Effect of arbitrary radiative heat-loss functions on jeans instability of partially: ionized plasma.

Pensia, Ram K. ; Patidar, Pradeep ; Shrivastava, V. 等

Introduction

Much theoretical effort has gone in to understanding the gravitational collapse of protostar but the question of gravitational instability of partially-ionized gaseous medium in the presence of radiative heat-loss function is of particular interest in cosmogony. Jeans [1] have discussed the condition under which the fluid become gravitational unstable under the action of its own gravity. A detailed contribution of the self gravitational instability with different assumption on the magnetic field and rotation has been given by Chandrashekhar [2]. In this connection, many investigators have discussed the gravitational instability of a homogeneous plasma considering the effects of various parameters. [Chhajlani and Sanghavi, [3] Elmegreen [4], Langer [5], and Bondyopadhya [6].]

Recently, Lima et al. [7] have studied the problem of Jeans gravitational instability and non-extensive kinetic theory. The problem of magneto-thermal instability of self-gravitating, viscous, Hall plasma in the presence of suspended particles has been investigated by Pensia et al. [8]. Khan and sheikh [9] have discussed the instability of thermally conducting self-gravitating system. The role of magnetic field in contraction and fragmentation of interstellar clouds has been studied by Pensia et al. (10). thus we find that the problem of Jeans gravitational instability is the important phenomena to understand gravitational collapse of the protostar.

Linearized Perturbation Equations

We assume that the two components of the partially-ionized plasma (the ionized fluid and the neutral gas) behave like a continuum fluid and their state velocities are equal. The effect of the magnetic field, field of gravity and the pressure on the neutral components are neglected. Also it is assumed that the frictional force of the neutral gas on the ionized fluid is of the same order as the pressure gradient of the ionized fluid. Thus, we are considering only the mutual frictional effects between the neutral gas and the ionized fluid. It is assumed that the above medium is permeated with a uniform magnetic field [??] (0, 0, H).

Thus the linearized perturbation equations governing the motion of hydromagnetic thermally conducting two components of the partially-ionized plasma are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

[[nabla].sup.2][delta][phi] = -4[pi]G[delta] G[delta][rho] (4)

1/[([gamma]-1)] [partial derivative][delta][rho]/[partial derivative]t - [gamma]/[([gamma]-1)] p/[rho] [partial derivative][rho]/[partial derivative]t + [rho]([L.sub.p][delta][rho] + [L.sub.t][delta]T) - [lambda][[nabla].sup.2][delta]T = 0 (5)

[delta]p/p = [delta]T/T + [delta][rho]/[rho] (6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

The parameters [phi], G, [lambda], R, c, [rho], [[rho].sub.d], [u.sub.c], [gamma], T, and p denote the gravitational potential, gravitational constant, thermal conductivity, gas constant, velocity of light, density of ionized component, density of neutral components ([rho] >> [[rho].sub.d]), collision frequency between two components, ratio of two specific heats, temperature and pressure, respectively.

The perturbations in fluid velocity, fluid pressure, fluid density, magnetic field, gravitational potential, temperature and the radiative heat-loss function are given as [??]([v.sub.x],[v.sub.y], [v.sub.z]), [delta]p, [delta][rho], h([h.sub.x], [h.sub.y], [h.sub.z]) [delta][phi], [delta]T and [delta]L respectively. In equation (5), [L.sub.[sigma],T] are the partial derivatives of the density dependent [([partial derivative]L/[partial derivative][rho]).sub.T] and temperature dependent [([partial derivative]L/[partial derivative]T).sub.[rho]] heat-loss functions respectively.

Dispersion relation

We assume that all the perturbed quantity vary as

exp {i([k.sub.x]x + [k.sub.z]z + [omega]t)}

where [omega] is the frequency of harmonic disturbances, [k.sub.x,z] are wave numbers in x and z direction, respectively, such that [k.sup.2.sub.x] + [k.sup.2.sub.z] = [k.sup.2] combining equation (5) and (6), we obtain the expression for [delta]p as

[delta]p = ([[alpha]+[sigma][c.sup.2]]/[[sigma]+[beta]])[delta][rho], [alpha] = ([gamma]-1)([L.sub.T]T - [L.sub.[rho]][rho] + [lambda][k.sup.2]T/[rho]),

[beta] = ([gamma]-1)([L.sub.T][T.sub.[rho]]/p + [lambda][k.sup.2]T/p) (9)

where [sigma] = i[omega], c = [([gamma]p/[rho]).sup.1/2] is the adiabatic velocity of sound in the medium

Using equation (2)-(9) in equation (1), we obtain the following algebraic equations for the amplitude components

([[sigma].sup.2] + [k.sup.2.sub.z][V.sup.2] + [Bv.sub.c]/[sigma]+[v.sub.c] [[sigma].sup.2]) [v.sub.x] + [ik.sub.x]/[k.sup.2] [sigma][[OMEGA].sup.2.sub.T]s=0 (10)

([[sigma].sup.2]) + [k.sup.2][V.sup.2] + [Bv.sub.c]/[sigma] + [v.sub.c] [[sigma].sup.2) [v.sub.y] = 0 (11)

([sigma] + [Bv.sub.c]/[sigma]+[v.sub.c] [sigma]) [v.sub.z] + [ik.sub.z]/[k.sup.2] [[OMEGA].sup.2.sub.T]s=0 (12)

([ik.sub.x][k.sup.2][V.sup.2])[v.sup.x] - ([[sigma].sup.3] + [Bv.sub.c]/[sigma] + [v.sub.c] [[sigma].sup.3] + [sigma][[OMEGA].sup.2.sub.T])s=0 (13)

where s = [delta][rho]/[rho] is the condensation of the medium, V = H/[(4[pi][rho]).sup.1/2] is the Alfven velocity, [c.sup.2] = [gamma][c'.sup.2] where c and c' are the adiabatic and isothermal velocities of sound. Also wee have assumed the following substitutions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The nontrial solution of the determinant of the matrix obtained from equation (10) to (13) with [v.sub.x], [v.sub.y], [v.sub.z], S having various coefficients that should vanish is to give the following dispersion relation.

EE'aM + E' [ak.sup.2.sub.x][V.sup.2] [sigma][[OMEGA].sup.2.sub.T] = 0 (14)

The dispersion relation (14) shows the combined influence of thermal conductivity and arbitrary radiative heat-loss functions on the self gravitational instability of a two components of the partially-ionized plasmas we find that in this dispersion relation the terms due to the arbitrary radiative heat-loss function with thermal conductivity have entered through the factor [[OMEGA].sup.2.sub.T].

Analysis of the dispersion relation Longitudinal mode of propagation

-For this case we assume all the perturbations longitudinal to the direction of the magnetic field i.e. ([k.sub.z] = k, [k.sub.x] = 0). This is the dispersion relation reduces in the simple form to give

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

We find that in the longitudinal mode of propagation the dispersion relation is modified due to the pressure of neutral particles, thermal conductivity and arbitrary radiative heat-loss functions. This dispersion relation has three independent factors, each represents the mode of propagation incorporating different parameters. The first factor of this dispersion relation equating to zero

[sigma] + (1 + B)[v.sub.c] = 0 (16)

This represents stable mode due to collision frequency. The second factor of equation (15) equating to zero, we obtain following dispersion relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

This dispersion relation for self-gravitating fluid incorporated effect of neutral particles, thermal conductivity and arbitrary radiative heat-loss function. It is evident from equation (17) that the condition of instability is independent of magnetic field. The dispersion relation (17) is a fourth degree equation which may be reduced to particular

cases so that the effect of each parameter is analyzed separately.

For thermally non-conducting, non-radiating, fully ionized fluid we have [alpha] = [beta] = [v.sub.c] = 0 the dispersion relation (17) reduces to

[[sigma].sup.2] + [[OMEGA].sup.2.sub.j] = 0 and k < [k.sub.j] = [(4[pi]G[rho]/[c.sup.2]).sup.1/2] (18)

The fluid is unstable for all Jeans wave number k < [k.sub.j]. It is evident from equation (18) that Jeans criterion of instability remains unchanged in the presence of neutral particles.

For non-arbitrary radiative heat-loss function but thermally conducting and self-gravitating fluid having neutral particles, the dispersion relation (17) reduces to

[[sigma].sup.4] + [[sigma].sup.3] ([R.sub.1] + [[OMEGA].sub.k]) + [[sigma].sup.2][[R.sub.1][[OMEGA].sub.k] + [[OMEGA].sup.2.sub.j1]]

+ [sigma]{[[OMEGA].sup.2.sub.j1]([v.sub.c] + [[OMEGA].sub.k])} + [v.sub.c][[OMEGA].sub.k] [[OMEGA].sup.2.sub.j1] = 0 (19)

From equation (19) we get

k < [k.sub.j1] = [(4[pi]G[rho]/[c'.sup.2]).sup.1/2] (20)

where [k.sub.j1] is the modified Jeans wave number for thermally conducting system. It is clear from equation (20) that the Jeans length is reduced due to thermal conduction [as [gamma] > 1], thus the system is destabilized. If we consider self-gravitating and thermally non-conducting plasma incorporated with neutral particles, arbitrary radiative heatloss function then the condition of instability is given as

k < [k.sub.j2] = [k.sub.j] [([gamma][L.sub.T]/[[L.sub.T]-[rho][L.sub.[rho]]/T]).sup.1/2] (21)

where, [k.sub.j2] is the modified critical wave number due to inclusion of arbitrary radiative heat-loss function. Comparing equation (18) and (21) we find that the critical wave number [k.sub.j2] is very much different from the original Jeans wave number [k.sub.j] and [k.sub.j2] depends on derivatives of the arbitrary radiative heat-loss function with respect to local temperature [L.sub.T] and local density [L.sub.[rho] in the configuration. It is clear from equation (21) that when the arbitrary radiative heat-loss function is independent of density of the configuration (i.e. [L.sub.[rho]] = 0), then [k.sub.j2] = [k.sub.j] i.e. critical wave number remains unaffected and if the arbitrary radiative heat-loss function is independent of temperature ([L.sub.T] = 0), then [k.sub.j2] vanishes.

The condition of instability of the system, when combined effect of all the parameters represented by the original dispersion relation (17) is given as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

Furthermore, if it is considered that the arbitrary radiative heat-loss function is purely density dependent ([L.sub.T] = 0) then the condition of instability is given as

K < [k.sub.j4] = [(4[pi]G[rho]/[c'.sup.2] + [[rho].sup.2][L.sub.[rho]]/[lambda]T).sup.1/2] (23)

It is evident from inequality (22) that the critical wave number is increased or decreased, depending on whether the arbitrary radiative heat-loss function is an increasing or decreasing function of the density.

Now equating zero the third factor of equation (15) and after solving we obtain dispersion relation as

[[sigma].sup.6] + [[sigma].sup.5] [A.sub.1] + [[sigma].sup.2] [A.sub.2] + [[sigma].sup.3] [A.sub.3] + [[sigma].sup.2] [A.sub.4] + [sigma] [A.sub.5] + [A.sub.6] = 0 (24)

where

[A.sub.6] = [V.sup.4] [k.sup.4] [v.sup.4.sub.c]

This dispersion relation shows the combined influence of magnetic field and the effect of the neutral particles, but this mode is independent of thermal conductivity, arbitrary radiative heat-loss function and self gravitation. This equation gives Alfven mode modified by the dispersion effect of the neutral particles.

Transverse mode of propagation

For this case we assume all the perturbations are propagating perpendicular to the direction of the magnetic field, for, our convenience, we take [k.sub.x] = k, and [k.sub.z] = 0, the general dispersion relation (16) reduces to

[a.sup.2][[sigma].sup.2] a ([sigma]a + [k.sup.2][V.sup.2]) + [sigma]a[[OMEGA].sup.2.sub. T]= 0 (25)

This is the general dispersion relation for transverse propagation shows the combined influence of magnetic field, self gravitation, thermal conductivity presence of neutral particles and arbitrary radiative heat-loss function on the self-gravitational instability of a two components of the partially-ionized plasmas. Dispersion relation (25) has two distinct factors, each represents different mode of propagation when equated to zero, separately. The first mode is same as discussed in the dispersion relation (16). The second factor of equation (25) equating zero and substituting the values of a and [[OMEGA].sup.2.sub.T] we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

This dispersion relation for transverse propagation shows the combined influence of thermal conductivity, arbitrary radiative heat-loss function on the self-gravitation instability of two components partially-ionized plasmas with the effect of neutral particles. The condition of instability is obtained from dispersion relation (26) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

The medium is unstable for wave number k < [k.sub.j5]. It may be noted here that the critical wave number involves, derivative of temperature dependent and density dependent arbitrary radiative heat-loss function, thermal conductivity of the medium and the magnetic field.

Conclusion

We have developed a model of wave propagation in an infinite self-gravitating magnetized two components of partially-ionized system incorporating thermal conductivity and arbitrary radiative heat-loss functions. The general dispersion relation is obtained, which is modified due to the presence of these parameters. This dispersion relation is reduced for longitudinal and transverse modes of propagation. We find that the Jeans condition remains valid but the expression of the critical Jeans wave number is modified. Owing to the inclusion of the thermal conductivity, isothermal sound velocity is replaced by the adiabatic velocity of sound. The effect of the collision with neutrals does not affect the instability condition of the system in both the longitudinal and transverse mode of propagation.

In the case of longitudinal propagation we obtain Alfven mode modified by the collisions with the neutrals. The thermal conductivity has a destabilizing influence. It is also found that the density dependent heat-loss function has a destabilizing influence on the instability of the system.

In the transverse mode of propagation, we find a gravitating thermal mode influenced by thermal conductivity and arbitrary radiative heat-loss functions. We also found the condition of instability and the expression of critical Jeans wave number both are modified due to the presence of magnetic field, thermal conductivity and arbitrary radiative heat-loss function. It is found that radiative critical wave number is same as original Alfven critical wave number when the arbitrary radiative heat-loss functions are independent of density of the medium.

References

[1] Jeans, J.H., 1929, "Astronomy And Cosmogony", Cambridge university press, Cambridge, UK.

[2] Chandrashekhar, S., 1961, "Hydrodynamics and Hydromagnetic Stability", Clarendon Press, Oxford.

[3] Chhajlani, R.K. and Sanghavi, R.K., 1985, "Effect Of Finite Larmor Radius On Magneto-Gravitational Instability Of Hall Plasma In The Presence Of Suspended Particles," Astrophy. Space Sci. 117, pp 35-46.

[4] Elmegree, B.G., 1989, "Molecular Cloud Formation By Gravitational Instability In A Clumpy Interstellar Medium", The Astrophysical journal 344, pp 306-310.

[5] Langer, W.D., 1978, "The Stability Of Interstellar Clouds Containing Magnetic Fields", The Astrophysical journal 225, pp 95-106.

[6] Bondyopadhya, R., 1972, "Heating Of Astrophysical Plasma Due To Rotation", Czech. J. Phys. B22, pp 1199-1201.

[7] Lima, J.A.S, Silva, R. and Santos, J. (2002)"Jeans Gravitational Instability And Non-Extensive Kinetic Theory", Astronomy and Astrophysics, 396, pp 309-314.

[8] Pensia, R.K., Ghorela V. and Chhajlani, R.K., 2009, "Magneto Thermal Instability Of Self-Gravitational Viscous Hall Plasma In The Presence Of Suspended Particle", Acta ciencia Indica, 25, pp 141-148.

[9] Khan, A. and Shaikh, S., (2010) "Instability Of Thermally Conducting Self-gravitating System", 1, pp 77-82.

[10] Pensia, R.K., Kumar, V. and Prajapat, V., 2010, "Role Of Magnetic Field In Contraction And Fragmentation Of Interstellar Clouds". Ultra scientist, 22(2), pp 253-258.

Ram K. Pensia, Pradeep Patidar, V. Shrivastava, Vikas Prajapt and Ashok K. Patidar

Post Graduate Department of Physics, Govt. P.G. College Neemuch, M.P., India

E-mail: rkpensia@rediffmail.com

Corresponding Author E-mail: vijayendra.shrivastava@hotmail.com