersonic at a distance of about

4000 AU. With further increase of the distance from the Sun the plasma and neutral (H

atoms) component parameters approach their values in the undisturbed LISM (M = 2). Itwas obtained in the calculations by Alexashov et al. (2004b) that the solar wind parametersare almost indistinguishable from those of the undisturbed LISM at distances of about 40

50 103 AU. These distances can be regarded as the heliospheric boundary in the tail region.It is also interesting to note that the jumps of the density and the tangential velocity of the

plasma component at the tangential discontinuity (HP) almost vanish at considerably smaller

heliocentric distances (of about 3000 AU).

5.6 Effect of Nonequilibrium of Pickup and Solar Protons in the Solar Wind

Since the mean free path of H atoms, which is mainly determined by the charge exchange

reaction with protons, is comparable with the characteristic size of the heliosphere, their dy-

namics is governed by the kinetic equation for the velocity distribution function fH(r, wH, t)

(see Sect. 4). The instantaneous assimilation of the interstellar and solar wind proton origin

with pickup protons is assumed in the model by Baranov and Malama (1993, 1995, 1996)

considered above. However, experimental data (e.g., Gloeckler et al. 1993; Gloeckler 1996;

Gloeckler and Geiss 1998) and theoretical estimations (e.g., Isenberg 1986) clearly show

that pickup ions must be considered as an individual component with the distribution func-

tion which is not coinciding with the distribution function of solar and interstellar proton ori-

gin. Even though some energy transfer from the pickup ions to the solar wind protons is now

admitted in order to explain the observed heating of the outer solar wind (Smith et al. 2001;

Isenberg et al. 2003; Richardson and Smith 2003; Chashey et al. 2003; Chalov et al. 2004b)it constitutes not more than 5% of the pickup ion energy.

An improved model of the solar wind interaction with the LISM flow was developed by

Malama et al. (2006) who considered the pickup protons as a separate charged population

with thermodynamic parameters which are different from those of the solar wind although

their bulk velocities are accepted to be equal (Vpui = Vp = V). In this case instead of (5)the Boltzmann equation for H-atom distribution function will have the form

wH fHr

+ Fr + FgmH

fHwH

= (ph + imp)fH(r, wH)

fH

i=p,pui

|wH wi|ex fi (r, wi )dwi

+

i=p,puifi (r, wH)

|wH wH|ex fH(r, wH)dwH. (23)

Here fp(r, wp) and fpui (r, wpui ) are the local distribution functions of protons and pickup

ions; wp , wpui and wH are the individual proton, pickup proton, and H-atom velocities in

the heliocentric rest frame, respectively; ex is the cross section of an H atom collision with

a proton accompanying by the charge exchange, ph

is the photoionization rate; mH

is the H

atom mass and imp is the H atom ionization rate by the electron impact.

Although the plasma and neutral components interact mainly by charge exchange, the

photoionization, the solar gravitation and radiation pressure forces, which are taken into

account in (23), are important at small heliocentric distances. Electron impact ionization

may be important in the inner heliosheath (region 2). As it is seen from (23) the distribution

function of H-atoms can be found if the distribution function of pickup ions is known (the


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46 V.B. Baranov

proton distribution function is assumed to be Maxwellian). Therefore, it is necessary to

have an equation for fpui (r, wpui ). Measurements of the pickup proton distribution function

onboard the Ulysses and ACE spacecraft showed that the pickup proton distribution function

is not Maxwellian though isotropic (see, in details, Chalov 2006). These data gave rise to

the conclusion that there is no thermodynamic equilibrium between the pickup and solarorigin protons though their bulk velocity being nevertheless equal. Since the pickup proton

distribution function is isotropic in the solar wind rest frame, the following angle-averaged

distribution function can be introduced

fpui (r, v) =1

4

fpui (r, w) sin d d. (24)

Here w = V + v, where v is the velocity of the pickup proton in the solar wind rest frame,and (v, , ) are coordinates of v in the spherical coordinate system. The equation for

fpui (r, v) can be written in the following general form taking into account velocity diffusionbut ignoring spatial diffusion, which is unimportant at the energies under consideration:

V f

pui

r= 1

v2

v

v2D

fpuiv

+ v

3

fpuiv

V+ S(r,v), (25)

where D(r, v) is the velocity diffusion coefficient. The source term S(r; v) reflects the birthand loss of the pickup protons due to charge exchange, photoionization and the ionization

by electron impact and can be written as

S(r, v)

=1

4 ion(v)fH(r, v +

V ) sin d d

14

fpui (r,v)H(v) sin d d. (26)

In this equation ion and H are ionization rates:

ion =

i=pui,p

fi (r, vi )|vi v|ex (|vi v|)v2i dvi sin i di di + ph + imp,

H= fH(r, wH)|wwH|ex (|wwH|) dwH.

Numerical solutions of (23)(26) together with the hydrodynamic equations for plasma com-

ponent

(V)= q;(V)V=p + F1;

5

2pV

V p = F2 + F2,e F1 V,

were self-consistently obtained by Malama et al. (2006). Here = mpnp + mp npui andp = pp + pe + ppui are the total mass density and total pressure of the plasma component,respectively. The effective thermal pressure of the pickup ion component and source terms

are determined by

ppui = 43

mpv

2fpui (r,v)v2 dv;

q = mpnH (ph + imp); nH(r)=

fH(r, wH)dwH;


25/50


Fig. 10 Strong discontinuity

surfaces calculated in accordance

with model by Baranov and

Malama (1993) (curves 1) and

with account for pickup proton

nonequilibrium (curves 2). (From

Malama et al. 2006)

F1 =

mp(ph + imp)wHfH(r, wH)dwH

+

mpvrelex (vrel)(wH w)fH(r, wH)

i=p,puifi (r, w)dwHdw, (27)

F2=

mp(ph+

imp)w2H

2

fH(r, wH)dwH

+ 12

mpvrelex (vrel)(w

2H w2)fH(r, wH)

i=p,pui

fi (r, w)dwHdw,

F2,e = nH(phEph impEion),

where vrel = |wH w| is the relative velocity of an atom and a proton, Eph is the meanphotoionization energy (4.8 eV), Eion is the ionization potential of H atoms (13.6 eV), fpis the Maxwellian distribution function of the solar wind and interstellar protons and fpui is

the distribution function of the pickup ions. The distribution function fpui was determined

by Malama et al. (2006) solving the pickup ion kinetic equation (25) while a complete

assimilation of the pickup ions into the solar wind plasma (Tpui = Tp ) was assumed byBaranov and Malama (1993, 1995, 1996).

Malama et al. (2006) used typical boundary conditions at the undisturbed LISM and

at the Earths orbit. However, the termination shock TS is collisionless and quasi-normal

one. In this case a conservation of the magnetic moment leads to the additional boundary

condition at the TS (see, for example, Fahr and Lay 2000), namely

f2,pui (r, w) = 1/2f1,pui(r,w/),

where is the compressed degree across the TS. A self-consistent solution of the problem

formulated above was obtained by Malama et al. (2006) for the case D = 0 correspondingto the quiescent solar wind in which a level of the magnetic field fluctuations is low. Some

of results, obtained by Malama et al. (2006), in Figs. 1012 are demonstrated.

As it is seen from Fig. 10 thermodynamic nonequilibrium of the pickup and solar origin

protons leads to a thinning of the inner heliosheath (region 2) which can be explained by de-


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48 V.B. Baranov

Fig. 11 Fluxes of H atom

population 2 at 1 AU (solid line).

Dashed line demonstrates the

fluxes calculated within the

framework of the model by

Baranov and Malama (1993).

(From Malama et al. 2006)

Fig. 12 Solar wind proton

temperature as a function on the

heliocentric distance: curves 1, 2

and 3 correspond to the

calculations for the case in which

5% of the thermal energy is

transferred from pickup ions to

the solar wind protons, to the

nonequilibrium model by

Malama et al. (2006) and to themodel by Baranov and Malama

(1993), respectively. (From

Malama et al. 2006)

creasing the total pressure of the charged component p = pp+pe +ppui . As compared withthe calculations performed in accordance with the model by Baranov and Malama (1993)

the TS is father from the Sun by 5 AU in the upwind direction whereas the HP is closer by

12 AU. In the downwind direction the heliocentric distance of the TS increases by 70 AU.

This refinement of the physical properties of the inner heliosheath in connection with non-equilibrium under consideration is important for planning experiments on the measurement

of the energetic H atom fluxes (H atoms of population 2) from this region at 1 AU onboard

the IBEX satellite. The results for these fluxes at 1 AU obtained by Malama et al. ( 2006)

are presented in Fig. 11. The model taking into account the thermodynamic nonequilibrium

predicts smaller fluxes for particles with energies smaller 1 keV as compared with the model

by Baranov and Malama (1993) and larger fluxes for H atoms of the population 2 with ener-

gies larger 1 keV. This effect can be explained by the formation of the first fluxes due to the

charge exchange of the interstellar H atoms and the solar origin protons whereas the second

fluxes are formed due to the charge exchange of the interstellar H atoms and pickup protons.

The calculated temperature of the solar wind protons as a function on the heliocentric

distance in upwind direction in Fig. 12 is presented. In accordance with the model by Bara-

nov and Malama (1993) a strong increase of the temperature at larger distance from the Sun

(curve 3 in Fig. 12) is due to the assumption on the instantaneous relaxation of the pickup

protons in the solar wind. The measurements of the solar proton temperature on board the

Voyager 2 spacecraft indicate that such a considerable heating of the solar wind does not oc-


27/50


cur. The solar proton temperature obtained within the model by Malama et al. ( 2006) (curve

2 in Fig. 12) decreases adiabatically up to the distance of about 20 AU, then decreases more

slowly than adiabatically, and starts to grow slowly at distances larger than 30 AU. This be-

haviour of the solar wind temperature is associated with the energy of electrons born in the

process of the photoionization (it is assumed in this model that Te = Tp ). The temperatureobtained by Malama et al. (2006) turns out to be lower than the solar wind temperature mea-sured onboard Voyager 2. This means that the energy supplied to the solar wind is greater

than the photoelectron energy. For this reason, the additional calculations were carried out by

Malama et al. (2006). It was assumed that there is an exchange of the thermal energy (about

5%) between the solar wind particles and pickup protons, whose temperature is much larger

than that of the solar wind protons, and this exchange does not depend on the heliocentric

distance. The temperature distribution in this case (curve 1 in Fig. 12) is in good agreement

with measurements onboard Voyager 2.

6 Comparisons of the Kinetic-Hydrodynamic Approach Results with the Results of a

Multi-Fluid Approximation

Results of the kinetic-hydrodynamic and kinetic-MHD models were considered in Sects. 4

and 5. In the last ten years multi-fluid models were developed by several scientific groups.

In these models the H atom motion of all populations as well as the plasma component were

described by hydrodynamic equations although the mean free path of H atoms is comparable

or more than the characteristic length of the problem considered (Knudsen number Kn 1for hydrogen atom collisions). However, it is known that hydrodynamic equations are correctif the distribution function is locally Maxwellian. Evidently, that the Maxwellian distribution

function can be supported only by collisions, i.e. at Kn 1. Therefore, multi-fluid modelshave no a reliable and physically correct theoretical basis although they are much simpler

than the kinetic-hydrodynamic models. It is shown below that results of multi-fluid models

can give rise to erroneous results and, therefore, to erroneous interpretations of experimental

data.

First multi-fluid models were constructed by Pauls et al. (1995), Zank et al. (1996) and

Zank and Pauls (1996) without comparisons with the results of the kinetic-hydrodynamic

model by Baranov and Malama (1993) although comparisons of the new simplified model

results with the results of previous models are obligatory. Authors of these models assumedthat every of H atom populations has the Maxwellian distribution function in the whole

regions of the flow. The hydrodynamic equations for H atoms have been used at this er-

roneous assumption. Results of the kinetic-hydrodynamic model by Baranov and Malama

(1993) were first compared with the results of multi-fluid model of Zank et al. (1996) in

Baranov et al. (1998) It was shown that the numerical solution of the considered problem in

the framework of the multi-fluid approach is physically and mathematically not real. These

comparisons also showed that the H atom temperature of all populations is not isotropic,

namely, TH = TH , where indexes and denote the radial and tangential tempera-ture, respectively, as distinguished from the isotropic temperature in the multi-fluid models

This effect demonstrates that the distribution function of H atoms is not Maxwellian. As an

example, a hydrogen atom radial and tangential temperatures (moments of H atom ther-

mal velocities) of the population 3 as a function on the heliocentric distance in Fig. 13 are

shown (Baranov et al. 1998). Any later, the distribution functions of the all H atom pop-

ulations were calculated by Izmodenov (2001) and Izmodenov et al. (2001) using Monte

Carlo method (Malama 1991). It was clearly shown that these distribution functions are not


28/50

50 V.B. Baranov

Fig. 13 Hydrogen atom radial and tangential temperatures of the population 3

+4 as a function on the

heliocentric distance demonstrating an anisotropy of the LISM H atom temperature. (From Baranov et al.

1998)

Maxwellian in the whole region of the flow, i.e. the hydrodynamic approach for the descrip-

tion of H atom motion is erroneously in the problem considered.

However, one-fluid and multi-fluid descriptions of the H atom component in the he-

liospheric interface were extensively developed later in quite large number of publications

(see, for example, McNutt et al. 1998, 1999; Wang and Belcher 1998, 1999; Fahr 2000;

Fahr et al. 2000) due to their relative simplicity as compared with the kinetic approach.A detailed comparison of results of these models (kinetic and multi-fluid) was made

by Alexashov and Izmodenov (2005). For the purpose of comparison with the fluid-kinetic

model by Baranov and Malama (1993) authors calculated source terms (see Sect. 4.1)

using four different multi-fluid models for description H atom component. In the model F1

they did not divide H atoms into populations and considered all H atoms as a single fluid

(one-fluid approach). This model corresponds to the description of H atoms by Fahr et al.

(2000). In the model F2 hydrogen atoms are separated into two populations, namely, into

atoms born in the solar wind plasma (populations 1 and 2) and atoms born in the interstellar

plasma component (populations 3 and 4). In model F3 three-fluid approach was considered,

namely, population 1, population 2 and populations 3 and 4 together. This model corre-

sponds to the H atom description chosen, for example, by Zank and Pauls (1996), Florinskii

et al. (2004). Finally, in the model F4 hydrogen atoms of all populations (populations 1

4) were considered as different fluids. Distribution functions of H atoms were, of course,

assumed to be Maxwellian because the models F1F4 are correct only at this assumption.

All calculations by Alexashov and Izmodenov (2005) were made at following boundary


29/50


Table 2 Density, bulk velocity and temperature of different populations of H-atoms in the distant solar wind

at 80 AU in upwind direction

Populations

1 2 3 4

Number density (cm3) K 0.26103 0.91103 0.058 0.076F4 0.27103 0.62103 0.054 0.059F3 0.30103 0.41103 0.14 F2 0.29103 0.14 F1 0.053

Bulk velocity (km s1) K +328 5.8 15.00 26.9F4 +335 16.2 20.0 28.0F3 +328 54 20.5

F2 +330 20.4 F1 29 8

Temperature (K) K 182 500 67 335 15 490 7100

F4 162 200 978 590 16 000 6760

F3 195 100 1 218 440 15 630

F2 194 100 14 870

F1 64 198

conditions at the Earths orbit

neE = npE = 7 cm3; VE = 375 km s1; TE = 51109 Kand at the LISM

V = 26.4 km s1; T = 6527 K; nH = 0.18 cm3; np = 0.06 cm3.Authors used the expressions of the source terms given by McNutt et al. (1998). Their

test calculations with the source terms given by Zank et al. (1996) gave rise to the same

results. As would be expected, the most differences were obtained for distributions of H

atom parameters.Density, bulk velocity and temperature of different populations (14) of H atoms at 80

AU in upwind direction are presented in Table 2. We see significant differences of some

results not only between kinetic (K) and fluid (F1F4) models but between fluid models as

well (especially between F3 and F4 models). It is important that the difference between the

results of F4 and K models is not negligible and could lead to incorrect interpretation of

observational data. In particular, from Fig. 14 we see the large differences between F3, F4

and K models in the distributions of the H atom number density, bulk velocity and temper-

ature of population 2 and the temperature of population 1. At the Earths orbit the values of

the population 2 parameters as obtained by both F3 and F4 models are drastically different

from those obtained in the kinetic model. Apart from the inner heliosheath the properties

of population 2 hydrogen atoms are even qualitatively different as one can see for the bulk

velocity of this population. This result is important because population 2 can be measured

at the Earths orbit as Energetic Neutral Atoms (ENAs). As it was noted above, NASA has

plans to launch Interstellar Boundary Explorer (IBEX). This spacecraft will study the he-

liospheric interface r

Documents

Kinetic-Hydrodynamic Models of the SolarWind