International Scholarly Research Notices

International Scholarly Research Notices / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 104941 |

Joginder S. Dhiman, Rekha Dadwal, "On the Magnetogravitational Instability of a Ferromagnetic Dust Cloud in the Presence of Nonuniform Rotation", International Scholarly Research Notices, vol. 2012, Article ID 104941, 6 pages, 2012.

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

Academic Editor: S. Bogovalov
Received10 Apr 2012
Accepted06 Jun 2012
Published16 Aug 2012


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=(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]) 𝑑→𝑢𝑑𝑡=−grad𝑝+𝜌grad𝜙+𝜇0→𝑀⋅∇→𝐻,(1)𝜕𝜌+𝜕𝑡→𝑢⋅grad𝜌+𝜌∇.→𝑢=0,(2)∇2𝜙=−4𝜋𝐺𝜌−𝜌0.(3) The ferrofluid magnetization →𝑀 satisfies Shliomis’ [16, 17] equation of magnetization, which is given by 𝑑→𝑀=1𝑑𝑡2∇×→𝑢×→𝑀−𝛼→𝑀−→𝑀0−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/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=𝑛𝑀1coth𝜓−𝜓𝐻0‖‖𝐻0‖‖,(5) where 𝑀 is the magnetic moment of single ferromagnetic particle and the nondimensional quantity 𝜓 is given by 𝜓=𝜇0(𝑀‖𝐻0‖/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=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=(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𝜕𝑢𝑟−𝑢𝜕𝑟𝜃2𝑟=𝜌𝜕𝜙−𝜕𝑟𝜕𝑝𝜕𝑟+𝜇0𝑀𝑟𝜕𝐻𝑟−𝑀𝜕𝑟𝜃𝐻𝜃𝑟,𝜌𝜕𝑢𝜃𝜕𝑡+𝑢𝑟𝜕𝑢𝜕𝑟𝜃+𝑢𝑟𝑢𝜃𝑟=𝜇02𝑀𝑟𝜕𝐻𝜃𝜕𝑟+𝐻𝑟𝜕𝑀𝜃𝜕𝑟+𝑀𝜃𝜕𝐻𝑟𝜕𝑟−𝐻𝜃𝜕𝑀𝑟+𝜕𝑟2𝑀𝜃𝐻𝑟𝑟,𝜌𝜕𝑢𝑧𝜕𝑡+𝑢𝑟𝜕𝑢𝑧=𝜇𝜕𝑟02𝑀𝑟𝜕𝐻𝑧𝜕𝑟+𝐻𝑟𝜕𝑀𝑧𝜕𝑟+𝑀𝑧𝜕𝐻𝑟𝜕𝑟−𝐻𝑧𝜕𝑀𝑟+𝑀𝜕𝑟𝑧𝐻𝑟−𝐻𝑧𝑀𝑟𝑟,𝜕𝑀𝑟𝜕𝑡+𝑢𝑟𝜕𝑀𝑟+1𝜕𝑟2𝑀𝜃𝜕𝑢𝜃+𝑢𝜕𝑟𝜃𝑟+𝑀𝑧𝜕𝑢𝑧𝑀𝜕𝑟+𝛼𝑟−𝑀0𝑀+𝛽𝜃𝑀𝑟𝐻𝜃−𝐻𝑟𝑀𝜃−𝑀𝑧𝐻𝑟𝑀𝑧−𝑀𝑟𝐻𝑧=0,𝜕𝑀𝜃𝜕𝑡+𝑢𝑟𝜕𝑀𝜃−1𝜕𝑟2𝑀𝑟𝜕𝑢𝜃+𝑢𝜕𝑟𝜃𝑟𝑀+𝛼𝜃−𝑀0𝑀+𝛽𝑧𝑀𝜃𝐻𝑧−𝐻𝜃𝑀𝑧−𝑀𝑟𝑀𝑟𝐻𝜃−𝐻𝑟𝑀𝜃=0,𝜕𝑀𝑧𝜕𝑡+𝑢𝑟𝜕𝑀𝑧−1𝜕𝑟2𝑀𝑟𝜕𝑢𝑧𝑀𝜕𝑟+𝛼𝑧−𝑀0𝑀+𝛽𝑟𝐻𝑟𝑀𝑧−𝑀𝑟𝐻𝑧−𝑀𝜃𝑀𝜃𝐻𝑍−𝐻𝜃𝑀𝑧=0,𝜕𝐻𝑧𝜕𝑟=0,𝜕𝐻𝜃+𝐻𝜕𝑟𝜃𝑟=0,𝜕𝐻𝑟+𝐻𝜕𝑟𝑟𝑟=−𝜕𝑀𝑟+𝑀𝜕𝑟𝑟𝑟,𝜕𝜌+𝜕𝑡→𝑢⋅grad𝜌+𝜌∇⋅→𝑢∇=0,2𝜙=−4𝜋𝐺𝜌−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);→𝐻=0,0,𝐻𝑧;→𝑀=𝑀𝑟,0,𝑀𝑧𝑝=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𝜕𝑟−𝐻𝑧𝜕𝑀𝑟+𝑀𝜕𝑟𝑟𝑟=0(13)𝛼𝑀𝑟−𝑀0+𝛽𝑀𝑧𝑀𝑟𝐻𝑧=0(14)−12𝑀𝑟𝜕𝑢𝜃+𝑢𝜕𝑟𝜃𝑟+𝛼𝑀0=0(15)𝛼𝑀𝑧−𝑀0−𝛽𝑀2𝑟𝐻𝑧=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; â†’ğ‘¢î…ž=î€·ğ‘¢î…žğ‘Ÿ,𝑟𝜔+ğ‘¢î…žğœƒ,ğ‘¢î…žğ‘§î€¸;â†’ğ»î…ž=î€·â„Žî…žğ‘Ÿ,â„Žî…žğœƒ,𝐻𝑧+â„Žî…žğ‘§î€¸;ğ‘î…ž=𝑝0+𝛿𝑝→𝑀=𝑀𝑟+ğ‘šî…žğ‘Ÿ,ğ‘šî…žğœƒ,𝑀𝑧+ğ‘šî…žğ‘§î€¸;ğœ™î…ž=𝜙0+𝛿𝜙;ğœŒî…ž=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𝜕𝑡−2𝜔𝑢𝜃=𝜌0𝜕𝛿𝜙−𝜕𝑟𝜕𝛿𝑝𝜕𝑟+𝜇0î‚µğ‘€ğ‘Ÿğœ•â„Žğ‘Ÿî‚¶,𝜕𝑟(22)𝜌0𝜕𝑢𝜃𝜕𝑡+𝑢𝑟𝜕𝜕𝑟(𝑟𝜔)+𝑢𝑟𝜔=𝜇02î‚µğ‘€ğ‘Ÿğœ•â„Žğœƒğœ•ğ‘Ÿâˆ’â„Žğœƒğœ•ğ‘€ğ‘Ÿî‚¶,𝜕𝑟(23)𝜌0𝜕𝑢𝑧=𝜇𝜕𝑡02î‚µâ„Žğ‘Ÿğœ•ğ‘€ğ‘§ğœ•ğ‘Ÿ+𝑀𝑧1ğ‘Ÿğœ•î€·ğœ•ğ‘Ÿğ‘Ÿâ„Žğ‘Ÿî€¸âˆ’ğ»ğ‘§1𝑟𝜕𝜕𝑟𝑟𝑚𝑟,(24)𝜕𝑚𝑟𝜕𝑡+𝑢𝑟𝜕𝑀𝑟+1𝜕𝑟2𝑀𝑧𝜕𝑢𝑧𝜕𝑟+𝑚𝜃𝜕𝜕𝑟(𝑟𝜔)+𝜔+𝛼𝑚𝑟+𝛽−𝑀2ğ‘§â„Žğ‘Ÿ+𝐻𝑧𝑀𝑧𝑚𝑟+ğ‘€ğ‘§ğ‘€ğ‘Ÿâ„Žğ‘§+𝐻𝑧𝑀𝑟𝑚𝑧=0,(25)𝜕𝑚𝜃−1𝜕𝑡2𝑚𝑟𝜕𝜕𝑟(𝑟𝜔)+𝜔+𝑀𝑟1𝑟𝜕𝜕𝑟𝑟𝑢𝜃+𝛼𝑚𝜃𝐻+𝛽𝑧𝑀𝑧𝑚𝜃−𝑀2ğ‘§â„Žğœƒâˆ’ğ‘€2ğ‘Ÿâ„Žğœƒî€¸=0,(26)𝜕𝑚𝑧𝜕𝑡+𝑢𝑟𝜕𝑀𝑧−1𝜕𝑟2𝑀𝑟𝜕𝑢𝑧𝜕𝑟+𝛼𝑚𝑧𝑀+ğ›½ğ‘Ÿğ‘€ğ‘§â„Žğ‘Ÿâˆ’2𝑀𝑟𝐻𝑧𝑚𝑟−𝑀2ğ‘Ÿâ„Žğ‘§î€¸=0,(27)ğœ•â„Žğ‘§ğœ•ğ‘Ÿ=0,(28)î‚µğœ•â„Žğœƒ+â„Žğœ•ğ‘Ÿğœƒğ‘Ÿî‚¶=0,(29)1𝑟𝜕𝜕𝑟𝑟𝑚𝑟1=âˆ’ğ‘Ÿğœ•î€·ğœ•ğ‘Ÿğ‘Ÿâ„Žğ‘Ÿî€¸,(30)𝜕𝜕𝑡𝛿𝜌+𝜌0𝜕𝑢𝑟+𝑢𝜕𝑟𝑟𝑟=0,(31)1𝑟𝜕𝑟𝜕𝑟𝜕𝛿𝜙𝜕𝑟=−4𝜋𝐺𝛿𝜌,(32) where (ğ‘¢î…žğ‘Ÿ,ğ‘¢î…žğœƒ,ğ‘¢î…žğ‘§), (â„Žî…žğ‘Ÿ,â„Žî…žğœƒ,â„Žî…žğ‘§), (ğ‘šî…žğ‘Ÿ,ğ‘šî…žğœƒ,ğ‘šî…žğ‘§), 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𝜕𝑡−2𝜔𝑢𝜃=𝜌0𝜕𝛿𝜙𝜕𝑟−𝑐2𝜕𝛿𝜌𝜕𝑟+𝜇0î‚µğ‘€ğ‘Ÿğœ•â„Žğ‘Ÿî‚¶.𝜕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(ğœŽğ‘¡),(35) where ğœŽ is frequency of the perturbation.

For this type of dependence of perturbation on 𝑟 and 𝑡, we have ğœ•ğœ•ğœ•ğ‘¡â‰¡ğœŽ,𝑑𝜕𝑟𝑓(𝑟)=𝑑𝑟𝑓(𝑟).(36) Using the above dependence, the perturbations equations (34) and (23)–(32) assume the following forms: ğœŽğ‘¢ğ‘Ÿâˆ’2𝜔𝑢𝜃−𝑑𝛿𝜙+𝑐𝑑𝑟2𝜌0𝑑𝛿𝜌−𝜇𝑑𝑟0𝜌0î‚µğ‘€ğ‘Ÿğ‘‘â„Žğ‘Ÿî‚¶ğ‘‘ğ‘Ÿ=0,(37)ğœŽğ‘¢ğœƒ+𝑢𝑟𝑑𝑑𝑟(𝑟𝜔)+𝑢𝑟1𝜔−2𝜇0𝜌0î‚µğ‘€ğ‘Ÿğ‘‘â„Žğœƒğ‘‘ğ‘Ÿâˆ’â„Žğœƒğ‘‘ğ‘€ğ‘Ÿî‚¶ğ‘‘ğ‘Ÿ=0,(38)ğœŽğ‘¢ğ‘§âˆ’12𝜇0𝜌0î‚µâ„Žğ‘Ÿğ‘‘ğ‘€ğ‘§ğ‘‘ğ‘Ÿ+𝑀𝑧1ğ‘Ÿğ‘‘î€·ğ‘‘ğ‘Ÿğ‘Ÿâ„Žğ‘Ÿî€¸âˆ’ğ»ğ‘§1𝑟𝑑𝑑𝑟𝑟𝑚𝑟=0,(39)(ğœŽ+𝛼)𝑚𝑟+𝑢𝑟𝑑𝑀𝑟+1𝑑𝑟2𝑀𝑧𝑑𝑢𝑧𝑑𝑟+𝑚𝜃𝑑𝑑𝑟(𝑟𝜔)+𝜔+𝛽−𝑀2ğ‘§â„Žğ‘Ÿ+𝐻𝑧𝑀𝑧𝑚𝑟+ğ‘€ğ‘§ğ‘€ğ‘Ÿâ„Žğ‘§+𝐻𝑧𝑀𝑟𝑚𝑧=0,(40)(ğœŽ+𝛼)𝑚𝜃−12𝑚𝑟𝑑𝑑𝑟(𝑟𝜔)+𝜔+𝑀𝑟1𝑟𝑑𝑑𝑟𝑟𝑢𝜃𝐻+𝛽𝑧𝑀𝑧𝑚𝜃−𝑀2ğ‘§â„Žğœƒâˆ’ğ‘€2ğ‘Ÿâ„Žğœƒî€¸=0,(41)(ğœŽ+𝛼)𝑚𝑧+𝑢𝑟𝑑𝑀𝑧−1𝑑𝑟2𝑀𝑟𝑑𝑢𝑧𝑀𝑑𝑟+ğ›½ğ‘Ÿğ‘€ğ‘§â„Žğ‘Ÿâˆ’2𝑀𝑟𝐻𝑧𝑚𝑟−𝑀2ğ‘Ÿâ„Žğ‘§î€¸=0,(42)ğ‘‘â„Žğ‘§ğ‘‘ğ‘Ÿ=0,(43)1ğ‘Ÿğ‘‘î€·ğ‘‘ğ‘Ÿğ‘Ÿâ„Žğœƒî€¸=0,(44)1𝑟𝑑𝑑𝑟𝑟𝑚𝑟1=âˆ’ğ‘Ÿğ‘‘î€·ğ‘‘ğ‘Ÿğ‘Ÿâ„Žğ‘Ÿî€¸,(45)ğœŒğœŽğ›¿ğœŒ+0𝑟𝑑𝑑𝑟𝑟𝑢𝑟=0,(46)1𝑟𝑑𝑟𝑑𝑟𝑑𝛿𝜙𝑑𝑟=−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 â„Žğ‘§=0,â„Žğœƒ=0,â„Žğ‘Ÿ=−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;⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣1ğœŽî€·ğœŽ2𝜇+𝑄−2𝜔0−𝑖𝑘0𝜌0𝑀𝑟𝜇00ğ‘ƒğœŽ000000ğœŽ02𝜌0𝑑𝑀𝑧𝑑𝑟−𝑖𝑘𝐵𝑧00𝑑𝑀𝑧𝑑𝑟0−𝑖𝑘𝑀𝑧2ğ‘ƒğœŽ+𝑁2𝛽𝐻𝑧𝑀𝑧0−𝑖𝑘𝑀𝑟20𝑃2ğœŽ+𝑅0𝑑𝑀𝑧𝑑𝑟0−𝑖𝑘𝑀𝑟2âŽ¤âŽ¥âŽ¥âŽ¥âŽ¥âŽ¥âŽ¥âŽ¥âŽ¥âŽ¥âŽ¥âŽ¥âŽ¥âŽ¦âŽ¡âŽ¢âŽ¢âŽ¢âŽ¢âŽ¢âŽ¢âŽ¢âŽ¢âŽ¢âŽ¢âŽ¢âŽ£ğ‘‡0ğœŽ+ğ›¼ğ‘¢ğ‘Ÿğ‘¢ğœƒğ‘¢ğ‘§ğ‘šğ‘Ÿğ‘šğœƒğ‘šğ‘§âŽ¤âŽ¥âŽ¥âŽ¥âŽ¥âŽ¥âŽ¥âŽ¥âŽ¥âŽ¥âŽ¥âŽ¥âŽ¦=0.(51)

Here, 𝐵𝑧=(𝑀𝑧+𝐻𝑧), 𝑄=𝑘2𝑐2−4𝜋𝐺𝜌0, 𝑁=𝛼+𝛽(𝑀𝑧2+𝐻𝑧𝑀𝑧), 𝑅=(𝛼+𝛽𝑀𝑧𝐻𝑧), and 𝑇=(𝑀𝑟𝑀𝑧−2𝑀𝑟𝐻𝑧).

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: ğœŽ6+𝐵1ğœŽ5+𝐵2ğœŽ4+𝐵3ğœŽ3+𝐵4ğœŽ2+𝐵5ğœŽ+𝐵6=0,(52) where, 𝐵1𝐵=(𝛼+𝑁+𝑅),2=8𝑃𝜔+4𝑄−𝑃2+4𝑁𝑅+4𝛼𝑁+4𝛼𝑅+𝑘2𝜇0𝜌0𝐵𝑧𝑀𝑧−4𝛽𝑇𝐻𝑧𝑀𝑟,𝐵3=𝜇𝑜𝜌04𝑁𝑄+4𝑄𝑅+4𝛼𝑄−𝛼𝑃2+4𝑁𝑅+8𝜔𝑁𝑃+8𝜔𝑃𝑅+8𝛼𝜔𝑃+𝑘2𝜇0𝜌0𝑅𝐵𝑧𝑀𝑧+𝛼𝑘2𝜇𝑜𝜌0𝑅𝐵𝑧𝑀𝑧+𝑘2𝛽𝜇𝑜𝜌0𝑅𝐵𝑧𝑀𝑧𝑀2𝑟−4𝛽𝑅𝑇𝑀𝑟𝐻𝑧,𝐵4=−𝑃2𝑄−2𝜔𝑃3+4𝜌0𝑁𝑄𝑅+4𝛼𝑁𝑄+4𝛼𝜌0𝑄𝑅+8𝜔𝑁𝑃𝑅+8𝛼𝜔𝑁𝑃+8𝛼𝜔𝑃𝑅−4𝛽𝑄𝑇𝑀𝑟𝐻𝑧−8𝜔𝛽𝑃𝑇𝑀𝑟𝐻𝑧+𝑘2𝜇0𝜌0𝑃2𝑀2𝑟+𝑘2𝜇0𝜌0𝑄𝐵𝑧𝑀𝑧+𝛼𝑘2𝜇0𝜌0𝑅𝐵𝑧𝑀𝑧+2𝜔𝑘2𝜇0𝜌0𝑃𝐵𝑧𝑀𝑧+𝛽𝑘2𝜇0𝜌0𝑅𝐻𝑧𝑀2𝑟,𝐵5=−2𝛼𝜔𝑃3−𝛼𝑃2𝑄+4𝛼𝑅𝑁𝑄+8𝛼𝜔𝑁𝑃𝑅−4𝛽𝑄𝑅𝑇𝑀𝑟𝐻𝑧−8𝜔𝛽𝑃𝑅𝑇𝑀𝑟𝐻𝑧+𝑘2𝜇0𝜌0𝑄𝑅𝐵𝑧𝑀𝑧+𝛼𝑘2𝜇0𝜌0𝑄𝐵𝑧𝑀𝑧+𝛼𝑘2𝜇0𝜌0𝑃2𝑀2𝑟+𝛽𝑘2𝜇0𝜌0𝑄𝐵𝑧𝐻𝑧𝑀2𝑟+2𝜔𝑘2𝜇0𝜌0𝑃𝑅𝐵𝑧𝑀𝑧+2𝛼𝜔𝑘2𝜇0𝜌0𝑃𝐵𝑧𝑀𝑧−2𝜔𝛽𝑘2𝜇0𝜌0𝑃𝐵𝑧𝐻𝑧𝑀2𝑟,𝐵6=𝜇0𝑘24𝜌0𝑅𝐵𝑧𝛼𝑀𝑧+𝛽𝐻𝑧𝑀2𝑟(𝑄+𝑃).(53) Equation (52) is of sixth degree in ğœŽ with all the coefficients real and the coefficients of ğœŽ6 and ğœŽ5 are clearly positive while the coefficients of ğœŽ4, ğœŽ3, ğœŽ2, ğœŽ, 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+2𝜔𝑑𝑟𝜔𝑟2<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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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.

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