Theory of the plasma sheet in the jovian magnetosphere

Theory of the plasma sheet in the jovian magnetosphere

Planer Space Sci., Vol. 25. pp.673 THEORY to 679. Pergamon Press, 1977. Printed in Northern Ireland OF THE PLASMA SHEET MAGNETOSPHERE IN THE J...

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Planer Space Sci., Vol. 25.

pp.673

THEORY

to 679. Pergamon

Press, 1977. Printed

in Northern

Ireland

OF THE PLASMA SHEET MAGNETOSPHERE

IN THE JOVIAN

H. GOLDSTEIN Ruhr-Universitat

Bochum Institut fur Theoretische

Physik, D 4630 Bochum, Federal Republic of Germany

(Received in jinal form 1.5 October 1976)

Abstract-The magnetic field in the middle magnetosphere of Jupiter was suggested to be the planetary dipole field plus a perturbation field due to a current sheet (Smith et al., 1974). Since no data of the low energy plasma are available the existence of a plasma sheet could not be confirmed directly. In this paper we show how the plasma pressure and density-can be derived from the magnetic field in the framework of a self-consistent theory. For the magnetic field mode1 proposed by Goertz et al. (1976~) we compute the isobars and isodensity lines and confirm the existence of a thin plasma sheet.

1. INTRODUCTION

Our present knowledge of the structure of the Jovian Magnetosphere is essentially based on the magnetometer measurements made by Smith ef al. on Pioneer 10 and 11 (Smith et al., 1974, 1975a, b). The authors suggested that the magnetic field in the middle magnetosphere (-20-50 R,) can be represented by the planetary dipole field and a perturbation field due to a current sheet. The symmetry is dominated by the rotation axis of the planet rather than the dipole axis. One therefore concludes that the jxB force is essentially compensated by the centrifugal force acting on the thermal plasma concentrated in the equatorial plane. However, no direct measurements of the thermal plasma density are available. So we are interested to see whether theoretical considerations can confirm the existence of the plasma sheet. At present two different approaches are available for a rotating magnetosphere. The first was proposed by Eviatar et al. (1964) for a nonrotating exosphere and was developed by Lemaire and Scherer (1974) (see also Goertz, 1976a). By Liouville’s theorem they derived the particle density choosing a Maxwellian velocity distribution at an ionospheric reference level. Reasonable assumptions are made about the density of trapped particles. The authors take into account the conservation of adiabatic invariants. If this restriction is dropped one obtains the well-known barometric model (e.g. Gledhill, 1967; Michel and Sturrock, 1974). The theory claims that beyond a critical radius (Alfvtn radius) the corotational velocity exceeds the local AlfvCn velocity. This implies super673

Alfvenic radial outflow of plasma with the possibility of internal shocks (Kennel1 and Coroniti, 1975). The value of the Alfven radius can be predicted if models of the plasma density and the magnetic field are used. However, near the critical radius, density and field mutually depend on each other and a self-consistent treatment is therefore required. Self-consistency is the goal of the second approach. Here an Alfven radius does not occur since a self-consistent theory implies the exact balance of magnetic, centrifugal and pressure forces at euery distance. In constrast to Lemaire’s theory the pressure is assumed to be isotropic. This simplification will be discussed later. Gleeson and Axford (1974,1976) (see also Goertz, 1976b) developed a theory for a rotating magnetosphere, which contains a very thin plasma sheet. For a reasonable current density they derived the magnetic field on the basis of a rough fit of the observed Jovian field and deduced the particle density for an assumed sheet thickness. In a more rigorous way we start from the magnetic field data presented by a model of Goertz et al. (1976c). As Gleeson and Axford we assume the magnetosphere as a static isotropic axisymmetric configuration. Then the current density, the plasma density and pressure can be derived in the framework of Vlasov’s theory. The results confirm the existence of a thin plasma sheet. 2.

SELF-CONSISTFAVT

THEORY

Since the plasma in the Jovian magnetosphere can be assumed to be collision-free we use Vlasov’s theory, i.e. Liouville equations for the one-particle

674

H. Go~~sram

distribution equations.

functions

af z+v

f coupled with Maxwell’s

diagonal and isotropic in the v, v, space, i.e. the equilibria are static and isotropic in the r, z plane.

af .$+;(E+vxB)-_=O av

j*(r,JI) =

c e/[email protected]

d”v

(6)

p*(r,$)=~m~~Fd’o. VxB=p,ze

J

vfd3e

V-E =$ej-fd”v ‘V.B

=0,

e and m denote particle charge and mass. c sums over the particle species. Here we suppose the magnetic field to be axisymmetric and meridional (B, = 0); (r, cp,z) are cylindrical coordinates. With these assumptions the Vlasov equations can be reduced by a method presented by Schmidt-Burgk (1965) and Schindler et al. (1973). An equilibrium solution of the Liouville equations is given by

fb, 4 = F(E,P,),

(1)

where the energy E =F(0:+oq2+uz2)+e+(r, and the generalized

momentum

pq = mu, + Nr,

z),

(2)

z),

are constants

of motion. The electric potential and the stream function JI are defined by

4

E=-V4

(3)

q%/ris the Q-Component of the vector potential. Note that a meridional magnetic field does not require r and z-components of the vector potential. JI is constant on a field line because of B * V$ = 0. Moments of the distribution functions, as the charge density u for instance, depend on r and z via 4, $ and r. We use the condition of vanishing charge density 0 = a(@, 4, r) = c e/F d3t),

By asterisks we distinguish functions of r and # from functions of r and z. From the definition of j and p in terms of the distribution functions we find that there is a simple relationship between p*, j* and the centrifugal force density denoted by fc* (cf. Appendix).

ap*(r, = i*(r, +)

a* ap*h JI)

~ ar

(5)

to eliminate I$ in terms of r and $. For the present choice of distribution functions the current density j has only a Q-COmpOnent and the pressure tensor is

(8)

r

= p*(r,$1T-

fC*(r, JI).

(9)

The bulk velocity V has only a Q-COmpOnent V,. p denotes the mass density. It is easy to prove, that the equations (8) and (9) guarantee momentum conservation (cf. Appendix). The continuity equation and energy transport equation are identically satisfied. The stream function + is determined by Ampere’s law

la* --a -ar ( r ar 1

in the Q-direction

(7)

-:C$=

pJ*(r,

#)_

(10)

Therefore, we have three equations (8~(10) for the four unknown functions Jr(r, z), p*(r, I,+), j*(r, JI) and fC*(r, I++),i.e. one function has to be prescribed. The related one-particle distribution functions can be derived from p* by a method outlined in the Appendix. If sufficient information about the distribution of the plasma pressure were available one might use a procedure similar to the theory of the Earth’s magnetosphere (Birn et al., 1975). Prescribing p*(r, $)> 0 with ap*/ar > 0 one would compute j* and f* from equations (8) and (9) and Jl(r, z) by integrating equation (10). In the case of Jupiter, little is known about the plasma population a priori. Therefore we proceed in a different way, starting from the observed magnetic field. The large rotation speed is actually a simplifying factor in this procedure. From Ampere’s law we find the current density j(r, z) as a function of r and z. Transforming into (r, $) coordinates we obtain j(r, z(r, JI)) = j*(r, +) and by a J, integration p*(r, 4). Differentiating p* with respect to r we find fc*(r,(I).However, p* is

675

Theory of the plasma sheet in the Jovian magnetosphere

not uniquely determined because we can add a function g(r), which is arbitrary except for the condition that p* and fc* must be positive. The condition fc*> 0 implies by equation (9) that the pressure increases radially on field lines. On the other hand it seems reasonable to assume that p does not increase radially in the equatorial plane. We can prescribe the pressure in the equatorial plane to determine g(r). For practical calculations we present p, j, and fc as functions of r and z. $-integrations are transformed into z-integrations. (11)

ik 2) = p(r, -7)= poW+

fcfr

,

%)

=

’ ih 2’) Wfr, z’> -----dz’ r a2

I

dr

r

ar

B, and B, are given by

~~r,z~=~~~-~[logch~z,~~+C],

(14)

1w B,(r, z) = --r a2

-&faogch(z/D)+C]

B,(r,

(15)

z)=fz

(12)

0

dpdr) ik 0) W(r, 2) -__~

$ and the field components the following equations.

[logch(z/D)+C]

(16) (17)

IE=O 1fi, be, a, C, D) = (4x rosy, 9 x 103y, 0.7,10,1).

-2

(7)

y]

dz’.

(13)

p&r) denotes the pressure in the equatoriaf plane. It is evident that we find self-consistent plasma quantities for an arbitrary magnetic field expressed by a stream function $. This method is not possible in the theory of the Earth’s magnetosphere, because it relies on the presence of a strong centrifugal force. Since second derivatives of tfr occur, ihe formulas (Il)-(13) have to be applied to models which give a reasonable fit not only of the field components but also of its spatial derivatives. 3. A PAR’MCULAR

MODEL

Analytic models of the Jovian magnetosphere are published by Barish and Smith (1975) and Goertz et al. (1976c). The model of Barish and Smith exhibits general features of the dayside field observed by Pioneers 10 and 11 whereas the model of Goertz et al. provides a fit of the field in the pre-dawn sector as measured during the ffyby of Pioneer 10. We apply our theory to the latter model. The dipole tilt towards the rotation axis and the q-component of the magnetic field are neglected. Taking lengths in units of R,, the stream function

(18)

M, denotes the Jovian dipole moment. So the magnetic field is represented by the planetary dipole field plus a perturbation field whose source is a ring shaped current sheet. The related self-consistent plasma quantities are obtained by the equations (ll)-(13) where we evaluate the integrals numerically. The isobars and isodensity lines for constant p. are shown in Figs. 1 and 2, respectively. We find that the pressure decreases rapidly outside the equatorial plane in the region O< (zJ < 2 and is almost constant for (t]>2. For constant z the pressure increases in positive r-direction. The asymptotic limit is po, the pressure in the equatorial plane. To derive the particle density we assume that the thermal plasma rotates rigidty with the planet and consists of protons and electrons only. Then we have

(19) n, m and o, denote the particle density, proton mass and angular velocity, respectively. For constant r the density has a maximum in the equatorial plane. For constant z it decreases in radial direction. The thermal plasma is therefore concentrated in a disk shaped region in the equatorial plane.

616

H.

GOLDSTEIN

p = p. -10’

pressure

pbar

FIG. 1. I~~BARSOFTHERMALPLASMAFORCONSTANTPRE~~~REINTHEEQUAT~RIALPLANE~~.

The r and z coordinates are given in units of RJ (1 pbar = 1 dynlcm’). The density decrease in radiai direction combined with the pressure increase implies a steep temperature gradient. Since this seems to be unreasonable, we restrict our model to a region in the middle magnetosphere between 20 and SO R+ The temperature gradient can be diminished if we introduce a decreasing pressure in the equatorial plane. The lower limit of the pressure gradient is given by the condition that the centrifugal force density and equivalently the particle density have to be positive. Since the minimal density for constant r in Fig. 2 was higher than 4 of the maximal dpo(r) density in the equatorial plane we choose dr= $j(r, O)B,(r, 0) taking only the two leading terms. We obtain. 9xlV _ __

PO(r)= pm+-

X

2OP 4x105 i 3X7Xr”.‘-3X4XP4

6*3x

lo4

1

[&bar].

(20)

The pressure in the region 0 5 z I 15, 20 5 r I 80 is positive if we set p_ = IO-’ pbar. The temperature gradient is then reduced by a factor of the order of 1. Figures 3 and 4 show the isobars and isodensity lines respectively. 4. CONCLUSIONS

We presented a self-consistent theory for a rotating magnetosphere, which was assumed to be in static isotropic equilibrium. The solution sustains, that for each volume element the forces are balanced exapproaches are selfactly, whereas previous consistent only in an integral sense. For any given set of magnetic field data or, alternatively, for an arbitrary field pressure, density, and temperature of the low energy plasma can easily be derived, if one knows its angular velocity and its pressure in the equatorial plane, Our method cannot be applied to the Earth’s magnetosphere since it relies on the presence of a strong centrifugal force. We therefore conclude, that self-consistency is a less stringent

L

II

a= 15

20

05

1

I 30

- 0.5

0

I 40

1 50

density FIG.~.

DI~~BU~O~

OF THERMAL

PLASMA

-1

I 60

, 70

r

n: lO”~rn-~

DENSITY FOR CONSTANT PLANE.

PRESSURE

IN THE EQUATORIAL

611

Theory of the plasma sheet in the Jovian magnetosphere

a=

-b:

-b

a=

pressure

p=p_+lO’pbar

FIG. 3. ISOBARSOFTHERMALPLASMAFORRAL)IWYDECREASINGPRESSUREINTWEEQUATORIALPLANE. The "+" and ~~_~~signs refer to the isobars below and above the isobar p = p_(a = -m)

respectively. Note that there is no accumulation df isobars at (r, z) = (20,2) as it might seem, since no isobars are drawn for p > pm+ 10m9 pbar. for rotating magnetospheres than for magnetospheres with translational symmetry. In an alternative treatment Lemaire and Scherer (1974) obtained the density distribution for a rotating collisionless model. However assumptions about the density of trapped particles are still necessary. The theory has the advantage of describing the coupling with the ionosphere and allowing an anisotropic pressure tensor. On the other hand the response of the magnetic field to particle forces was neglected. This is of course legitimate for small radii where the magnetic field dominates. The region of strongly disturbed magnetic field lines, however, requires a self-consistent treatment. As was shown the total density including trapped particles can be derived. We assumed an isotropic pressure tensor though collisions are infrequent. Such conditions may prevail if wave particle interactions are condition

a=

1

0.5

0

present (Melrose, 1967; Goertz, 1976a). In fact in the quiet plasma sheet of the Earth’s magnetosphere a nearly isotropic plasma tensor was observed (Hones, 1973). As a first approximation we therefore expect similar conditions in the Jovian magnetosphere. Our theory was applied to the model of Goertz et al. (1976c) proposed for the pre-dawn sector of the Jovian magnetosphere. Assuming the plasma rotating rigidly with the planet and exerting constant or radially decreasing pressure in the equatorial plane we computed the isobars and isodensity lines. Maxima of pressure and density in the equatorial plane confirmed the existence of a thin plasma disc. Here it might be interesting to add that Kupo et al. (1976) reported the presence of sulfur ions concentrated in a semi-disk of about l-2 RJ in thickness around the equatorial plane.

- 0.5

60 density FIG.~.

DISTRIBUTION

OF THERMAL

70

r

n: 10’ mm3

PLASMA DENSITY FOR RADIALLY EQUATORIALPLANE.

DECREASING

PRESSURE

IN THE

678

H. GOLDSTEIN

A necessary consequence of our static model is the fact that the plasma temperature increases with radial distance by an unreasonably high factor. The reason is that the jx B force, which compensates the centrifugal force in the disc, is a rapidly decreasing function of r. Therefore, similar ditliculties necessarily arise in the other models using trapped plasma (e.g. Gledhill, 1967; Piddington, 1969; Michel and Sturrock, 1974; Gleeson and Axford, 1976). Introducing a pressure gradient dp,/dr in the equatorial plane the temperature gradient was diminished only by a factor of the order of 1, since the pressure p0 has to be compatible with the condition that the centrifugal force and the pressure have to be positive everywhere. The temperature variation can be reduced by restricting the model to a smaller region (e.g. 2&50 R,) or by taking the angular velocity of the plasma as a function decreasing with r. Particle measurements, however, indicate that the plasma corotates with Jupiter (Trainor et al., 1974; Van Allen et al., 1975). Moreover, rigid rotation corresponds to local thermodynamic equilibrium. We expect that some of our assumptions (e.g. absence of directed flow) break down beyond some critical radius. A qualitative theory of the outer region is not available. Smith et al. (1975b) discuss the existing qualitative models in the light of experimental results. So far we have not considered instability phenomena. It might be of interest to discuss the roles played by tearing modes (Coppi 1966; Schindler et al. 1973) and interchange motions of flux tubes (Melrose 1967). We finally remark that the conclusions of our theory are essentially based on the assumptions that our results are not significantly changed by the asymmetry caused by the solar wind, the tilt of the dipole axis and the spiralling of the field lines. The magnetic field data published by Smith suggest a need for a modified model field valid for all sectors of the middle magnetosphere. A final test of our results requires measurements of density and pressure of the low energy plasma. Acknowledgements-1 wish to thank Prof. K. Schindler for his interest and many helpful suggestions. This work was supported by the Deutsche Forschungsgemeinschaft through the Sonderforschungsbereich “Plasmaphysik Bochum/Jillich”.

Coppi, B., Lava], G. and Pellat, R. (1966). Phys. Reu. Lett. 16, 1207. Eviatar, A., Lenchek, A. M. and Singer, S. F. (1964). Phvs. Fluids 7, 1775. Gledhill, J. A. (I967). Nature, Land. 214J, 155. Gleeson. L. J. and Axford. W. I. (19’74). EOS 55. 404. Gleeson; L. J. and Axford W. I. (I976). J. geophyi. Res. 81, 3403. Goertz, C. K. (1976a). J. geophys. Res. 81, 2007. Goertz, C. K. (1976b). J. geophys. Res. 81, 3368. Goertz, C. K., Jones, D. E., Randall, B. A., Smith, E. J. and Thomsen, M. F. (1976c), J. geophys. Res. 81, 3393. Hones, E. W., Jr., Asbridge, J. R., Bame, S. J. and Singer, S. (1973). J. geophys. Res. 78, 5463. Kennel, C. F. and Coroniti, F. V. (1975). The Magnerospheres of Earth and Jupiter (Ed. V. Formisano) p. 451. Reidel, Dordrecht. Kupo, I., Mekler, Y. and Eviatar, A. (1976). Asrrophys. _I. Lett. 205, L 51. Lemaire, J. and Scherer, M. (1974). Space Sci. Reu. 15, 591. Melrose, D. B. (1967). Planet. Space Sci. 15, 381. Michel, F. C. and Sturrock, P. A. (1974). Planet. Space Sci. 22, 1501. Piddington, J. H. (1969). Cosmic Electrodynamics, p. 200. Wiley, New York. Schindler, K., Pfirsch, D. and Wobig, H. (1973). Plasma Phys. 15, 1165. Schmeidler, W. (1950). Inregralgieichungen mit Anwendungen in Physik und Technik, Vol. 1, pp. 79, 113. Akademische Verlagesellschaft, Leipzig. Schmidt-Burgk, J. (1965). Max-Planck-Institut fiir Physik und Astrophysik, Report No. MPI-PAF/Pl, 3165. Smith, E. J., Davis, L., Jr., Jones, D. E., Coleman, P. J., Jr., Colburn, D. S., Dyal, P., Sonett, C. P. and Frandsen, A. M. A. (1974). .I. geophys. Res. 79, 3501. Smith, E. J. Davis, L., Jr., Jones, D. R., Coleman. P. J., Colburn, D. S., Dyal, P. and Sonett, C. P. (1975a). Science 188, 451. Smith, E. J., Davis, L., Jr. and Jones, D. E. (1975b). Jupiter’s magnetic field and magnetosphere. To appear in Jupiter, the giant planet, Proc. of the Tuscan Conference. Titmarsh, E. C. (1948). Introduction to the Theory of Fourier Integrals, pp. 314, 315. Clarendon Press, Oxford. Trainor, J. H., McDonald, F. B., Teegarden, B. J., Webber, W. R. and Roelof, E. C. (1974). J. aeoohvs. _ _ - Res. 79, 3600. Van Allen, J. A., Randall, B. A., Baker, D. N., Goertz, C. K., Sentman, D. D., Thomsen, M. F. and Flindt, H. R. (1975). Science 188, 459. APPENDIX

Here we outline some properties of the distribution function F(e,p) and its moments p, j, (T and p defined by

REFERENCES

Barish, F. D. and Smith, R. A. (1975). Geophys. Res. Len. 2, 269. Birn, J., Sommer, R. and Schindler, K. (1975). Astrophys. Space Sci. 35, 389.

p+(r,+,4)=zmJgFd3u j+(r, q, 0) = Ce Ju?Fd’u

(Al)

Theory of the plasma sheet in the Jovian magnetosphere

619

tion (A12) term by term vanishes as one easily verifies

u+cr, $, c#)= &jF d3u

(A3) taking into account $ = 0 and V = (0, V,, 0).

zmkd3u

p’(r, &4)=

(A4)

with w2 = IJ,’+ vzz. The + denotes functions of the three variables r, +, 4. Since F depends on u, and u, via wz we have d’o = ?r dw* dv,. We find

= -&[Fd”v=

-a,

(As)

V.(pV)=O

(All)

V=-V.q+j.E-V.pV).

(A12)

U denotes the internal energy density and q the heat flow vector which has a q-component only. Thus we have confirmed the self-consistency of our solution. From the partial pressures p,(r, JI) a one-particle distribution function can be found. Y denotes the particle species. Following Schmidt-Burgk (1965) we substitute the velocity components by E, p- and the angle a = rg-’ u,/u,. Since the integrand is independent of a, we obtain a factor r by the a-integration.

and similarly aP+ _=a*

i

(‘46)

r

x [+c$)-(y’)‘]+y,

(A13)

(A7) with If the pressure tensor is isotropic (p = p,,) we obtain:

x

for

x20

[xl+ = [ 0 for x
=z

and $

(A9)

vp*& $)=$

I%, P,),

= 5.

(AI4)

where

(A6) and (A7) guarantee momentum conservation. Vp*(r, $)=$

If the electric potential 4 is known as a function of r and I+$(A13) represents an integral-equation for E In the case of exact neutrality we obtain

e,+$VJ v,*%,+$

vlj.

(AIO)

The first term on the R.H.S. is the centrifugal force density, the second is the jxB term. In the continuity equation (Al 1) and the energy equa-

aj=eJI

T= 2mr’.

Equation (A14) can be formally solved for F by integral transformations, since_ the kernel is a function of the difference of p* and JI and of the product of e and P (cf. Schmeidler, 1950; Tit&march, 1948).