A revised theory of the charge-exchange coupling between
plasma-gas counterflows in the heliosheath
H.J. Fahr
Institute for Astrophysics and Space Research, University of Bonn, Auf dem Huegel 71, D-53121 Bonn, Germany
Abstract. Various hydrodynamic models meanwhile were presented in the literature giving views of the interaction of the
heliospheric plasma bubble with the counterflowing partially ionized interstellar medium in gradually increasing degrees of
consistency. In these models the solar and interstellar hydrodynamic flows of neutral atoms and protons are coupled by mass-,
momentum-, and energy- exchange terms due to charge exchange collisions between ionized and neutral particle species. In a
simplified case by which we describe the main physics of the penetration of an H-atom flow through the well known plasma
wall ahead of the heliopause, we show that the exchange terms used in the up-to-now hydrodynamic treatments unavoidably
lead to an O-type critical point at the sonic point of the H-atom flow. At this point no continuation of the integration of the
hydrodynamic set of differential equations is possible. As we show the way out of this problem is given by a more accurate
formulation of the momentum exchange term for quasi- and sub-sonic H-atom flows. With a momentum exchange term
derived from basic kinetic Boltzmann principles we instead arrive at a characteristic equation with an X-type critical point
allowing for a contiunuous solution from supersonic to subsonic flow conditions. Under these new auspices the already often
treated problem of the penetration of interstellar H-atoms into the inner heliosphere has to be revised since under these newly
derived, more effective plasma - gas friction forces substantially different results are to be expected in this context.
CHARGE-EXCHANGE COUPLING OF
DIFFERENTIAL PLASMA - GAS
MOTIONS
In problems like the mutual interaction of plasma and Hatom gas flows in the heliosheath due to the large local
Knudsen numbers Kn , in principle a kinetic treatment
of charge-exchange induced coupling processes is required. In such kinetic approaches the distribution function f H r v of the H-atom gas for the stationary case
is described by a Boltzmann-Vlasov integro-differential
equation (see e.g. Ripken and Fahr, 1983, Osterbart and
Fahr, 1992, Baranov and Malama, 1993, Pauls and Zank,
1996, Fahr, 1996), McNutt et al. (1998,1999), Bzowski
et al. (1997, 2000) given by:
v
r v 3
fH r v
3
fp
d fH r v
ds
(1)
v ‘ and σ vrel is the velocity-dependent charge exand change cross-section.
Changing over from Equ.(1) to a set of hydrodynamic
moment equations by introducing the moments Φ 0 m p ;
Φ1 m p v ; Φ2 12 m p v2 , then first leads to the following equation for the average mass exchange: Φ0 Q
Q0 0 , i.e. as evident no net mass gain will
0
occur under pure charge exchange reactions. To evaluate the exchange terms for the higher moments i = 1,
2 some knowledge of the distribution functions for protons, f p and H-atoms, f H is needed. Writing these distribution functions as functions of the lowest hydrodynamic moments themselves using shifted Maxwellians
with isotropic temperatures T p and TH (see e.g. Holzer,
1972, Ripken and Fahr, 1983, Fahr and Ripken, 1984,
Isenberg, 1986, Pauls and Zank, 1996, Fahr, 1996, Lee,
1997) then permits to present the above expressions in
the forms:
fH r v ‘vrel v v ‘σ vrel d 3 v‘
Φ1
and:
fp r v ‘vrel v v ‘σ vrel d 3 v‘
Φ2
where r and v are the relevant phase-space variables, ds is the increment of the line element on the associated dynamical particle trajectory, v rel denotes the relv
ative velocity between collision partners of velocities σrel
vrel m p n p nH V H
σrel
vrel m p n p nH (2)
(3)
Pp PH
1 2
VH Vp
ρ p ρH
2
where nH VH PH and n p Vp Pp are density, bulk velocity, and pressure of the H-atoms
and of the protons, re
spectively, σrel σ vrel is the actual charge exchange
1
γ
1
CP679, Solar Wind Ten: Proceedings of the Tenth International Solar Wind Conference,
edited by M. Velli, R. Bruno, and F. Malara
© 2003 American Institute of Physics 0-7354-0148-9/03/$20.00
129
V p
cross section, and vrel is the double-Maxwellian average of the relative velocity between protons and H-atoms
as given e.g. by Holzer (1972):
128
9π
Pp
ρp
PH
ρH
Vp2
VH
(4)
2.4
3
2
THE HYDRODYNAMIC EQUATIONS
1.8
The problem manifest ahead of the heliopause resembles the passage of an H-atom gas through a fixed quasistatic plasma structure downstream of an outer interstellar shock (see Baranov and Malama, 1993, Zank,
1999, Fahr et al., 2000). In a one-dimensional first order
approach along the stagnation line (z-axis!) this plasma,
due to its low sonic Mach number, can be approximated
as incompressible and nearly stagnating. For the H-atom
flow one then obtains the following set of equations:
1.6
0.5
0.3
0.1
0.2
2
0.05
0.4
d
ρ V 0
dz H H
ρH VH
d
V
dz H
d
P
dz H
d
V
dz H
σrel Vrel n p ρH ρH VH2
2
γ
1
σrelVrel n p ρH VH
1
(5)
γ PH
γ 1
Pp
ρp
(6)
PH
ρH
(7)
VH2
2
where γ is the polytropic index both of the protons
and the H-atoms, σ rel σ Vrel is the relevant charge
exchange cross section for protons and H-atoms interacting with an average relative velocity V rel equal to:
Vrel
128
9π
Pp PH
ρ p ρH
VH2 1.4
Vrel Λ ∆ρ Pp
γ PH
0.2
(9)
PH 12 C0VH
γ 1
CoVH
(10)
Vrel ΛC0
C0
0.6
0.8
FIGURE 1. Shown as a function of the plasma wall coordinate x z D, D being the linear extent of the plasma wall,
is the H-atom bulk velocity VH in units of [km/s] (left ordi
nate) and the H-atom pressure PH in units of 10 13 dyne cm2
(right ordinate) calculated for various values of the parameter
Λ Dσrel n p . The H-atom flow in all cases enters the plasma
wall at x 0 with a velocity of VH 0 25 km/s. Curves
reaching the singular point xc where γ PH c C0VH c stop at
this point.
Starting the integration at ζ 0 with supersonic H2 γ P ρ
atom flow velocities, i.e.with VH0
H0 H0 , one at
first, i.e. for small values of ζ , obtains reasonable results
(see Figure 1) .
Arriving, however, at some critical point ζ ζ c 0
where locally γ PHc C0VHc is valid the integration of the
above system of equations cannot be continued anymore.
The singularity at ζ ζ c hereby cannot be avoided by
taking care that at this point both the numerator and the
denominator of Equ.(10) vanish, since putting γ PHc C0VHc and introducing this into the numerator, the following requirement had to be fulfilled at ζ c :
γ
Pp
ρp
2
γ γ 1
2
130
(12)
THE CHARGE-EXCHANGE INDUCED
PLASMA-GAS FRICTION
As explicitly derived in Fahr (2002) the charge-exchange
induced momentum loss rate is calculated with the expression:
Q1 v
H
(11)
As evident this requirement, however, cannot be fulfilled by real values of VHc !
d
V dζ H
1
x
and the differential equation for the pressure:
d
P
dζ H
0.4
VHc and the use of the normalized coordinate ζ defined by
z = ζ D and of the auxiliary quantity Λ Dλ Dσ rel n p
(ζ 0 and ζ 1 mark the two borders of the plasma
wall) one then obtains the following characteristic equation:
d
V
dζ H
0
0
With the mass flow constant :
1
(8)
C0 ρ0VH0 ρH VH
4
0.05
2.2
v [10 km/s]
5
p[10^-13 dyne/cm^2]
vrel
2.6
π2 n pnH m p
2
2KTp
mp
π
∞
o
0
x cos ϑ σ vrel (13)
2xH x cos ϑ exp
x2H x2
x2 x2 dx sin ϑ d ϑ
0
momentum exchange (arb. units)
On the other hand the charge-exchange induced momentum gain rate has to be evaluated from the following
expression:
Q
1 vH 2π n pnH m p
2KTp v
mp H
π
0
∞
0
σ vrel (14)
2xH x cos ϑ exp
x2H x2
x x dx sin ϑH d ϑH
2
2
-2
0.2
-4
0.5
-6
1.0
-8
1.5
-10
-12
alpha =
2.0
-14
The net momentum exchange rate resulting from the
sum of the two above expressions evaluates to (see Fahr,
2002):
π
H
Q1 x H
H
ΠC
MH C MH 2xMH cosh
∞
0
Σ x exp
2xMH x2 MH2 (16)
sinh 2xMH dx
Here the following notations have been used:
ρ V
1
α TH Tp ; MH2 γHP H ; g1 15
1 σB
H
rel
;
1
π 2B
g2 g1
σrel Xrel
where:
Xrel 964π 1 α M 2p
and the charge exchange cross section as taken from
Maher and Tinsley (1977) is given by:
2
σrel A B log Xrel Now it can be shown that expression (15) valid for
moderate and small Mach numbers M H further evaluates
to:
Π π
Q1 sub 9g1 MH
7Π
g M
3 1 H
(17)
2g2MH 5α g1 MH 2α g1MH3 Q1 super ΠMH
4π
1 α α MH2
9
0.8
1
1.2
Mach number
1.4
1.6
1.8
2
-1
-2
-3
0.2
-4
0.5
-5
1.0
-6
1.5
-7
alpha =
2.0
-8
-9
0
0.2
0.4
0.6
0.8
1
1.2
Mach number
1.4
1.6
1.8
2
FIGURE 3. Shown as function of the H-atom flow Mach
number MH is the net momentum exchange rate Q1 sup er for
various values of the temperature ratio α TH Tp .
equal Mach numbers MH the revised rates Q1 sub always
are larger than those Q 1 super taken in conventional theories. This evidently means that at moderate and small
Mach numbers the plasma-gas friction force was substantially underestimated.
THE CHARACTERISTIC EQUATION
whereas, compared to the above expression, in the up
to now literature always the following expression was
used:
0.6
0
where the function C MH is defined by:
π
8MH2
0.4
(15)
2g2MH 5α g1MH 2α g1MH3 9g1MH
0.2
FIGURE 2. Shown as function of the H-atom flow Mach
number MH is the net momentum exchange rate Q1 sub for
various values of the temperature ratio α TH Tp .
momentum exchange (arb. units)
Q
1 x H
0
With the expression given by (17) and with γ 1 γ 1
we now obtain instead of Equ.(10) the characteristic
equation in the following form:
(18)
Figures 2 and 3 show the two different representations,
Q1 sub and Q1 super , respectively, of the momentum transfer rate between H-atoms and protons as function of the
Mach number MH and they clearly demonstrate that at
131
dVH
dz
σ Vrel n p ∆ρ Pp
PH
γ PH
1
2 C0VH γ 1 γ VH Q1 sub
C0VH
(19)
The critical point condition for a vanishing numerator
of Equ.(19) now writes in the following form:
γ 2 π2 73 g1
π
1 1
1 γ γ
2
9g1
64
9π
REFERENCES
(20)
2
2g2 5α g1 α g1 γ2 1 α M 2p
As an example, adopting numerical values of the parameters after Fahr (2000), for TP 20 000 K, α 04 and VH 26 km s 1 , γ 53 we have Vrel 415 km s 1 , σrel 64 10 15 cm 2 , M p 143 and,
when putting in values for g 1 and g2 and the crosssectional constants A and B given for the charge exchange reaction between H-atoms and protons by Maher
and Tinsley (1977) one obtains the request that:
14
9
156
1 39
clearly meaning that this requirement can be fulfilled
and a continuous integration of the set of differential
equations now is possible.
CONCLUSIONS
As evident from works by Baranov and Malama (1993)
Baranov et al. (1998), Fahr (2000) or Fahr et al. (2000)
the Mach numbers MH of the relative flows between the
LISM proton plasma and the LISM H-atom fluid in the
heliosheath region in all cases are smaller than 2. Consequently and strictly speaking, in this region the specific momentum exchange term in the form of Equ.(17)
instead of Equ. (18) had to be used. Since the effective momentum exchange rate described by this expression at identical thermodynamical conditions is greater
than that described by the conventionally used term given
by Equ.(18) one can presume that the adaption of the
LISM H-atom flow to the nearly stagnating LISM proton
plasma in this interface region ahead of the heliopause
is operating faster, i.e. the LISM H-atoms will be much
more effectively decelerated down to the strongly reduced LISM proton flow velocity.
The use of an inadequate momentum exchange term
may thus lead to the erroneous claim for much higher
LISM proton or H-atom densities to reach the same
amount of reduced LISM H-atom bulk flow velocity.
Hence more reliable quantitative interpretations of the
H-atom flow through the heliosphere in terms of needed
LISM parameters (see papers by Scherer and Fahr, 1996,
Scherer et al., 1997, 1999, Izmodenov et al., 1997, Izmodenov, 2000) should be obtained with the application
of the new term given in Equ. (17). LISM parameters
claimed on the basis of these earlier interpretations may
thus need substantial revisions.
132
. Baranov, V.B., and Y.G.Malama, J.Geophys.Res.,157, 1993
. Baranov, V.B., and Y.G.Malama, Space Sci.Rev., 78, 305,
1996
. Baranov, V.B., Izmodenov, V.,and Y.G.Malama,
J.Geophys.Res.,103, 9575,1998
. Bzowski, M., Fahr, H.J., Rucinski, D., and Scherer, H.,
Astron.Astrophys., 326, 396, 1997
. Bzowski, M., Fahr, H.J., and Rucinski, D., Astrophys.J., 544,
496, 2000
. Fahr, H.J., Solar Physics, 30, 193, 1973
. Fahr, H.J., Space Sci.Rev., 78, 199, 1996
. Fahr, H.J., Astrophys.Space Sci., 274, 35, 2000
. Fahr, H.J., Astrophys.Space Sci., 2002, in press
. Fahr, H.J., and Ripken, H.W., Astron.Astrophys., 139, 551,
1984
. Fahr, H.J., and Rucinski, D., Astron.Astrophys., 350, 1071,
1999
. Fahr, H.J., Kausch, T., and Scherer, H., Astron.Astrophys.,
357, 268, 2000
. Holzer, T.E., J.Geophys. Res., 77, 5407, 1972
. Holzer, T.E., and Leer, E.G., Astrophys.Space Sci., 24, 335,
1973
. Izmodenov, V.V., Lallement, R., and Malama, Y.G.,
Astron.Astrophys., 317, 1 93, 1997
. Izmodenov, V.V., Astrophys.Space Sci., 274, 55, 2000
. Isenberg, P.A., J.Geophys.Res., 91, 9965, 1986
. Khabibrakhmanov, I.K., Summers, D., Zank, G.P., and Pauls,
H.L., Astrophys. J., 469, 921, 1996
. Lee, M.A., in ”Stellar Winds”, ed.by R.Jokipii et al.,
pp.857-886, 1997
. Maher, L., and Tinsley, B., J.Geophys.Res., 82, 689, 1977
. McNutt, R.L., Lyon, J., and Goodrich, C.C., J.Geophys.Res.,
103, 1905, 1998
. McNutt, R.L., Lyon, J. Goodrich, C.C., and Wildberg, M.,
Solar Wind Nine, CP471, 823, 1999
. Osterbart, R., and Fahr, H.J., Astron.Astrophys., 264, 260,
1992
. Pauls, H.L., and Zank, G.P., J.Geophys.Res., 101, 17081,
1996
. Pauls, H.L., Zank, G.P., and Williams, L.L., J.Geophys.Res.,
100, 21595, 19 95
. Ripken, H.W., and Fahr, H.J., Astron.Astrophys., 122, 181,
1983
. Scherer, H., and Fahr, H.J., Astron.Astrophys., 309, 957,
1996
. Scherer, H., Fahr, H.J., and Clarke, J.T., Astron.Astrophys.,
325, 745, 199 7
. Williams, L.L., Zank, G.P., and Matthaeus, W.H.,
J.Geophys.Res., 100, 17059 , 1995
. Whang, Y.C., Astrophys.J., 468, 947, 1996
. Whang, Y.C., J.Geophys.Res., 103, 17419, 1998
. Zank, G.P., Space Science Rev., 89, 413, 1999