d3

PHYSICS OF PLASMAS

VOLUME 5, NUMBER 10

OCTOBER 1998

Effects of dust temperature and fast ions on gravitational instability in a self-gravitating magnetized dusty plasma A. A. Mamun

Department of Physics, Jahangirnagar University, Savar, Dhaka, Bangladesh Received 8 May 1998; accepted 6 July 1998

A theoretical investigation has been made of effects of dust temperature, fast nonthermal ions, and external magnetic field on gravitational instability in a self-gravitating magnetized hot dusty plasma,

which consists of extremely massive, negatively charged hot dust fluid, nonthermally distributed ions, and Boltzmann distributed electrons, where the gravitational force is comparable to or greater than the electrostatic force. The effects of dust temperature, fast ions, and external magnetic field have been found to modify the criterion for this gravitational instability. It has been shown that the growth rate of this gravitational instability decreases with dust temperature, fast ions, and external

magnetic field, but increases with the number of free electrons, with the ratio of ion temperature to electron temperature, and with the ratio of dust mass to dust charge. The implications of this result in situations are briefly discussed. © 1998 American Institute of Physics. some astrophysical S1070-664X 98 02410-0

studied the effects of dust temperature, fast ions, and external

magnetic field on the gravitational instability . /known as 0 Jeans instability1 in a hot magnetized self-gravitating dusty

I. INTRODUCTION

There has been a rapidly growing interest in understanding different types of electrostatic modes in dusty plasmas,1–4 which are very common in laboratory, space, and astrophysical plasma environments, such as asteroid zones, planetary rings, cometary tails, interstellar medium, earth’s environment, etc. It has been found both theoretically and experimentally that the presence of static charged dust grains modifies the existing plasma wave spectra, 5–7 whereas the dust charge dynamics introduces new eigenmodes.8–15 The low frequency dust-acoustic mode,8,15 where the dust particle mass provides the inertia and the pressures of iner tialess ions and electrons provide the restoring force, is one of them. Motivated by these theoretical and experimental studies,8,15 we have reported some features of dust-acoustic

waves in a two-component dusty plasma consisting of a negatively charged cold dust fluid and Maxwellian16 and non-Maxwellian17 distributed ions. These theoretical investi gations, where the gravitational force is neglected, are only valid in that plasma regime in which the electrostatic force is much more greater than the gravitational force. Accordingly, Panday et al.18,19 have studied the dynamics of a self gravitating unmagnetized dusty plasma in the regime with ! " # ! $ %2 & ' G ( m d / Z d e ) O (1), where G is the universal gravitational ( constant, m ! d is the dust grain mass, and Z ! d e is the magnitude % of the dust grain charge. Recently, Verheest et al.20 have investigated gravitational instability in an unmagnetized self gravitating dusty plasma. As the effects of fast particles, dust temperature, and external magnetic field, which have not ) been considered in these earlier investigations,18–20 drasti(cally modify the electrostatic modes, in the present work we have considered a self-gravitating, hot, magnetized dusty plasma consisting of a hot dust fluid, nonthermal ions * ions

with fast particles+ ,, and Boltzmann distributed electrons,

where the gravitational force is comparable to or greater than the electrostatic or electromagnetic force,3,21 –23 and have 1070-664X/98/5(10)/3542/5/$15.00

plasma. The work is organized as follows. The governing equa tions have been given in Sec. II. The dispersion relations for the electrostatic mode propagating parallel and perpendicular to the magnetic field have been derived in Sec. III. Finally, a ) brief discussion has been presented in Sec. IV. 2

II. GOVERNING EQUATIONS 3

We consider a three-component dusty plasma, which consists of extremely massive, negatively charged hot dust 4 fluid, nonthermally distributed ions, and Boltzmann distrib5uted electrons, in the presence of an external static magnetic : field 6 B7 0 8 z,, where z is a unit vector along the 9z direction . Thus, at equilibrium we have (

; n 7 < #Z ! ;n ! 7 = ;n > 7 ,, i0 d d0 e0

? 1@

where ;n 7 ,, ;n ! 7 ,, and ;n > 7 are the unperturbed ion, dust, and i0 d0 e0 electron number densities, respectively, and Z ! is the number d

of electrons residing on the dust grain. The dynamics of

these negatively charged dust grains is governed by8,17,18 A B ! Nd C D E

t

B G F N ! d U! d HJI 0,

M G ! Ud G N D OQP U! d

t

R GU! S

K

TVUXW Y Z GU! [ z\^] cd d

d

`

a

5 c !d b B 1/3 3 N!

B ! N d ,,

2L

_ b e d3

d

3542

© 1998 American Institute of Physics

Phys. Plasmas, Vol. 5, No. 10, October 1998

f % 2gVh B ! ikj 7 l m npokq i Nd 0e y z

4

x

b |w}

1r

b uwv

1t 3

y 4

'ˆÂ ! Ou

d

] %2

D ^

t

2 _

'ˆ; ! On

where M ˆ mFn ing y 10z we

Ï ; ! nd Ð D Ñ

t

Ö Â ! ud × D Ø

t

û

2 üpýÿþ

! >

?

d

A

11

B

12C

354768:9

;n ! d D ,,

t

P

JLK MON

d

b

`a

a

ˆ M

c5d7efgh

% " o D % p % qFr 2 2 2

b

5 3

! ;n ! d d i

Q

j

k

0,

) Y 2cdsut2 . Operating v 13w by

/ t obtain

a

L

b

t

5 3

!

d

%

x

13l

% 2 and us-

% ˆ ;n ! M ˆ 2 M 0. d

2

14

Now, performing Fourier transformation of 9 and 14 we arrive at the dispersion relation for the low frequency elec mode characterized by the angular frequency trostatic

where is normalized to ª pd and the propagation vector k

1 where k is normalized to Dd! : %

2

%

¢¡

±

2

£F¤

%

¥ % ¦ Y 2cd k 2

1 ²³ b µ 1´ 3

§L¨

¶

b

a

5 3

1 ·F¸ i ¹

% ! k 2 d ©

7

%

2 2

k

2 »½¼¬¾ º

0

%

ª¬«

k

%

%¯ ° Y 2cd k 2

F®

2 2 ¿FÀ

k

Á

Y 2cd k 2 Â k 2 . Ã

15Ä

This represents a dispersion relation for an obliquely propagating electrostatic mode ÅÇÆ ,, kÈ in a self-gravitating magne tized hot dusty plasma with fast ions and free electrons. We now consider this low frequency mode in two special cases of interest, namely, parallel propagation (k É"Ê 0 ) and per ËÌ pendicular propagation (k 0).

ÃÄ Å

6

Â ! Ò ud 0,

iñ

The dispersion relation for the low frequency electrostatic mode, which propagates along the magnetic field Ó"Ô 0 ), can be expressed from Õ 15Ö as B7 0 ( k

7

Â å æ^ç ÙÛÚÝÜÊÞàß^áXâ Y cd ã u! d ä z ð

Ñ Ò

A. Parallel propagation Í k ÎÐÏ 0

ÓÔ Õ

7 òkó 0

1

1 ôÊõ b ÷ 1ö 3

b

a

ê ë é8

5 è ! ;n ! , , 3 d d ø ,,

ß

1 àFá i â

7

0

ã

%ä

2

k

b å

10 ê

k

1 ÛÜ b Þ 1Ý 3

ù9

" D$

a

Ú2 ×ÙØ

ú

Now, adding Y cd z (8) and time derivative ( / t ) of 8 ,,

we can express parallel and perpendicular components of Âu! d as

@

5 b 3 =

021

.

;n ! . d

e with T > e being the electron tem perature in energy units ; ! T ! "/( #Z ! T $) with T ! )being the d d d i " d k $ dust fluid temperature in energy units ; 7 0 k /(1 ) and $ " 1/(1 ¡ ) with ¢¤£ ;n > e7 0 / ;n i07 ; ¥ is a parameter determin1 ing the number of nonthermal ions24,25 present in our gravitating dusty plasma model; ¦¨§ G ( m ! "/ #Z ! e $) % 2 with G )bed d ing the universal gravitational constant. The time and space variables are in the units of the dust plasma period © ª « 1 pd ¬ ( m ! "/4 ;n ! 7 Z ! 2 e % 2 $) 1/2 and the Debye length ® ! ¯ ( T "/ i d d0 d Dd $ " $ ´ 4 ° Z ! d ;n ! d7 0 e 2 ) 1/2,, respectively; ± Y cd ² ( Z ! d eB 7 0 / m ! d ³c )/ ª pd is the dust cyclotron frequency normalized to µ ª pd in which ³c is the speed of light in vacuum. It should be noted here that if we put ¶p· 0, i.e., % neglect ¸ ,, which is clearly explained by ¹ Cairns et al.,, 24 the last term in º 4» will represent ion distri) bution of the Boltzmann type.

ÆVÇ ÈÊÉ

5 3

$ "!

;

B ! ¾ N d 1 ¿ ; n ! d ,, G ! À Á Â ! U 0 u ,,

b

a 5

where B N ! is the dust particle number density normalized to d ;n ! 7 ; GU! is the dust fluid velocity normalized to the dustd0 d ) " $ acoustic speed C ! d ( Z ! d T i / m ! d ) 1/2 with T i being the ion tem ) perature in energy units and m ! being the mass of negad tively charged dust particulates; is the electrostatic wave "

)

2

t

B ! N d ,,

2 Q

a

Â ! ud

D

e ~ ,,

2

1{ 3

1

4s

A. A. Mamun

5 3 æ

ðLñ

! k ç2 èêé . d

ë

16ì

0, and ò ! d ó 0, It should be pointed out that when íïî 0, this equation reduces to the dust-acoustic wave dispersion P relation which can be expressed ô in dimensional form, i.e.,

with õÙöø÷ "/ ù ª and k úüû k ýÇþ ! ÿ as pd Dd

3544


Z FIG. 1. [ \ is plotted against ] ikj 1.0, l m n 0.1 o curve 1p , q m r i

i

2De>

"

;n 7 i0 Z!d ; 7 T>e n>e 0 ;n 7 m!d i0 1 ; 7 n>e 0

a b c mv d e f u w

gh

for 0.25, d 0.01, cd 0.05, k 0.01, 0.5 curve 2 , and i 1 curve 3 .

k

where

^`_ s

A. A. Mamun

$

t

1

1

T>e Ti

k 2 1

x

2 De>

cd

,,

T > e /(4 ;n > e7 0 e 2 ) . This is exactly the same form of the dispersion relation for the dust-acoustic mode derived by Rao et al.8 It is obvious from 16 that for the parallel propa gation the mode depends on dust temperature, fast ions, and free electrons but not on the external magnetic field B7 0 . Therefore, depending on the relative values of and

7 2 1 i 0 k

!

1 b 1 3

b

a

5 3

% " ! d k #2

m

b

0.1, i 0.1, 0.1 curve 3 .

d

0.01,

e

cd

Q

R

S

U V

T

W

X

Y

k % 2

cd

the effects of dust temperature and fast particles nonthermal ions play the stabilizing role. It is clear that as the value of increases, the growth rate increases very rapidly. Figures 1–3 show how the growth rate of the unstable mode changes

with ,, ,, ,, and ! . It is seen that the growth rate of the i d perturbation mode decreases with the number of fast par

1/2

;n 7 T i0 i ;n > 7 T > e0 e

Z FIG. 2. y \ is plotted against z for {}| 1.0, ~ 0 curve 1 , e 0.05 curve 2 , and e

,,

this mode will be stable or unstable. Thus, the condition for the mode to be unstable can be written as

k '2

$&%

a

5 176 3 ! d k 4 2 %1 2 () + , 1 - . i/ 70 0 k 2 * and the growth rate 798 normalized to : ª ; of this unstable pd b

1 b 1 3

5 3

mode is given by

=

?&@

k % A2

BC K G H iI 70 J k 2 D E F 1

1 b 1 3

L b

a

5 3

% M ! d k N2

.

O 18P

This means that the effects of gravitational force, ion tem-

perature, and free electrons make the mode unstable, whereas

Z FIG. 3. \ is plotted against b for } 1.0, 0.1, m 0.1, & 0 curve 1 , e ¡ 0.05 ¢ curve 2£ , and ¤ e ¥ 0.1 ¦ curve 3§ . d

cd

i

cd

0.25,

e

cd


¨ª©¬«

A. A. Mamun

Pthe effects of fast ions, dust temperature, and external mag-

$ ticles and with the dust temperature ( ! d ) , but increases

with the ion temperature ( $) and with the number of free i

®

¯±°³² .

electrons

netic field play stabilizing role, i.e., try to make the mode

´ µ·¶ ¸0¹

Ustable and counter the gravitational condensation of the dust Tgrains. It is seen that the growth rate V >of the unstable mode, Pthe mode satisfying the condition W 20X 0, decreases with the

B. Perpendicular propagation k

The dispersion relation for the low frequency electrostatic mode, which propagates perpendicular to the magnetic field B7 0 ( k 0 ), can be expressed from 15 as

º

»½¼

a

Â Y2cd Ã

ÀÁ

5 b 3

Ä ! d k Å 2 Æ

¾ ¿ k Ç 2

magnitude of the external magnetic field, with the number of

Mnonthermal ions, and with dust temperature, but increases Ywith the ratio of grain mass to grain charge (m C 5 D/ EZ 5 Fe 2) , with d d Pthe ratio of ion temperature to electron temperature, and with

Ó

Pthe number of free electrons. Z It may be pointed out here that in planetary rings, Ywhere [ \2 ] ^_5 ` 5 k 2 a B( b 2 5 D/ cc 2 d ^_5 e 5 k 2f 2 5 2) g 1 h Ywith i j 5 3 d Ad Ad Dd k B l D/ m 4 n cdon 5 l 3Cm 5 dp 0, the dust temperature or the external mag0 d0 d

Ò . 1 ÈÉ Ñ 2 7 Ð b Ë Ì 1 ÍÎ i Ï 0 k 1Ê 3 Ô 19Õ %

Mnetic field has negligible effect on gravitational condensation >of the dust grains. On the other hand, in interstellar medium, Ywhere q \2 r ^_5 s 5 k G 2 t B( u 2 5 D/ cc G 2 v ^_5 w 5 k G 2x 2 5 2) y 1, the effects of 3 d 3 d Ad Dd cd

This mode, which is for the perpendicular propagation, de$ pends on the external magnetic field (B7 0 ) , dust temperature ! $ $ ,, ion temperature ( i ) , and free electrons ( d ) , fast ions . It is clear that this mode will be stable or unstable de pending on the relative values of and

Ö Û±ÜÝ

×ªØ¬Ù

ß Y2cd à %

a

5 b 3

Ú

% ã á ! d k â 2

Þ

k % ä 2

åæ è é 7 í î2 1 êë i ì 0 k ç

1 b 1 3

ï&ð ñ Y2 ò cd

a

5 b 3

ó ôõ

and the growth rate

! k2 d

.

k % ö 2

÷ø 7 ÿ 2 ù ú û 1 üý i þ 0 k

1 b 1 3

zexternal magnetic field and dust-temperature are of great im;portance. Z

It should be mentioned that if we neglect the effects of fast | ions and 5dust temperature, and external magnetic field i.e., }~ 0, d 0, and \ cd 0 0, our present G results com;pletely agree with the recent published results.20 Z O It may be stressed here that the results of the present investigation should be useful not only for the consequence for the gravitational condensation of the dust grains in planzetary system and interstellar medium, but also in understandO ing the features of electrostatic disturbances in space plasmas Ywhere extremely massive, negatively charged hot dust fluid, nonthermally distributed ions, and thermally distributed elecPtrons are the plasma species.

{

Therefore, the condition for the mode to be unstable can be

written as

3545

%

20

of this unstable mode is given by k 2 a Y 2 b 5 ! k 2 cd 3 d 1 % . b 1 i 7 0 k 2 1 3

21

This implies that the effects of gravitational force, ion tem$ perature ( i ) , and free electrons "!$# also make this mode 5unstable, whereas the effects of the external magnetic field % $ $ ( Y ) , dust temperature ( & ! d ) , and the fast particles ')(+* play cd the stabilizing role. Figures 1–3 show the variation of the growth rate of this unstable mode with the external magnetic , 2 3field Y cd ,, number of5 2 fast particles -).+/ 0, ion temperature ( 1 i ), dust temperature ( 4 d ) , and the number of free electrons 687:9 . It is seen that as the number of fast particles, the dust tem;perature or the magnitude of the external magnetic field inof the perturbation mode deor I d d i increases, the growth rate increases.

ACKNOWLEDGMENTS

The author is grateful to Professor P. K. Shukla, Profes-

Usor R. A. Cairns, Dr. R. Bingham, Professor M. Salimullah, and Professor L. Stenflo for their stimulating influence and helpful suggestions.

J

º

IV. DISCUSSION

The effects of gravitational force, external magnetic O field, fast L MnonthermalN ions, dust temperature, ion temperaPture, and free electrons on the low frequency electrostatic modes in a self-gravitating magnetized dusty plasma with Sfast Q nonthermalR ions and Boltzmann distributed electrons have been investigated. It is observed that the effects of Tgravitational force, number of free electrons, and ion tem;perature make these electrostatic modes unstable, whereas

K

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3546


A. A. Mamun

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