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

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 [1–8] 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=(𝑢_𝑥,𝑢_𝑦,𝑢_𝑧) be the velocity, 𝑀^→=(𝑀_𝑥,𝑀_𝑦,𝑀_𝑧) be the ferrofluid magnetization under the magnetic field 𝐻^→=(𝐻_𝑥,𝐻_𝑦,𝐻_𝑧). For the treatment of this physical configuration for gravitational instability, the conventional set of ferrohydrodynamic equations are given by (cf. [16, 17]) 𝑑^→𝑢𝑑𝑡=−grad𝑝+𝜌grad𝜙+𝜇₀^→𝑀⋅∇𝐻^→,(1)𝜕𝜌𝜕𝑡+^→𝑢⋅grad𝜌+𝜌∇.^→𝑢=0,(2)∇²𝜙=−4𝜋𝐺𝜌−𝜌₀.(3) The ferrofluid magnetization 𝑀^→ satisfies Shliomis’ [16, 17] equation of magnetization, which is given by 𝑑𝑀^→𝑑𝑡=21∇×^→𝑢×𝑀^→−𝛼𝑀^→−𝑀^→₀−𝛽^→𝑀×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/𝑘_𝑏𝑇 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 𝑀^→₀=(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=𝑛𝑀coth𝜓−1𝜓‖‖𝐻𝐻⁰₀‖‖,(5) where 𝑀 is the magnetic moment of single ferromagnetic particle and the nondimensional quantity 𝜓 is given by 𝜓=𝜇₀(𝑀‖𝐻₀‖/𝑘_𝑏𝑇_𝑏), where 𝑘_𝑏 is the Boltzman constant and 𝑇_𝑏 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^→=𝜇₀^→𝐻+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=(𝑢_𝑟,𝑢_𝜃,𝑢_𝑧) and 𝑀^→=(𝑀_𝑟,𝑀_𝜃,𝑀_𝑧) 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𝜕𝑢^𝑟𝜕𝑡+𝑢^𝑟𝜕𝑢_𝑟=𝜌𝜕𝜙𝜕𝑟−𝑢𝑟^𝜃2𝜕𝑟−𝜕𝜕𝑝𝑟+𝜇⁰𝑀^𝑟𝜕𝐻_𝑟𝜕𝑟−𝑀^𝜃𝐻_𝜃𝜌𝜕𝑢_𝜃𝑟,𝜕𝑡+𝑢^𝑟𝜕𝜕𝑟𝑢^𝜃+𝑢^𝑟𝑢_𝜃=𝜇₀𝑟2𝑀^𝑟𝜕𝐻_𝜃𝜕𝑟+𝐻^𝑟𝜕𝑀_𝜃𝜕𝑟+𝑀^𝜃𝜕𝐻_𝑟−𝐻_𝜃𝜕𝑟𝜕𝑀_𝑟𝜕𝑟+2𝑀^𝜃𝐻_𝑟𝜌𝜕𝑢_𝑧𝑟,𝜕𝑡+𝑢^𝑟𝜕𝑢_𝑧=𝜇𝜕𝑟₀2𝑀^𝑟𝜕𝐻_𝑧𝜕𝑟+𝐻^𝑟𝜕𝑀_𝑧𝜕𝑟+𝑀^𝑧𝜕𝐻_𝑟𝜕𝑟−𝐻^𝑧𝜕𝑀_𝑟+𝑀𝜕𝑟_𝑧𝐻_𝑟−𝐻_𝑧𝑀_𝑟𝜕𝑀_𝑟𝑟,𝜕𝑡+𝑢^𝑟𝜕𝑀_𝑟𝜕𝑟+12𝑀^𝜃𝜕𝜕𝑢𝑟^𝜃+𝑢𝑟^𝜃+𝑀^𝑧𝜕𝑢_𝑧+𝛼𝑀_𝑟𝜕𝑟−𝑀₀−+𝑀𝛽_𝑧𝑀_𝜃𝑀_𝑟𝐻_𝜃−𝐻_𝑟𝑀_𝜃𝜕𝑀_𝜃𝐻_𝑟𝑀_𝑧−𝑀_𝑟𝐻_𝑧=0,𝜕𝑡+𝑢^𝑟𝜕𝑀_𝜃+𝛼𝑀𝜕𝑟_𝜃−12𝑀^𝑟𝜕𝜕𝑢𝑟^𝜃+𝑢𝑟^𝜃−𝑀₀−+𝑀𝛽_𝑟𝑀_𝑧𝑀_𝜃𝐻_𝑧−𝐻_𝜃𝑀_𝑧𝜕𝑀_𝑧𝑀_𝑟𝐻_𝜃−𝐻_𝑟𝑀_𝜃=0,𝜕𝑡+𝑢^𝑟𝜕𝑀_𝑧𝜕𝑟−12𝑀^𝑟𝜕𝑢_𝑧𝜕𝑟+𝛼𝑀^𝑧+𝛽𝑀_𝑟−𝑀₀−𝑀_𝜃𝐻_𝑟𝑀_𝑧−𝑀_𝑟𝐻_𝑧𝑀_𝜃𝐻𝜕𝐻_𝑍𝑧−𝐻_𝜃𝑀_𝑧=0,𝜕𝐻_𝜃𝜕𝑟=0,𝜕𝐻_𝑟𝜕𝑟+𝐻𝑟^𝜃=0,𝜕𝜌𝜕𝑟+𝐻𝑟^𝑟=−𝜕𝜕𝑀𝑟^𝑟+𝑀𝑟^𝑟,𝜕𝑡+^→𝑢⋅grad𝜌+𝜌∇⋅^→∇𝑢=0,₂𝜙=−4𝜋𝐺𝜌−𝜌₀.(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)𝑝;=𝑝₀𝐻^→=0,0,𝐻_𝑧;𝑀^→=𝑀_𝑟,0,𝑀_𝑧;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𝜕𝜕𝜙𝑟⁰+𝑟𝜔²(12)𝑀_𝑟𝜕𝐻_𝑧𝜕𝑟−𝐻^𝑧𝜕𝜕𝑀𝑟^𝑟+𝑀𝑟^𝑟=0(13)𝛼𝑀_𝑟−𝑀₀+𝛽𝑀_𝑧𝑀_𝑟𝐻_𝑧=0(14)−12𝑀^𝑟𝜕𝜕𝑢𝑟^𝜃+𝑢𝑟^𝜃+𝛼𝑀⁰=0(15)𝛼𝑀_𝑧−𝑀₀−𝛽𝑀^2𝑟𝐻_𝑧=0(16)𝜕𝐻_𝑧𝜕𝑟=0(17)𝜕𝜕𝑀𝑟^𝑟+𝑀𝑟^𝑟=0(18)1𝑟𝜕𝜕𝑟𝑟𝜕𝜕𝜙𝑟⁰=0(19) Equation (17) implies that 𝐻_𝑧=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; 𝑢𝑝^→_=𝑢^𝑟,𝑟𝜔+𝑢^𝜃,𝑢^𝑧;𝐻^→=ℎ^𝑟,ℎ^𝜃,𝐻_𝑧+ℎ^𝑧;=𝑝₀+𝜙_𝛿𝑝𝑀^→=𝑀_𝑟+𝑚^𝑟,𝑚^𝜃,𝑀_𝑧+𝑚^𝑧;=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𝜕𝜕𝑢𝑡^𝑟−2𝜔𝑢^𝜃=𝜌⁰𝜕𝛿𝜙𝜕𝑟−𝜕𝜕𝛿𝑟𝑝+𝜇⁰𝑀^𝑟𝜕ℎ_𝑟𝜕𝑟,(22)𝜌₀𝜕𝜕𝑢𝑡^𝜃+𝑢^𝑟𝜕𝜕𝑟(𝑟𝜔)+𝑢^𝑟𝜔=𝜇⁰2𝑀^𝑟𝜕ℎ_𝜃𝜕𝑟−ℎ^𝜃𝜕𝑀_𝑟𝜕𝑟,(23)𝜌₀𝜕𝑢_𝑧𝜕𝑡=𝜇2⁰ℎ^𝑟𝜕𝑀_𝑧𝜕𝑟+𝑀^𝑧1𝑟𝜕𝜕𝑟𝑟ℎ^𝑟−𝐻_𝑧1𝑟𝜕𝜕𝑟𝑟𝑚^𝑟,(24)𝜕𝑚_𝑟𝜕𝑡+𝑢^𝑟𝜕𝑀_𝑟𝜕𝑟+21𝑀^𝑧𝜕𝑢_𝑧𝜕𝑟+𝑚^𝜃𝜕𝜕𝑟(𝑟𝜔)+𝜔+𝛼𝑚^𝑟+𝛽−𝑀^2𝑧ℎ_𝑟+𝐻_𝑧𝑀_𝑧𝑚_𝑟+𝑀_𝑧𝑀_𝑟ℎ_𝑧+𝐻_𝑧𝑀_𝑟𝑚_𝑧=0,(25)𝜕𝑚_𝜃𝜕𝑡−12𝑚^𝑟𝜕𝜕𝑟(𝑟𝜔)+𝜔+𝑀^𝑟1𝑟𝜕𝜕𝑟𝑟𝑢^𝜃+𝛼𝑚_𝜃+𝛽𝐻_𝑧𝑀_𝑧𝑚_𝜃−𝑀^2𝑧ℎ_𝜃−𝑀^2𝑟ℎ_𝜃=0,(26)𝜕𝑚_𝑧𝜕𝑡+𝑢^𝑟𝜕𝑀_𝑧𝜕𝑟−12𝑀^𝑟𝜕𝑢_𝑧𝜕𝑟+𝛼𝑚^𝑧+𝛽𝑀_𝑟𝑀_𝑧ℎ_𝑟−2𝑀_𝑟𝐻_𝑧𝑚_𝑟−𝑀^2𝑟ℎ_𝑧=0,(27)𝜕ℎ_𝑧𝜕𝑟=0,(28)𝜕𝜕ℎ𝑟^𝜃+ℎ𝑟^𝜃=0,(29)1𝑟𝜕𝜕𝑟𝑟𝑚^𝑟=−1𝑟𝜕𝜕𝑟𝑟ℎ^𝑟,(30)𝜕𝜕𝑡𝛿𝜌+𝜌⁰𝜕𝜕𝑢𝑟^𝑟+𝑢𝑟^𝑟=0,(31)1𝑟𝜕𝜕𝑟𝑟𝜕𝜕𝛿𝑟𝜙=−4𝜋𝐺𝛿𝜌,(32) where (𝑢^𝑟,𝑢^𝜃,𝑢^𝑧), (ℎ^𝑟,ℎ^𝜃,ℎ^𝑧), (𝑚^𝑟,𝑚^𝜃,𝑚^𝑧), 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𝜕𝜕𝑢𝑡^𝑟−2𝜔𝑢^𝜃=𝜌⁰𝜕𝛿𝜙𝜕𝑟−𝑐²𝜕𝛿𝜌𝜕𝑟+𝜇⁰𝑀^𝑟𝜕ℎ_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 𝜕𝜕𝑡≡𝜎,𝜕𝜕𝑟𝑓(𝑟)=𝑑𝑑𝑟𝑓(𝑟).(36) Using the above dependence, the perturbations equations (34) and (23)–(32) assume the following forms: 𝜎𝑢_𝑟−2𝜔𝑢_𝜃−𝑑𝛿𝜙𝑑𝑟+𝜌𝑐²₀𝑑𝛿𝜌𝑑𝑟−𝜇𝜌⁰₀𝑀^𝑟𝑑ℎ_𝑟𝑑𝑟=0,(37)𝜎𝑢_𝜃+𝑢_𝑟𝑑𝑑𝑟(𝑟𝜔)+𝑢^𝑟𝜔−12𝜇𝜌⁰₀𝑀^𝑟𝑑ℎ_𝜃𝑑𝑟−ℎ^𝜃𝑑𝑀_𝑟𝑑𝑟=0,(38)𝜎𝑢_𝑧−12𝜇𝜌⁰₀ℎ^𝑟𝑑𝑀_𝑧𝑑𝑟+𝑀^𝑧1𝑟𝑑𝑑𝑟𝑟ℎ^𝑟−𝐻_𝑧1𝑟𝑑𝑑𝑟𝑟𝑚^𝑟=0,(39)(𝜎+𝛼)𝑚_𝑟+𝑢_𝑟𝑑𝑀_𝑟𝑑𝑟+12𝑀^𝑧𝑑𝑢_𝑧+𝛽−𝑀_2𝑧𝑑𝑟+𝑚^𝜃𝑑𝑑𝑟(𝑟𝜔)+𝜔ℎ_𝑟+𝐻_𝑧𝑀_𝑧𝑚_𝑟+𝑀_𝑧𝑀_𝑟ℎ_𝑧+𝐻_𝑧𝑀_𝑟𝑚_𝑧=0,(40)(𝜎+𝛼)𝑚_𝜃−12𝑚^𝑟𝑑𝑑𝑟(𝑟𝜔)+𝜔+𝑀^𝑟1+𝛽𝐻_𝑧𝑟𝑑𝑑𝑟𝑟𝑢^𝜃𝑀_𝑧𝑚_𝜃−𝑀^2𝑧ℎ_𝜃−𝑀^2𝑟ℎ_𝜃=0,(41)(𝜎+𝛼)𝑚_𝑧+𝑢_𝑟𝑑𝑀_𝑧𝑑𝑟−21𝑀^𝑟𝑑𝑢_𝑧+𝛽𝑀_𝑟𝑑𝑟𝑀_𝑧ℎ_𝑟−2𝑀_𝑟𝐻_𝑧𝑚_𝑟−𝑀^2𝑟ℎ_𝑧=0,(42)𝑑ℎ_𝑧𝑑𝑟=0,(43)1𝑟𝑑𝑑𝑟𝑟ℎ^𝜃=0,(44)1𝑟𝑑𝑑𝑟𝑟𝑚^𝑟=−1𝑟𝑑𝑑𝑟𝑟ℎ^𝑟,(45)𝜎𝛿𝜌+𝜌⁰𝑟𝑑𝑑𝑟𝑟𝑢^𝑟=0,(46)1𝑟𝑑𝑑𝑟𝑟𝑑𝑑𝛿𝑟𝜙=−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 𝑑𝑑𝑟≡−𝑖𝑘.(49) Equations (43)–(45) upon using (49) yield ℎ_𝑧=0,ℎ_𝜃=0,ℎ_𝑟=−𝑚_𝑟.(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;⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣1𝜎𝜎²+𝑄−2𝜔0−𝑖𝑘𝜇⁰𝜌₀𝑃0𝜎0𝜎0𝜇₀0𝑀_𝑟0000𝑑𝑀_𝑧2𝜌₀𝑑𝑑𝑀𝑟^𝑧−𝑖𝑘𝐵^𝑧00𝑑𝑟0−𝑖𝑘𝑀2^𝑧𝜎+𝑁𝑃2𝛽𝐻^𝑧𝑀_𝑧0−𝑖𝑘𝑀^𝑟𝑑𝑀_𝑧20𝑃2𝜎+𝑅0𝑑𝑟0−𝑖𝑘2𝑀^𝑟𝑇0𝜎+𝛼⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑢_𝑟𝑢_𝜃𝑢_𝑧𝑚_𝑟𝑚_𝜃𝑚_𝑧⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦=0.(51)

Here, 𝐵_𝑧=(𝑀_𝑧+𝐻_𝑧), 𝑄=𝑘²𝑐²−4𝜋𝐺𝜌₀, 𝑁=𝛼+𝛽(𝑀_𝑧2+𝐻_𝑧𝑀_𝑧), 𝑅=(𝛼+𝛽𝑀_𝑧𝐻_𝑧), and 𝑇=(𝑀_𝑟𝑀_𝑧−2𝑀_𝑟𝐻_𝑧).

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, 𝐵𝐵₁=(𝛼+𝑁+𝑅),₂=8+𝑃𝑘𝜔₂+4𝑄−𝑃²+4𝑁𝑅+4𝛼𝑁+4𝛼𝑅𝜇₀𝜌₀𝐵_𝑧𝑀_𝑧−4𝛽𝑇𝐻_𝑧𝑀_𝑟𝐵₃,=𝜇^𝑜𝜌₀+8𝛼4𝜔𝑁𝑃𝑄++𝑘4₂𝑄𝑅+4𝛼𝑄−𝛼𝑃²+4𝑁𝑅+8𝜔𝑁𝑃+8𝜔𝑃𝑅𝜇₀𝜌₀𝑅𝐵_𝑧𝑀_𝑧+𝛼𝑘²𝜇_𝑜𝜌₀𝑅𝐵_𝑧𝑀_𝑧+𝑘²𝛽𝜇^𝑜𝜌₀𝑅𝐵_𝑧𝑀_𝑧𝑀^2𝑟−4𝛽𝑅𝑇𝑀_𝑟𝐻_𝑧𝐵₄,=+−8𝑃𝜔²𝑁𝑄𝑃−𝑅2+𝜔𝑃8𝛼³𝜔+𝑁4𝑃𝜌₀+𝑁8𝑄𝛼𝑅𝜔𝑃+𝑅4𝛼−𝑁4𝑄𝛽𝑄+𝑇4𝑀𝛼𝜌_0𝑟𝑄𝑅𝐻_𝑧−8𝜔𝛽𝑃𝑇𝑀_𝑟𝐻_𝑧+𝑘²𝜇₀𝜌₀𝑃²𝑀^2𝑟+𝑘²𝜇₀𝜌₀𝑄𝐵_𝑧𝑀_𝑧+𝛼𝑘²𝜇₀𝜌₀𝑅𝐵_𝑧𝑀_𝑧+2𝜔𝑘²𝜇₀𝜌₀𝑃𝐵_𝑧𝑀_𝑧+𝛽𝑘²𝜇₀𝜌₀𝑅𝐻_𝑧𝑀^2𝑟𝐵₅,=−−42𝛽𝛼𝑄𝜔𝑅𝑃𝑇³𝑀−𝛼_𝑟𝑃²𝑄+4𝛼𝑅𝑁𝑄+8𝛼𝜔𝑁𝑃𝑅𝐻_𝑧−8𝜔𝛽𝑃𝑅𝑇𝑀_𝑟𝐻_𝑧+𝑘²𝜇₀𝜌₀𝑄𝑅𝐵_𝑧𝑀_𝑧+𝛼𝑘²𝜇₀𝜌₀𝑄𝐵_𝑧𝑀_𝑧+𝛼𝑘²𝜇₀𝜌₀𝑃²𝑀^2𝑟+𝛽𝑘²𝜇₀𝜌₀𝑄𝐵_𝑧𝐻_𝑧𝑀^2𝑟+2𝜔𝑘²𝜇₀𝜌₀𝑃𝑅𝐵_𝑧𝑀_𝑧+2𝛼𝜔𝑘²𝜇₀𝜌₀𝑃𝐵_𝑧𝑀_𝑧−2𝜔𝛽𝑘²𝜇₀𝜌₀𝑃𝐵_𝑧𝐻_𝑧𝑀^2𝑟𝐵₆,=𝜇⁰𝑘²4𝜌₀𝑅𝐵_𝑧𝛼𝑀_𝑧+𝛽𝐻_𝑧𝑀^2𝑟(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: 𝑘²𝑐²+2𝜔𝑑𝑑𝑟𝜔𝑟²<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.