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Pulsational mode behaviour in complex non-extensive non-thermal viscous astrophysical fluids Barsha Bhakta, Munmi Gohain and Pralay Kumar Karmakar EPL, 119 (2017) 25001
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July 2017 EPL, 119 (2017) 25001 doi: 10.1209/0295-5075/119/25001
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Pulsational mode behaviour in complex non-extensive non-thermal viscous astrophysical fluids Barsha Bhakta, Munmi Gohain and Pralay Kumar Karmakar(a) Department of Physics, Tezpur University - Napaam-784028, Tezpur, Assam, India received 16 June 2017; accepted in final form 12 September 2017 published online 5 October 2017 PACS PACS PACS
52.27.Lw – Dusty or complex plasmas; plasma crystals 04.40.-b – Self-gravitating systems; continuous media and classical field in curved spacetime 95.30.Lz – Hydrodynamics
Abstract – A semi-analytic evolutionary model is classically constructed to see the pulsational mode dynamics of gravitational collapse in a hydrostatically bounded complex non-thermal astrocloud on the Jeans scales of space and time. The multi-fluidic model consists of non-extensive electrons and ions, and massive dust grains along with partial ionization in flat space-time. A linear Fourier-based normal mode analysis around the defined static (homogeneous) equilibrium reduces the basic cloud equations into a quartic (biquadratic) dispersion relation with a unique set of multi-parametric coefficients. It is interestingly found that non-thermal associations in the cloud pave the way for faster normal mode propagation. The neutral dust viscosity plays a decisive role towards a transition of the pulsational mode from a non-dispersive to a dispersive form. It is also observed that the viscosity has a stabilizing influence on the cloud. The dust-charge variation is noted to play an insignificant role on the stability. The results may be significant in understanding the dynamic non-homological cloud collapse leading to a hierarchical initiation of a bounded structure formation in diverse astro-cosmic anti-equilibrium environs. c EPLA, 2017 Copyright
Introduction. – The interstellar complex dust molecular clouds (DMCs) support a wide variety of collective waves, oscillations and instabilities [1–5]. It is an interesting and challenging topic of research investigations because of their important roles played in stellar and other bounded structure formations [3–8]. The DMC constituent dust grains are mainly composed of graphite, silicate and other metallic compounds varying from micron to sub-micron scale in geometrical size [1,2]. The importance of the excitation processes in a complex grainy plasma dynamics and associated wave instabilities are also realizable on the laboratory scales under micro-gravity conditions, like in thermonuclear fusion reactors, Q-machine, and so forth [4]. It may be further noted that the electric charge of dust particulates plays an important role in modifying diverse collective wave-instability phenomena starting from laboratory to astrophysical scales [5,7–11]. Out of all the existing naturalistic normal modes in the above complex astro-environments, the pulsational mode of gravitational collapse (PMGC) of self-gravitating (a) E-mail:
[email protected] (corresponding author)
unbounded dusty plasmas in different configurations has recently drawn growing attention among many researchers [6–11]. The PMGC stems from the mutualistic gravito-electrostatic coupling in the presence of partially ionized massive dust grains [6]. In other words, it is the resultant outcome of the conjoint periodic interplay of the astrofundamental Jeans mode (gravitational, inward, by Newtonian particles) and acoustic mode (electrostatic, outward, by Coulombic particles). This composite conjugational mode is responsible for a tremendous energy through stellar and quasi-stellar structure formation processes on the Jeans scales of space and time [6–8]. This plausible tapestry is yet to be inclusively construed by astro-plasma researchers for long-sought investigative explorations of star-formational mechanics as a real bistep perennial process of contraction-expansion instead of the usual uni-step one of sheer contraction of the clouds. It may be noted that various multi-space satellite measurements of plasma velocity distributions in different astrophysical circumstances, like in the solar wind, in the planetary magnetospheres, magneto sheaths, interstellar media and so forth, have revealed the non-Maxwellian
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Barsha Bhakta et al. particle distributions (q-nonextensive, non-equilibrium) as the most naturalistic ones [12–15]. This is because of the large-scale non-local and gradient effects inherent in the long-range interparticle interaction forces in such astroenvirons. As a consequence, in such inhomogeneous situations, the plasmas are to be treated in the framework of non-extensive thermo-statistics, as initiated by Tsallis [12]. The q-nonextensivity (entropic index) in such configurations is quite important. It gives a measure of the thermo-gravitation coupling [13,14], as q = 1 + (∂x Ut /∂x Ug ); where Ut is the randomizing thermal (kinetic) energy and Ug is the organizing gravitational (potential) energy of a constitutive particle. The nonextensivity parameter plays an important role in shaping the propagatory features of wave structures, such as the Mach number, polarity, pulse width, and so forth [16,17]. As far as seen, nobody has, however, so far reported the behaviour of the PMGC in a complex non-thermal nonextensive viscous astrocloud and the subsequent structure formation processes. In this motivated work, we propose a theoretical semianalytic model to investigate the PMGC of a complex viscous anti-equilibrium astrocloud on the Jeans scales of space and time. The model is devised in the presence of q-nonextensive electrons and ions, kinematic viscosities of neutral and charged dust. A Fourier-based plane-wave analysis is used methodologically to derive a generalized dispersion relation (quartic in degree). Finally, we develop a constructive scheme for numerical analysis to explore the stability behaviour of a complex astrocloud. The stabilising and destabilising factors are identified and explored in detail followed by future scope. Physical model and formulation. – We consider an unbounded complex viscous astrocloud fluid composed of q-nonextensive (non-thermal) lighter electrons and ions, and charged massive dust grains with partial ionization. The dust grains are electrically charged due to the plasma environment (via contact electrification processes) under random collision effects by the electrons and ions over the grain surfaces [4,8]. The charged and neutral grains, which are assumed to be identical micro-spheres, are treated as two distinct viscous fluids. The translational and rotational Brownian motions of the dust grains, caused by the equipartition of energy between the solid phase of dust grains and the gaseous phase of the background plasma [8], are ignored for simplicity. The developed model excludes complications, like dust distribution, magnetic field, gravitational effects by distant astrophysical objects, etc. The various constituents of the cloud are assumed to maintain a global quasi-neutral homogeneous hydrostatic equilibrium at least in the zeroth order. Thus, our basic model setup describing the complex cloud is governed by non-thermal fluid dynamical equations in the form of fundamental conservation principles of particle flux, net force, total energy, and so forth [18]. Thus, the q-nonextensive non-thermal electronic and ionic
population distributions in unnormalized form with all the conventional notations [5,15] describing the cloud are, respectively, given as 1/1−qe eϕ , (1) ne = neo 1 + (1 − qe ) Te 1/1−qi eϕ ni = nio 1 − (1 − qi ) , (2) Ti where the different notations, ne (ni ) represent the electron (ion) population densities, qe (qi ) the non-thermal entropic indices of electrons (ions), ne0 (ni0 ) the electron (ion) equilibrium number densities, Te (Ti ) the electron (ion) temperatures (in eV), and ϕ the electrostatic potential developed by conjoint charge density fields. Now, the continuity and momentum equations, with all the customary notations [18–22], for the constitutive cold neutral dust grains can, respectively, be written as ∂ ∂ndn + (ndn vdn ) = 0, ∂t ∂x ∂vdn ∂vdn ∂ψ ∂ 2 vdn + vdn =− + ηdn , ∂t ∂x ∂x ∂x2
(3) (4)
where vdn is the neutral dust velocity, ndn is the neutral dust population density, ψ is the gravitational potential created by the collective material density fields, and ηdn is the neutral dust kinematic viscosity. Similarly, the continuity and momentum equations [18–22] for the charged dust fluid are, respectively, cast as ∂ ∂ndc + (ndc vdc ) = 0, ∂t ∂x ∂vdc qd ∂ϕ ∂ψ ∂vdc + vdc =− − ∂t ∂x md ∂x ∂x ∂ 2 vdc −fcn (vdc − vdn ) + ηdc , ∂x2
(5)
(6)
where md is the dust mass, vdc is the charged dust velocity, ndc is the charged dust population density, fcn is the binary collision rate (linear collision frequency) of linear momentum transfer between the charged and neutral dusts, and ηdc is the charged dust kinematic viscosity. Finally, the model closure is obtained with the help of the Poisson equations [19–22] for the electrostatic potential and gravitational potential distributions caused by the non-zero density fields, respectively, given as ∂2ϕ = 4πe (ne − ni − qd ∗ ndc /e) , ∂x2 ∂2ψ = 4πGmd (ndc + ndn − nd0 ) , ∂x2
(7) (8)
where e is the electronic charge, qd = Zd e is the electrical charge acquired by each of the identical dust grains and G is the Newtonian universal gravitational constant via which gravitational interactions are realized in the physical world.
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Pulsational mode behaviour in complex non-extensive non-thermal viscous astrophysical fluids ∂Mdn1 ∂Ndn1 + = 0, ∂τ ∂ξ ∂Mdn1 ∂ Ψ1 ηdn ∂ 2 =− + Mdn1 , ∂τ ∂ξ cs λJ ∂ξ 2 ∂Ndc1 ∂Mdc1 + = 0, ∂τ ∂ξ ∂ Φ1 ∂ Ψ1 ∂Mdc1 = −Zd + ∂τ ∂ξ ∂ξ 1 ∂2 1 −fcn [Mdc1 − Mdn1 ] + ηdc 2 Mdc1 , ωJ cs λJ ∂ξ 2 2 2 ωpe ωpi ∂ Φ1 = Φ1 − Φ1 ∂ξ 2 ωJ ωj 2 ωpd (Zdo Qd ) Ndc1 , − ωJ 2 2 ωJc ωJn ∂ 2 Ψ1 = N − Ndn1 . dc1 ∂ξ 2 ωJ ωj
The analysis focally aims at the scale-invariant (scalefree) micro-physical insights of the pulsational mode dynamics and its stability in the linear order. Therefore, the basic governing equations, eqs. (1)–(8), are normalized, respectively, as Ne = [1 + (1 − qe ) Φ]
1/1−qe
,
(9)
1/1−q
i , (10) Ni = [1 − (1 − qi ) Φ] ∂Ndn ∂ + (Ndn Mdn ) = 0, (11) ∂τ ∂ξ ∂Mdn ∂Mdn ∂2 ∂Ψ 1 + Mdn =− + ηdn 2 Mdn , ∂τ ∂ξ ∂ξ cs λJ ∂ξ (12) ∂ ∂Ndc + (Ndc Mdc ) = 0, (13) ∂τ ∂ξ ∂Mdc ∂Φ ∂Ψ ∂Mdc + Mdc = −Zd − ∂τ ∂ξ ∂ξ ∂ξ 1 ∂2 1 − fcn (Mdc − Mdn ) + ηdc 2 Mdc , (14) ωJ cs λJ ∂ξ 2 2 ωpe ωpi ∂2Φ = N − Ni e ∂ξ 2 ωJ ωj 2 2 ωpd Zd e Ndc , (15) − ωJ 2 2 ωJc ωJn ∂2Ψ = N − Ndn − 1. (16) dc ∂ξ 2 ωJ ωJ
The standard astrophysical normalization scheme adopted here deals with the different customary notations in filtered form [13–16]. For instant perception, the normalization details are symbolically cast as ne t ni x , τ = −1 , Ne = , Ni = , λJ n n ωJ e0 i0 ndc vdn vdc ndn , Ndc = , Mdn = , Mdc = , Ndn = ndn0 ndc0 css css eϕ Φ= , Ψ = ψ c2ss , ωJ = 4π ρd G, Te T css , λJ = . ρd = (mdn ndn0 + mdc ndc0 ) , css = md ωJ
ξ=
We are interested in the evolutionary dynamics of slightly perturbed solutions of the cloud. Thus, all the relevent cloud parameters describing the dynamics of electrons, ions, neutral dust grains and charged dust grains are linearly perturbed about the defined static homogeneous equilibrium as Ne = 1 + Ne1 ,
Ni = 1 + Ni1 ,
Ndn = 1 + Ndn1 ,
Ndc = 1 + Ndc1 , Mdn = 0 + Mdn1 , Φ = 0 + Φ1 , Ψ = 0 + Ψ1 .
Mdc = 0 + Mdc1 ,
(17)
Ni1 = − Φ1 ,
(18)
(20) (21)
(22)
(23) (24)
Now, we apply the Fourier-based plane-wave analysis [7,8] of the diversified fluctuations as ∼ exp −i(Ωτ − Kξ), where Ω is the Jeans-normalized angular frequency and K is the Jeans-normalized angular wave number of the perturbations. Accordingly, in the Fourier space (Ω, K), the linear differential operators now get transformed as ∂/∂ξ ≡ iK and ∂/∂τ ≡ −iΩ. Thus, the Fouriertransformed set of eqs. (19)–(24) can be respectively given as Mdn1 =
Ω K
Ndn1 ,
(25)
1 (iΩ) Mdn1 = (iK) Ψ1 + ηdn K 2 Mdn1 , (26) cs λJ Ω (27) Mdc1 = Ndc1 , K (iΩ) Mdc1 = Zd (iK) Φ1 + (iK) Ψ1 1 1 (Mdc1 − Mdn1 ) + ηdn K 2 Mdc1 , (28) + fcn ωJ cs λJ
2 2 2 ωpe ωpi ωpd 2 1 Φ1 + Φ1 − Zd e Ndc1 , Φ1 = − 2 K ωJ ωJ ωJ (29)
Ψ1 = −
1 K2
ωJc ωJ
2
Ndc1 +
ωJn ωJ
2
Ndn1 .
(30)
Substituting the expressions from eqs. (25), (27) and (30) in eqs. (26) and (28), one obtains
After an analytical exercise of simplification, the linearized form of eqs. (9)–(16) can be respectively derived as Ne1 = Φ1 ,
(19)
25001-p3
Ndn1 = −i
ωJc ωJ
2
× iΩ2 + i
Ndc1
ωJn ωJ
2
1 ηdn K 2 Ω − cs λJ
−1
,(31)
Barsha Bhakta et al.
Φ1
+ iΩ2 + i
−
−
fcn Ω × ωj
ωJn iΩ + i ωJ 2 2 2
ωJn ωJ ωJn ωJ
2
11
ωJc ωJ
2
iΩ2 + i
1 ηdn K 2 Ω cs λJ
−
1 cs ω J
i
ηdn K 2 Ω 2
ωJn ωJ
−1
.
ωJc ωJ 2
−
1 cs λJ
* = dn * = dn * = dn
10 Real frequency
Ndc1 = Zd iK
2
fcn Ω − ωJ
9 8
(a)
5 10 15
7 6 5 4
0
1
2
3
4 5 6 Wave number
7
8
9
10
9
10
25
ηdn K 2 Ω Imaginary frequency
(32)
Now, substituting the expression from eq. (31) in eq. (28) in accordance with the method of decomposition [6,7], one finally gets the generalized dispersion relation (quartic) as D (Ω , K) ≡ Ω4 + A 1 Ω3 + A 2 Ω2 + A 3 Ω + A4 = 0, (33) where, the different involved coefficients are given as
2 2 ∗ 2 2 ωpe ωpi 2 A1 = i ηdn ωJ K + +K ωJ ωJ 2 2 fcn ωpi ωpe fcn 2 − − − fcn ωJ K ωJ ωJ 2 −1 ωJc 2 (ωpe + ωpi + ωJ K) , × −1 − ωJn
2 2 ωJc ωpe ωJc ωpi 2 2 A2 = −i + + ωJc K ωJ ωJ −1 fcn 2 (ωpe + ωpi + ωJ K) × i ∗ 2 ωJ ηdn K −1 2 3 2 2 (ωpe + ωpi + ωJ K) Zd eK , − ωpd
2 ωJn ωJc ωJc fcn 1 A3 = i ∗ K2 + ωJ ηdn ωJ 2 2 2 ωJn ωpi ωpe 2 + ω f K + × J cn ∗ ωJ ωJ ηdn 2 −1 ωJn 2 (ωpe + ωpi + ωJ K) × , ωJ
2 2 (ωJn ) (ωJc ) , A4 = 4 (ωJ ) ∗ where ηdn = ηdn /cs λJ is the normalized (filtered to the Jeans viscosity [19], given as cs λJ ) kinematic viscosity associated with the neutral dust fluid. It may clearly be mentioned here that, as a reliability checkup, if all the factors considered here are piecewise removed, eq. (32) is clearly in conformity with the previous results by others for static dust-charge [6] and non-static dust-charge [7].
Results and discussions. – A fourth-degree (quartic) dispersion relation (eq. (32)) is obtained with the help
* = dn * = dn * = dn
20 15
5 10 15
10 5 0
(b) 0
1
2
3
4 5 6 Wave number
7
8
Fig. 1: (Colour online) Profile of the normalized (a) real frequency part (Ωr ) and (b) imaginary frequency part (Ωi ) with the variation in the normalized wave number (K). The vari∗ = 5 (blue, ous lines indicate the diversified situations with ηdn ∗ ∗ solid line), ηdn = 10 (red, dashed line), and ηdn = 15 (black, dotted line), respectively. The fine details are given in the text.
of the Fourier technique in order to see the pulsational fluctuation dynamics in a complex astrocloud. An analytic inspection reveals that the pulsational propagatory properties are controlled by the various plasma-dependent multi-parametric involved coefficients. It is then numerically analysed to explore the basic mode characteristic features as in figs. 1, 2. Figure 1 displays the profile patterns of the normalized (a) real frequency part (Ωr ) and (b) imaginary frequency part (Ωi ) of the pulsational mode with the variation in the normalized wave number (K). The various lines in∗ = 5 (blue, solid dicate the diversified situations of ηdn ∗ ∗ line), ηdn = 10 (red, dashed line), and ηdn = 15 (black, dotted line), respectively. The different input values from the astronomical literature [1–8] used here are Zd = 100, e = 1.60 × 10−19 C, ωpe = 1.793 × 10−9 rad s−1 , ωpi = 1.552 × 10−9 rad s−1 , ωpd = 4.009 × 10−12 rad s−1 , ωJn = 1.29 × 10−3 rad s−1 , ωJc = 1.29 × 10−4 rad s−1 , ωJ = 4.09 × 10−11 rad s−1 and fcn = 8.85 × 10−4 Hz. We interestingly see that the real frequency of the mode increases with rise in the kinematic viscosity of the neutral dust fluid, and vice-versa (fig. 1(a)). It further implicates a fantastic mode feature that the non-thermal associations pave the way for faster normal mode propagation in comparison with that found in a thermalized cloud situation [6,7]. A unique transition from non-dispersive to dispersive nature is found to exist depending on the neutral dust fluid kinematic viscosity (fig. 1(a)). In contrast, the growth rate decreases with increase in the neutral dust kinematic viscosity, and vice-versa (fig. 1(b)). As a result, it can be inferred that the neutral fluid viscosity has a stabilizing influence on the slightly perturbed non-thermal cloud, and, hence, in the pulsational dynamics.
25001-p4
Pulsational mode behaviour in complex non-extensive non-thermal viscous astrophysical fluids
Real frequency
-1.2 -1.4
Zd= 10
-1.6
Zd= 100
analysis spectrally offers a decisive tool to characterize the extremal behaviours of the pulsational mode under investigation via the complex two-step gravito-electrostatic interplay (inward gravitational contraction and outward electrostatic expansion). It is confessed here that the evolutionary conjugational dynamics in the adopted complex cloud model does not show any significant sensitivity to the judicious variations of the non-extensivity parameters [15–17]. The wave-characterizing profiles remain similar to those in fig. 1, but with some negligible microquantitative changes.
Zd= 1000
-1.8 -2 -2.2 -2.4 -2.6
(a) 0
1
2
3
4 5 6 Wave number
7
8
9
10
25 Imaginary frequency
Z = 10 d
20
Z =100 d
Z =1000 d
15 10 5
(b) 0
0
1
2
3
4 5 6 Wave number
7
8
9
10
Fig. 2: (Colour online) Profile of the normalized (a) real frequency part (Ωr ) and (b) imaginary frequency part (Ωi ) with variation in the normalized angular wave number (K). The various lines correspond to the diversified situations of Zd = 10 (blue, solid line), Zd = 100 (red, dashed line), and Zd = 1000 (black, dotted line), respectively. The different input values ∗ = 10 kept fixed. used are the same as in fig. 1, but with ηdn
In fig. 2, we analogously depict the profile structures of the normalized (a) real frequency part (Ωr ) and (b) imaginary frequency part (Ωi ) of the fluctuations with variation in the normalized angular wave number (K). The various lines now link to the diversified situations of the equilibrium dust-charge number as Zd = 10 (blue, solid line), Zd = 100 (red, dashed line), and Zd = 1000 (black, dotted line), respectively. The different input values used are the ∗ = 10 kept fixed. There is no same as in fig. 1, but with ηdn apparent macroscopic visible distinction of the lines due to their mutualistic overlapping with the dust-charge variation. It is further demonstrated that the variation of the unperturbed dust-charge plays no significant role in the propagatory characteristics of the pulsational mode dynamics, viz. its real frequency part (fig. 2(a)) and growth rate (fig. 2(b)), in the cloud. It, therefore, indicates that the dust charge fluctuations do not introduce any relevant modification to the pulsational mode stability in the proposed non-thermal non-extensive cloud configuration. This is against the charge-varying thermalized cloud reported previously [7], where dust-charge fluctuations have been demonstrated to act as a damping agency to the pulsational mode evolution. It can, thus, be noted that, despite the dust-charge variation, the non-thermal nature of the cloud resists the pulsational mode to damp out. Our numerical analysis parallelly reveals an important feature about the excitation mechanism of the investigated pulsational mode. In the Jeansian limit (gravitational, K → 0), the modal growth goes maximum. However, the growth becomes asymptotically minimum in the electrostatic regime (acoustic, K → ∞), as clearly evident in figs. 1, 2. Thus, we can say that the
Conclusions. – A normal mode analysis of the pulsational mode conjugational dynamics in a boundless complex non-thermal non-extensive viscous astrocloud relative to a defined homogenous hydrostatic equilibrium configuration is semi-analytically carried out on the astrophysical scales of space and time. It is seen from the investigation that the real frequency of the mode increases with a rise in the kinematic neutral dust viscosity, and viceversa. It furthermore implicates an interesting point that the non-thermal associations in the cloud pave the way for faster normal mode propagation against that found in a thermalized situation. A unique transition from nondispersive to dispersive nature is found to exist depending on the neutral dust fluid kinematic viscosity (fig. 1(a)). In contrast, the growth rate decreases with an increase in the kinematic viscosity of the neutral dust fluid, and vice-versa (fig. 1(b)). As a result, the neutral fluid viscosity has a stabilizing influence on the slightly perturbed non-thermal cloud. It is further demonstrated that the variation of the equilibrium dust-charge plays no significant role in the propagatory characterestics of the pulsational mode dynamics, viz. its real frequency part (fig. 2(a)) and growth rate (fig. 2(b)), in the cloud. All these wave dynamical features are practically realizable in partially ionized dust molecular clouds with a special mention to the H-II regions [1,8]. The results may, hopefully, be helpful in the basic understanding of the dynamic self-gravitational collapse mechanism leading to a non-homological large-scale bounded structure formation in diversified astro-cosmic fluid environs. It is finally admitted that the analysis needs to be further ameliorated for a realistic astronomical non-local depiction with the proper inclusion of heat transport equation, translatory and rotatory dust Brownian motions, tidal force fields, Coriolis effects, interstellar inhomogeneous magnetic field, etc. ∗∗∗ We gratefully recognize the insightful comments and creative suggestions raised by the respected anonymous referees leading to the scientific improvement of the manuscript. The financial support received from SERB, DST, GoI, Grant No. 021/2011, is thankfully acknowledged.
25001-p5
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