Space charge sheath formation in a magnetized plasma flow around space bodies

Acta Astronautica Vol. 10, No. 2, pp. 91-97, 1983 Printed in Great Britain.

0094-5765/83/020091d)7503.0010

Pergamon Press Ltd.

SPACE CHARGE SHEATH FORMATION IN A MAGNETIZED PLASMA FLOW AROUND SPACE BODIESt A. V. GUREVICHand N. T. PASHCHENKO Intercosmos Council, USSR Academy of Science, Leninskii Pr. 14, 117901 Moscow V-71, U.S.S.R. (Received 13 January 1982; revised version received 3 June 1982)

Abstract--Space charge sheath formation in a magnetized plasma flow around space bodies is considered for the cases of high surface conductivity, free electron emission, dipole magnetic field. The results are applied to explain different phenomena in the ionosphere and magnetosphere of the Earth and planets such as polar aurora and kilometric radiation of the Earth and decametric radiation of the Jovian satellite Io. I. INTRODUCTION

Comparing (4) and (2) one can see that the potential difference in a space charge sheath is determined by formula (2) only in the case when R ~
A body in plasma does not remain electrically neutral while colliding with it, electrons are absorbed and ions recombinate; this results in an electric charge, and at the body surface an electrical field appears:~ E~ = -grad ~p~.

R >>PHi

(1)

an induction potential (4) plays the main role. It grows when the size of the streamlined body increases, therefore a potential ~s, and, hence, the potential difference in a space charge sheath can be essential. For example, ~B - 100 kV for magnetized plasma of solar wind around the Earth, ~oB- 400 kV for the Jovian satellite Io. In such cases plasma electrons and ions can be highly accelerated by the electrostatic field of the space charge sheath. Thus, a space charge sheath arising in a case of magnetized plasma flow around conducting space bodies can be a source of accelerated particles. Electrons and ions accelerated in the space charge sheath can ionize gas, generate optical luminescence in the upper atmosphere and generate radiofrequency radiation. The aim of the present paper is to consider the conditions under which a space charge sheath arises.

It the general case, potential ~ diffffers from plasma potential cp; therefore a charge sheath is formed at the boundary between body and plasma: in this sheath potential varies from ~ up to ~0p.Such a sheath is called Langmuir space charge sheath[l]. In the case of bodies which do not move or move in nonmagnetized plasma a potential difference A¢ is of the order of: A ~ = ¢~ - ¢p ~ KTe/e, KT~e,

(2)

K being a numerical factor - 1 + 511, 2, 3]. Later ~, will be assumed to be zero, q~o = 0, i.e. potential q~ means a potential difference between plasma and body. If a body moves in magnetized plasma, the situation is essentially changed. The magnetic field frozen in plasma around the body induces an electric field (in the reference system connected with the body): EB = - / [VB].

2. BOUNDARY CONDITION FOR MAGNETIZED PLASMA FLOW AROUND A BODY

The polarization field on a conducting body surface, ~p~,is described by the following equation[4, 5]:

(3)

An induction field EB generates electrical currents on the conducting body, and this leads to electrical' polarization of the body surface, ¢s. A polarization field shields an induction one; this results in a potential difference A~p = ~os between moving magnetized plasma and body surface, and this potential difference can reach the magnitude of full induction potential: ~P, - ~s

VBRo =

C

(5)

Divs[X(V~s - E)] = j., = \-EH, 2~]"

(6) (7)

For steady-state flow the following conditions must be satisfied:

J,: fjn ds,

(4)

(8)

which means that the total current from plasma to body is equal to zero and, hence, there is no electrical charge on the body. Electrical current (6) generated on the conducting body by induced field EB is a source of inductive plasma-

tPaper presented at the XXXIInd Congress of the International Astronautical Federation, Rome, Italy, 7-12 September 1981. ~tSee nomenclature in Appendix at the end of the paper. 91

92

A. V. GUREVICH and N. T. PASHCHENKO

body interaction[6, 7]. As a result of this interaction the perturbations of magnetic field B and flow velocity V arise• In fact, the relation (6) is a boundary condition in the theory of inductive plasma-body interaction. It is important, however, that under the following conditions: 41rI ~ 1, 4~'E VA ~ 1, VA Bok/(41rNoM), c Bo

- ' - ' cT ~

<~

(9)

=

the perturbations of induction field are small. Under the conditions (9) an induction field EB can be taken in the form: Ea = - 1 [VoBo].

(10)

Note that induction field perturbations are also small for the bodies which are not very large, when Bo2

~ 1, it = ire i = •

c O)oe.i

, ~Oo~= ~/(4~re2Nolm), (11)

OJo, = V (41re2 No/ M)• Here A = A~ when emission currents are absent, A = A~ for full electron emission (see below). Thus, when conditions (9) or (11) are satisfied one can consider the induction field Es to be given and not depending on perturbations induced by body motion in plasma. Current j, is determined by electron or ion currents of absorption and emission• In the case of slowly moving (Vo<~ X/(T/M)) or resting bodies, an absorption current [2, 3], is equal to: ; fexp(- e~oslZ) ~o, > 0, jio = v f _ L r , eNo, j, =,,o[ 1 ', ~,s -< O, \2¢rM] (12) It is assumed that ions recombinate at the body surface[8]. For the case of supersonic body motion (Vo >>X/(TelM)) the ion current is: j _ feNo(Vn) at M(Von)2/2e > ~0s, '- [ 0 at M(Von)212e < ~,,.

(13)

Electron absorption current is:

j~q =

_ ; fexp(e~od T,), at ~, < O, j~o [ I at ~,s -> 0,

~/[ T~ ~ eNo(BOn). jeo = \2¢rm /

(14)

It is supposed here that body velocity is less than thermal electron velocity:

Vo ~/(TJm).

(15)

Natural cosmic bodies with free emission are stars and planets with their own ionosphere (for example, Earth, Jovian satellite Io).

Emission current for small values of ~, ~< Tde is equal to:

., . f 1 at ~p,-<0, 1, = l~o~[exp(_ e~o,IT~) ~p~> O.

(16)

For large values of 9, ~> T i e the perturbations induced by electron acceleration in a space charge sheath are high and they can influence emission and absorption currents. Let us emphasize that electron current j~o is two orders greater than ion current j~o,therefore in the case of free electron emission the ion currents are nonsignificant. Surface conductivity of the planets with their own ionosphere is an integral ionosphere conductivity[4]. For small potentials ~,~ a surface conductivity does not depend on potential, but in the case of high potentials the electrons accelerated in a space charge sheath can ionize neutral moleuules and change integral ionosphere conductivity. These strongly non-linear processes can essentially influence the space distribution of polarization potential. Some particular examples of space charge sheath formation will be considered below.

3. HIGHLY CONDUCTING SPHERE IN A MAGNETIZED PLASMA FLOW

Let us consider supersonic matnetized plasma flow (13), (15) around a sphere with radius R. To simplify the problem we assume that plasma velocity Vo is normal to the magnetic field Bo. Let x-axis lie in the Bo direction, the z-axis in Vo direction. Hall conductivity is assumed to be negligibly small, so the surface conductivity tensor 2 is diagonal and characterized by the Padersen component 2~, only• Let us assume the surface conductivity 2~, to be rather high, so that: j~maxcR

-

e = X~,VoBo ~ 1.

(17)

The solution of eqn (6) under condition (17) is readily found by the series expansion with respect to a small parameter e~. Considering the zero approximation, one can find from (6) that the polarization field E, is directed along the y-axis as well as the induction field (parallel to [VJo], and potential ~0, is determined by the following equation: d~o-z~= - VoBol c, dy

(18)

from which we have; ~o, = ~po- VoBoylc = - VoBo(y - yo)lc.

(19)

Constant yo = C~oolVoBo is determined from condition (8)• Let us assume that there is no emission--all the electrons colliding with the body surface are absorbed and

93

Sheath formation around space bodies ions recombinate. The whole electron current is equal to: L =

]~ds=2

X/(o-Y)I~(Y)

Y.

q~s/qa8

(20)

It is taken into account here that electrons and ions (under condition (13)) collide with the front part of the body surface[2, 3]. Considering the body to be large enough (5), one can find from (20), (19), (15): i~

./[T~\ ....

4e

=-3---x~VkmJNOii-yo)

3/2-,/2

K

.

D

-

~'o

Y

VoSoRo ¢

(21) Fig. 1. Distribution of polarization potential.

The total ion current to the front part of the body surface (13), in the same approximation (15), is equal to:

I~ = ,rR2No Voe.

(22)

4. BODIES WITH FREE ELECTRON EMISSION

Using condition (8) one finds: /y~2/3

Yo = - R(1 - a), a = [,~]

/mVo2,~m

zrk---~-]

.

(23)

A potential distribution ~ps on the front body surface is shown in Fig. 1. At point yo a polarization potential es = 0 (19) and, hence, a potential at the body is equal to plasma potential. There is no space charge sheath at this point. For

R<_ y < - R ( l - a ) , potential ~os> 0. Electrons from plasma reach this region and are accelerated in the space charge re#on up to the energies $' = e¢s. Maximal energy of accelerated electrons at the body edge (y = - R) is: ....

-~ a e

for large bodies this potential can be rather high; this will lead to essential fluxes of charge particles accelerated up to energies of the order of e~s.

VoBoR/e.

(24)

Let us consider bodies with free electron emission from the surface. Assume that body conductivity is not very high:

X VoBo

e, = - - w - - - ~ l . CJeoK

(26)

Usually such conditions take place for planets with ionosphere. First we assume that the streamlined body is a plate s located along the plasma flow Vo, the magnetic field Bo is normal to the plate surface (Fig. 2). Assume, as before, that plate surface conductivity is characterized by Padersen Component 2~, only; it is constant or depends on coordinate y (x-axis is parallel to [VoBo]): 2~. = 2~,(y)A, Xn = 0. Then we obtain from (6) the polarization potential:

d

dq~,

c

(27)

In the region

R>-y>-R(1-a) a potential is negative, this region is filled by ions. Their maximal energy is: g, max - - ~ VoBoR.

(25)

In the case of free emission considered here a current is determined according to (14), (16); the relatively small ion current (12), (13) can be neglected. We look for a solution of eqn (27) in a series form with respect to small parameter e: ~os = ~,o+ e¢~, + . . .

(28)

It follows from (27), (28) that, in zero approximation, The maximal value of potential ~0s (i.e. maximal potential difference is a space charge sheath) is close to the total induction potential (4). Note that if a small electron flux (le < L) is emitted from the region y > 0 of the body surface the electrons are accelerated in a space charge sheath and then move in plasma along the magnetic field with energy of the order of g~. . . . One can consider a body to be a source of highly energetic electrons. The conditions considered in this section take place when large satellites and rockets having conducting surfaces move in the ionosphere. Arising polarization potential is of the order of ~p~~ 0.5RVo. One can see that

j.(q~so) = O,

Fig. 2. Plasma flow around a plate.

(29)

94

A. V. GUREVICHand N. T. PASttCHENKO This gives:

and, hence, ~O~o= 0.

(30)

Here the condition (8) and particular expressions for emission and absorption currents (14), (16) are used. From a physicali point of view, the results obtained seem to be quite reasonable--under the conditions (26) a potential at every point of the body is a result of local balance between incident and going away charges and for free electron emission it is close to plasma potential, i.e. ~O,o.In the next approximation with respect to E, we obtain from (27), (14), (16):

dy from which we have

~o,,-

7",VoBo dE m ec dy (y)"

(32)

Hence, in the region of smooth plate conductivity variation a polarization potential is very small, ~o~t'~ Tie, and induction effects do not influence the structure of space charge sheath. One can say that in this case there is no induction space charge sheath at all. The situation is changed near a sharp plate boundary Y = Yo, where expansion (28) is not valid. There, in the vicinity of y = yo(z), we have a special boundary layer. Assuming that 2~ is not changed in the boundary layer: X~, = X~.(yo) = 2,o,

(33)

and (e~odT.)> 1, so that emission and absorption currents have their optimal values j,o, we write eqn (27) in the form:

ay~ t

\ay~

c /j

(34)

C=

X sin i//Vo2Bo2 X sin $VoBo. 2c'~o , y,o Cjeo '

ym is a thickness of boundary region, constant-C the maximum of the potential:

~o. . . . =

X/ZrEo sin @VoBo2X/m X/(2 T~)c2eNo

(39)

Thus, in a boundary layer near the plate edge a space charge sheath is formed with maximal potential differences Cm~x(39). Potential and current distributions in a space charge sheath for a plasma flow around a plate are given in Fig. 3. Space charge sheath formation is quite clear from the physical point of view. Current I =-E(VoBolc) flowing along a plate surface must be transformed at the edge of the plate into a longitudinal current flowing along the magnetic field. But longitudinal current density j. is bounded, j,
VoBoY,o/C= q~y,o/R.

(40)

So we see that in the case considered a potential of a space charge sheath is equal to the induction potential multiplied by the ratio of a boundary layer thickness y,o to a body size. Let us emphasize that in the case of free electron emission considered here the electrons accelerate in a space charge sheath up to energies:

~m,x= X/(2) X~*°VoBoX/m sin @ c2NoX/L ,

(41)

Here y, = yo(z)- y; the angle between the boundary and y-axis is taken into account: sin ~F = Inyl. Then: [1,y~>0

2~.(y,)= X~.oO(y,),fl(y,) = t0, y, < 0.

(35)

°l I

~P~/~P~ ~ , ~ Y~c

'q'

Integrating eqn (34) in the region y, > 0 we have:

Y

voBoRo

SOH:--

q~

jeo y,2 22~.osin O

VoBoyl+ C. c

d~s

¢,(Y,o)=°, ~ (Y,o)=°.

C

(36)

Here one uses the fact that, because of (35), the condition ((d~os/dy,)- ( VoBo/c))--, 0 when y, ~ 0 is satisfied. Constant C is determined from the matching with the solution (30) far from the boundary: for some yl = Y,o a potential ~,~ and its derivative d¢s/dy must vanish: (37)

(38)

~ d ' i 0 rrl2 ~ 2 ~

2:o"

L SIR

JIL Fig. 3. Potential and current distribution around a plate.

Sheath formation around space bodies moving towards the body (for ~¢~> 0) as well as towards the plasma (for , , < 0).

95

Here yt : yo0¢)- y and conductivity X~ = X~(y) = X~,ll(y,).

(48)

5. THE SPHERE WITH FREE ELECTRON EMISSION

Let us consider now a plasma flow around a sphere with free electron emission under the same conditions (26). In this case eqn (6) takes the form[4]:

The solution of eqn (47) in region Yt > 0 is found in the same way as in Section 4. It has the form: ~Os= R2 j~o(y - y,o)2/2~;

1

0 f sin 0 -

si--n0 ~ ~ . ~ m o - ~ - ~

a~o.

+ X . sin¢)VoBo/c +~-~

En

c~Os sin 0 (E~. cos

R sin---~c~¢ sin I

here:

R s i n l a0 ÷ R s i n 0 a~

- (X~, sin ~o- 2 n cos ~¢)VoBolc } = j.(~o~)R.

(42)

Here O and ~o are spherical coordinates, O measures a deviation from the magnetic field direction Bo (z-axis), a n d , from the electric induction field [VoBo] (y-axis); I is the angle between a plane tangent to the body surface and the magnetic field direction. In our case:

Ylo =

V°B°X~'Ic°s ~°1

~o,, : (T~VoBo/ecR){(l/sin 0) ~0(X~, sin ~ + Xn cos ~p)

(44)

According to the condition (26) potential ~o,t is very small, , , . < Te[e, everywhere on the sphere. There is no induction space charge sheath in this case. But the solution obtained is not valid near the equator ( 0 ~ ,r]2, cos O~0) where the expression for ,s~ diverges. Here a special boundary layer arises. Considering this layer we take into account that y = sin 0 cos ~,

Ar = - RAO : RX/(2y,o/lCOS ~J),

(51)

and maximal polarization potential: X~ cos ~oVoBo 4c2jeo

(52)

Thus, in the flow around a sphere in near equatorial zone a region of space charge arises with a potential ~, of the order of Y~o~oB/R, where ~s = VoBoR[c is the maximal induction potential. Hydrodynamical perturbations arising due to body motion in plasma were not taken into account, but they can essentially influence a polarization potential ~,. First of all Padersen current flows from the sphere surface into plasma only by Alfv~n wave radiation[5, 7]. Radiation characteristics lie in th, xz-plane, their angle with x-axis is equal to a = arctg(VA/Vo). They are shown in Fig. 4. At the sphere front surface where an angle between tangent and x-axis exceeds a the radiation conditions, i.e. the conditions for current to flow away, are changed and, hence, a polarization field is also changed. Similar perturbations arise near the rear part of the sphere surface where plasma concentration can be essentially decreased because of particle interaction with the sphere[2, 3]. As a result the polarization potential and boundary layer thickness are changed because they

(45)

and for the points at the equator Y = Yocos q~.

(50)

(43)

As before, the current ./, resulting from electron absorption and emission currents is determined by formulas (14), (16). One looks for a solution of eqn (42) under the condition (26) as a series with respect to ,~ (28). As before in (30), in zero approximation ~¢, = 0. In the next approximation we have

tg 0 + - ~ (X~, cos ¢ - EH sin ~)}

2jeocR

The thickness of the boundary layer region on the sphere is:

~os(O)= ~max sin I = cos 0.

(49)

(46)

Here, as usual, y-axis is directed along the induction field. Considering a thin layer in the vicinity of yo and neglecting, as before, Hall conductivity we obtain (starting from (42)) the following equation for the electric field in a boundary layer

(47)

a=O ~ctg

vo

-

Fig. 4. Flow pattern around a sphere.

96

A. V. GUREVICHand N. T. PASHCtlENKO

depend on No (51), (52), more exactly, on Nom~,, where Nomi, is the minimal plasma concentration on a magnetic force line starting from a given point on the surface. Therefore one can expect that the polarization potential would grow at the rear part of the sphere surface. The case considered simulates magnetized plasma flow around Jovian satellite Io. The satellite moves in Jovian magnetosphere in a plasma torus. Plasma concentration in a torus center reaches No = 1 to 4. 103cm -3, and decreased towards torus edges, where N o - 10 cm-319, 10]. Electron temperature T, is ~8eV, plasma velocity Vo=56km/s, magnetic field Bo = 0.02 H, Io radius R = 1800 kin. The satellite has its own ionosphere which is a result of Io volcano activity. Its integral conductivity Y,- 10mho. Currents flowing through satellite and leading to induction interaction and Alfv6n wave radiation were observed experimentally at "Voyager I"[11]. Let us estimate the polarization potential at Io. According to (52) we have for the plasma torus center: g0max~ (0.1 + 0.5) kV,

of the phenomena occurring in polar regions of a magnetosphere. The ionosphere plays the role of a conductive plate surface. Compare this theory with phenomena observed. Electric field intensity on the Earth in the region of polar caps is approximately constant and directed along the induction field (3). A boundary layer zone, so called "polar oval", arises near the polar cap boundary[12]. Here the main longitudinal currents flow which close Padersen currents in polar caps (see Fig. 2). Distribution of the main part of longitudinal current (simple and double currentt) over the oval agrees rather well with the Fig. 3b[13, 14]. In polar oval a polarization potential es arises. Its experimentally observed value (¢ - 1 to 10 kV) is in agreement with (39), Potential distribution q~sin a boundary layer (36), (38) (see Fig. 3a) corresponds to well-known "inverted V" distribution which was experimentally observed[15]. Electrons and ions accelerated by the electrostatic field of the space charge sheath are the main sources of polar aurora and kilometric radiation of the Earth [16, 17]. More detailed considerations of all these effects is out of the frame of this paper.

and for the torus edges: REFERENCES ~0max ~ 5 0 k V .

Take into account also that polarization potential at the rear part of satellite surface can be increased due to Nom~n decreasing. It follows from this that satellite Io can accelerate electrons up to energies of order of a few keV due to electrostatic polarization potential in space charge sheath; this effect is increased during Io approaching the torus edges. Accelerated electron flux is of the order of j - 10" cm 2 s-t, their total energy flux is of the order of l0 t° to 1012kW. It seems to be possible that electrons accelerated in the space charge region would essentially contribute to generation of decametric radiation splashes which, as it is known, correlate with Io motion and are induced by electrons accelerated up to energies of orders of - I0 kV. 6. BODIESWITHDIPOLEMAGNETICFIELDS Now let us discuss the effects of plasma flow around the bodies having their own dipole magnetic fields. This field shields the body from incident plasma and forms an isolated closed region--the magnetosphere. But far from the body this field is weak, therefore an external field of incident plasma encloses it and goes through regions in the dipole axis vicinity at south and north poles. These regions of magnetic force lines going to external plasma are called "polar caps". It seems that polar caps would represent two sides of a plate through which external magnetic force lines pass. Therefore the considered theory of conductive plate polarization in magnetized plasma flow around the plate can be a rough model ~Both Earth polar caps are electricallyconnected throughclosed magnetosphere and ionosphere. Symmetrization, i.e. flow along closed lines, of the in-flowingcurrents leads to double current arising[4, 14].

1. I. Langmuir.The interaction of electron and positive ion space charges in cathode sheaths. Phys. Rev. 33, 954--989(1929). 2. Ya. L. Alpert, A. V. Gurevich and L. P. Pitaevsky. Space physics with artificialsatellites. N.Y., Consult. Bureau 0965). 3. A.V.Gurevich, L.P.Pitaevskyand V.V.Smirnova. SpaceSci. Rev. 9, 805-871 (1969). 4. A. V. Gurevich,A. U Krylov and E. E. Tseditina.Electric field in the Earth's magnetosphereand ionosphere. Space Sci. Rev. 19, 59-160 (1976). 5. A. V. Gurevich and N. T. Paschenko. Inductive interaction of movingbodies with ionosphericplasma. Acta Astronautica 8, 663-674 (1981). 6. E. N. Fedorov, A. V. Gurevich and A. L. Krylov. Inductive interaction between conductive bodies and magnetized plasma. ZRETF 75, 2132-2140(1978). 7. A. V. Gurevich, N. T. Paschenko. Inductive interaction of electroconductivebodies with ionospheric and interplanetary plasma. Acta Astronautica 8, 47-56 (1981). 8. M. Kaminski. Atomic and ionic impact phenomena on metal surface. Springer Verlag, Berlin (1965). 9. F. Bagenal, I. W. Belcher, H. S. Bridge, G. K. Goertz, A. J. Lazarus, R. L. McNutt, I. D. Scudder,G. L. Siscoe,E. C. Sittler I. D. Sullivan, V. M. Vasyliuhas and C. M. Yeates. Plasma observation near Jupiter: Initial results from Voyager 1. Science 204, 987-991 (1979). 10. I.W. Belcher, H. S. Bridge, R. J. Buttler, A. J. Lazarus, A. M. Mavretic, G. L. Siscoe, I. D. Sullivan and V. M. Vasyliunas. The plasma experiment on the Voyager Mission. Space Sci. Rev. 21,259-287 (1977). II. M. M. Acuna, K. W. Benannon, L. F. Burlaga, R. P. Lepping, N. T. Ness and F. M. Neubauer. Magnetic field studies at Jupiter by Voyager 1: Preliminaryresults. Science 284,982-987 (1979). 12. S.I. Akasofu. Polar and Magnetospheric Substroms. D. Reidel, Podrecht (1968). 13. T. Iijima and T. A. Potembra. The amplitudedistribution of a field-aligned currents at northern high latitude observed by Triad. J. Geophys. Res. 81, 5971-5977(1976). 14. A. V. Gurevich, A. L. Krylov and A. S. Udler. Electric fields and currents in highlatitude ionosphere.Prepr. IZMIRAN, No. 20 (194) (1977). 15, L. A. Frank. Plasma entry into the Earth's magnetosphere.In: Crit. Porbl. Magnetosph. Phys. Proc. Syrup. COSPAR, IAGA and URST, Madrid, 5365 (1972).

Sheath formation around space bodies 16. B. Hultqvist. On the production of a magnetic-field-aligned electric field by the interaction between the hot magnetuspheric plasma and the cold ionosphere. Planet.SpaceSci. 19, 749-759 (1971). 17. C. K. Goertz. Double layers in plasma. Rev. Geophys. and Space Phys. 17, 418-436 (1979).

x, y, z et PH~ ,p ~o

coordinates energy small parameter ion Larmor radius potential surface conductivity plasma frequency

APPENDIX

Nomenclature B c E gs I J j, M

MA n R T

VA

magnetic field light velocity electric field induction electric field surface current total current from plasma normal current density ion mass Aifv~n-Mach number normal to the body typical body size temperature AIfv~n velocity

Subscripts 0 s ,o B p H min max i e

unperturbed values surface values polarization induction plasma Padersen component Hall component minimal values maximal values ion electron

Upper subscripts a, e mean absorption and emission values.

97