EffectofRadiativeHeat …downloads.hindawi.com/journals/isrn.astronomy...2 ISRN Astronomy and Astrophysics Along with this, in above discussed problems, the eﬀect of ﬁnite ion

International Scholarly Research NetworkISRN Astronomy and AstrophysicsVolume 2012, Article ID 420938, 14 pagesdoi:10.5402/2012/420938

Research Article

Effect of Radiative Heat-Loss Function and Finite Larmor RadiusCorrections on Jeans Instability of Viscous Thermally ConductingSelf-Gravitating Astrophysical Plasma

Sachin Kaothekar1, 2 and R. K. Chhajlani1

1 School of Studies in Physics, Vikram University, Madhya Pradesh, Ujjain 456010, India2 Department of Physics, Mahakal Institute of Technology, Madhya Pradesh, Ujjain 456664, India

Correspondence should be addressed to Sachin Kaothekar, [email protected]

Received 3 April 2012; Accepted 15 June 2012

Academic Editors: C. W. Engelbracht, A. Ferrari, F. Fraschetti, I. Goldman, and C. Meegan

Copyright © 2012 S. Kaothekar and R. K. Chhajlani. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

The effect of radiative heat-loss function and finite ion Larmor radius (FLR) corrections on the self-gravitational instability ofinfinite homogeneous viscous plasma has been investigated incorporating the effects of thermal conductivity and finite electricalresistivity for the formation of a star in astrophysical plasma. The general dispersion relation is derived using the normal modeanalysis method with the help of relevant linearized perturbation equations of the problem. Furthermore the wave propagationalong and perpendicular to the direction of external magnetic field has been discussed. Stability of the medium is discussed byapplying Routh Hurwitz’s criterion. We find that the presence of radiative heat-loss function and thermal conductivity modify thefundamental Jeans criterion of gravitational instability into radiative instability criterion. From the curves we see that temperaturedependent heat-loss function, FLR corrections and viscosity have stabilizing effect, while density dependent heat-loss functionhas destabilizing effect on the growth rate of self-gravitational instability. Our result shows that the FLR corrections and radiativeheat-loss functions affect the star formation.

1. Introduction

The problem of self-gravitational instability is widely investi-gated due to its relevance to the fragmentation of interstellarmedium and its role in star formation. Also, the self-gravitational instability of molecular clouds is connected tothe cloud collapse and star formation. Hayashi [1] has dis-cussed the problem of evolution of protostars and discussesthe different phase of formation of protostar in connectionwith variation of temperature and density. Shu et al. [2]have investigated the problem of star formation in molecularclouds and concluded that the star formation occurs mainlyin four phases. In this problem, the self-gravitational insta-bility of dust and gas plays important role. Draine and Mckee[3] have studied the theory of interstellar shocks. Mckee andOstriker [4] have discussed the theory of star formation.They concluded that the key dynamical processes involvedin star formation are turbulence, magnetic fields, and self-gravity. In this connection, the gravitational instability

of infinite homogeneous self-gravitating magnetized androtating plasma is also discussed by Chandrasekhar [5].Several authors (Pacholczyk and Stodolkiewicz [6], Nayyar[7], and Shaikh et al. [8]) have investigated the problemof gravitational instability of plasma with different physicalparameters such as viscosity, finite electrical conductivity,Hall current, thermal conductivity, magnetic field, and rota-tion. Yang et al. [9] have investigated the problem of large-scale gravitational instability and star formation. Borah andSen [10] have investigated the gravitational instability ofpartially ionized molecular clouds considering the effects ofelectrons, ions, and charged dust grains. Avinash et al. [11]have studied the dynamics of self-gravitating dust cloudsand the formation of planetesimals. Thus we find that alarge number of problems are discussed for self-gravitatingdusty and nondusty plasma with different parameters undervarious assumptions due to its importance in star formationand in many astrophysical situations.

2 ISRN Astronomy and Astrophysics

Along with this, in above discussed problems, the effectof finite ion Larmor radius is not considered. In many astro-physical situations such as in interstellar and interplanetaryplasmas the approximation of zero Larmor radius is notvalid. Several authors (Roberts and Taylor [12], Jeffery andTaniuti [13], Jukes [14], and Vandakurov [15]) have point-ed out the importance of finite ion Larmor radius (FLR)effects in the form of magnetic resistivity, on the plasmainstability. Sharma [16] has shown the stabilizing effect ofFLR on gravitational instability of rotating plasma. Bhatiaand Chhonkar [17] have investigated the stabilizing effectof FLR on the instability of a rotating and self-gravitatingplasma. Herrnegger [18] has studied the effects of collisionand gyroviscosity on gravitational instability in a two-com-ponent plasma and concluded that the critical wave numberbecomes smaller with increasing gyroviscosity for finiteAlfven numbers and showed that Jeans criterion is changedby FLR for wave propagating perpendicular to magnetic field.Vaghela and Chhajlani [19] have investigated the stabilizingeffect of FLR on magneto-gravitational stability of resistiveplasma through porous medium with thermal conduction,but they neglect the effect of radiative heat-loss function ongravitational instability. Chhajlani and Parihar [20] have car-ried out the stabilizing effect of FLR on magnetogravitationalinstability of anisotropic plasma with generalized polytropelaws. Ferraro [21] has shown the stabilizing effect of FLR onmagneto-rotational instability. Recently Sandberg et al. [22]have investigated the stabilizing effect of FLR on the coupledtrapped electron and ion temperature-gradient modes. Morerecently Devlen and Pekunlu [23] have studied the effect ofFLR on weakly magnetized, dilute plasma. Thus FLR effectis an important factor in discussion of self-gravitationalinstability and other hydrodynamic instability.

In addition to this it is well known that thermal andradiative effects do play an important role in the stabilityinvestigations. The thermal instability arising due to variousheat-loss mechanisms may be the cause of astrophysical con-densation and the formation of large and small objects.Several authors (Field [24], Hunter [25], Ibanez [26], Kimand Narayan [27], Radwan [28], Menou et al. [29], andInutsuka et al. [30]) have investigated the phenomenon ofthermal instability arising due to heat-loss mechanism inplasma. Shadmehri and Dib [31] have discussed the thermalinstability in magnetized partially ionized plasma withcharged dust particles and radiative cooling function. Shaikhet al. [32] have investigated the Jeans gravitational instabilityof thermally conducting plasma in a variable magnetic fieldwith Hall current, finite conductivity, and viscosity, but theyneglect the effect of FLR corrections and radiative heat-lossfunction on gravitational instability. Aggarwal and Talwar[33] have discussed magnetothermal instability in a rotatinggravitating fluid, but they neglect the effect of FLR correctionon radiative instability. Bora and Talwar [34] have inves-tigated the magnetothermal instability with finite electricalresistivity and Hall current, but they neglect the effect of FLRcorrections and viscosity on radiative instability. Recently El-Sayed and Mohamed [35] have discussed the gravitationalinstability of rotating viscoelastic partially ionized plasma inthe presence of oblique magnetic field and Hall current.

More recently Kaothekar and Chhajlani [36] have carried outthe problem of gravitational instability of radiative plasmawith FLR corrections.

In the light of above work, we find that all these authorshave studied the problem of gravitational instability withdifferent combinations of these parameters, but none studiedthe joint effect of all the parameters together with radiativeeffects. We also find that none of the above authors has triedto explore Jeans condition for viscous and nonviscous plasmawith FLR corrections, radiative heat-loss function, and ther-mal conductivity. The problem given by Kaothekar andChhajlani [36] is briefly discussed with permeability param-eter, but in present paper we have made a detailed analysisof the problem of self-gravitational instability with all theconsidered physical parameters, without including the effectof permeability to give a better insight in real problem.Therefore in the present work self-gravitational instability ofmagnetized plasma with FLR corrections, radiative heat-lossfunction, viscosity, thermal conductivity, and finite electricalresistivity for self-gravitating configuration is studied. Wealso wish to explore the importance of viscous and non-viscous system and its impact on the self-gravitationalinstability of plasma in connection with FLR correction,radiative heat-loss function, thermal conductivity, and finiteelectrical resistivity. The stability of the system is discussedby applying Routh-Hurwitz criterion. The above work isapplicable to formation of stars in astrophysical plasma.

2. Basic Equations ofthe Problem and Perturbation

Let us consider an infinite homogeneous, self-gravitating,radiating, thermally conducting, and viscous plasma offinite electrical resistivity, including the effect of finite ionLarmor radius (FLR) corrections in the presence of magneticfield H(0, 0,H). The equations of the problem with theseeffects are written as

dudt= −1

ρ∇p +∇U − ∇ · P

ρ+

14πρ

(∇×H)×H + υ∇2u,

1γ − 1

dp

dt− γ

γ − 1p

ρ

dρ

dt+ ρL−∇ · (λ∇T) = 0,

p = ρRT ,

dρ

dt+ ρ∇ · u = 0,

∇2U = − 4πGρ,

∂H∂t= ∇× (u×H) + η∇2H,

∇ ·H = 0,

(1)

where p, ρ, υ,T ,η, λ,U ,G,R, and γ denote the fluid pressure,density, kinematic viscosity, temperature, electrical resis-tivity, thermal conductivity, gravitational potential, gravi-tational constant, gas constant, and ratio of two specific

ISRN Astronomy and Astrophysics 3

heats respectively. L(ρ,T), the radiative heat-loss function,depends on local values of density and temperature of thefluid. The operator (d/dt) is the substantial derivative givenas (d/dt) = (∂/∂t + u · ∇).

The components of pressure tensor P, considering thefinite ion gyration radius for the magnetic field along z-axisas given by Roberts and Taylor [12], are

Pxx = −ρυ0

(∂uy

∂x+∂ux∂y

), Pyy = ρυ0

(∂uy

∂x+∂ux∂y

),

Pzz = 0, Pxy = Pyx = ρυ0

(∂ux∂x

− ∂uy

∂y

),

Pxz = Pzx = −2ρυ0

(∂uy

∂z+∂uz∂y

),

Pyz = Pzy = 2ρυ0

(∂uz∂x

+∂ux∂z

).

(2)

The parameter υ0 has the dimensions of the kinematicviscosity and defined as υ0 = ΩLR

2L/4, where RL is the ion-

Larmor radius, and ΩL is the ion gyration frequency.The perturbation in fluid pressure, density, temperature,velocity, magnetic field, heat-loss function, and gravitationalpotential is given as δp, δρ, δT , u(ux,uy ,uz), h(hx,hy ,hz), L,and δU , respectively. The perturbation state is given as

p = p0 + δp, ρ = ρ0 + δρ,

T = T0 + δT , u = u0 + u (with u0 = 0),

H = H0 + h, L = L0 + L (with L0 = 0),

U = U0 + δU.

(3)

Suffix “0” represents the initial equilibrium state, which isindependent of space and time.

3. Dispersion Relation

We linearize the basic set of MHD equations by substituting(3) in (1)-(2), and we assume the variation of perturbedquantities as

exp(ikxx + ikzz + iσt), (4)

where σ is the frequency, and kx and kz are the wavenumbers of the perturbations along x and z axes. Using(4) in linearized perturbed form of (1) with (2), we getfour equations governing the perturbations of velocity and

the condensation of medium. These four equations can bewritten in the following determinant form:

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

P F 0ikxk2

Ω2T

−F Q −2υ0kxkz 0

0 2υ0kxkz Mikzk2

Ω2T

ikxV 2k2

dikxυ0

(k2x + 4k2

z

)0 −R

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

×

⎡⎢⎢⎢⎣uxuy

uzs

⎤⎥⎥⎥⎦ = 0.

(5)

We have made following substitutions:

iσ = ω, R = ω2 + ωυk2 + Ω2T ,

F = υ0(k2x + 2k2

z

), d = (

ω + ηk2),

M = ω + υk2, P =M +V 2k2

d,

Q =M +V 2k2

z

d, Lρ =

(∂L

∂ρ

)T

,

V 2 = H2

4πρ, Ω2

T =Ω2

I + ωΩ2j

ω + B,

Ω2j = c2k2 − 4πGρ, Ω2

I = k2A− 4πGρB,

LT =(∂L

∂T

)ρ, A = (

γ − 1)(

TLT − ρLρ +λk2T

ρ

),

B = (γ − 1

)(TρLTp

+λk2T

p

),

(6)

where c = (γp/ρ)1/2 is the adiabatic velocity of sound in themedium, c′ = (p/ρ)1/2 is the isothermal velocity of soundin the medium, and s = δρ/ρ is the condensation of themedium.

The general dispersion relation can be obtained from thedeterminant of matrix of (5) is

[ω + υk2 +

V 2k2

ω + ηk2

](ω2 + ωυk2 +

Ω2I + ωΩ2

j

ω + B

)

×{

4υ20k

2xk

2z +

(ω + υk2)×

[ω + υk2 V 2k2

z

ω + ηk2

]}

− (k2x + 4k2

z

)[ω + υk2 +

V 2k2

ω + ηk2

]

×(

2υ20k

2xk

2z

k2

)(Ω2

I + ωΩ2j

ω + B

)

+ υ20

(k2x + 2k2

z

)2[ω2 + ωυk2 +

Ω2I + ωΩ2

j

ω + B

](ω + υk2)


+

(Ω2

I + ωΩ2j

ω + B

)(V 2

ω + ηk2

)× 2υ2

0k2xk

2z

(k2x + 2k2

z

)

− (ω + υk2)υ2

0k2x

k2

(Ω2

I + ωΩ2j

ω + B

)(k2x + 2k2

z

)(k2x + 4k2

z

)

− (ω + υk2)

×[ω + υk2 +

V 2k2z

ω + ηk2

](V 2k2

x

ω + ηk2

)(Ω2

I + ωΩ2j

ω + B

)

− 4υ20k

4xk

2z

(V 2

ω + ηk2

)(Ω2

I + ωΩ2j

ω + B

)= 0.

(7)

The dispersion relation (7) represents the simultaneousinclusion of FLR corrections, radiative heat-loss function,thermal conductivity, viscosity, magnetic field, and finiteelectrical conductivity on self-gravitational instability ofplasma. In absence of radiative heat-loss function, the generaldispersion relation (7) is identical to that of Vaghela andChhajlani [19] for (ε = 1 and K1 = ∞) in their case. Inabsence of radiative heat-loss function, thermal conductivity,finite electrical resistivity, and viscosity the general dispersion

relation (7) is identical to Sharma [16] for nonrotationalcase. In absence of FLR corrections and radiative heat-loss function the general dispersion relation (7) is identicalto Shaikh et al. [32] neglecting Hall current in theircase. In absence of FLR corrections and viscosity dispersionrelation (7) is identical to Bora and Talwar [34] neglectingHall current and electron inertia in their case. Also inabsence of FLR corrections, viscosity and finite conductivitydispersion relation (7) reduces to that obtained by Field [24]for nongravitating medium. The dispersion relation (7) isdifferent from (9) of Kaothekar and Chhajlani [36].

Thus we have obtained the modified dispersion relationof self-gravitational instability including the combined effectof FLR corrections, radiative heat-loss function, viscosity,thermal conductivity, finite electrical resistivity, and mag-netic field. Now we discuss the general dispersion relation (7)for longitudinal and transverse wave propagation.

4. Discussion

4.1. Longitudinal Propagation (kx = 0, kz = k). For thiscase, we assume all the perturbations are longitudinal tothe direction of magnetic field (i.e., kz = k, kx = 0). Thedispersion relation (7) reduces to

(ω + υk2)×

⎧⎨⎩4υ2

0k4 +

[ω + υk2 +

V 2k2

ω + ηk2

]2⎫⎬⎭

×⎡⎣ω2 + ωυk2 +

ω(c2k2 − 4πGρ

)ω +

(γ − 1

)(TρLT/p + λk2T/p

)

+

(γ − 1

){k2(TLT − ρLρ + λk2T/p

)− 4πGρ

(TρLT/p + λk2T/p

)}ω +

(γ − 1

)(TρLT/p + λk2T/p

)⎤⎦ = 0.

(8)

The above dispersion relation shows the combined effectof FLR corrections, radiative heat-loss function, thermalconductivity, viscosity, magnetic field strength, and finiteelectrical resistivity on self-gravitation instability of plasma.On multiplying all the components of (8), we get thedispersion relation, which is an equation of degree eight inω, and it is cumbersome to write such a lengthy equation.If we remove the effect of FLR corrections and viscosity inthe above relation, then we recover the relation given byBora and Talwar [34] excluding Hall current and electroninertia in their case. Hence the above dispersion relation isthe modified form of equation (21) of Bora and Talwar [34]due to the inclusion of FLR corrections and viscosity in ourcase and by neglecting Hall current and electron inertia intheir case for longitudinal propagation in dimensional form.But the condition of instability is unaffected by the presenceof FLR correction and viscosity. Thus we conclude that FLRcorrections and viscosity have no effect on the conditionof instability, but presence of these parameters modifies thegrowth rate of instability in the present case. Hence these arethe new findings in our case than that of Bora and Talwar

[34]. Also on comparing our dispersion relation (8) withdispersion relation (20) of Vaghela and Chhajlani [19] wefind that two factors are the same, but the third factor isdifferent and gets modified because of radiative terms.

The dispersion relation (8) has three different compo-nents, and our aim is to take out the physics involved in eachcomponent. So we discuss each component separately. Thefirst component of the dispersion relation (8) gives

ω + υk2 = 0. (9)

This represents a damped mode modified by the presenceof viscosity of the medium. Thus viscous force is capableof stabilizing the growth rate of the considered system. Theabove mode is unaffected by the presence of FLR correction,magnetic field strength, finite electrical resistivity, ther-mal conductivity, radiative heat-loss function, and self-gravitation.


The second factor of (8) on simplification gives

ω4 + 2(υk2 + ηk2)ω3

+[(υk2 + ηk2)2

+ 2(V 2k2 + ηk2υk2) + 4υ2

0k4]ω2

+[2(υk2 + ηk2)× (

V 2k2 + ηk2υk2) + 8ηk2υ20k

4]ω+(V 2k2 + ηk2υk2)2

+ 4η2k4υ20k

4 = 0.

(10)

The above dispersion relation represents the Alfven modemodified by the presence of FLR corrections, viscosity,and finite electrical resistivity. But it is independent ofthermal conductivity, radiative heat-loss function, and self-gravitation. Equation (10) is a four-degree equation inpower of ω having its all coefficients positive which is anecessary condition for the stability of the system. To achievethe sufficient condition the principal diagonal minors ofHurwitz matrix must be positive. On calculating we get allthe principal diagonal minors positive. Hence (10) alwaysrepresents stability.

The third component of the dispersion relation (8) onsimplifying gives

ω3 +

[υk2 +

(γ − 1

)(TρLTp

+λk2T

p

)]ω2

+

[(γ − 1

)(TρLTp

+λk2T

p

)υk2 +

(c2k2 − 4πGρ

)]ω

+

{k2(γ − 1

)(TLT − ρLρ +

λk2T

ρ

)

− 4πGρ(γ − 1

)(TρLTp

+λk2T

p

)}= 0.

(11)

The above equation represents the combined influenceof radiative heat-loss function, thermal conductivity, and vis-cosity on the self-gravitational instability of plasma. But thereis no effect of FLR corrections, finite electrical conductivity,and magnetic field on the self-gravitational instability of theconsidered system. If the constant term of cubic equation(11) is less than zero, this allows at least one positive realroot which corresponds to the instability of the system. Thecondition of instability obtained from constant term of (11)is given as

k2

(TLT − ρLρ +

λk2T

ρ

)< 4πGρ

(TρLTp

+λk2T

p

).

(12)

This condition of instability is modified form of Jeanscondition by inclusion of thermal conductivity and radiativeheat-loss functions. The above condition of instability isindependent of FLR corrections, finite electrical conduc-tivity, magnetic field strength, and viscosity. The aboveinequality (12) is same as obtained by Bora and Talwar [34],and by Kaothekar and Chhajlani [36]. But Kaothekar andChhajlani [36] have also considered the effect of permeabilityin their analysis. Here in the present problem we neglect theeffect of permeability to give better insight in real problem.Thus in this mode the dispersion relation and the growthrate is modified due to FLR corrections and viscosity, but thecondition of radiative instability is unaffected by the presenceof FLR corrections and viscosity. The condition of instability(12) can be solved to yield the following modified criticalJeans wavelength:

λJ1 =

⎡⎢⎢⎢⎢⎣

8π{(4Gρ/c′2 +ρ2Lρ/πλT − ρLT/πλ

)±[(

4Gρ/c′2 +ρ2Lρ/πλT − ρLT/πλ)2

+(16Gρ2/πλc′2

)LT

]1/2}⎤⎥⎥⎥⎥⎦

1/2

.(13)

The medium is unstable for wave length λ > λJ1. Hereit may be noted that the modified critical wave lengthinvolves the derivatives of temperature-dependent, density-dependent heat-loss functions and thermal conductivity ofthe medium.

In dispersion relation (8) first and second factors showwave propagation, but third factor shows instability. Thusto discuss the effect of each parameter on the growth rateof instability, we solve (11) numerically by introducing thefollowing dimensionless quantities:

k∗ = kc(4πGρ

)1/2 , ω∗ = ω(4πGρ

)1/2 ,

υ∗ = υ(4πGρ

)1/2

c2,

L∗T =(γ − 1

)ρTLT

p(4πGρ

)1/2 , L∗ρ =(γ − 1

)ρLρ

c2(4πGρ

)1/2 ,

λ∗ =(γ − 1

)Tλ

(4πGρ

)1/2

pc2.

(14)

The nondimensional form of (11) in terms of self-gravitationis given as

ω∗3 +(υ∗k∗2 + L∗T + λ∗k∗2)ω∗2

+[υ∗k∗2(L∗T + λ∗k∗2) + k∗2 − 1

]ω∗

+

{k∗2

[1γ

(L∗T +λ∗k∗2)−L∗ρ

]−(L∗T +λ∗k∗2)}=0.

(15)


00.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Wave number (k∗)

L∗T = 0L∗T = 2L∗T = 5

Gro

wth

rat

e (ω∗ )

Figure 1: Growth rate (positive values of ω∗) against wave numberk∗ for three values of parameter L∗T = 0.0, 2.0, 5.0, keeping the otherparameters fixed L∗ρ = 1.0, λ∗ = 1.0, υ∗ = 1.0.

Here for calculations we take the numerical values cor-responding to the conditions in interstellar molecular cloudsfor star formation as

ρ = 1.7× 10−21 kg m−3

G = 6.658× 10−11 (kg)−1m3s−2

c2 = 2.25× 108 m2s−2

V 2 = 5.0× 108 m2s−2.

(16)

And the parameters are taken as nondimensional.Numerical calculations were performed to determine theroots of ω∗ from dispersion relation (15), as a function ofwave number k∗ for several values of the different parametersinvolved taking γ = 5/3. Out of the three modes only onemode is unstable for which the calculations are presented inFigures 1–3, where the growth rate ω∗ (positive real valueof ω, after multiplying by 1016) has been plotted againstthe wave number k∗ (after multiplying by 1020) to showthe dependence of the growth rate on the different physicalparameters such as temperature-dependent heat-loss func-tion L∗T (after multiplying by 108), density-dependent heat-loss function L∗ρ (after multiplying by 108), and viscosity υ∗

(after multiplying by 1024).

It is clear from Figure 1 that growth rate decreases withincreasing temperature-dependent heat-loss function. Thusthe effect of temperature-dependent heat-loss function isstabilizing. From Figure 2 we conclude that growth rateincreases with increasing density-dependent heat-loss func-tion. Thus the effect of density-dependent heat-loss function

0 0.5 1 1.5 2 2.5 3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Wave number (k∗)

Gro

wth

rat

e (ω∗ )

L∗ρ = 0L∗ρ = 2L∗ρ = 5

Figure 2: Growth rate (positive values of ω∗) against wave numberk∗ for three values of parameter L∗ρ = 0.0, 2.0, 5.0, keeping the otherparameters fixed λ∗ = 1.0, L∗T = 1.0, υ∗ = 1.0.

0

0.2

0.4

0.6

0.8

1

1.2 1.4 1.6 1.80 0.2 0.4 0.6 0.8 1

v∗ = 5

v∗ = 0v∗ = 2

Wave number (k∗)

Gro

wth

rat

e (ω∗ )

Figure 3: Growth rate (positive values of ω∗) against wave numberk∗ for three values of parameter υ∗ = 0.0, 2.0, 5.0, keeping the otherparameters fixed L∗ρ = 1.0, λ∗ = 1.0, L∗T = 1.0.

is destabilizing. From Figure 3 we conclude that growth ratedecreases with increasing viscosity of the medium. Thus theeffect of viscosity is stabilizing.

To discuss the stability of the system, if constant termof cubic equation (11) is greater than zero, then all thecoefficients of (11) must be positive. Equation (11) is a third-degree equation in the power of ω having its coefficientspositive, which is a necessary condition for the stability ofthe system. To achieve the sufficient condition the principal


diagonal minors of Hurwitz matrix must be positive. Theprincipal diagonal minors are

Δ1 ={υk2 +

(γ − 1

)(TρLTp

+λk2T

p

)}> 0,

Δ2 ={υk2

[Δ1(γ − 1

)(TρLTp

+λk2T

p

)+ c2k2 − 4πGρ

]

+(γ − 1

)k2ρLρ

}> 0,

Δ3 = Δ2

{k2(γ − 1

)(TLT − ρLρ +

λk2T

ρ

)

−4πGρ(γ − 1

)(TρLTp

+λk2T

p

)}> 0.

(17)

Since Ω2j > 0, Ω2

I > 0 and γ > 1, it is clear that all the Δs arepositive hence system represented by (11) is stable system.

Now we wish to examine the effect of radiative heat-lossfunction in the considered system with some simplifications,and at the same time we wish to investigate the physicsinvolved in such simplifications in the present problem.

4.1.1. For Viscous Medium

(1) Effect of Radiative Heat-Loss Function

For nonradiative medium (LT ,ρ = 0, υ = λ /= 0) (11) be-comes

ω3+

(υk2 +

γλk2

ρcp

)ω2 +

(υk2 γλk

2

ρcp+ c2k2 − 4πGρ

)ω

+γλk2

ρcp

(c′2k2 − 4πGρ

)= 0.

(18)

The condition of instability from constant term of (18) is(c′2k2 − 4πGρ) < 0, or

k2 <4πGρc′2

(19)

if

k2J2 =

4πGρc′2

, or λJ2 = c′[

π

Gρ

]1/2

. (20)

The medium is unstable for wave length λ > λJ2,where λJ2 is the modified Jeans wave length for thermallyconducting medium. On comparing (18) and (11) we seethat there is no new mode coming due to inclusion ofradiative heat-loss functions, but the condition of instabilitygets modified in form of (11). Also it is interesting tonote that the thermal conductivity term is separated fromcondition of instability when we neglect the effect of heat-loss function in (11).

For nonthermal and nonradiating medium (λ = LT ,ρ =0, υ /= 0) the condition of instability obtained from (11) is

(c2k2 − 4πGρ

)< 0, (21)

λJ = c

[π

Gρ

]1/2

. (22)

The medium is unstable for wave length λ > λJ , whereλJ is the Jeans wave length. The condition of Jeans insta-bility given by (21) is identical to Chandrasekhar [5]. Oncomparing (12) and (21) it is clear that due to the presenceof radiative heat-loss functions and thermal conductivitythe fundamental criterion of Jeans gravitational instabilitychanges into radiative instability criterion. Also, we concludethat one mode is increased because of inclusion of radiativeheat-loss functions.

4.1.2. For Nonviscous Medium

(1) Effect of Radiative Heat-Loss Function

For inviscid and nonradiating medium (υ = LT ,ρ = 0,λ /= 0) (11) becomes

ω3 +γλk2

ρcpω2 +

(c2k2 − 4πGρ

)ω

+γλk2

ρcp

(c′2k2 − 4πGρ

)= 0.

(23)

The condition of instability is (c′2k2 − 4πGρ) < 0, it isdiscussed in (19) and (20). On comparing (23) and (11)we see that there is no new mode which comes due toinclusion of radiative heat-loss functions, but the conditionof instability gets modified in form of (12). In both the caseswhether the medium is viscous or nonviscous, the conditionof instability is the same; hence, we conclude that viscosity ofthe medium has no effect on the condition of instability, andit only modifies the growth rate of the instability for viscouscase.

For inviscid, thermally nonconducting and nonradiatingmedium (υ = λ = LT ,ρ = 0) the condition of instabilityobtained from (11) is (c2k2 − 4πGρ) < 0, and it is discussedin (21) and (22). We see that inclusion of viscosity cannotchange the condition of instability, but it modifies the growthrate of instability.

Thus we conclude that for longitudinal wave propagationas given by (8) the system is unstable only for modifiedJeans condition, and else it is stable. Also the modified Jeanscriterion remains unaffected by FLR corrections, viscosity,magnetic field, and finite electrical resistivity, but radiativeheat-loss function and thermal conductivity modifies theJeans expression and the fundamental Jeans instabilitycriterion becomes radiative instability criterion. The growthrate of the system is modified by the presence of radiativeheat-loss functions, thermal conductivity, and viscosity. Thegrowth rate of Alfven mode is modified by the presenceof viscosity, finite electrical resistivity, and FLR corrections.


From the curves we find that temperature-dependent heat-loss function and viscosity have stabilizing influence, whereasdensity-dependent heat-loss function has a destabilizinginfluence on the self-gravitational instability of plasma.

4.2. Transverse Propagation (kx = k, kz = 0). For this case,we assume that all the perturbations are transverse to thedirection of magnetic field (i.e., kx = k, kz = 0). Thedispersion relation (7) reduces to

(ω + υk2)2

⎧⎨⎩ωυ2

0k4 +

(ω + υk2)

⎡⎣ω2 + ωυk2 +

ωV 2k2

ω + ηk2+

ω(c2k2 − 4πGρ

)ω +

(γ − 1

)(TρLT/p + λk2T/p

)

+

(γ − 1

){k2(TLT − ρLρ + λk2T/p

)− 4πGρ

(TρLT/p + λk2T/p

)}ω +

(γ − 1

)(TρLT/p + λk2T/p

)⎤⎦⎫⎬⎭ = 0.

(24)

We find that for transverse mode of propagation the dis-persion relation is modified due to the presence of FLR cor-rections, radiative heat-loss function, thermal conductivity,viscosity, finite electrical resistivity, and magnetic field. Thedispersion relation (24) has two different components. Thefirst component of the dispersion relation (24) represents adamped mode modified by the presence of viscosity of themedium and is discussed in (9).

The second component of the dispersion relation (24) onsimplifying gives

ω5 +

[2υk2 + ηk2 +

(γ − 1

)(TρLTp

+λk2T

p

)]ω4

+

{2υk2

[ηk2 +

(γ − 1

)(TρLTp

+λk2T

p

)]

+ υ2k4 + η k2(γ − 1)(TρLT

p+λk2T

p

)

+V 2k2 + υ20k

4 + c2k2 − 4πGρ

}ω3

+

{2ηk2υk2(γ − 1

) ×(TρLTp

+λk2T

p

)

+

[ηk2 +

(γ − 1

)(TρLTp

+λk2T

p

)](υ2k4 + υ2

0k4)

+ V 2k2

[υ k2 +

(γ − 1

)×(TρLTp

+λk2T

p

)]

+(c2k2 − 4πGρ

)(υk2 + ηk2)

+(γ − 1

)[k2

(TLT − ρLρ +

λk2T

ρ

)− 4πGρ

×(TρLTp

+λk2T

p

)]}ω2

+

{υk2

[ηk2υk2(γ − 1

)(TρLTp

+λk2T

p

)

+ V 2k2(γ − 1)

×(TρLTp

+λk2T

p

)+ ηk2(c2k2 − 4πGρ

)]

+ ηk2υ20k

4(γ − 1)(TρLT

p+λk2T

p

)

+(γ − 1

)[k2

(TLT − ρLρ +

λk2T

ρ

)

−4πGρ

(TρLTp

+λk2T

p

)](υk2 +ηk2)}ω

+ηk2υk2

×[k2(γ − 1

)(TLT − ρLρ +

λk2T

ρ

)

−4πGρ(γ − 1

)(TρLTp

+λk2T

p

)]= 0.

(25)

The above fifth degree equation in ω represents thecombined influence of FLR corrections, radiative heat-lossfunction, thermal conductivity, finite electrical resistivity,viscosity, and magnetic field on self-gravitational instabilityof plasma. When constant term of (25) is less than zero,this allows at least one positive real root which correspondsto the instability of the system. The condition of instabilityobtained from constant term of (25) is given as

(γ − 1

)[k2

(TLT − ρLρ +

λk2T

ρ

)

−4πGρ

(TρLTp

+λk2T

p

)]< 0.

(26)

The above condition of instability is independent of FLRcorrection, finite electrical resistivity, viscosity, and magneticfield strength and is discussed in (12). Thus from (25), we


find that the modified Jeans condition of instability remainsthe same whether we consider the effect of FLR correctionsor not, but, owing to the presence of FLR corrections, thegrowth rate of radiative instability is modified. This resultcan be easily explained by graphical representation as shownin Figure 6; by increasing values of FLR, we see that thegrowth rate of radiative instability decreases. Hence FLRshows stabilizing influence on the growth rate of radiativeinstability.

In several astrophysical plasma situations, such as ininterstellar molecular clouds the formation of objects ismainly due to the unstable modes. Thus to discuss the effectof each parameter on the growth rate of unstable modes, wesolve (25) numerically by using the relation given in (14)with the following additional dimensionless quantities:

V∗2 = V 2

c2, η∗ = η

(4πGρ

)1/2

c2,

υ∗0 =υ0(4πGρ

)1/2

c2.

(27)

For calculations we take the numerical values corre-sponding to the conditions in interstellar molecular cloudswhich are shown in (16), and the rest of values are takenas nondimensional. Numerical calculations were performedto determine the roots of ω∗ as a function of wave numberk∗ for several values of the different parameters involvedtaking γ = 5/3. Out of the five modes, only one mode isunstable for which the calculations are presented in Figures4–6, where the growth rate ω∗ (positive real value of ω,after multiplying by 1016) has been plotted against the wavenumber k∗ (after multiplying by 1020) to show the depen-dence of the growth rate on the different physical parameterssuch as temperature-dependent heat-loss function L∗T (aftermultiplying by 108), density-dependent heat-loss functionL∗ρ (after multiplying by 108), viscosity υ∗ (after multiplyingby 1024), and FLR corrections υ∗0 (after multiplying by 1024).

It is clear from Figure 4 that growth rate decreases withincreasing temperature-dependent heat-loss function. Thusthe effect of temperature-dependent heat-loss function isstabilizing. From Figure 5 we conclude that growth rateincreases with increasing density-dependent heat-loss func-tion. Thus the effect of density-dependent heat-loss functionis destabilizing. One can observe from Figure 6 that thegrowth rate decreases with increasing FLR corrections. Thusthe effect of FLR corrections is stabilizing. If the viscouseffect of plasma is removed, then the stabilizing rate of FLRcorrections increases and the peak value of the curve alsoincreases. Hence we conclude that both FLR corrections andviscosity stabilize the system.

To discuss the stability of the system given by dispersionrelation (25), if constant term of equation (25) is greater thanzero, then all the coefficients of the equation (25) must bepositive. Equation (25) is a fifth degree equation in the powerof ω; hence, it will involve a lengthy algebra to calculate theprincipal diagonal minors of Hurwitz matrix. Since (25) haspositive real coefficients, having Ω2

j > 0, Ω2I > 0 and γ > 1, so

it is sufficient to show that alternate Δs are positive; hence,system represented by (25) is a stable system.

0

0.2

0.4

0.6

0.8

1

0.2 0.4 0.6 0.8 1 1.20

Wave number (k∗)

L∗T = 0L∗T = 1.5L∗T = 3

Gro

wth

rat

e (ω∗ )

Figure 4: Growth rate (positive values of ω∗) against wave numberk∗ for three values of parameter L∗T = 0.0, 1.5, 3.0, keeping the otherparameters fixed υ∗ = 0.5, L∗ρ = 0.0, λ∗ = 1.0, υ∗0 = 1.0, V∗ =1.0, η∗ = 0.5.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7

Wave number (k∗)

Gro

wth

rat

e (ω∗ )

L∗ρ = 0L∗ρ = 1.5L∗ρ = 3

Figure 5: Growth rate (positive values of ω∗) against wave numberk∗ for three values of parameter L∗ρ = 0.0, 1.5, 3.0, keeping the otherparameters fixed υ∗0 = 1.0, λ∗ = 1.0, υ∗ = 0.5, V∗ = 1.0, η∗ =0.5, L∗T = 0.5.

Now we wish to examine the effect of FLR correctionsand radiative heat-loss function on the considered systemwith some simplifications such as viscous and nonviscousmedium, and at the same time we wish to investigatethe physics involved in such simplifications in the presentproblem.


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

0.2

0.4

0.6

0.8

1

1.2

v∗ = 0 : v0∗ = 0

v∗ = 0 : v0∗ = 1.5

v∗ = 0 : v0∗ = 3

v∗ = 0.9 : v0∗ = 1.5

v∗ = 0.9 : v0∗ = 3

Wave number (k∗)

Gro

wth

rat

e (ω∗ )

Figure 6: Growth rate (positive values of ω∗) against wave numberk∗ for several values of parameter υ∗0 keeping the other parametersfixed L∗ρ = 0.5, L∗T = 0.5, λ∗ = 1.0, V∗ = 1.0, η∗ = 0.5. υ∗ = 0.0:(1) υ∗0 = 0.0, (2) υ∗0 = 1.5 (3) υ∗0 = 3.0. υ∗ = 0.9: (4) υ∗0 = 1.5, (5)υ∗0 = 3.0.

4.2.1. For Viscous Medium

(1) Effect of Radiative Heat-Loss Function with FLR Cor-rections

For nonradiative medium, but in the presence of FLRcorrections (LT ,ρ = 0, υ = λ = V = η = υ0 /= 0) the conditionof instability obtained from the constant term of (25) is givenas (

c′2k2 − 4πGρ)< 0. (28)

The above (28) is identical to that of Vaghela andChhajlani [19]. We see that there is no new mode due toinclusion of radiative heat-loss function, but the growth rateof instability gets modified due to inclusion of radiative heat-loss functions and FLR corrections. Also it is interesting tonote that, on neglecting the effect of radiative heat-loss func-tion in (25), the thermal conductivity parameter (i.e., λ) isseparated from instability condition as shown in (28). So weconclude that on neglecting the effect of radiative heat-lossfunction the condition of instability is independent of boththe radiative heat-loss function and thermal conductivity ofthe medium, but the growth rate of instability is affected bythermal conductivity and FLR corrections.

For nonradiative and nonthermal medium, but in pres-ence of FLR corrections (LT ,ρ = λ = 0, υ = V = η = υ0 /= 0)the condition of instability obtained from (25) is given as(

c2k2 − 4πGρ)< 0. (29)

We find that one mode is increased due to inclusionof radiative heat-loss functions. On comparing (26) and(29), it is clear that owing to the presence of thermalconductivity and radiative heat-loss functions the funda-mental criterion of Jeans gravitational instability changes

into radiative instability criterion. In case of transverse modeof propagation, for viscous medium, FLR corrections haveno effect on the condition of instability for radiating ornonradiating medium, but the growth rate of instabilityis modified by FLR corrections; that is, FLR stabilizes thesystem in all the cases. The joint stabilizing effect of FLRcorrections and viscosity decreases the peak value of growthrate of the system shown in Figure 6.

4.2.2. For Nonviscous Medium

(1) Effect of FLR Corrections

For inviscid medium, but in presence of FLR corrections(υ = 0,η = LT ,ρ = V = λ = υ0 /= 0) (25) becomes

ω4 +

[ηk2 +

(γ − 1

)(TρLTp

+λk2T

p

)]ω3

+

[ηk2(γ − 1

)(TρLTp

+λk2T

p

)+ V 2k2

+υ20k

4 + c2k2 − 4πGρ

]ω2

+

{υ2

0k4

[ηk2 +

(γ − 1

)(TρLTp

+λk2T

p

)]

+ V 2k2(γ − 1)

×(TρLTp

+λk2T

p

)+ ηk2(c2k2 − 4πGρ

)

+

[k2(γ − 1

)(TLT − ρLρ +

λk2T

ρ

)

−4πGρ(γ − 1

)(TρLTp

+λk2T

p

)]}ω

+ ηk2

{υ2

0k4(γ − 1

)(TρLTp

+λk2T

p

)+ k2(γ − 1

)

×(TLT − ρLρ +

λk2T

ρ

)

−4πGρ(γ − 1

)(TρLTp

+λk2T

p

)}= 0.

(30)

The above equation (30) is a modified form of Chhajlaniand Parihar [20] by inclusion of radiative heat-loss function,thermal conductivity, and finite electrical resistivity. Thecondition of instability obtained from constant term of (30)is given as

{k2

(TLT − ρLρ +

λk2T

ρ

)

−(4πGρ− υ20k

4)(TρLTp

+λk2T

p

)}< 0.

(31)


From the above condition of instability given by (31) weconclude that FLR correction tries to stabilize the radiativeinstability. Also on comparing (25) and (30) we see thatinclusion of viscosity removes the effects of FLR correctionfrom condition of instability. So in both the cases either thesystem is viscous or nonviscous, FLR correction stabilizesthe growth rate of radiative instability, as shown in Figure 6.Thus in presence of FLR correction the condition of insta-bility given by Bora and Talwar [34] is modified as given by(31). In absence of FLR corrections (30) is identical to Boraand Talwar [34]; thus, the dispersion relation is modifieddue to the inclusion of effects of FLR in the present system.Thus we find that due to the presence of FLR correctionsJeans criterion of instability is modified, and also the growthrate of instability in this case is modified with the inclusionof FLR correction in the present MHD set of equations.The above condition of instability (31) can be written innondimensional form as{(

λ∗k∗2 + L∗T)[

υ∗02k∗4 +

k∗2

γ− 1

]− k∗2L∗ρ

}< 0. (32)

In absence of FLR corrections the above condition ofinstability is the same as given by Bora and Talwar [34]and by Hunter [25] for an electrically nonconducting gas.We conclude that condition of instability given by Bora andTalwar [34] is modified by inclusion of FLR corrections inour case. Also the condition of instability given by (31)is the modified form of Field [24] by inclusion of FLRcorrection and self-gravitation. Thus the present result isthe improvement of Field [24], Hunter [25], and Bora andTalwar [34]. Condition (32) involves the derivatives L∗T andL∗ρ of the heat-loss function with respect of local temperatureand density, and υ∗0 is FLR correction factor in the system.It may be noted that for a density-independent heat-lossfunction (L∗ρ = 0), which increases with temperature (L∗T >

0), condition (32) gives instability if k∗2 < ( γ − 1/γυ∗02).

However if instead, the heat-loss function decreases withtemperature (L∗T < 0), and the instability comes to k∗2 lyingbetween (γ − 1/γυ∗0

2) and {|L∗T |/λ∗}.For a purely density-dependent heat-loss function (L∗T =

0), condition (32) gives instability if

k∗2 <12

⎧⎪⎨⎪⎩±

⎡⎣( 1

γυ∗02

)2

+4

υ∗02

(1 +

L∗ρλ∗

)⎤⎦1/2

−(

1

γυ∗02

)⎫⎪⎬⎪⎭.

(33)

The above equation (33) applies for L∗ρ > 0 or |L∗ρ | < λ∗

if the heat-loss function, respectively, increases or decreaseswith increase in density.

(2) Effect of Radiative Heat-Loss Function with FLRCorrections

For nonradiating and inviscid medium, but in presenceof FLR (LT ,ρ = υ = 0, η = λ = υ0 = V /= 0) the condition ofinstability obtained from (25) is given as

k2(c′2 + υ2

0k2)< 4πGρ. (34)

The critical Jeans wave number and critical Jeans wave lengthare given as

k2J3 =

{−c′2 ±

(c′4 + 16πGρυ2

0

)1/2}

2υ20

. (35)

Or

λJ3 =⎡⎣ 8π2υ2

0{−c′2 ± (

c′4 + 16πGρυ20

)1/2}⎤⎦

1/2

. (36)

Condition (34) is identical of Vaghela and Chhajlani [19]for (ε = 1) in their case. From (34) we conclude that FLRhave stabilizing effect on the instability of the system. Wefind that one mode is increased due to inclusion of viscosity,and we conclude that inclusion of viscosity removes the effectof FLR corrections from the condition of instability. Alsoon comparing (31) and (34), we conclude that condition ofinstability given by Vaghela and Chhajlani [19] is modified byinclusion of radiative effects and the growth rate of instabilitygiven by (31) is modified by radiative heat-loss function andFLR corrections.

For inviscid, nonradiating and thermally nonconductingmedium, but in presence of FLR (υ = LT ,ρ = λ = 0, η =υ0 /= 0), the condition of instability obtained from (25) isgiven as

{(c2 + υ2

0k2)k2 − 4πGρ

}< 0. (37)

The critical Jeans wave number or critical Jeans wave lengthis given as

k2J4 =

{−c2 ± (

c4 + 16πGρυ20

)1/2}

2υ20

. (38)

Or

λJ4 =⎡⎣ 8π2υ2

0{−c2 ± (

c4 + 16πGρυ20

)1/2}⎤⎦

1/2

. (39)

From (37) we find that Jeans condition of gravitationalinstability gets modified by FLR corrections. On comparing(37) and (31) we conclude that inclusion of thermal con-ductivity and radiative heat-loss function modifies both theJeans condition of instability and growth rate of instabilityin presence of FLR corrections. Hence the fundamentalJeans criterion of gravitational instability gets modifiedinto radiative instability criterion by inclusion of thermalconductivity and radiative heat-loss functions in the presenceof FLR corrections.

(3) Effect of Electrical Conductivity with FLR Correc-tions

For inviscid and infinitely conducting medium, but inpresence of FLR (υ = η = 0,LT ,ρ = V = λ = υ0 /= 0), the


condition of instability obtained from constant term of (25)is given as{

k2

(TLT − ρLρ +

λk2T

ρ

)

−(4πGρ − υ20k

4 −V 2k2)(TρLTp

+λk2T

p

)}< 0.

(40)

From the above condition of instability given by (40) weconclude that FLR correction and magnetic field strength tryto stabilize the radiative instability. Also on comparing (26)and (40) we see that inclusion of viscosity and finite conduc-tivity removes the effect of FLR correction and magnetic fieldfrom condition of instability. So in both the cases whetherthe system is finitely conducting or infinitely conductingmagnetic field and FLR correction stabilizes the growth rateof radiative instability. The above condition of instability isidentical to Kaothekar and Chhajlani [36]. Kaothekar andChhajlani [36] have considered the effect of permeability,but, in the present problem, we have not considered theeffect of permeability, so we conclude that the conditionsof instability for permeable or nonpermeable medium aresame, but the growth rate of the system is modified by thepresence of permeability. The above condition of instability(40) can be written in nondimensional form as{(

λ∗k∗2 + L∗T)[

υ∗02k∗4 + k∗2

(1γ

+ V∗2

)− 1

]

−k∗2L∗ρ

}< 0.

(41)

In absence of FLR corrections the above condition ofinstability is identical to Bora and Talwar [34] excludingelectron inertia in that case. We conclude that condition ofinstability given by Bora and Talwar [34] is modified byinclusion of FLR corrections in our case. Thus the presentresults are the improvement of Bora and Talwar [34] due toFLR corrections.

From condition of instability (41) it may be noted thatfor a density-independent heat-loss function (L∗ρ = 0),which increases with temperature (L∗T > 0), it gives instabilityif k∗2 < (1/υ∗0

2){1− [(1/γ) + V∗2]}. However if instead, theheat-loss function decreases with temperature (L∗T < 0), theinstability comes to k∗2 lying between (1/υ∗0

2){1 − [(1/γ) +V∗2]} and {|L∗T |/λ∗}.

For a purely density-dependent heat-loss function (L∗T =0), condition (41) gives instability if

k∗2 <12

⎧⎪⎨⎪⎩±

⎡⎣ 1

υ∗04

(1γ

+ V∗2

)2

+4

υ∗02

(1 +

L∗ρλ∗

)⎤⎦1/2

− 1

υ∗02

(1γ

+ V∗2

)⎫⎪⎬⎪⎭.

(42)

The above equation (42) applies for L∗ρ > 0 or |L∗ρ | < λ∗,if the heat-loss function, respectively, increases or decreaseswith increase in density.

(4) Effect of Nongravitation and FLR Corrections

For inviscid, nongravitating, infinitely conducting, radi-ating, thermally conducting, magnetized medium with FLRcorrections (υ = η = G = 0, υ0 = V = LT ,ρ = λ /= 0) (25)becomes

ω3 +

[(γ − 1

)(TρLTp

+λk2T

p

)]ω2

+[V 2k2 + υ2

0k4 + c2k2]ω

+(γ − 1

)k2

[(V 2 + υ2

0k2)

×(TρLTp

+λk2T

p

)+

(TLT−ρLρ+

λk2T

ρ

)]=0.

(43)

The condition of instability obtained from constant term of(43) is given as

[(υ2

0k2 + V 2)(TρLT

p+λk2T

p

)

+

(TLT +

λk2T

ρ− ρLρ

)]< 0.

(44)

From above equation (44) we see that FLR correctionand magnetic field strength try to stabilize the system. Ifwe neglect the effect of FLR and magnetic field the abovedispersion relation (43) is identical to Field [24]. In thepresent case we have considered the effects of FLR correctionand magnetic field, but Field [24] has not considered theseeffects. Thus in the present analysis both the dispersionrelation and the condition of instability get modified due tothe presence of FLR corrections and magnetic field strength.On comparing (40) and (44) we see that consideration ofself-gravitation modifies the thermal instability criterion intoradiative instability criterion. Also from (25) it is clear thatthe growth rate of the dispersion relation given by Field[24] is modified due to the presence of FLR correction,viscosity, magnetic field, finite electrical conductivity, andself-gravitation in our present case. Hence these are the newresults in our present problem than that of Field [24].

Thus we conclude that for transverse wave propagationthe Jeans criterion is affected by FLR corrections, radiativeheat-loss functions, thermal conductivity, viscosity, magneticfield strength, and finite electrical resistivity. From curves wefind that FLR corrections, temperature-dependent heat-lossfunction, and viscosity have stabilizing influence, whereasdensity-dependent heat-loss function has destabilizing influ-ence on the self-gravitational instability of plasma.

5. Conclusion

We have dealt with the self-gravitation instability of aninfinite homogeneous viscous electrically and thermallyconducting fluid including the effects of FLR corrections andradiative heat-loss function for star formation. The general


dispersion relation is obtained, which is modified due to thepresence of considered physical parameters. This dispersionrelation is reduced for longitudinal and transverse modeof propagation to the direction of the magnetic field. Wefind that Jeans criterion remains valid and gets modifiedbecause of FLR corrections, radiative heat-loss function,thermal conductivity, and magnetic field. We also find thatthe presence of thermal conductivity and radiative heat-loss function modifies the fundamental Jeans criterion ofgravitational instability into radiative instability criterion.The effect of viscosity parameter is found to stabilize thesystem in both the longitudinal and transverse mode ofpropagation. For longitudinal wave propagation magneticfield, viscosity, finite electrical resistivity, and FLR correctionhave no effect on Jeans criterion, but FLR corrections,viscosity, and finite electrical resistivity modify the growthrate of Alfven mode. For transverse wave propagation FLRcorrections stabilize the growth rate of the system in all thecases, but it modifies the condition of instability only for thecase of nonviscous medium. Also, magnetic field stabilizesthe system but finite conductivity removes the effect ofmagnetic field there by destabilizing the system. Numer-ical calculation shows stabilizing effect of temperature-dependent heat-loss function, FLR corrections, and viscosity,where as destabilizing effect of density-dependent heat-lossfunction is on the self-gravitational instability.

Acknowledgments

The authors are highly thankful to Prof. and Head Dr. S. K.Ghosh, S. S. in Physics and Dr. T. R. Thapak, Hon’ble V. C.Vikram University Ujjain (MP) for their constant encour-agements. S. Kaothekar is grateful to Er. Praveen Vashishtha,Chairman of Mahakal Institute of Technology for continuoussupport and guidance.

References

[1] C. Hayashi, “Evolution of protostars,” Annual Review of Astro-nomy and Astrophysics, vol. 4, pp. 171–192, 1966.

[2] F. H. Shu, F. C. Adams, and S. Lizano, “Star formation in mole-cular clouds,” Annual Review of Astronomy and Astrophysics,vol. 25, pp. 23–81, 1987.

[3] B. T. Draine and C. F. McKee, “Theory of interstellar shocks,”Annual Review of Astronomy and Astrophysics, vol. 31, pp. 373–432, 1993.

[4] C. F. McKee and E. C. Ostriker, “Theory of star formation,”Annual Review of Astronomy and Astrophysics, vol. 45, pp. 565–687, 2007.

[5] S. Chandrasekhar, Hydrodynamics and Hydromagnetic Stabil-ity, Clarendon Press, Oxford, UK, 1961.

[6] A. G. Pacholczyk and J. S. Stodolkiewicz, “On the gravitationalinstability of some magnetohydrodynamical systems of astro-physical interest,” Acta Astronomica, vol. 10, pp. 1–29, 1960.

[7] N. K. Nayyar, “Magnetogravitational instability of an infinitemedium with finite electrical and thermal conductivity,” Zeit-schrift fur Astrophysik, vol. 52, pp. 266–271, 1961.

[8] S. Shaikh, A. Khan, and P. K. Bhatia, “Stability of thermallyconducting plasma in a variable magnetic field,” Astrophysicsand Space Science, vol. 312, pp. 35–40, 2007.

[9] C.-C. Yang, R. A. Gruendl, Y.-H. Chu, M.-M. Mac Low, and Y.Fukui, “Large-scale gravitational instability and star formationin the large magellanic cloud,” The Astrophysical Journal, vol.671, pp. 374–379, 2007.

[10] A. C. Borah and A. K. Sen, “Gravitational instability ofpartially ionized molecular clouds,” Journal of Plasma Physics,vol. 73, no. 6, pp. 831–838, 2007.

[11] K. Avinash, B. Eliasson, and P. K. Shukla, “Dynamics of self-gravitating dust clouds and the formation of planetesimals,”Physics Letters, Section A, vol. 353, pp. 105–108, 2006.

[12] K. V. Roberts and J. B. Taylor, “Magnetohydrodynamic equa-tions for finite Larmor radius,” Physical Review Letters, vol. 8,pp. 197–198, 1962.

[13] A. Jeffery and T. Taniuti, MHD Stability and ThermonuclearContainment, Academic Press, New York, NY, USA, 1966.

[14] J. D. Jukes, “Gravitational resistive instabilities in plasma withfinite Larmor radius,” Physics of Fluids, vol. 7, no. 1, pp. 52–58,1964.

[15] L. V. Vandakurov, “On the theory of stability of a plasma ina strong longitudinal magnetic field, variable along the axisof symmetry,” Journal of Applied Mathematics and Mechanics,vol. 28, no. 1, pp. 77–89, 1964.

[16] R. C. Sharma, “Gravitational instability of rotating plasma,”Astrophysics and Space Science, vol. 29, pp. L1–L4, 1974.

[17] P. K. Bhatia and R. P. S. Chhonkar, “Larmor radius effects onthe instability of a rotating layer of a self-gravitating plasma,”Astrophysics and Space Science, vol. 115, no. 2, pp. 327–344,1985.

[18] F. Herrnegger, “Effect of collisions and gyroviscosity on gravi-tational instability in a two-component plasma,” Journal ofPlasma Physics, vol. 8, pp. 393–400, 1972.

[19] D. S. Vaghela and R. K. Chhajlani, “Magnetogravitationalstability of resistive plasma through porous medium with ther-mal conduction and FLR corrections,” Contributions to PlasmaPhysics, vol. 29, pp. 77–89, 1989.

[20] R. K. Chhajlani and A. K. Parihar, “Magnetogravitationalinstability of anisotropic plasma with finite ion Larmor andgeneralized polytrope law,” Contributions to Plasma Physics,vol. 34, pp. 669–681, 1994.

[21] N. M. Ferraro, “Finite Larmor radius effects on the magne-torotational instability,” The Astrophysical Journal, vol. 662, pp.512–516, 2007.

[22] I. Sandberg, H. Isliker, and V. P. Pavlenko, “Finite Larmorradius effects on the coupled trapped electron and ion temp-erature gradient modes,” Physics of Plasmas, vol. 14, Article ID092504, 2007.

[23] E. Devlen and E. R. Pekunlu, “Finite Larmor radius effectson weakly magnetized, dilute plasmas,” Monthly Notices of theRoyal Astronomical Society, vol. 404, pp. 830–836, 2010.

[24] G. B. Field, “Thermal instability,” The Astrophysical Journal,vol. 142, pp. 531–567, 1965.

[25] J. H. Hunter, “The role of thermal instabilities in star forma-tion,” Monthly Notices of the Royal Astronomical Society, vol.133, pp. 239–245, 1966.

[26] M. H. Ibanez S., “Sound and thermal waves in a fluid withan arbitrary heat-loss function,” The Astrophysical Journal, vol.290, pp. 33–46, 1985.

[27] W.-T. Kim and R. Narayan, “Thermal instability in clusters ofgalaxies with conduction,” The Astrophysical Journal, vol. 596,pp. 889–902, 2003.

[28] A. E. Radwan, “Variable streams self-gravitating instability ofradiating rotating gas cloud,” Applied Mathematics and Com-putation, vol. 148, no. 2, pp. 331–339, 2004.


[29] K. Menou, S. A. Balbus, and H. C. Spruit, “Local axisymmetricdiffusive stability of weakly magnetized, differentially rotating,stratified fluids,” The Astrophysical Journal, vol. 607, no. 1, pp.564–574, 2004.

[30] S. I. Inutsuka, H. Koyama, and T. Inoue, “The role of thermalinstability in interstellar medium,” in Proceedings of the AIPConference on Magnetic Fields in the Universe: from Laboratoryand Stars to Primordial Structures, vol. 784, pp. 318–328, Angrados Reis, Brazil, December 2004.

[31] M. Shadmehri and S. Dib, “Magnetothermal condensationmodes including the effects of charged dust particles,” MonthlyNotices of the Royal Astronomical Society, vol. 395, pp. 985–990,2009.

[32] S. Shaikh, A. Khan, and P. K. Bhatia, “Jeans’ gravitationalinstability of a thermally conducting plasma,” Physics Letters,Section A, vol. 372, no. 9, pp. 1451–1457, 2008.

[33] M. Aggarwal and S. P. Talwar, “Magnetothermal instability in arotating gravitating fluid,” Monthly Notices of the Royal Astro-nomical Society, vol. 146, pp. 235–242, 1969.

[34] M. P. Bora and S. P. Talwar, “Magnetothermal instability withgeneralized Ohm’s law,” Physics of Fluids B, vol. 5, no. 3, pp.950–955, 1993.

[35] M. F. El-Sayed and R. A. Mohamed, “Gravitational instabilityof rotating viscoelastic partially ionized plasma in the presenceof an oblique magnetic field and Hall current,” ISRN Mechan-ical Engineering, vol. 2011, Article ID 597172, 8 pages, 2011.

[36] S. Kaothekar and R. K. Chhajlani, “Gravitational instabilityof radiative plasma with finite Larmor radius corrections,”Journal of Physics: Conference Series, vol. 365, no. 1, Article ID012053, 2012.

Submit your manuscripts athttp://www.hindawi.com

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



AstronomyAdvances in

International Journal of


Superconductivity


Statistical MechanicsInternational Journal of


GravityJournal of


AstrophysicsJournal of


Physics Research International


Solid State PhysicsJournal of

Computational Methods in Physics

Journal of



Soft MatterJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com

AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of

Documents

EffectofRadiativeHeat …downloads.hindawi.com/journals/isrn.astronomy...2 ISRN Astronomy and Astrophysics Along with this, in above discussed problems, the eﬀect of ﬁnite ion