EFFECTS OF HALL CURRENTS ON JEANS GRAVITATIONAL INSTABILITY OF ROTATING UNBOUNDED PLASMA

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1 PLASMA PHYSICS EFFECTS OF HALL CURRENTS ON JEANS GRAVITATIONAL INSTABILITY OF ROTATING UNBOUNDED PLASMA AIYUB KHAN, S.S. TAK, PRIYADARSHI PATNI Department of Mathematics and Statistics, Jai Narain Vyas University, Jodhpur, 5, India, and Department of Computer Science, Lachoo Memorial College of Science & Technology, Jodhpur,, Received March 6, 9 The combined influence of the effects of Hall currents, magnetic resistivity and viscosity has been studied on the gravitational instability of rotating homogeneous unbounded plasma in an oblique magnetic field. The solution has been obtained through the normal mode technique and the dispersion relation has been derived. It is shown that Jean s criterion for gravitational instability remains unchanged. Solving numerically the dispersion relation for conditions prevailing in an astrophysical situation, it is found that the Coriolis force, viscosity, Hall currents and finite conductivity have stabiliing influence on the instability of the plasma of disturbance. Key words: Gravitational instability, Hall current, rotation, viscosity, oblique magnetic field, Jean s criterion.. INTRODUCTION Plasma instability is by far the most researched subject in fluid dynamics. Dating back to the early 9s, the first significant criterion with regards to the gravitational instability of plasma was put forward by Jeans []. According to the Jeans criterion, an infinite homogeneous self-gravitating atmosphere is unstable for all wave number k less than the Jeans wave number k G j =, where is S the density, S is the velocity of sound in gas and G is the gravitational constant. The Jean s instability problems have been studied by several researchers under the separate or simultaneous effects of different physical parameters. A comprehensive account of these studies has been given by Chandrasekhar []. Several authors have studied this problem from different physical points of view. Tassoul [] has studied the effects of thermal conductivity, viscosity, rotation and electrical conductivity on the gravitational instability of a homogeneous isotropic plasma. Gliddon [] has studied gravitational instability of aniosotropic plasma by using the Chew-Goldberger-Low (CGL) equations. Bhatia [5] has etended the study to the case of a rotating an isotropic plasma and shown that stability depends on both magnetic field and rotation. The combined effects of uniform rotation, Hall Rom. Journ. Phys., Vol. 56, Nos., P. 9, Bucharest,

2 Aiyub Khan, S.S. Tak, Priyadarshi Patni currents, finite conductivity and FLR on gravitational instability have been studied by Bhatia [6]. Lehnert [7] considered the stability of the plasma interacting with a neutral gas. Kumar and Srivastava [8] have investigated the gravitational instability of a partially ionied plasma carrying a uniform magnetic field with Hall effects. Cade [9] has studied an applicability problem on the Jeans criterion to stationary self-gravitating cloud. Bhatia and Chhonkar [] have studied the combined influence of Coriolis forces and viscosity on plasma stability in the absence of Hall currents and concluded that the viscosity has a stabiliing influence on the system. Bhatia and Khan [] have investigated the Jeans gravitational instability of a thermally conducting plasma carrying an oblique magnetic field with viscosity and Hall effects. In all such investigations, carried out separately under varying assumptions, it was found that the condition of instability has been determined by the Jeans criterion with some modifications introduced by the inclusion of the various parameters. Ali and Bhatia [] have studied the effects of Hall currents and magnetic resistivity on the gravitational instability of partially ionied plasma in presence of an oblique magnetic field. In recent years, Jeans stability problems have been analyed in the framework of nonetensive Tsallis [] statistics and its associated kinetic theory. Lima et al. [] and Du [5] have studied the instability of self gravitating system in the framework of Tsalli s statistics. More recently, Shaista et al. [6] and [7] have investigated the Jeans gravitational instability problem of a thermally conducting plasma in the presence of nonetensive effects. A new Jeans criterion is derived, which depends eplicitly on the nonetensive parameter q. The standard values are obtained in the limiting case q =. In all these studies of Jeans problem whether in the absence or presence of nonetensive effects, the instability in presence of general rotation of a plasma endowed with an oblique magnetic field and Hall currents has not been analyed. It would, therefore, be of interest to eamine the influence of rotation on the Jeans gravitational instability of unbounded cosmic plasma endowed with an oblique magnetic field in the presence of effects of Hall currents and magnetic resistivity. In the present paper, we have etended the work of Ali and Bhatia [] to include the effects of viscosity and Coriolis forces when the rotating plasma is fully ionied.. PERTURBATION EQUATIONS We consider the motion of an infinitely etending homogenous gravitating finitely conducting viscous rotating plasma in the presence of effects of Hall currents. The considered fluid is embedded by a uniform oblique magnetic field H = ( H,, H ) with rotation Ω = ( Ω,, Ω).Under these assumptions, the linearied perturbation equations are:

3 Gravitational instability of rotating unbounded plasma u + Ω u = δp+ h H + δ + µ ²u + µ.u t h = ( u H) ( h) H + η ²h t Ne ( ) ( ) φ ( ) t ( δ) + (.u ) = () () () δφ = Gδ () δ p= S δ (5) In the above equations, the perturbations in density, pressure p, velocity u, magnetic field H and the gravitational potential φ are denoted by the symbols δ, δ p, u( u, v, w), h( h, hy, h) and δφ. In these equations η is the magnetic resistivity, N is the electron number density, e is the charge of an electron, µ is the coefficient of viscosity, G is the gravitational constant and S is the velocity of sound. We analye the disturbance into normal modes by assuming that all the perturbed quantities depend on space co-ordinates, and time t as: ep( ik sinθ + ik cosθ + i n t) (6) k = ksin θ,, kcosθ is the wave number of perturbation making an angle θ where ( ) with the -ais and n (may be comple) is the frequency of the perturbation. On using epression (6) for all the perturbed quantities in equations () (5), we obtain si equations in velocity u and magnetic field h which can be written in the matri form as: [ A][ B ] = (7) where [ A ] is the sith order square matri and [ B] = [ uvwh,,,,, ] T hy h. The elements of the matri [ A ] are: i A = in M + k + A sin θ ν sin θ n, A = Ω, im k = + ν sin θ cos θ n 5, A = ikh cosθ,, ( ) A6 ikh sin θ, A = = ( ) A = Ω, A ( in ν k ) = +, A = Ω,

4 Aiyub Khan, S.S. Tak, Priyadarshi Patni A =, ( ) A5 = ik N, A 6 =, im A = + k ν sinθ cosθ n, A = Ω, im A = in cos θ + νk + cos θ, n A = ( ikh ) cos θ, A 5 =, A6 = ( ikh ) sinθ, ( ) A = ikh cos θ, ( η ) A in k = +, 5, A =, = ( ) A k N 5 = cos θ, Ne A = A ( ikn ) 5, A ikh cos θ, A 6 =, = A 5 =, k k = ( N θ ), A55 = ( in+ ηk ), A = ( N θ ) A5 Ne cos A6 = ( ikh ) sin θ, 6, A = A ( N θ ) 56 Ne sin, A6 = ikh sin θ, A = ( ) 6, 65 Ne sin, where M = ( S k G ), N = ( H sinθ + H cos θ ) and kinematic viscosity. k = A66 = ( in+ ηk ), µ ν = is the coefficient of. DISPERSION RELATION The vanishing of A gives the dispersion relation which splits into two factors which gives: and in + η k = (8) CW + CW + CW + CW + CW + CW + C = (9) Where W is the growth rate introduced by n= iw and the coefficients Ci ( i=,,,,,5,6) are given by:

5 5 Gravitational instability of rotating unbounded plasma ( ) C6 = i sinθ + cos θ, 7 C5 = k ν sinθ + ηsinθ + ν cos θ + ηcos θ, 5 ν sinθ + η sinθ + ( νη) sinθ + ν cos θ 6 k { νη cos θ} + k N + η cos θ + ( νη) cos θ + cos θ ( Ne) C = i N ( H ) N H +, + k cos θ + sinθ + cos θ + M ( sinθ + cos θ) + 8( Ω +Ω) sinθ + ( νη ) sinθ + cos θ ( Ω Ω ) ν sinθ + ( νsinθ) η + 7( ν η) sinθ + cos θν 6 k 7 N + ( ην) cos θ+ ( νη) cos θ+ ν cos θ ( Ne) ν N + k ( H + H ) cos θ + ν cos θ 8 8 Mνsinθ + νsinθ( Ω +Ω ) + ν ( sin θcosθ) ΩΩ + Mηsinθ + 6ηsin θ( Ω +Ω ) + Mν cos θ C = + k cos θν ( ) cos θν ( cos θ sin θ) + Ω Ω + Ω Ω + Mηcos θ + 8ηcos θ( Ω Ω) 8 η 7 ν + k { νηcos θ} + ( H + H ) cos θ + N cos θ η 7 N η + N cos θ + ν sinθ + N sinθ,

6 Aiyub Khan, S.S. Tak, Priyadarshi Patni 6 C νηsinθ + νη sinθ + νηsinθ + νη cos θ+ νηcos θ 8 k N + ν cos θ ( Ne) ν ν νη cos θ+ ( H + H ) cos θ+ N cos θ 6 νη νη ν + k + ( H + H ) cos θ + N cos θ + N cos θ 7 νη ν ν + N ( cos θ + sinθ) + N sinθ + N sin θcos θ 9 M( ν + η )( sinθ + cos θ) + 8η sinθ( Ω +Ω ) + Mνηsinθ νη sinθ = i ( Ω +Ω ) + νη ΩΩsin θ cosθ, + k + ν ( Ω Ω ) cos θ + νηcos θ + 8νη( Ω Ω) cos θ 8 MN + νη cos θ ( Ω cos θ Ω sin θ ) + cos θ ( ) Ne N + ( H + H ) cos θ MN ( Ω ) H +ΩH ( cos θ( + cos θ) + sinθ) + cos θ k ( ΩΩ) HH cos θ Msinθ( Ω +Ω ) + 8MΩΩsin θcosθ + + M cos θ( Ω cos θ Ω sin θ )

7 7 Gravitational instability of rotating unbounded plasma 5 N k νη ( cos θ+ sinθ) + ν cos θ ( Ne) N νη N νη cos θ+ ( H + H ) cos θ+ ( νη) ( cos θ+ sinθ) 8 k N + ( νη) sin θcos θ 9 8 Mνη ( cos θ + sinθ ) + ( νη )( Ω +Ω) sinθ 8 + ( νη ) ΩΩsin θ cosθ + Mνη( cos θ + sinθ) 6 C = + k + νη ( Ω Ω) cos θ νη cos θ ( Ω cos θ Ωsin θ ) N + Mν cos θ ( Ne ) N N + ν ( H + H ) cos θ + ν cos θ MN MN N + k ηcos θ( + cos θ) + ηsinθ ( Mν) sin θcos θ, 8Mη( Ω +Ω ) sinθ + 6Mη( ΩΩ) sin θcosθ + k Mηcos θ( Ω cos θ Ωsin θ )

8 6 Aiyub Khan, S.S. Tak, Priyadarshi Patni 8 C 5 MN 8 MN k νsinθ ik Mν η ( sinθ + cos θ) + ν cos θ ( Ne) 6 MN MN + k νη sinθ νη sin θ cos θ Mη ( Ω +Ω ) sinθ + 8Mη ΩΩsin θcosθ = + k + Mη cos θ( Ωsin θ Ω cos θ) ( N + M Ω ) MN +Ω cos θ + cos θ ( ) Ne MN 5 + k 8 Ω sin cos Ω θ θ ( Ne ) Equation (8) gives. DISCUSSION n= iη k () which corresponds to viscous type of damped mode modified by finite conductivity. The dispersion relation, given by equation (9), is quite complicated, particularly as the coefficients are lengthy epressions involving the parameters characteriing the physical effects considered. It is therefore, not easily feasible to ascertain the influence of these physical effects on the growth rate or otherwise of the unstable mode of disturbance. We have therefore carried out the numerical calculations of growth rate of the unstable mode of values of the physical parameters. Products of the roots are negative when Sk G <. Therefore at least one root of W is negative real as equation (9) is a polynomial of sith degree in W and there must be at least one real root of W. The negative real root of W corresponds to an unstable mode. The plasma is consequently unstable when Sk G <, which is precisely the Jeans criterion. Thus, we find that Jeans criterion for gravitational instability remains unchanged in the presence of effects of Hall current, magnetic resistivity, viscosity and Coriolis force, when the homogeneous plasma is permeated by an oblique magnetic field. However, when Sk G >, the equation () has either all positive real roots or a positive root and pair of comple conjugate roots. The positive real roots correspond to stable

9 9 Gravitational instability of rotating unbounded plasma 7 mode. The comple roots also lead to stable mode since the equation satisfied by Re W turns out to be one which has its coefficient alternately positive and negative and consequently Re W is always positive. In order to study the influence of various physical parameters on the growth rate of an unstable mode, we have performed numerical calculations of the dispersion relation (9) to locate the roots of W against k (wave number) for several values of parameters. For these calculations the numerical values which correspond to the conditions in galaies have been given by Schmidt [8] and Pacholcyk and Stodolkiewic [9] as follows: =.7 Kgm, G = ( Kg) m S, 8 S =.5 m S, 8 V = 5 m S. a For these conditions the critical wave number for the gravitational instability is of the order of - m. We, therefore, take much less of - as the value of the wave number. We have therefore calculated the roots of the dispersion relation (9) for different values of the parameters ν, η, Ne, Ω characteriing respectively viscosity, finite conductivity, Hall current, and rotation, taking multiple of - as the values for the wave number... VISCOSITY To analye viscosity effects, we have plotted wave number (of order - m - ) against growth rate (s - ) for varying values of viscosity (ν =,,,, 5 ms - ) in Fig. (a, b, c). The other parameters are taken as Hall current Ne =. A, finite conductivity η =. m s - and rotation Ω =. s -, Ω =. s -. We notice that as the value of the viscosity increases the value of growth rate decreases for a fied wave number. Hence, we conclude that the increase in viscosity tends to stabilie the system... HALL CURRENT To analye effects of Hall current, we have plotted wave number (of order - m - ) against growth rate for varying values of Hall current ( Ne =,, 6 A) in Fig. (a, b, c). The other parameters are taken as viscosity ν =.5 ms -, finite conductivity η = m s - and rotation Ω =. s -, Ω =. s -. We notice that as the value of the Hall current increases the value of growth rate decreases for a fied wave number. Hence, we conclude that the increase in Hall current tends to stabilie the system.

10 8 Aiyub Khan, S.S. Tak, Priyadarshi Patni Plot of Growth rate against wave number for Viscosity υ =,,,, 5 H =.5, H =., Ne =, Ω =., Ω =. Effect of Viscosity υ, θ = π/6 8 6 υ= υ= υ= υ= υ= Fig. a. Effect of Viscosity υ, θ = π/ 8 6 υ= υ= υ= υ= υ= Fig. b. Effect of Viscosity υ, θ = π/ υ= υ= υ= υ= υ= Fig. c.

11 Gravitational instability of rotating unbounded plasma 9 Plot of Growth rate against wave number for Hall Current Ne =,, 6 H =.5, H =., η =, Ω =., Ω =. Effect of Hall Current Ne, θ = π/ Ne= Ne= Ne= Fig. a. Effect of Hall Current Ne, θ = π/ Growth Rate W 5 Ne= Ne= Ne= Fig. b Ne= Ne= Ne=6 Effect of Hall Current Ne, θ = π/ Fig. c.

12 Aiyub Khan, S.S. Tak, Priyadarshi Patni.. FINITE CONDUCTIVITY For analying the effects of finite conductivity, we have plotted wave number (of order - m - ) against growth rate for varying values of finite conductivity (η =,, m s - ) in Fig. (a, b, c). The other parameters are taken as viscosity ν =.5 ms -, Hall current Ne =. A and rotation Ω =. s -, Ω =. s -. We notice that as the value of the finite conductivity increases the value of growth rate decreases for a fied wave number. Hence, we conclude that the increase in finite conductivity tends to stabilie the system. Plot of Growth rate against wave number for Finite Conductivity η =,, H =.5, H =., Ne =, υ=.5, Ω =., Ω =. Effect of Finite Conductivity η, θ = π/6 5 η= η= η= Fig. a. Effect of Finite Conductivity η, θ = π/ η= η= η= Fig. b.

13 Gravitational instability of rotating unbounded plasma Effect of Finite Conductivity η, θ = π/ 8 6 η= η= η= Fig. c... ROTATION To analye effects of rotation, we have plotted wave number (of order - m - ) against growth rate for varying values of rotation ( Ω = 5,, 5 s - ) and Ω =. Ω in Fig. (a, b, c). The other parameters are taken as Hall current Ne =. A, viscosity ν = 5. ms - and finite conductivity η = 5. m s -. We notice that as the value of the rotation increases the value of growth rate decreases for a fied wave number. Hence, we conclude that the increase in rotation tends to stabilie the system. Plot of Growth rate against wave number for Rotation Ω = 5,, 5 H =.5, H =., η=5, Ne = 5, υ=5, Ω = Ω Growth Rate W Effect of Rotation Ω, θ = π/6 Ω=5,5 Ω= Ω=5,5,,,6,8,,,6,8 Fig. a.

14 Aiyub Khan, S.S. Tak, Priyadarshi Patni Ω=5 Ω= Ω=5 Effect of Rotation Ω, θ = π/ Fig. b. Effect of Rotation Ω, θ = π/ 9 8 Ω=5 Ω= Ω= Fig. c. 5. RESULT The Jeans gravitational instability of a thermally conducting plasma permeated by a oblique magnetic field has been analyed. It is concluded that viscosity, rotation, Hall currents and finite conductivity have a stabiliing influence on the growth rate of the system. Acknowledgments. This work was carried out as part of a major research project (F. No. 6- /8 (SR)) awarded by University Grants Commission (UGC), India to the one of the author Aiyub Khan. The financial assistance from UGC is gratefully acknowledged. The authors are grateful to the referee for useful suggestions, which have helped in improving the presentation and quality of the paper.

15 5 Gravitational instability of rotating unbounded plasma 6. REFERENCES. Jeans, J.H., Phil. Trans. Roy. Soc. London A, 99,, 9.. Chandrasekhar S., Hydrodynamic and Hydromagnetic Stability, Clarendon Press, Oford, 96.. Tassoul J., Acad r Belgique (Science), 9, 95, 96.. Gliddon J.E.C., Astrophys. J., 5, 58, Bhatia P.K, Physics of Fluids,, 65 65, Bhatia P.K, Nuovo Cimento. 59, 8 5, Lehnert B., Cosmic Electrodyn., 97, Kumar N. and Srivastava J., Astrophysics and Space Science, 7b, 6, Cade V.M., Astronomy and Astrophysics, 5,, 99.. Bhatia P.K. and Chhonkar R.P.S., Astrophysics and Space Science,, 5 9, Bhatia P.K. and Khan A., Contribution Plasma Phys. 5, 65 76, Ali A. and Bhatia P.K., Astrophysics and Space Science, 95, 89, 99.. Tsallis C., Possible generaliation of Boltmann-Gibbs statistics, 5, 79 87, Lima J.A.S., Silva R. and Santos J., Astronomy & Astrophysics, 96, 9,. 5. Du Jiulin, Physics Letters A,, 7 5,. 6. Shaikh S., Khan A. and Bhatia P.K., Z. Naturforsch. 6a, 75 8, Shaikh S., Khan A. and Bhatia P.K., Physics Letters A, 7, 5 57, Schmidt M., B.A.N.,, 5 5, (nr. 68), Pacholcyk A.G. and Stodolkiewic J. S., Acta Astronomica (Polska Akademia Nauk),, 9, 96.