Magnetorotational instability in dense electron–positron–ion plasmas

We in this manuscript analyzed the magnetorotational instability (MRI) by using a multi-component quantum fluid model with the effect of spin magnetization in a differentially rotating degenerate electron–positron–ion (e–p–i) quantum plasma. The electrons and positron having the same mass but opposite charge are taken to be degenerate whereas ions are considered as classical owing to their large inertia. The general dispersion relation is derived and a local dispersion relation for MRI is obtained by applying MHD approximations. To obtained MRI and to analyze the results numerically, reduced dispersion relation is derived using the local approximations. The obtained results are applied to the astrophysical situations exist there in the interiors of White Dwarfs and neutron stars. Contribution from spin magnetization and the number densities of electrons and positrons plays a vital role in the dynamics and can alter the instability. The increase in the electron number density, hence spin magnetization enhances the growth rate of the mode and leads the system to instability which results in the core collapse of certain massive stars.


Magnetorotational instability in dense electron-positron-ion plasmas S. Usman 1* & A. Mushtaq 2
We in this manuscript analyzed the magnetorotational instability (MRI) by using a multi-component quantum fluid model with the effect of spin magnetization in a differentially rotating degenerate electron-positron-ion (e-p-i) quantum plasma.The electrons and positron having the same mass but opposite charge are taken to be degenerate whereas ions are considered as classical owing to their large inertia.The general dispersion relation is derived and a local dispersion relation for MRI is obtained by applying MHD approximations.To obtained MRI and to analyze the results numerically, reduced dispersion relation is derived using the local approximations.The obtained results are applied to the astrophysical situations exist there in the interiors of White Dwarfs and neutron stars.Contribution from spin magnetization and the number densities of electrons and positrons plays a vital role in the dynamics and can alter the instability.The increase in the electron number density, hence spin magnetization enhances the growth rate of the mode and leads the system to instability which results in the core collapse of certain massive stars.
In degenerate plasmas, the electrons are closely packed together with the maximum allowed density by quantum mechanics at a given pressure.The quantum effects in such a situation play a vital role when the charged particles de-Broglie wavelength De (= ℏ/m e v te ) stands comparable to the scale length of the system e.g.interparticle distances n −1/3 , where ℏ, m e , n, and v te are the reduced planck's constant, mass of electron, equilibrium particle density and particle thermal speed respectively.These type of plasmas can be found in the interior of densest astrophysical object 1 (degenerate stars).There are mainly three classes of the degenerate stars known as white dwarfs (WDs), neutron stars (NS), and black holes (BH).The WDs are supported against the collapse by the electron degeneracy pressure, while the neutron stars are largely supported by neutron degeneracy.Black holes are the completely collapsed stars as they collapsed to a singularity.It is worthy to notice that the properties of a quantum plasma present in the interiors and surrounding of these degenerate stars, alters significantly from a classical plasma.
There has been an increasing interest in describing collective quantum effects in plasmas using quantum fluid theory (QFT).The regime of interest is, when the de Broglie wavelength of the charge carriers is comparable to the dimensions of the system, then quantum mechanical diffusion and tunneling cannot be ignored.This effect is described in QFT through the Bohm potential.The Bohm potential first appeared in Madelung's 1926 alternative to the Schrödinger equation 2 and has been re-derived in various ways 3 in QFT, the intrinsically quantum term is the Bohm potential, which describes quantum mechanical diffusion or tunneling.Motivated by application to solid state plasmas, the response tensor for a completely degenerate nonrelativistic electrons gas has been known since the 1950s 4 .The generalization to a fully relativistic quantum treatment for electrons and positrons, including the nonlinear response tensors, as well as the linear response tensor is available, and has been called quantum plasmadynamics (QPD) 5 .All relativistic quantum effects are included in QPD (like degeneracy, quantum recoil, spin, etc.).QPD is a more rigorous theory than QFT, and all the results of QFT should be derivable from QPD.However, the relationship between these two approaches is not immediately obvious.A notable difference in comparing them is that QFT is formulated in coordinate space (t,x), whereas QPD is formulated in Fourier space (ω,k).Quantum mechanical diffusion or tunneling does not appear explicitly in a Fourier space formulation, but it must be included in the known quantum effects.
Since the statistical description changes from Maxwell Boltzmann's to the Fermi Dirac, many attempts were made [6][7][8] to develop a quantum fluid theory.To address the situation of degenerate plasma regimes, quantum magnetohydrodynamic (QMHD) model was formulated by Haas in 2005 9 .The QHD equations are derivable from the electromagnetic Wigner equation 10 by taking the relevant moments of the distribution function, one can magnetosphere 47,48 , Jet compositions (pair electron-positron plasma), by detecting radio emission from quasars using VLBA 49 and in Payload for Antimatter Matter Exploration and Light-nuclei Astrophysics (PAMELA) experiments 50 and by the process of neutronization and by thermal emission 51 .For more details experimental observations see 52,53 .
In addition to the electrons and positrons, a small fraction of ions has also been observed in the recent observation by the advanced satellites for astrophysics and cosmology 54 , the plasma is an admixture of electrons, positrons and ions.With the introduction of ions as an ingredient to the usual e-p plasma the response of the plasma greatly change.The positrons have enough lifetime that the normal two-species (e-p) plasma becomes a three-species electron-positron-ion (e-p-i) plasma.Naturally, positrons are an ingredient that is present everywhere in compact astrophysical objects and is therefore, the existence of dense e-p plasmas are expected there.A high pair annihilation rate due to a very large densities of electrons and positrons is expected there in these dense objects.However, some investigations 55,56 have been made for some density ranges concerning WDs, where the rates of annihilation can be ignored and positrons have enough much lifetime to contribute in collective plasma phenomena 57 .Due to unit mass ratio, in many respects e-p plasma behave in a different way from usual e-i plasmas.This feature makes the investigation of e-p plasma extremely worthy, both for the fundamental physics and for astrophysical interests.The presence of positron in these astrophysical surroundings is certain, especially regarding electron-positron release phenomena via the neutrino annihilations and neutrino absorptions on to the nuclei.Till now, in study of MRI instability the role of positron has usually been neglected.In order to understand the dynamics, it is important to study the state of such a plasma and the dynamics of these species by employing a configuration of rotating object and suitable non-relativistic model (fluid model).
In the recent past decades a number of theories to describe quantum plasma and hydrodynamic stability in magnetized plasmas are developed with its importance to astrophysical environments [58][59][60] .One of the different instabilities arise in rotating astrophysical dense object is Magnetorotational Instability (MRI).It is a type of MHD instability initially addressed by Velikhov 61 in 1959 and then after confirmed by Chandrasekhar 62 in 1960 while studying the Taylor Couttee flow in the concentric differentially rotating cylinders.For almost three decades MRI is out of the context from the main stream research until Balbus and Hawley 63 in 1991 applied the concept to the accretion disks around a massive central objects.They showed that the growth rate of the MR instability is independent of the magnetic field strength, even a low magnetic field can change the stability of the system.These disks are stable hydrodynamically but they are unstable magnetohydrodynamically, leading to the disk turbulence and transport of angular momentum [64][65][66] .MRI is also expected to act as a dynamo in the accretion disks 67 .Hereinafter, there is a growing interest in MR instability applications concerning the astrophysical problems in various magnetized accretion disks [68][69][70] .Different models, various analytical explanations and numerical analysis has been performed to explain the dynamics of MRI in different situations i.e., Single 63 and two fluid model 71 was developed with effect of magnetized and un-magnetized plasma, The effect of viscosity in rotating plasma 72 and rotating dusty plasmas including dissipation 73 .The Incompressible magnetohydrodynamics simulations is presented 74 in spherical geometry with explicit diffusivities where the differential rotation is forced at the outer boundary.More recently, Nonlinear development of MRI in circularly magnetized eccentric disks 75 , impact of the MRI on the evolution of massive stars 76 , smoothed particle magnetohydrodynamics method 77 with the geometric density average force expression and the mean field dynamo effect on MRI 78 are extensively presented.The growth rate of MRI in circumstellar disks 79 is investigated with the effect of changes in the strength and direction of the magnetic field and reported that the MRI active region possibly exists with a weak magnetic field.The vertical shear instability in poorly ionized, and magnetized protoplanetary discs MRI in all three frequency ranges (low, intermediate and high) of weakly ionized electron-ion-neutral (e-i-n) and (e-p-i) plasma has been investigated by using the classical multi-fluid approach 80,81 .The purpose of this manuscript is to examine the instability in these regimes in a multi-fluid framework under the influence of quantum correction term in the form of spin magnetization force.Other correction terms e.g.relativistic correction terms, Quantum Bohm potential, pressure degeneracy and exchange correlation effect are not yet included in this work and planed to be included in future to develop a full quantum description of MRI mechanism in dense objects.
MRI for now can be considered as an important candidate in the core collapse of degenerate stars and for many other dynamical behaviors.In this work we examined MRI in three species (e-p-i) dense plasma by introducing quantum correction terms to the equation of motion governing the dynamics.Generally, in dense astrophysical objects, ions provides inertia, where the electrons and positrons are considered to follow the electron/ positron degeneracy pressure to support them against the gravitational collapse.Solving the QMHD equations coupled to the Maxwell's equations we derived the generalized dispersion relation.The quantum contribution from the ions are ignored because of its large mass in comparison to the electron and positron.Their quantum behavior depends upon degeneracy parameter which is larger than unity for quantum case.The dispersion relation is limited to certain MHD conditions to obtain a reduced dispersion relation.The electron and positron densities and spin magnetization effects reveals some important consequences on the instability growth rate.We in this work are intended to make a mathematical and numerical investigations of MRI by looking into the quantum viewpoint of dense astrophysical objects.
This manuscript is arranged as, In "Model Equations and dispersion relation", the basic quantum hydrodynamic equation of motion for e-p-i plasma along with the Maxwell's equations are presented.Based on the model equations, the dispersion relation for the e-p-i plasma is obtained.In "Reduced dispersion relation", the reduced dispersion relation is obtained with certain MHD limitations.Section "Results and discussion" contain the detailed devoted results and discussion, and Finally, in "Conclusions", the conclusions of the work are presented.

Model equations and dispersion relation
We consider an axisymmetric, collisionless, fully degenerate and quasi-neutral electron-positron-ion (e-p-i) plasma embedded in homogenous external magnetic field B = Bẑ .Using the standard cylindrical geometry (r, θ, z), the plasma rotates in the azimuthal θ direction with an angular frequency � = �(r) .The equilibrium quantities are respectively given as B 0 = (0, 0, B 0 ), E 0 = (E 0 , 0, 0) , v j0 = (0 , r�, 0) and P j0 = P j0 (r) .The dynamics of such a system is governed by continuity and multi-fluid hydrodynamic momentum equation 82 expressed as where n j is the particles number density of jth (= i, e, p) particle which allow us to write the quasi-neutrality condition as n i0 = n e0 − n p0 .(ρ = mn) is the particle density, v j and P j is the fluid velocity and thermal pressure, respectively.q j , E and B are the electric charge, electric field and magnetic field, respectively.For the degenerate electrons and positron we use Fermi pressure as P Fj = (3π 2 ) representing the Fermi velocity and for the massive non-degenerate ions, one can use the classical pressure as P i = γ i n i k B T i with γ i is the polytropic index.On the left-hand side of the equation, we have the continual deriva- tive of the velocity field and the gradient of pressure.On the right-hand side of the Euler equa- tion, we present the force fields of different natures.The first term is the Lorentz force term, The second term represents the quantum Bohm potential.The last term on the right hand side represents the effect of spin magnetization force.The parameter µ j = q 2m j c represents the magnetic moment of jth particle and B 1 stands for the perturbed magnetic field.We can define the electron magnetic moment as µ e = −µ B , with µ B =| q 2m j c | being the Bohr magneton.being the reduced plank's constant.The spin evolution equation for the spin-1/2 quantum plasma can be written as ds dt = 2µ (s × B) .Under the MHD limitations (ω ≤ ω ci ≤ ω ce ) , the spin inertia can be neglected well below the electron cyclotron frequencies, gives the spin equation of motion with solution 26 Here η(α j ) = tanh(α j ) is the Langevin parameter with α j = µ B B 0 k B T Fj , and T Fj = is the degenerate Fermi temperature of the j th species.The above set of continuity and momentum equations are coupled to max- well's equations in the form and where J = j=e,i q j n j v j + cJ Me + cJ Mp is the current density with J Me = ∇ × M e and J Mp = ∇ × M p being the spin magnetization current densities of electrons and positrons, respectively.The magnetization density vector is Me =µ B n e tanh(α) B and c = (ε 0 µ 0 ) − 1 2 is the speed of light in vacuum.In a cylindrical coordinates system, the perturbed magnetic and electric fields are B 1 = ( B r , B θ , B z ) and E 1 = ( E r , Ẽθ , E z ), and velocity v j1 = ( v jr , v jθ , v jz ) .While P j and n j are the perturbed pressure and perturbed number density, respectively.Each perturbed profile is considered to be proportional to e −iωt+ik z z , where ω is the wave frequency and k z is the wave number directed along z-axis.Due to the incompressibility the mass conservation is reduced to ∇ • v j = 0 , gives rise to v jz = i L k z v jr .The perturbed Poisson's equation ∇ • E 1 = 0 , resulting in Êz = i L k z Ẽr .Form the divergence free property of the magnetic field ∇ • B 1 = 0 gives rise to B z = i L k z B r and from the perturb Faraday's law we can get Êθ = ω k z B r and B θ = Ẽr .Here we define the operator L = 1 r + ∂ r .For instance neglecting the contribution of quantum Bohm potential in the momentum equation and only incorporating the contribution from spin magnetization, the linearized equation ( 2) in component (r, θ, z) form can be expressed as  8), ( 9) and (10) we can write the corresponding electron, positron and ion equations of motion as and The quantum contributions to the momentum equation associated with the ions have been neglected because of their heavier mass in comparison to the electron and positron.Here ω ce = eB 0 m e is the electron cyclotron frequency associated with external magnetic field, c = eB 0 m i stands for ion gyrofrequency and p = eB 0 m p stands for positron gyrofrequency and ω is the wave frequency.The local approximations are adopted, assuming ∂ r ≃ ik r and k r r ≫ 1, where k r is the radial wave number.Thus ∂ r L ≃ −k 2 r and k = (k 2 r + k 2 z ) 1/2 is the total wave number.The perturbed magnetic field can be determined by using where S t = S e − S p with S e = η e (α)µ e n e and S p = −η p (α)µ p n p .By using the problem geometry the ion per- turbed velocities ṽir and ṽiθ are obtained from Eq. ( 17) as From θ-component of the positron equation of motion, we can find the positron velocities ṽpr and ṽpθ as and The dispersion relation corresponding to Eqs. (11-16) and (18-21) can be obtained as where V A = B 2 0 m i n i µ 0 and ω A = k z V A are the Alfvén speed and frequency, respectively.Here we denote The formula refers in Eq. ( 22) is of complex nature and complicated to investigate analytically in the present form.It contains the information about MRI in high, intermediate and low frequency regimes.

Reduced dispersion relation
Equation ( 22) reveals the contribution of spin magnetization force to the dispersion of wave depending on the magnetic field B strength and orientation of the electron and positron.The obtained DR (22) for the given multi-species (e-p-i) rotating plasma system is too complicated to analyze directly.To understand the spin contribution of both the plasma ingredients (electron and positron) we limit ourselves to the low frequency or longer wavelength MHD approximations i.e. kV A ≪ c , ω ≪ � c , ≪ c assuming � c ω ce = m e m i ≃ 0, α 0 ≃ 1, and β ≃ −1 , the DR ( 22) can be expressed as Equation ( 23) the reduced relation for the MRI in three component ideal MHD model with the effect of spin magnetization correction.If the effect of spin magnetization is set to be zero ( S t = 0 ) in Eq. ( 23), the classical dispersion relation for two fluid model recovers 63 .In some magnetized plasmas, the contribution of spin magnetization is small to the total magnetic field.In a low temperature and high density plasmas like that in the locality of magnetars and pulsars, the contributions appears due to the fact that the component of spin force is in line to the ambient magnetic field.For the higher values of magnetic field B the magnetization energy shows some important consequences on the dynamics of the system.To demonstrate the instability, we can write the DR (22)   25) is a biquadratic equation in ω .The MRI growth rate (γ = −iω) can be determined by using the following relation 22) www.nature.com/scientificreports/Here δ = k z k .We obtained a biquadratic equation given in Eq. ( 25) describing four different MR modes.Indeed, some of the modes can be unstable in some different conditions, but here in our current work we have studied only one purely unstable mode, giving the growth rate of MRI expressed in Eq. (26).

Results and discussion
To probe the impact of different plasma parameters like magnetic field B, particle density n (electrons and positrons) and spin magnetization η , we evaluate Eq. ( 26) numerically to investigate the growth rate of MR modes.For this purpose, we have taken some typical degenerate plasma parameters related to some compact objects e.g.white dwarf and Neutron stars 13,[83][84][85] , the particle number densities are in the range (∼ 10 30 ) m −3 , magnetic field B strength is of the order ∼ megatesla to teratesla and the temperatures lie in the range (10 5 -10 7 ) K.
We plotted the dependence of growth rate of unstable MR mode γ against wave vector k z in Fig. 1 to study the effect of the background magnetic field B. We have obtained two curves (blue 1.5 × 10 5 T) and (red 1.2 × 10 5 T) for different magnetic field strength in the presence of the magnetization effect η from both electron and positron.It is clearly shown that the magnetic field enforce the growth rate of the mode towards stability.The spin terms are of particular significance and importance for low temperature and strong magnetized plasmas, when the spin are aligned with the field.We here in this work stress that the spin term can have more influence than the other terms in equation.As a consequence it turns out that the spin force can be important even when the magnitude of the imposed magnetic field is smaller than the usual J × B force.In order to demonstrate this property, we have studied the growth rate of MR instability, when the strength of magnetic filed B increases the growth rate can become stable for some of the possible orientations of k z .MRI take place in a weak magnetic field regions and enhancing the field by producing field amplification.The field will grow because of the MRI dynamo action until it reaches a saturation field limit.The details of the dynamo action is beyond the scope of this work.Strong magnetization effect is observed from both the ingredients of plasma where the field strength is higher, consequently impose a stability on the system.This instability always occurs in the vicinity of a low magnetic field where the toroidal field components dominate.This leading to the rapid growth of the B field whose characteristic time scale is of the order of fluid rotational period.The instability broadly occurs in the core collapse which is consider to be the dominant mechanism of magnetic flux production, has the capacity to strong enough to affect.If the field is not strong enough it cause the explosion in massive stars.Our calculations clearly presenting, that only a low or weak magnetic field B can affects the stability properties of the system.The spin effects become noticeable even when the external magnetic field B 0 is below the quantum critical magnetic field strength (∼ 10 10 ) T.
In Fig. 2, the growth rate γ of the MR mode is plotted against the wavevector k z in the absence and in the presence of contribution of spin magnetization from positron.By considering S p = 0 , enhances the growth rate and system is less stabilized and is more stabilized in the presence of the contribution from both electron and positron.It is clear from the figure that increasing positron concentration in dense astrophysical (e-p-i) plasmas impose stabilizing effects on the system.Increase in the electron number density n e , the effect of spin magnetization and consequently the instability of the system enhances as shown in Fig. 3.The plot having three different curves (red, blue and green), representing the variation in density gradient which modifies significantly the instability growth rate.This result is the confirmation of the diamagnetic behavior of plasma.Stellar plasmas becomes degenerate at high densities soon after the evolved star leaves the main sequence of formation and the structure readjusts.Degeneracy is important in white dwarfs stars and also in the central region of evolved stars because of the large densities found there.This result reveal our previous finding that the increasing electron densities destabilize the system and the growth rate consequently increases, also addressing the fact that the particle (electron) degeneracy pressure exceeds the external imposing fields and can take to the collapse of massive objects.It is clear that the degenerate electron gas cannot support a star with mass larger than the Chandrasekhar mass ( 1.4M ⊙ ).Conversely the increase in the positron number densities put stabilizing effects on the system.It is shown in Fig. 4 that, the presence of light positive species, i.e., positrons, can significantly modify the instability growth rate.

Conclusions
In this current work, we have examined MRI in three component (e-p-i) plasmas using QHD model in a differentially rotating magnetized degenerate plasma.The DR is obtained with the contribution of spin magnetization force from electron and positron.Spin contributions has a significant importance in a high density, low temperature and highly magnetized plasmas that can be found in WDs.The DR has a complex nature and have the informations in all the frequency ranges.To briefly understand the dynamics of the system we limited  To analyze the growth rate γ of the instability we numerically solved the reduced dispersion relation and by using various astrophysical plasma (WD) parameters we plotted the dependence of growth rate γ to the wave vector k z .We obtained four different plots for the growth rate to study the effect of different parameters like magnetic field B, magnetization effect η , electron number density n e and positron number densities n p .We concluded that the magnetic field strength has stabilizing effects on the growth rate.As the instability always takes place in the vicinity of a weak magnetic field, amplifies the field strength by the action of magnetic dynamos and pinches the system against the run away from stability.The electron spin magnetization force and the increasing electron number density enhance the growth of the MR mode and the system run towards instability.At very high number densities corresponding to MR instability results in the core collapse in many stars.On the other hand, the positron number density putting a stabilizing effect on the system.Therefore, the contribution from electron and positron fluids validates our consideration of their quantum mechanical effects in this model.The results of our findings presented here can be of particular importance for multi-species dense astrophysical environments.) is observed as the growth rate γ decrease with the increasing value of n p0 .

Figure 2 .
Figure2.The normalized growth rate γ versus wave vector k z .The typical value of = 10 3 and κ 2 < 0 , with electron and positron densities given in Fig.1and α 0 ≃ 1, and β ≃ −1 .Neglecting the contribution of positron spin magnetization effect η p = 0 (dotted blue curve), showing the increasing trend of growth rate γ .The stability is observed when the spin magnetization of both the species are taken into effect.Dotted red curve show the combine effect of both electron and positron magnetization.

Figure 4 .
Figure 4.The plot shows the normalized growth rate γ against wave vector k z for various values of positron number density n p0 .Assuming the same typical parameters, = 10 3 and κ 2 < 0 , with fixed electron number density n e0 = 10 30 .The dotted curve (blue) n p0 = 1 × 10 30 .The variation in the red curve ( n p0 = 1.5 × 10 30) is observed as the growth rate γ decrease with the increasing value of n p0 .