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ISRN Astronomy and Astrophysics
Volume 2012 (2012), Article ID 104941, 6 pages
http://dx.doi.org/10.5402/2012/104941
Research Article

On the Magnetogravitational Instability of a Ferromagnetic Dust Cloud in the Presence of Nonuniform Rotation

Department of Mathematics, Himachal Pradesh University, Summer Hill, Shimla 171005, India

Received 10 April 2012; Accepted 6 June 2012

Academic Editors: S. Bogovalov, C. W. Engelbracht, and R. Hawkes

Copyright © 2012 Joginder S. Dhiman and Rekha Dadwal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The effect of a nonuniform magnetic field on the gravitational instability of a nonuniformly rotating infinitely extending axisymmetric cylinder in a homogenous ferromagnetic medium has been studied. The propagation of the wave is allowed along radial direction. A general dispersion relation, using the normal mode analysis method on the perturbation equations of the problem, is obtained. It is found that Bel and Schatzman criterion determines the gravitational instability of this general problem. Thus, it appears that the effect of non-uniform magnetic field on the gravitational instability as discussed by (Dhiman and Dadwal, 2010) is marginalized by the magnetic polarizability of ferrofluid.

1. Introduction

The gravitational instability (GI) or the Jeans instability causes the collapse of interstellar gas clouds and subsequent star formation. It occurs when the internal gas pressure is not strong enough to prevent gravitational collapse of a region filled with matter. GI plays a dominant role in determining the star formation properties of galaxies. The problem of instability of self-gravitating large gas clouds was first considered by Jeans [1] and underlined the importance of self-gravitating instabilities in astrophysics because of their crucial role in understanding collapse, formation, and evolution of interstellar molecular clouds, star formation, galactic structure and its evolution, and so forth. For a latest and broader review on the subject of gravitational instability, one may refer to [18] and references therein.

Theoretical studies have shown that the magnetic field plays a vital role in self-gravitating star-forming regions and in the evolution of interstellar clouds into self-gravitating star forming regions. The problem of magnetogravitational instability of interstellar rotating medium is of considerable importance in connection with the protostar and star formation in magnetic dust clouds. In astrophysics, the problems that are considered generally assume magnetic field/rotation to be uniform: however, this idealization of the uniform character in theoretical investigations is valid only for laboratory purposes (cf. Larson [8] and references within), because in the interstellar interior and atmosphere, the magnetic field/rotation may be variable and may together alter the nature of the instability. Limited efforts have been put to investigate the GI by considering the magnetic field or rotation to be variable. Bel and Schatzman [9] studied the effect of nonuniform rotation on the onset of gravitational instability and obtained a modified expression for the Jeans criterion, which is now known as the Bel and Schatzman criterion. Recently, Dhiman and Dadwal [10, 11] also studied the simultaneous effect of nonuniform magnetic field and rotation on the gravitational instability of gaseous medium and obtained some general qualitative results.

In astronomy, the interstellar medium (ISM) is the matter that exists in the space between the star systems in a galaxy. This matter includes gas in ionic, atomic, and molecular form, dust, and cosmic rays. It fills interstellar space and blends smoothly into the surrounding intergalactic space. The dust, which is composed of small solid particles by segregation in the interstellar clouds, is a very important component of the ISM. In the recent past, there have been dramatic changes in the conception of the interstellar medium, and recent observational and numerical works have suggested that interstellar medium (ISM) plays an important role in the star formation. Spitzer [12] has shown that solid particles in space including dust grains are eclectically charged, relative to surrounding plasma. Further, the efficiency and timescale of stellar birth in Galactic molecular clouds strongly depend on the properties of interstellar medium (ISM).

Ferrofluids are colloidal liquids made of nanoscale ferromagnetic, or ferrimagnetic, particles suspended in a carrier fluid (usually an organic solvent or water). A ferrofluid flow in the presence of an applied magnetic field is accompanied by an intertwinement of hydrodynamic and magnetic interactions (Rosenwieg [13]). A ferrofluid is stable and does not solidify even in the presence of a high magnetic field and is commonly characterized by its strong tendency to magnetize in the direction of the magnetic field. Jones and Spitzer [14] provided a model for the existence of gas-dust interstellar mediums with a highly pronounced property of magnetic polarizability. This can be assumed due to a super paramagnetic dispersion of the fine ferromagnetic grains suspended in a gaseous cloud of molecular hydrogen. Mamun and Shukla [15] observed the usual Jeans instability in a self-gravitating dark interstellar molecular cloud containing ferromagnetic dust grains and baryonic gas clouds and supported the existence of ferromagnetic dust particles in a magnetically supported dark interstellar self-gravitating interstellar molecular cloud.

The present study is primarily motivated by the investigations of Mamun and Shukla [15] regarding the instability of a self-gravitating dark interstellar molecular cloud containing ferromagnetic dust grains. Our aim here is to investigate the gravitational instability of an infinite axisymmetric cylinder of a homogenous nonuniformly rotating medium containing the ferromagnetic dust particles in the presence of nonuniform magnetic field. The mathematical analysis followed in this paper is precisely the same as that of [9, 10]. Since the system of linearized perturbation equation governing the problem contains variable coefficients, the sufficient condition for GI is derived using the local instability approach as adopted in [9].

2. Mathematical Model and Basic Equations

Consider an infinite homogeneous, interstellar self-gravitating molecular cloud containing ferromagnetic dust particles. The system is permeated with the simultaneous action of a nonuniform rotation and a nonuniform magnetic field. In a rectangular coordinate system, let 𝑢=(𝑢𝑥,𝑢𝑦,𝑢𝑧) be the velocity, 𝑀=(𝑀𝑥,𝑀𝑦,𝑀𝑧) be the ferrofluid magnetization under the magnetic field 𝐻=(𝐻𝑥,𝐻𝑦,𝐻𝑧). For the treatment of this physical configuration for gravitational instability, the conventional set of ferrohydrodynamic equations are given by (cf. [16, 17]) 𝑑𝑢𝑑𝑡=grad𝑝+𝜌grad𝜙+𝜇0𝑀𝐻,(1)𝜕𝜌+𝜕𝑡𝑢grad𝜌+𝜌.𝑢=0,(2)2𝜙=4𝜋𝐺𝜌𝜌0.(3) The ferrofluid magnetization 𝑀 satisfies Shliomis’ [16, 17] equation of magnetization, which is given by 𝑑𝑀=1𝑑𝑡2×𝑢×𝑀𝛼𝑀𝑀0𝛽𝑀×𝑀×𝐻.(4) In the above equations 𝑝, 𝜙, 𝜇0, 𝜌, and 𝐺, respectively, denote the pressure, gravitational potential, the inverse Brownian time constant for particle diffusion in the ferrofluid, the magnetic permeability, density, the gravitational constant, 𝛼=3𝜂𝑉/𝑘𝑏𝑇 is the Brownian time of rotational particle diffusion and 𝛽=1/6𝜂𝜑, where 𝜂 is the dynamic viscosity of the carrier fluid and 𝜑=𝑛𝑉 is the volume fraction of magnetic grains in the liquid. Here, 𝑛 is the number density and 𝑉 the volume of a single particle. Also, 𝜌0 and 𝑀0=(0,0,𝑀0) are the equilibrium values of density and magnetization and 𝑀0 the equilibrium magnetization of ferrofluid is related to the equilibrium magnetic felid 𝐻0 by 𝑀0=𝑛𝑀1coth𝜓𝜓𝐻0𝐻0,(5) where 𝑀 is the magnetic moment of single ferromagnetic particle and the nondimensional quantity 𝜓 is given by 𝜓=𝜇0(𝑀𝐻0/𝑘𝑏𝑇𝑏), where 𝑘𝑏 is the Boltzman constant and 𝑇𝑏 is temperature of the ferrofluid. Further, the Poisson’s equation (3) is considered so as to avoid the “Jeans Swindle” (cf. Speigel and Thiffeault [7]).

In Chu formulation of electrodynamics (see Penfield and Haus [18]), the magnetic field 𝐻, magnetization 𝑀, and magnetic induction 𝐵 are related by 𝐵=𝜇0𝐻+𝑀.(6) We know that the magnetic field 𝐻 satisfies the following Maxwell’s equations:×𝐻=0,(7)𝐵=0.(8)

Equation (6), using (8) yields 𝐻=𝑀.(9)

In the present analysis, we shall consider an infinite axisymmetric cylinder of homogeneous, infinitely conducting self-gravitating ferromagnetic dust cloud under the simultaneous effect of a nonuniform rotation and a nonuniform magnetic field, therefore, transforming the above fundamental equations in cylindrical coordinates (𝑟,𝜃,𝑧). Denoting 𝑢=(𝑢𝑟,𝑢𝜃,𝑢𝑧) and 𝑀=(𝑀𝑟,𝑀𝜃,𝑀𝑧) by the components of velocity 𝑢 and ferrofluid magnetization 𝑀 along the radial 𝑟, the transverse 𝜃, and the axial 𝑧 directions, respectively. The cylinder is assumed to be rotating about its axis (𝑧-axis) with nonuniform angular velocity 𝜔. The propagation of wave is taken along the radial direction “r” of the cylinder, hence 𝜕/𝜕𝑟 is the only nonzero component of the gradient.

Following Dhiman and Dadwal [10], the basic equations (1)–(4), (7) and (9) under these assumptions take the following forms in cylindrical polar coordinates 𝜌𝜕𝑢𝑟𝜕𝑡+𝑢𝑟𝜕𝑢𝑟𝑢𝜕𝑟𝜃2𝑟=𝜌𝜕𝜙𝜕𝑟𝜕𝑝𝜕𝑟+𝜇0𝑀𝑟𝜕𝐻𝑟𝑀𝜕𝑟𝜃𝐻𝜃𝑟,𝜌𝜕𝑢𝜃𝜕𝑡+𝑢𝑟𝜕𝑢𝜕𝑟𝜃+𝑢𝑟𝑢𝜃𝑟=𝜇02𝑀𝑟𝜕𝐻𝜃𝜕𝑟+𝐻𝑟𝜕𝑀𝜃𝜕𝑟+𝑀𝜃𝜕𝐻𝑟𝜕𝑟𝐻𝜃𝜕𝑀𝑟+𝜕𝑟2𝑀𝜃𝐻𝑟𝑟,𝜌𝜕𝑢𝑧𝜕𝑡+𝑢𝑟𝜕𝑢𝑧=𝜇𝜕𝑟02𝑀𝑟𝜕𝐻𝑧𝜕𝑟+𝐻𝑟𝜕𝑀𝑧𝜕𝑟+𝑀𝑧𝜕𝐻𝑟𝜕𝑟𝐻𝑧𝜕𝑀𝑟+𝑀𝜕𝑟𝑧𝐻𝑟𝐻𝑧𝑀𝑟𝑟,𝜕𝑀𝑟𝜕𝑡+𝑢𝑟𝜕𝑀𝑟+1𝜕𝑟2𝑀𝜃𝜕𝑢𝜃+𝑢𝜕𝑟𝜃𝑟+𝑀𝑧𝜕𝑢𝑧𝑀𝜕𝑟+𝛼𝑟𝑀0𝑀+𝛽𝜃𝑀𝑟𝐻𝜃𝐻𝑟𝑀𝜃𝑀𝑧𝐻𝑟𝑀𝑧𝑀𝑟𝐻𝑧=0,𝜕𝑀𝜃𝜕𝑡+𝑢𝑟𝜕𝑀𝜃1𝜕𝑟2𝑀𝑟𝜕𝑢𝜃+𝑢𝜕𝑟𝜃𝑟𝑀+𝛼𝜃𝑀0𝑀+𝛽𝑧𝑀𝜃𝐻𝑧𝐻𝜃𝑀𝑧𝑀𝑟𝑀𝑟𝐻𝜃𝐻𝑟𝑀𝜃=0,𝜕𝑀𝑧𝜕𝑡+𝑢𝑟𝜕𝑀𝑧1𝜕𝑟2𝑀𝑟𝜕𝑢𝑧𝑀𝜕𝑟+𝛼𝑧𝑀0𝑀+𝛽𝑟𝐻𝑟𝑀𝑧𝑀𝑟𝐻𝑧𝑀𝜃𝑀𝜃𝐻𝑍𝐻𝜃𝑀𝑧=0,𝜕𝐻𝑧𝜕𝑟=0,𝜕𝐻𝜃+𝐻𝜕𝑟𝜃𝑟=0,𝜕𝐻𝑟+𝐻𝜕𝑟𝑟𝑟=𝜕𝑀𝑟+𝑀𝜕𝑟𝑟𝑟,𝜕𝜌+𝜕𝑡𝑢grad𝜌+𝜌𝑢=0,2𝜙=4𝜋𝐺𝜌𝜌0.(10) In these equations, the operators and 2now are respective operators in cylindrical coordinates.

3. Equilibrium State and Perturbation Equations

Following the physical models of Mamun and Shukla [15] and Kumar et al. [19], the magnetization is taken along 𝑟 and 𝑧 directions.

The equilibrium state under discussion is clearly characterized as follows: 𝑢=(0,𝑟𝜔,0);𝐻=0,0,𝐻𝑧;𝑀=𝑀𝑟,0,𝑀𝑧𝑝=𝑝0;𝜙=𝜙0;𝜌=𝜌0(11) Using (11) in (10) and using the fact that propagation of wave is along 𝑟 direction and thus 𝜕/𝜕𝑟 is the only nonzero component, we obtain the following basic solution: 𝜕𝑝0𝜕𝑟=𝜌0𝜕𝜙0𝜕𝑟+𝑟𝜔2(12)𝑀𝑟𝜕𝐻𝑧𝜕𝑟𝐻𝑧𝜕𝑀𝑟+𝑀𝜕𝑟𝑟𝑟=0(13)𝛼𝑀𝑟𝑀0+𝛽𝑀𝑧𝑀𝑟𝐻𝑧=0(14)12𝑀𝑟𝜕𝑢𝜃+𝑢𝜕𝑟𝜃𝑟+𝛼𝑀0=0(15)𝛼𝑀𝑧𝑀0𝛽𝑀2𝑟𝐻𝑧=0(16)𝜕𝐻𝑧𝜕𝑟=0(17)𝜕𝑀𝑟+𝑀𝜕𝑟𝑟𝑟=0(18)1𝑟𝜕𝑟𝜕𝑟𝜕𝜙0𝜕𝑟=0(19) Equation (17) implies that 𝐻𝑧=Constant(20) Since pressure and density are uniform initially, therefore it is clear from (12) that the gravitational potential is balanced by centrifugal force.

Let us allow the small perturbation in the initial state described by (11) as following; 𝑢=𝑢𝑟,𝑟𝜔+𝑢𝜃,𝑢𝑧;𝐻=𝑟,𝜃,𝐻𝑧+𝑧;𝑝=𝑝0+𝛿𝑝𝑀=𝑀𝑟+𝑚𝑟,𝑚𝜃,𝑀𝑧+𝑚𝑧;𝜙=𝜙0+𝛿𝜙;𝜌=𝜌0+𝛿𝜌.(21) Using (21) in (10), ignoring the terms of second and higher orders in the perturbations, and using (12)–(20), we have the following linearized perturbed equations: 𝜌0𝜕𝑢𝑟𝜕𝑡2𝜔𝑢𝜃=𝜌0𝜕𝛿𝜙𝜕𝑟𝜕𝛿𝑝𝜕𝑟+𝜇0𝑀𝑟𝜕𝑟,𝜕𝑟(22)𝜌0𝜕𝑢𝜃𝜕𝑡+𝑢𝑟𝜕𝜕𝑟(𝑟𝜔)+𝑢𝑟𝜔=𝜇02𝑀𝑟𝜕𝜃𝜕𝑟𝜃𝜕𝑀𝑟,𝜕𝑟(23)𝜌0𝜕𝑢𝑧=𝜇𝜕𝑡02𝑟𝜕𝑀𝑧𝜕𝑟+𝑀𝑧1𝑟𝜕𝜕𝑟𝑟𝑟𝐻𝑧1𝑟𝜕𝜕𝑟𝑟𝑚𝑟,(24)𝜕𝑚𝑟𝜕𝑡+𝑢𝑟𝜕𝑀𝑟+1𝜕𝑟2𝑀𝑧𝜕𝑢𝑧𝜕𝑟+𝑚𝜃𝜕𝜕𝑟(𝑟𝜔)+𝜔+𝛼𝑚𝑟+𝛽𝑀2𝑧𝑟+𝐻𝑧𝑀𝑧𝑚𝑟+𝑀𝑧𝑀𝑟𝑧+𝐻𝑧𝑀𝑟𝑚𝑧=0,(25)𝜕𝑚𝜃1𝜕𝑡2𝑚𝑟𝜕𝜕𝑟(𝑟𝜔)+𝜔+𝑀𝑟1𝑟𝜕𝜕𝑟𝑟𝑢𝜃+𝛼𝑚𝜃𝐻+𝛽𝑧𝑀𝑧𝑚𝜃𝑀2𝑧𝜃𝑀2𝑟𝜃=0,(26)𝜕𝑚𝑧𝜕𝑡+𝑢𝑟𝜕𝑀𝑧1𝜕𝑟2𝑀𝑟𝜕𝑢𝑧𝜕𝑟+𝛼𝑚𝑧𝑀+𝛽𝑟𝑀𝑧𝑟2𝑀𝑟𝐻𝑧𝑚𝑟𝑀2𝑟𝑧=0,(27)𝜕𝑧𝜕𝑟=0,(28)𝜕𝜃+𝜕𝑟𝜃𝑟=0,(29)1𝑟𝜕𝜕𝑟𝑟𝑚𝑟1=𝑟𝜕𝜕𝑟𝑟𝑟,(30)𝜕𝜕𝑡𝛿𝜌+𝜌0𝜕𝑢𝑟+𝑢𝜕𝑟𝑟𝑟=0,(31)1𝑟𝜕𝑟𝜕𝑟𝜕𝛿𝜙𝜕𝑟=4𝜋𝐺𝛿𝜌,(32) where (𝑢𝑟,𝑢𝜃,𝑢𝑧), (𝑟,𝜃,𝑧), (𝑚𝑟,𝑚𝜃,𝑚𝑧), 𝛿𝑝, 𝛿𝜌, and 𝛿𝜙 are the respective perturbations from basic state in velocity, magnetic field, magnetization, pressure, density, and gravitational potential vector. In the above equations, the dashes have been dropped for convenience in writing.

Since we have considered that the fluctuations in pressure and density take place adiabatically, therefore 𝛿𝑝=𝑐2𝛿𝜌.(33) Using this equation of state (33) for adiabatic medium, (22) reduces to 𝜌0𝜕𝑢𝑟𝜕𝑡2𝜔𝑢𝜃=𝜌0𝜕𝛿𝜙𝜕𝑟𝑐2𝜕𝛿𝜌𝜕𝑟+𝜇0𝑀𝑟𝜕𝑟.𝜕𝑟(34)

4. Gravitational Instability

In order to investigate the stability of the forgoing stationary state, we shall consider the dependence of the perturbation on 𝑟 and 𝑡 of the form 𝜓(𝑟)exp(𝜎𝑡),(35) where 𝜎 is frequency of the perturbation.

For this type of dependence of perturbation on 𝑟 and 𝑡, we have 𝜕𝜕𝜕𝑡𝜎,𝑑𝜕𝑟𝑓(𝑟)=𝑑𝑟𝑓(𝑟).(36) Using the above dependence, the perturbations equations (34) and (23)–(32) assume the following forms: 𝜎𝑢𝑟2𝜔𝑢𝜃𝑑𝛿𝜙+𝑐𝑑𝑟2𝜌0𝑑𝛿𝜌𝜇𝑑𝑟0𝜌0𝑀𝑟𝑑𝑟𝑑𝑟=0,(37)𝜎𝑢𝜃+𝑢𝑟𝑑𝑑𝑟(𝑟𝜔)+𝑢𝑟1𝜔2𝜇0𝜌0𝑀𝑟𝑑𝜃𝑑𝑟𝜃𝑑𝑀𝑟𝑑𝑟=0,(38)𝜎𝑢𝑧12𝜇0𝜌0𝑟𝑑𝑀𝑧𝑑𝑟+𝑀𝑧1𝑟𝑑𝑑𝑟𝑟𝑟𝐻𝑧1𝑟𝑑𝑑𝑟𝑟𝑚𝑟=0,(39)(𝜎+𝛼)𝑚𝑟+𝑢𝑟𝑑𝑀𝑟+1𝑑𝑟2𝑀𝑧𝑑𝑢𝑧𝑑𝑟+𝑚𝜃𝑑𝑑𝑟(𝑟𝜔)+𝜔+𝛽𝑀2𝑧𝑟+𝐻𝑧𝑀𝑧𝑚𝑟+𝑀𝑧𝑀𝑟𝑧+𝐻𝑧𝑀𝑟𝑚𝑧=0,(40)(𝜎+𝛼)𝑚𝜃12𝑚𝑟𝑑𝑑𝑟(𝑟𝜔)+𝜔+𝑀𝑟1𝑟𝑑𝑑𝑟𝑟𝑢𝜃𝐻+𝛽𝑧𝑀𝑧𝑚𝜃𝑀2𝑧𝜃𝑀2𝑟𝜃=0,(41)(𝜎+𝛼)𝑚𝑧+𝑢𝑟𝑑𝑀𝑧1𝑑𝑟2𝑀𝑟𝑑𝑢𝑧𝑀𝑑𝑟+𝛽𝑟𝑀𝑧𝑟2𝑀𝑟𝐻𝑧𝑚𝑟𝑀2𝑟𝑧=0,(42)𝑑𝑧𝑑𝑟=0,(43)1𝑟𝑑𝑑𝑟𝑟𝜃=0,(44)1𝑟𝑑𝑑𝑟𝑟𝑚𝑟1=𝑟𝑑𝑑𝑟𝑟𝑟,(45)𝜌𝜎𝛿𝜌+0𝑟𝑑𝑑𝑟𝑟𝑢𝑟=0,(46)1𝑟𝑑𝑟𝑑𝑟𝑑𝛿𝜙𝑑𝑟=4𝜋𝐺𝛿𝜌.(47)

Since, the above equations involve the variable coefficients, we shall therefore investigate the local stability of the above system in the neighborhood of 𝑟=𝑟0. For this, let us assume that the perturbations have a periodic form in the neighborhood of 𝑟=𝑟0, as 𝑓exp(𝑖𝑘𝑟),(48) where 𝑘 is the wave number. For this type of dependence, we have 𝑑𝑑𝑟𝑖𝑘.(49) Equations (43)–(45) upon using (49) yield 𝑧=0,𝜃=0,𝑟=𝑚𝑟.(50) Now, substituting (49) in (37)–(42), (46), (47) and using (50), we obtain a system of algebraic equations (with coefficients, in the vicinity of 𝑟=𝑟0) for amplitudes, marked with bars, which can be put in the matrix notation as following;1𝜎𝜎2𝜇+𝑄2𝜔0𝑖𝑘0𝜌0𝑀𝑟𝜇00𝑃𝜎000000𝜎02𝜌0𝑑𝑀𝑧𝑑𝑟𝑖𝑘𝐵𝑧00𝑑𝑀𝑧𝑑𝑟0𝑖𝑘𝑀𝑧2𝑃𝜎+𝑁2𝛽𝐻𝑧𝑀𝑧0𝑖𝑘𝑀𝑟20𝑃2𝜎+𝑅0𝑑𝑀𝑧𝑑𝑟0𝑖𝑘𝑀𝑟2𝑇0𝜎+𝛼𝑢𝑟𝑢𝜃𝑢𝑧𝑚𝑟𝑚𝜃𝑚𝑧=0.(51)

Here, 𝐵𝑧=(𝑀𝑧+𝐻𝑧), 𝑄=𝑘2𝑐24𝜋𝐺𝜌0, 𝑁=𝛼+𝛽(𝑀𝑧2+𝐻𝑧𝑀𝑧), 𝑅=(𝛼+𝛽𝑀𝑧𝐻𝑧), and 𝑇=(𝑀𝑟𝑀𝑧2𝑀𝑟𝐻𝑧).

For the nontrivial solution of the system (51), the determinant of the matrix of coefficients should vanish. Thus, on expanding the coefficient matrix and equating the real part equal to zero, we get the following dispersion relation: 𝜎6+𝐵1𝜎5+𝐵2𝜎4+𝐵3𝜎3+𝐵4𝜎2+𝐵5𝜎+𝐵6=0,(52) where, 𝐵1𝐵=(𝛼+𝑁+𝑅),2=8𝑃𝜔+4𝑄𝑃2+4𝑁𝑅+4𝛼𝑁+4𝛼𝑅+𝑘2𝜇0𝜌0𝐵𝑧𝑀𝑧4𝛽𝑇𝐻𝑧𝑀𝑟,𝐵3=𝜇𝑜𝜌04𝑁𝑄+4𝑄𝑅+4𝛼𝑄𝛼𝑃2+4𝑁𝑅+8𝜔𝑁𝑃+8𝜔𝑃𝑅+8𝛼𝜔𝑃+𝑘2𝜇0𝜌0𝑅𝐵𝑧𝑀𝑧+𝛼𝑘2𝜇𝑜𝜌0𝑅𝐵𝑧𝑀𝑧+𝑘2𝛽𝜇𝑜𝜌0𝑅𝐵𝑧𝑀𝑧𝑀2𝑟4𝛽𝑅𝑇𝑀𝑟𝐻𝑧,𝐵4=𝑃2𝑄2𝜔𝑃3+4𝜌0𝑁𝑄𝑅+4𝛼𝑁𝑄+4𝛼𝜌0𝑄𝑅+8𝜔𝑁𝑃𝑅+8𝛼𝜔𝑁𝑃+8𝛼𝜔𝑃𝑅4𝛽𝑄𝑇𝑀𝑟𝐻𝑧8𝜔𝛽𝑃𝑇𝑀𝑟𝐻𝑧+𝑘2𝜇0𝜌0𝑃2𝑀2𝑟+𝑘2𝜇0𝜌0𝑄𝐵𝑧𝑀𝑧+𝛼𝑘2𝜇0𝜌0𝑅𝐵𝑧𝑀𝑧+2𝜔𝑘2𝜇0𝜌0𝑃𝐵𝑧𝑀𝑧+𝛽𝑘2𝜇0𝜌0𝑅𝐻𝑧𝑀2𝑟,𝐵5=2𝛼𝜔𝑃3𝛼𝑃2𝑄+4𝛼𝑅𝑁𝑄+8𝛼𝜔𝑁𝑃𝑅4𝛽𝑄𝑅𝑇𝑀𝑟𝐻𝑧8𝜔𝛽𝑃𝑅𝑇𝑀𝑟𝐻𝑧+𝑘2𝜇0𝜌0𝑄𝑅𝐵𝑧𝑀𝑧+𝛼𝑘2𝜇0𝜌0𝑄𝐵𝑧𝑀𝑧+𝛼𝑘2𝜇0𝜌0𝑃2𝑀2𝑟+𝛽𝑘2𝜇0𝜌0𝑄𝐵𝑧𝐻𝑧𝑀2𝑟+2𝜔𝑘2𝜇0𝜌0𝑃𝑅𝐵𝑧𝑀𝑧+2𝛼𝜔𝑘2𝜇0𝜌0𝑃𝐵𝑧𝑀𝑧2𝜔𝛽𝑘2𝜇0𝜌0𝑃𝐵𝑧𝐻𝑧𝑀2𝑟,𝐵6=𝜇0𝑘24𝜌0𝑅𝐵𝑧𝛼𝑀𝑧+𝛽𝐻𝑧𝑀2𝑟(𝑄+𝑃).(53) Equation (52) is of sixth degree in 𝜎 with all the coefficients real and the coefficients of 𝜎6 and 𝜎5 are clearly positive while the coefficients of 𝜎4, 𝜎3, 𝜎2, 𝜎, and the constant term may be negative depending on 𝐵2, 𝐵3, 𝐵4, 𝐵5, and 𝐵6. As per the criteria of Guillemin [20] for the signs of the roots, we get that if 𝐵6 is negative, then the constant term of (52) is negative, which is sufficient to give the required condition of instability. Therefore, we obtain the following condition for the onset of gravitational instability for the present problem: 𝑘2𝑐2𝑑+2𝜔𝑑𝑟𝜔𝑟2<4𝜋𝐺𝜌0,(54) which is the same result as obtained by Bel and Schatzman [9].

5. Conclusions

In the present analysis, we have investigated the effect of ferromagnetization in the presence of nonuniform magnetic field on the gravitational instability of a homogenous, ferromagnetic dust cloud in the presence of nonuniform rotation. The inequality (54) implies that the Bel and Schatzman criterion determines the gravitational instability of this general problem, thus we can conclude that the ferromagnetism in the presence of magnetic field has no effect on the gravitational instability on the self-gravitating cloud. Comparing this result with result obtained by Dhiman and Dadwal [10] for the gravitational instability of a homogeneous gaseous medium in the presence of nonuniform magnetic field for nonferromagnetic case, it appears that the effect of magnetic field on the criterion for instability has been marginalized by the magnetic polarizability of ferrofluid. Further, from inequality (54) for 𝜔=0, we can deduce that gravitational instability is governed by the Jeans criterion, in the absence of nonuniform rotation.

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