first -56-

-56-

Third Cycle in Astronomy THE SOLAR WIND (25 November 1975) - M. SCHERER

6.- Theoretical models for corona and solar wind (kinetic models)

1. Introduction

The first evaporative or exospheric model for the solar wind was proposed by Chamberlain in 1960. On the analogy of the escape of neutral particles from a planetary exosphere, Chamberlain suggested that the radial expansion of the solar corona results from the thermal evaporation of the hot coronal protons. Assuming a Pannekoek-Rosse1and electric potential distribution in the collisionfree exosphere

(located above 2.5 solar radii) Chamberlain calculated a solar wind flow speed of about 20 km sec -1 -at 1 A.U.

Since Parker's hydrodynamic calculations predicted a highly

supersonic flow speed a controversy started between Parker and Chamberlain.

Finally direct observations of the solar wind showed the existence of a

-57-

continuous supersonic plasma flo", 2nd the exospheric description of the solar winu \.;as considered as an academic solution vi thout any interes t for the solar corona.

Later on Jensen (1963) and Brandt and Cassinelli (1966) proposed more elaborated exospheric models. They still used the Pannekoek-Rosse1and field (Pannekoek, 1922; Rosse1and, 1924). It was only at the beginning of this decade that the validi ty of the Pannekoek-Rosseland field in the ion-exosphere lvas questioned. The Pannekoek-.Rosseland field which prevents charge separation in a collision dominated plasIVa under hydrostatic

equilibrium conditions, cannot be the true electric field in the exosphere where the gravitational. charge separation becomes less important than

the charge,separation of thermal origin.

The Pannekoek-'Rosselanu field decelerates the electrons and

and electric forces on electrons and protons become equal. Therefore, this field leads to equal densities of electrons and protons. The flux of the electrons however would be (m

1m )

1/2

:t

43 times larger than the

p e

escape flux of the protons. Since this would cause a continuous charge deposition on the sun the true electric field must be larger to decelerate

the electrons and to accelerate .the protons more strongly than the

Pannekoek-Rosseland field does. This electric field must be determined in a self consistent way so that the quasi"neutrality condition n (r) = n (r)

e p

and the zero"current condition F (r) = F (r) is satisfied in the exosphere

e p

(F : electron flux; F

e p proton flux). Figures I and 2 illustrate the

difference between a self consistent calculated electric field and potential, and the Panne~oek-Rosseland field and potential.

C)

E

a.

...

:1:1

-58-

2.5

2.0

w 1.5

Q)

1.0

05 --- ^.

1 2

5 10 20

50 100

200

HELIOCENTRIC DISTANCE[R _s]

Fi g • 1. - The ratio of the parallel electric force to the gravitational force acting on a proton in the solar wind. The solid line corresponds to the kinetic model I of Lemaire and Scherer I I 972c).

The dots correspond to empiricaJ values deduced from Pottasch's [1960) observed coronal density distribution. The dashed line corresponds to Pannekoek-Rosseland's electrostatic poteotial distribution.

~ 0

-e-

UJ I

~

-e-

UJ

.. -200

~ ^'0

-

,>,

- l

«

..- -1.00 Z

..-

UJ

o

Q..

U 0::: _ 600

..-

U UJ

- l

UJ

~ ~

^$

"', _~~.",

^.

·

^...

- PR

· - ^---

i ---- ___ _

• •

·

• •

• •

·

·

• ^•

·

^•

:

·

^•

_•...

...

.. ...

.. ..

2 5 10

.. .. ....

RADIAL DISTANCE

.. ... _...

... S

...•...•...

20 50 100 200

[RaJ

Fig. 2.- The el~ctric potential in some exospheric models of the solar corona.

Th~ da3h~d line (PR) illustrates the PannckockRosscland's disrrlbuti0n the dotted curve (5) gives Sen's exosphQric model for a baropause

ro = 6.05 R.; and a temperature T(r o ) ⁼ l06"K ; the soLid line (LSa) corrcspond;.,'/with the kinet ie models 0f LE'lnaire and SChcl-E'r for the same baropause conditions_

I U1

~

I

-60-

2. DESCIUPTIOK OF A t::.IHETIC HODEL

Guidin£, ceuter 81lproximation - Non-rotating sun

- Spherical sYIDIl1etric outflow

- The collisionfree exosphere 1S separated from the collision dominated region by a sharply defined surface : t:le exobase.

-- r,adial lilagnetic field li(t) ^'V r-2

This last assumption 1S not restrictive, the exospheric I,louel reLiains valid for the Liore bcneral case in which the magnetic field strength

~

is a monotonic decrcasint; function along a field line. For the sun hm!ever,the r.abnetic field lines of the spiral field Dodel are radial if solar rotation is neglected.

Accordint, to the sinble-particle picture of a plas£ia there exis t t:w invariants of the I~tion

- the total energy

1 2-r -+ -+

m v (r) + rn • (r) + Ze ~(r)

2 g

¢ (r) = --+ G g

I

= .+ 1

r

constant (1)

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~(t) is the electrostatic potentiel due to a very snlall charge separation of thermal origin. Because the flow speed of the solar \vind is substantially below the electron thermal velocity, the electrostatic potential will reflect most·of the electrons and cp(r) will be a monotonic decreasing function of

, ~ +

radial distance. The electric field, E

= -

^V~, \yhich is directed outwards, accelerates the protons and decelerates the electrons.

- the magnetic moment or 1st adiabatic invariant

or

2 ~

m vi.(r)

2 B(~)

=

const.

2 ~ . 2 ~)

v (r) un o(r B(r) ~

=

const ⁽²⁾

where e(r) is the pitch angle; i.e. the angle between the velocity vector and the lnagnetic field. Equations (1) and (2) define unambiguously the single particle trajectories in the collisionfree exosphere.

Those trajectories can be classified into four classes :

1. The ballistic particles emerge from the exobase but are reflected because they have not enough kinetic energy to escape ;

2. The escaping particles emerge from the exobase and are lCtst in the interplanetary space,

3. The trapped particles have a periodically motion between two mirror points in the exosphere. The lower mirror point is a magnetic mirror point; the .upper reflection is due to the fact that the kinetic energy of those particles is not large enough to pass the potential barrier ••

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4. The inco~ng particles : these particles come from infinity and are reflected in the exosphere or they reach the exobase and are

lost in the collision dominated reeion. For the solar wind this class of particles is almost emptr..

To determine the different regions in phase space corresponding to those classes we apply (1) and (2) between r and r > r •

o 0

Hence we obtain :

(1) 2 2

ljJ(r) - ljJ(r )]

....

v (r ) = v (r) + [

0 0

[ Ze

<per) ] with !per)

=

^{2 .4J (r) + -}

g m

!

(2) . 2

e(r )

=

G(r ,r) . 2 e(r)

.... _{Sln Sln}

0 0

B(r ) v (r) ² B(r )

with G(r ,r) "'" ^{0 0} > 1 for

2 r < r

0 B(r) v (r ) nCr) ⁰

0

If we assume that the potential energy is a monotonic function along a magnetic field line we have to consider two cases.

This will be the case ~or the protons and is illustrated in Fig. 3a. All particles at the exobase with a velocity vector directed outwardly are accelerated continuously and will escape. There will be no ballistic nor trapped particles.

This .will be the case for the electrons and is illustrated in Fig. 3b. Escaping particles have to overcome a potential barrier and

-63-

r

q, (r)

_,II

(r ) ________________________________________ _

~ 0 . .

Fig. 3a

q,(r)-q,(ro)

-q,(~)- --- --- ---

Fig. 3b

- -

I

-

^~ II ^Q

r

-b4-

therefore their velocity at the exobase must be larger than the minimum escape velocity v (r) - (- ;(r)] 1/2

•

Particles with velocities vCr ) < v (r ) :: ( ;(r) - ;(r )] 1/2

o a 0 0

will never reach the level r because they have not enough kinetic energy.

Particles with velocities v (r ) _a < vCr ) < v.b(r ) :: v (r

)1

0 0 0 a 0

( 1 - B(r)/B(r

»)

and pitch

o angles 0 < O(r ) < 0 (r )

=

arc sin { IG(r .r») 1/2 }

o m o o

will reach r. If they have a pitch angle 0 (r ) ^< e(r ) ^< 2w

- e (r ), however,

m o o m ⁰

they will be reflected by a magnetic mirror point before reaching r.

Particles with velocities vCr ) ^> vb(r ) will always reach level r

o 0

where their velocities and pitch angles satisfy the following inequalities

vCr) > vb(r) -

B(r) B(r )

o

o < OCr) < 0 (r)

=

arc sin { [G(r .r)] - 1/2 }

m 0

In this case there are also trapped particles. All possible classes of particles at the level r are illustrated in Fig. 4.

The velocity distribution is a solution of the collisionles8 Boltzmann equation or Vlasov's equation which states that the distribution function f(!, ;) is constant along phase space trajectories. The phase space trajectories are characterized by constants of motion. Therefore any function of the constants of motion is a solution of Vlasov's equation.

OWing to the truncation procedure that excludes the incomming particle.

it is obvious- that the function f[ r 0' ;(r 0» adopted in kinetic calculations

..q

...

OQ

....

_e(r)

=

^'It

2

Vb(r)

V.rn

~(7)

\

^,~

-<\\\,/

~~~

e:," /"/

.~/

".~/

",,/

vcr) g--=~ -I ESCAPING PARTICLES

Illli111!1

TRAPPED PARTICLES

b·-:::}j:] BALLISTIC PARTLClES 0 INCOMING PARTICLES

I 0'\

U1 I

-66-

is not necessarily a realistic microscopic representation of the actual velocity distribution at the exobase. It is however always possible to build up a function f [r , ~(r)] so that the s first moments coincide

o 0

with the s corresponding lOOments of the actual velocity distribution at the exobase.

For example a linear combination of truncated Maxwellians

+ - Sj _{v (ro)} 2

f [r , vCr )] ;: I: c. e

o 0 • J

J

is appropriate since kinetic models usually have been developed for Maxwellian distribution functions. The parameters c. and 6. can be

J J

determined ~o that the dens~ty, the flux, the temperature, the anisotropy or any higher order momentum coincide with the actual values at the exobase.

This method avoids zero order discontinuities (jumps) at the exobase. but first order discontinuities (gradient-discontinuities) can not be avoided because of the sharp transition between the collision dominated and

collisionfree regions.

In what follows we will assume a Pseudo-Haxwell-iloltzmann distribut:lon

f[ r • vCr )} _o + _{. 0} ⁼ e

were the incoming particles are missing. This is illustrated in Fig. 5.

Therefore

nCr ) ; Nand T(r ) ; T

o 0

PROTONS

^-67-

ELECTRONS

f [r o. v ( r

^{0 )]}

ESCAPING

Fig. 5.-

-68-

The velocity distribution in the collisionfree region (r ^> r ) follows o

from Liouville's theorem:

N(m/211 k'f) 3/2 e -Q exp

I -

^m

2kT

m wi th

l.ll. ~101lJelltl) ot lilt: .list(ibulil'l! JiIIKll"lI

- Conduction flux

-< - ~.

11(\) '''' , I I I , v} •

"

oJ '\f

) V) d v

J • _{, I"}

I' (r)

...

111 V f ( r ^{t v)} " v

II ^I.

I ,

p _(I:') 2

'2 ^{OJ ! vI. f ( 1: •}

!

!

^lD

I

²

£(r) '" v v f(r,

2 , II

w(r) - F(r)/n(r)

T (r) .. p.t.. (r) /k n (r)

.L

•. j.

v)

-+ v)

PI,(r) - m vCr) F(r)

Til (r) - - - - k nCr)

T(r)

-"3

I [T.,(r) + 2 T.L (r»)

d j v

d3 v

- dr) + vCr) (Ill vCr) F(r) -

'2

3 Pit (r) - P.1 (r»

-69-

The explicit formula of the escape flux shows that the proton flux is independant of the electrostatic potential, whereas the electron flux d'epends only on the height of the potential uarrier [ .- 1jJ(r

o)] ,i.e. on w(r) and not on fer). Therefore the zero-current condition

o

Fe(r) • Fp(r) can be used to calculate - .e(ro) or ^~(ro).

The local quasi-neutrality condition n (r)

=

n (r) can then

e p

be used to calculate the electrostatic potential distribution,~(r), from which the electric field E = -

d;P~~)

can be deduced.

~Ioreover

we can also calculate the average temperatures and terr;,perature anisotropies, the energy fluxes and conduction fluxes, etc ... ,since the self-consistent electrostatic potential distribution is knO\Vll in the collisionfree exosphere.

3. APPl-ICATION TO QUIET SDLAR HIND CONDITIONS

To calculate a solar wind model by means of the previous exospheric approximation we firs t have to determine theexobase level.

This is the surface where the mean free path 1 becon'es equal to the density scale heigth ll.

The proton mean free path ^{~s g~ven} by

T 2 V n _e In A _pp

[ cm ]

-70-

where the Coulonb logarithm is practically constant and equal to 25 for the temperatures and densities considered.

TIle density distribution n can be deduced from observations of e

the solar corona (Pottasch, 1960) for r < 20 solar radii. Hence we can (

II In ne)-l

also calculate the scale height H == -

----=-

_dr • These results are shO\-!n in Fig.

6,

^curve 1 for the density and curve 2 for the scale heitht.

For a given proton temperature T ; the proton exobase follows

p

from the condition 1 (r ) =H(r ); or inversely if the exobase level is 1" 0 0

known we can determine the proton temperature by means of

log T (r ) ==

p 0

1 2

log n (r ,) e 0

1 + -

2

log H(r ) + 0.57 o

where n is given in cm-3 and H in km. Curve 3 in.Fig. 6 shows T (r )

e p 0

obtained fromPottasch's electron density distribution.

The mean free path for the elections is given by

1 - 0.416 (T

IT

)2 1

e e p p

and the electron exobase follows from the condition 1 (r ) _e

=

_H(ro)'

0

In what follows we will assume that the electron exobase coincides with the proton exobase. This yields the constraint I _e

=

^I _p ^or

T (r.) • 0.645 T (r ) •

p o e ⁰

-71-

10'

1'a.o

103~ __ ~ __ ~~~~~~ ____ ~~

1 2 5 10 20

HEIGHT ABOVE SOLAR LIMB: h [R~

Fig. 6. - Curve I shows the equatorial electron density distribution (cm"3) in the solar corona observed during an eclipse near minimum in the sunspot cycle as reported by Pottasch (1960) ; Curve 2 gives the corresponding density scale height, U, in km ; Curves 3 and 4 i1Iu~trate respectively the proton and electron 'temperatures at the exobase as a function of the exobase altitude ho expressed

in solar ra dii.

The electron temperatures corresponding to Pottasch's density distribution are shown in Fig. 6, curve 4.

Fig. 6 can be used to determine the boundary conditions for the exospheric solar wind models. Indeed if we choose. an electron temperature at the exobase we can determine the exobase level r by curve 4; the density

o nero) by curve 1 and the proton temperature by curve 3.

In Fig. 7 and 8 we plotted respectively the solar wind flow speed and density at 1 AU and the average electron and proton temperature at I AU calculated for different electron temperatures at the exobase. From the

results we conclude thal there exists a positive correlation between the

flow speed and the proton temperature at 1 AU whereas the electron temperature at 1 AU is nearly independent from the bulk velocity at 1 AU. This positive correlation which Is also supported by solar wind measurements is illustrated in Fig. 9.

Since the proton velocity distribution at the exobase can be highly asymmetric, we also considered kinetic models in which the proton velocity distribution at the exobase is given by

f [r , vCr

» ,.

N(m/2'1fkT)312 exp [ -

o 0

m

2kT

(-+ v - u -+)2

J

where ~ is a velocity vector parallel to the magnetic field. Since the incoming particles are missing u is not the flow speed at the exobase

b~t a parameter which can be chosen in order to obtain a given flux or bulk velocity at roo

Fig. 10 and 11 illustrate the influence of u on the solar wind p

flow speed, density and average electron and proton temperature at 1 AU.

Finally, in Table 1 we compare the results from a calculated solar wind model with the observations reported by Hundhausen [1968, 1970, 1972a.b).

-73-

450

t::::I

I

U

<1.1

VI

t;;t

E

400 20 ^I

..x

E

L.J ~

w w

3

^c

::)

::)

«

«

₃₅₀

1 5

...

... «

«

_>-

...

>-

...

^tI)

-. _Z

U

0 _..J 300

to

^UJ ₀

UJ

>

n E

0 ^z

⁰

:::.:::

a:::

..J

...

. ::) u

III UJ

250 5 _UJ ^..J

1.0 1.5 2.0 2.5 3.0

ELECTRON

~TEMPERATURE:

^T_e^(r._o

)Do

^6oK]

Fig. 7.- The bulk velocity WE (left hand scale), and the electron denBity nE(right hand scale) at IAU

for Up

=

_Ue ⁼ 0 versus the electron temperature at the baropause. The observed (Obs) quiet solar wind flow speed and density are also plotted.

-74-

'6r---~---~---~---

__

~14

_o ....,

L.::J o

UJ

"

I - 12 v

I - 10

<{

V1

w

a:: ::>

I - 8

«

a::

w

0..

~

~

⁶

w

<!)

<{

a::

w

> ,

<{

1.0 1.5 2.0 2.5 30

. 6 ELECTRON TEMPERATURE: Te

t

^ro)[l0

oK]

Fig. 8.- The average electron and proton temperatures at 1 AU and for Up

=

_Ue

=

0, versus the electron temperature at the baropallse. The observed (Obs) quiet soLar wind values are also plotted.

-75-

10~----~--~---~---~

::l

8

<C

I -

-

<C

w

0:::

::J

6

I -

<C

0:::

ll.. UJ

~ W l -

I.

0

z

I -

0

a:::

ll..

w 2

<!)

<C

a:::

w >

<C

6

0---

250 300

3~

400

BULK VELOCITY AT 1 A.U. [kmseC~

Fi g. 9. - Correlation between the solar wind velocities and average proton temperatures at I AU. The solid dots correspond to Vela 3 measurements (Hundhausen et al., 1970). The shaded area corresponds to Explorer 34 data illustrated by the relation: <T p>~~ (0.036 ± 0.003)wE . (5.54 ± 1.50) proposed by Burlaga and Ogilvie [1970 I. Point 1 gives the quiet solar wind condition [H\.Indhausen, 19'72a 1 ; Points 2 to 6 give the results obtained from the kinetic and semi·kinetic models reported in Table II from the second to the sixth linc. The solid line shows the relationship deduced by Lemaire [1971 J and discussed in the text.

-76-

1-1 I I'M"1

0 cD

U)

E

I

E

~ UJ

t-=...

^c

3

^{LU35O 15 ::J}

«

::::>

I -

<{

«

- 0- >-

t-<{

- - - -

^{I -}

( 300

>-

t- U 0 ....J

>

UJ

- -

- --

-

--

10 ^(/)

z w

0

z

0

- cr:

- - ^--

~ 250 5 ^{I -}U

....J

::J ^W _....J

III _W

0 20 40 60 80

Up [k m sec-I]

Fig. 10.- The bulk velocity We (left hand scale), and the

electron density nE (right hand scale) at I AU, versus Up. The dashed and solid lines correspond to an

electron temperature at the baropause of respectively 1.4 x 1060K and 1.6 x 106^oK. The dots illustrate the model LSb which corresponds with the barnpause conditions ro

=

6.6

1\0,

Te(r o ) '" 1.52 x 1060K, Tp(ro)

=

0.984 x 106^oK, Ue :::: 0, Up '" 14 km sec-I, and ne(r o)

=

3.1 x 104 cm- J .

-77-

16r----r---r---r---r---~---~

~ o

...;t

,011.

l.r!=J w

1\

t- V

::;, 12

<(

-

t-

<!

t3

¹⁰

0:::

:::>

t-<!

0:: UJ 0.. 8

.~ UJ

....

UJ

<!) 6

<!

0:::

W

>

<(

4

- ^--

----

o

2.0

-

...

-- -

^...

80

Fig. 11.- The average electron and proton temperatures at I AU, versus Up. The dashed and the solid lines correspond respectively to an electron temperature at the baropause of 1.4 x 106⁰K and 1.6 x 106°K. The dots illustrate the model LSb which corresponds with the baropause conditions ro

=

^{6.6 R ' ) I}

Te(r o ) : 1.52 x i06o K, Tp(ro)

=

O.984xl06o_K, U_e = 0, Up

=

14 km sec-l, and ne(r o ) ⁼ 3. I x 104 cm- 3 .

-78-

TABLE I Results at 1 AU for the solar wind model r

=

6.6 solar radii,

0

'f (r ) 1.52 106

" T (r ) = 0.934 x ^{W 6 ..} n(r ) =

= X h. ^{J h.} ,

e 0 p 0 0

3.1 x 104 cm ^{- )}

,

u _e

=

⁰ ; u _P

=

14 km sec -1 The observations are

.

revorted by Cundhausen (1963, 1970, 1972a, b).

At 1 AU Hodel abs~rvations

w [km sec. -1

J

³²⁰ 300 - 350

- )

'WI [ cm ] 7.2 8.7 + 4.6

-2 -1 ^,.., lOS

"

F [cm sec

J

2.3 x IOu 1.2 x - 4.6 ^{x 10°}

4 (14 + 5) x 104

T [ K ] 1l.7 x 10

e

[ K ] 4

(4.4 .::. 1.8) x 104

T 4.8 x 1Q

P

(T"

ITJ. )

^{e 3.05} 1.1 - 1.2

(TII/Tl) p 164 2 + 1

e:: [erg cm -·2 sec ] -1 0.2 ~ 0.24

1-' -2 -1 -2 0(10-2)

c

[erg cm sec ] 5.1 x 10 e

Except for the terrlerature anisotropies the agreement i~ quite Lood.

The s&:al1er calculated ten.perature ani;:;otropy for the electrons cor-pared to t!le protons is a consequence Qf the large nur..ber of trapped electrons present at 1 AU. Since the calculated anisotropies are rr.uch lareer than the observed ones it is suggested that some scattcrinl:, n:echanisn must modify the pitch angle distribution bett-'ecn the exobase And 1 AU.

-79- REFErJmCES

BPJUiDT, J.C.) and J.P. CASSINELLI, Interplanetary bas, 11, An exospheric model of the solar \-Ilnd, !'£~E~§', ~, 47'--63, 1966.

BURLAGA, L.F., and K.U. OGILVIE, l!eating of the solar uind, !}§'~E2P.!!Y.~_' __ ~"

159, 659--670, 1970.

ClWIDERLAIN, J.l-1., Interplanetary gas, 2, Expansion of a model solar corona, !}§.~.r:~E!!l~.'._:!:') 131,47--56,1960.

HOLUlEG, J. V., Co11isionless solar- wind, 1, Cons tant electron temperature, :!!..J~~9.2~~:t.~ ~ .. ~~~ "

21,

2403',2418, 1970.

1I0LLWEG, J. V., Collisionless solar \.,ind, 2, Varinb Ie electron temperature,

{~_~_~~~~_~~~., 76,7491-7502, 1971.

HUNDH,,\USEN)' A.J., Direct observations of solar-t·dnd particles, §E~.£~_~.£i.

E~~~, ~, 690-749, 1968.

HUNDHAUSEN, A.J., Composition and dynamics of the solar wind plasma, B~y~_9~2Eh:i~ .... __ ~2~£~_E!!:i~" ~, 729-'311, 1970.

HUNDl~USEN, A.J., Composi tion and dynamics of the solar wind plasma. in Solar-Terrestrial Physics/1970, part 2, edited by E.R. Dyer, pp. 1-31, D. Reidel ~ Dordrecht, Netherlands, 1972a.

HUNDllAUSEfI, A.J., Coronal Expansion and Solar. Wind, 238 pp., Springer, Berlin, 1972b.

lIUNDi:;AUSEN, A.J •• S.J. BA~m, J.R. ASBRIDGE, and S.J. 3YDORIAK, Solar ,.,ind proton properties ; Vela 3 observations from July 1965 to

June 1967, .J:.:.._~QI!.Qy2.:..J~:~2."

21,

4643-4657, 1970.

JENSEN, E., l1ass losses throu~h evaporation from a completely ionized atmosphere with applications to the solar corona, ~§l._~f<.?P.!!Zs.

~?£~., Q, 99-126, 19G3.

-80-

JOCKERS, K., Solar wind models based on exospheric theory, ~~!~~.

!~~!~~!!l~" ~, 219--239, 1970.

LEER, E., and T.E. HOLZER, Collisionless solar wind protons: A comparison of kinetic and hydrodynamic descriptions,

:!_. __

~~e.l~l~.."_~~·' ]2,

4035-4051, 1972.

LEMAIRE, J., and fl. SCHERER, Sin:ple model for an ion-exosphere 1n an open

LEHAIRL, J., and lIe SCHERER, Kinetic models of the solar wind, ~_. __ ~~p'~y'~.

~~., 76, 7479-7490, 1971b.

LE~;J!.IRE, J., and H •. SCnCr£L:., Ion-exosphere with asyn:metric velocity c.1istributioll, !:~y'~: .... !,l~~~~, 15, 760-766, 1972a.

LEMAIRE, J., and

,

ti. SCHERER, COlI'.portements asymptotiques d 'un modele cinetique du vent solaire, ~~!_~~

__

g!.."_~~~_~

__

~<:!'l:d .•. ~~y'." .. ~~!g., 5S, lll2--1134, 19721..

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