Magnetosonic shock waves in dense astrophysical electron–positron–ion plasmas

Physics Letters A 377 (2013) 2105–2110

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Physics Letters A www.elsevier.com/locate/pla

Magnetosonic shock waves in dense astrophysical electron–positron–ion plasmas S. Hussain a,b,∗ , S. Mahmood a,b , A. Pasqua c a b c

Theoretical Plasma Physics Division (TPPD), PINSTECH, P.O. Nilore, Islamabad 44000, Pakistan Department of Physics and Applied Mathematics (DPAM), PIEAS, P.O. Nilore, Islamabad 44000, Pakistan Physics Department, University of Trieste, Via Valerio 2, 34127 Trieste, Italy

a r t i c l e

i n f o

Article history: Received 2 January 2013 Received in revised form 17 May 2013 Accepted 10 June 2013 Available online 12 June 2013 Communicated by F. Porcelli Keywords: Quantum plasmas Magnetohydrodynamics Shock waves Korteweg–de Vries–Burgers Electron–positron–ion Compact stars

a b s t r a c t Multifluid quantum magnetohydrodynamic model (QMHD) is used to investigate small but finite amplitude magnetosonic shock waves in dense) electron–positron–ion (e–p–i) plasmas. The Korteweg– de Vries–Burgers (KdVB) equation is derived by using reductive perturbation method. It is noticed that variations in the positron density modify the profile of magnetosonic shocks in dense e–p–i plasmas significantly. The numerical results are also presented by taking into account the dense plasma parameters from published literature of astrophysical conditions, in compact stars. © 2013 Elsevier B.V. All rights reserved.

1. Introduction The study of nonlinear waves in dense electron–positron plasmas is important due to its existence in astrophysical conditions in compact stars such as neutron stars, pulsars and white dwarfs. The positron, which has same mass of electron but opposite electric charge, is used for a wide range of applications, including the study of atomic and molecular physics [1] antihydrogen formation [2], plasma physics [3] and the characterization of materials and surfaces [4]. Electron and positron plasmas are believed to exist in the early universe, [5,6] in active galactic nuclei (AGN), [7] and in pulsar magnetospheres [8,9] for this reason, a better comprehension of their main characteristics will be very fruitful for astrophysical applications (AGN and pulsar magnetosphere are often used to test relativistic effect which otherwise cannot be tested in terrestrial laboratories). It has been investigated both theoretically and experimentally that characteristic behavior of nonlinear waves changes when positrons are also present besides usual electron and ions in the plasmas [10]. Magnetosonic wave propagates in plasmas in the direction perpendicular to the magnetic field causing compression and rarefac-

*

Corresponding author at: Theoretical Plasma Physics Division (TPPD), PINSTECH, P.O. Nilore, Islamabad 44000, Pakistan. Tel.: +92 51 3008344330; fax: +92 51 9248808. E-mail address: [email protected] (S. Hussain). 0375-9601/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2013.06.010

tions of both the lines of magnetic force and the plasma conducting fluid. This wave has properties similar to an electromagnetic wave, since time varying magnetic field is in the perpendicular the direction of wave propagation and the time varying electric field is perpendicular to both the direction of wave propagation and external magnetic field. Therefore, the time varying transverse electric and magnetic field cause an “E × B” drift which causes rarefactions and compressions of magnetic field lines and plasma density in the direction of wave propagation [11]. It is well known that magnetosonic shock waves are excited when dissipation is present in the plasma and this dissipation phenomenon can occur through Landau damping, kinematic viscosity among the plasma constituents, as well as collisions between charged and neutral particles present in the system. The shock wave structures have been observed in many astrophysical, geophysical and laboratory plasma situations [12]. These nonlinear structures are produced by the supernova explosions, by the outflow of certain gases present in hot stars, by solar flares and in the solar wind upstream of planetary magnetospheres. Jehan et al. [13] studied the cylindrical magnetosonic shock waves in a dissipative three component plasma system comprising of electrons, positrons and ions. They investigated that high values of plasma β and positron concentration diminish the shock structures in e–p–i plasmas. The study of quantum mechanical effects becomes more important when the de Broglie wavelength λ B = h¯ /(2π mk B T ) associated with the charged particle is comparable to inter molecular distance

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S. Hussain et al. / Physics Letters A 377 (2013) 2105–2110

− 13

∼ n0

(where n0 is particle density, T is the temperature and m is the mass of the particle) i.e., n0 λ3B  1. Haas [14] introduced the quantum magnetohydrodynamic model (QMHD) to study quantum mechanical effects in magnetized plasma with the inclusion of quantum statistical (due to Fermionic nature of electrons) and Bohm potential (due to quantum tunneling effects) terms. Mushtaq and Vladimirov [15] studied arbitrary magnetosonic solitary waves in spin half degenerate quantum plasmas. The authors used QMHD model to derive Sagdeev potential and revealed the fact that the width of magnetosonic solitary wave gets broadened while amplitude remains the same with the increase in the value of quantum parameter. Masood et al. [16] investigated both linear and nonlinear properties of obliquely propagating magnetoacoustic shock waves in dense e–p–i magnetoplasma using the QMHD. The authors derived the nonlinear Kadomtsev–Petviashvili–Burgers (KPB) equation for two-dimensional fast and slow magnetosonic shock waves which was found to admit both rarefactive and compressive shock structures. They found that the strength of the slow magnetosonic shock decreases with the increasing of the positron concentration. However, fast magnetosonic wave does not get affected by the variation in the positron concentration. Hussain and Mahmood [17] studied the magnetosonic shock wave structures in dense electron–ion plasma by using two fluid QMHD. The authors have investigated the strength of the shock wave with dense plasma parameters. They noticed that the strength decreases when the value of magnetic field intensity increases. In this Letter, we want to study the effects produced by positron concentration on the profile of magnetosonic shock structures in dissipative, dense e–p–i plasmas by using multi-fluid QMHD. The kinematic viscosity of inertial ions is taken into account and the quantum mechanical effects of positrons and electrons are included in the model. The set of nonlinear dynamic equations for ions, electrons and positrons with Maxwell equations are presented in Section 2. The KdVB equation is derived using reductive perturbation method and its shock solution with tanh method is presented in Section 3. The numerical results are discussed by using the parameters of dense plasmas in Section 4, and finally the results are discussed in Section 5.

μ

where ν = m n is the kinematic viscosity of the ions, μ is the i i dynamic viscosity of the ions [18,19] in a dense plasma, mi is the mass of the ion and c is the speed of light. The continuity and momentum equations for inertialess electron fluid are given by:

∂ ne + ∇ · (ne ve ) = 0 (3) ∂t     √ 2 1 1 1 0 = −e E + (ve × B) − ∇ p Fe + ∇ √ ∇ 2 ne (4) c

ne

In order to study the excitation of magnetosonic ion acoustic shock wave structures in multicomponent e–p–i dissipative magneto-quantum plasma, ions are assumed classical due to their large inertia, while quantum mechanical effects are taken into account for electron and positron fluids. The kinematic viscosity of the ions is taken into account due to the fact that nondegenerate ions in dense plasmas are in strongly coupled state because Coulomb interaction between ions is much larger than the ion kinetic energy [18,19]. The quantum mechanical effects in the model are included in momentum equations of electrons and positrons through quantum statistical (due to fermionic nature of electrons) and Bohm potential (due to quantum tunneling effects) terms. The electric field lies in the xy plane and magnetic field is directed along z-axis. The propagation vector is directed along the x-axis (i.e., ∇ = (∂x , 0, 0)). The nonlinear set of dynamic equations for ion fluid can be written as:

The continuity and momentum equations for inertialess positron fluid can be written as:

∂np + ∇ · (n p v p ) = 0 ∂t      2 1 1 1 2 ∇ p Fp + ∇ √ ∇ np 0 = e E + (v p × B) − c

np

2m p

(1)

∇×E=−

(6)

np

∇×B=

1 ∂B

(7)

c ∂t

4π c

j+

1 ∂E

(8)

c ∂t

where E is the electric field vector, B is the magnetic field vector, j is the current density, k B is Boltzmann constant and c is the speed of light. For low frequency waves, the quasi-neutrality condition is described as (ni + n p )  ne ; and in equilibrium, we have (ni0 + n p0 ) = ne0 . The set of dynamic equations given in (1)–(8) in normalized form can be rewritten as follows:

∂ ni + ∇ · (ni vi ) = 0 ∂t ∂ vi ∂ 2 vi + (vi .∇)vi = E + (vi × B) + η 2 ∂t ∂x ∂ ne + ∇ · (ne ve ) = 0 ∂t

(9) (10) (11)

5

ne

2

(12)

ne

∂np + ∇ · (n p v p ) = 0 ∂t 



0 = E + (v p × B) −

(13) 2β 5

5 3

σ

∇n p np

H 2p

+

2

   1 ∇ √ ∇2 np

(14)

np

Instead, the Maxwell equations in the normalized form are given by:

∇×E=−

∂B ∂t

∇ × B = n i vi +

(15) p

(1 − p )

np vp −

1

(1 − p )

ne ve + α

∂E ∂t

(16)

The following normalizations have been used in Eqs. (9)–(16): n Ω r = V i R, t = t Ωi , B = BB , E = mVeEΩ , n j = n j ( j = e , i , p). The A

0

A

i

j0

ion cyclotron frequency, ion sound speed and Alfvén velocity have e B0 , mi c

Vs =

k B T Fe , mi

VA = √

B0 , 4π mi ni0

respec-

tively. The other parameters such as plasma beta, i.e. β = quantum parameter for electrons, i.e. H e =

(2)

(5)

The Faraday’s induction and the Ampere’s law are given by:

been defined as Ωi =

  ∂ vi e 1 + (vi .∇)vi = E + (vi × B) + ν ∇ 2 vi ∂t mi c

ne

5     2β ∇ne3 √ H2 1 + e ∇ √ ∇ 2 ne 0 = − E + (ve × B) −

2. Basic set of equations

∂ ni + ∇ · (ni vi ) = 0 ∂t

2me

Ωi and (me mi )1/2 V 2A

α=

V s2

V 2A

,

V 2A c2

have been defined. The Fermi pressure for electron quantum fluid

S. Hussain et al. / Physics Letters A 377 (2013) 2105–2110

is given by p Fe =

mv 2Fe 2 3 5ne0

( 12 me v 2Fe

ture of the electrons are related by = k B T Fe ). The Fermi temperature and equilibrium density of electrons are related by 2

2 (3π 2 n ) 3

e0 k B T Fe = in a degenerate electron gas. The ratio of Fermi 2me temperature of positron and electron quantum fluid is defined as

σ=

T Fp , T Fe

while positron to electron equilibrium density ratio is de-

fined as p = Hp =

n p0 . ne0

(1 )

5

ne3 , where Fermi velocity and tempera-

The positron quantum parameter is defined as

Ωi and the Fermi temperature and equilibrium den(m p mi )1/2 V 2A

−V m

ion kinematic viscosity

η=

ν Ωi V 2A

2m p

and normalized

(1 )

∂ v ex ∂ ne + =0 ∂ξ ∂ξ

(1 ) (1 )

(22)

(1 )

(1 )

E x + v ey +

2β ∂ n e 3

=0

∂ξ

(23)

(1 )

E y − v ex = 0

(24)

The 3/2 order terms of positron continuity equation and x and y components of momentum equation of positrons give the following set of equations: (1 )

2

2 (3π 2 n p0 ) 3

sity of positrons are related as k B T Fp =

2107

−V m

has been defined.

(1 )

∂np ∂ v px + =0 ∂ξ ∂ξ

(25)

(1 )

(1 )

(1 )

E x + v py −

3. Derivation of Korteweg–de Vries–Burgers equation

(1 )

2β σ ∂ n p 3

=0

∂ξ

(26)

(1 )

E y − v px = 0 In order to derive the one-dimensional KdVB equation for nonlinear wave propagating along x-axis, we have used standard reductive perturbation method [20] and introduce the following stretched independent variables ξ and τ given by:

ξ =

1/2

(x − V m t )

τ =

3/2

t

(17)

where represent a small expansion parameter which lies in the range (0 < < 1) and V m is the normalized phase velocity of the magnetosonic wave which will be determined later on. We expand the perturbed quantities ni , ne , n p , v ix , v ex , v px , B Z in terms of in the following form:

⎡ ⎡

⎤

⎡ ⎤

(1 )

ni

⎤

⎡

(2 )

ni

⎤

⎢ (1 ) ⎥ ⎢ (2 ) ⎥ ni 1 ⎢ v ix ⎥ ⎢ v ix ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ v ix ⎥ ⎢ 0 ⎥ ⎢ (1 ) ⎥ ⎢ (2 ) ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ne ⎥ ⎢ ne ⎥ ⎢ ne ⎥ ⎢ 1 ⎥ ⎢ (1 ) ⎥ ⎢ (2 ) ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ np ⎥ ⎢ ⎥ ⎢ np ⎥ ⎢ 1 ⎥ 2 ⎢ np ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥+ ⎢ ⎥ + ⎢ (2 ) ⎥ + · · · ( 1 ) ⎢ v ex ⎥ ⎢ 0 ⎥ ⎢ v ex ⎥ ⎢ v ex ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ v px ⎥ ⎢ 0 ⎥ ⎢ (1 ) ⎥ ⎢ (2 ) ⎥ ⎣ B ⎦ ⎣1⎦ v v ⎢ ⎥ ⎢ ⎥ px px z ⎢ ⎥ ⎢ ⎥ ⎣ B (1 ) ⎦ ⎣ B (2 ) ⎦ 0 Ey (1 )

v

⎞

⎛

(1 ) ⎞

v iy

(18)

(1 )

Ex

The lowest order terms of x component of Ampere’s law are described as:

(1 − p )

(1 ) (1 )

ne v ex = 0

(29)

3/2 order terms of y component of Ampere’s laws

p

1 1 + 23 ( 1− p σ + 1− )β p 1+α

(31)

In the limiting case of absence of positrons (i.e., n p0 = 0), the Fermi temperature of positron T Fp → 0, then Eq. (31) becomes the normalized phase speed for magnetosonic wave in degenerate e–i plasmas [17], which propagates in the perpendicular direction where the condition α  1 holds. (1) (1) (1) (1) From above set of equations, the variables v ex , v ix , ni , ne ,

(2 ) ⎞

v iy

(2 )

Ex

(1)

(1)

(1)

(1)

(1)

(1)

(1)

n p , B (1) , E x , E y , v iy , v ey , v py can be written in terms of v px as follows: (1 )

(1 )

(1 )

(1 )

E y = v ix = v ex = v px

(32)

(1 )

(1 )

ni

(1 )

(1 )

= ne = n p =

∂ n(i 1) ∂ v (ix1) + =0 ∂ξ ∂ξ

(19)

B (1 ) =

−V m

∂ v (ix1) = E (x1) + v (iy1) ∂ξ

(20)

E (1 ) = −

(1 )

n p v px −

Using the lowest order set of equations given in (22)–(30), the normalized phase velocity of the magnetosonic wave in dense e–p–i plasmas is obtained as follows:

−V m

(1 )

1

(1 ) (1 )

(1 − p )

The lowest yields:

Vm =

For weak dissipation of ions, we have η = 1/2 η0 . Using the above perturbation scheme in Eqs. (9)–(16) and collecting the lowest order terms, i.e. the 3/2 order terms of ion continuity and x and y components of ion momentum equation, yield to the following set of equations:

E y − v ix = 0

p

(1 ) (1 )

ni v ix +



iy ⎜ (1 ) ⎟ ⎜ (2 ) ⎟ ⎜ v ey ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = 3/2 ⎜ v ey ⎟ + 5/2 ⎜ v ey ⎟ + · · · ⎜ (1 ) ⎟ ⎜ (2 ) ⎟ ⎝ v py ⎠ ⎝ v py ⎠ ⎝ v py ⎠

Ex

(28)

(30)

Ey

⎛

(1 )

∂Ey ∂ B (1 ) − Vm =0 ∂ξ ∂ξ

(1 )

Instead, for v iy , v ey , v py , E x , we have

⎛

order terms of Faraday’s law give:

∂v ∂ B (1 ) p 1 (1 ) (1 ) (1 ) (1 ) v py − ne v ep − α V m 1x = 0 + v iy + ∂ξ (1 − p ) (1 − p ) ∂ξ

(2 )

Ey

The lowest

(27)

3/ 2

v px

(33)

Vm

(1 )

(21)

The 3/2 order terms of electron continuity equation and x and y components of momentum equation of electrons give the following set of equations:

v px

(34)

Vm (1 )

∂ v px (1 + α ) ∂ξ Vm

(35)

(1 )

(1 )

v iy = −

 (1 )

v ey =

α V m ∂ v px (1 + α ) ∂ξ Vm 1+α

−

2β 3V m

(36)



(1 )

∂ v px ∂ξ

(37)

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S. Hussain et al. / Physics Letters A 377 (2013) 2105–2110



Vm

(1 )

v py =

1+α



2β σ

−

3V m

(1 )

(2)

∂ v px ∂ξ

(38)

The next higher order terms (∼ 5/2 ) of ion continuity and x and y components of ion momentum equation give the following set of equations:

(1 )

(1 )

(39)

(2)

(1 )

(2 )

(1 ) (1 )

E y − v ix − v ix B

+ Vm

∂ v iy

=0

∂ξ

(1 )



(1 )

(2 )

(40)

+

(41)

−

(2 )

(2 )

(1 ) (1 )

E x + ne E x +

B=

(2 )

∂ v ex ∂ ∂ ne ∂ ne (1 ) − Vm + + ne(1) v ex =0 ∂τ ∂ξ ∂ξ ∂ξ 2β ∂ n e 3

∂ξ

H e2 ∂ 3ne

∂ 3ξ

4

(2 )

(1 ) (1 )

− E y + v ex B

(43) (2 )

+ v ex = 0

2 β

1

3 Vm 1 − p



(2 )

(2 )

(45)

(2 )

(2 )

(1 ) (1 )

Ex + np Ex −

+

H 2p ∂ 3n(p1)

∂ 3ξ

4

(2 )

2β σ ∂ n p 3

(1 )

(2 )

(1 ) (1 )

+ B (1) v py + v py + n p v py

∂ξ

=0

(1 )

(46) (2 )

− E y + v px B (1) + v px = 0

(47)

The next higher order terms (∼ 5/2 ) of Faraday’s induction law yields: (2 )

∂ B (1 ) ∂ B (2 ) ∂ E y − Vm + =0 ∂τ ∂ξ ∂ξ

p

v ix +

1

(2 )

(1 − p )

v px −

(2 )

(1 − p )

v ex

(49)

∂ B (2 ) p p (1 ) (1 ) (2 ) n p v py + v py + n(i 1) v (iy1) + v (iy2) + ∂ξ (1 − p ) (1 − p ) 1

(1 − p )

(1 ) (1 )

ne v ey −

1

(1 − p )

(2 )

v ey − α V m

H2

1+α

2 β

1

+ −

1





2 β 3 Vm

σ

3 Vm

 3V m −

1 )β  1− p

+

Vm 1+α



2 β 3 Vm

σ +

 −1 ( p σ + 1) Vm

1+α

Vm

1− p

1+α

2 β

1

3 Vm 1 − p



2 β

4

Vm

3 Vm 1 − p

−

1+α

+

−

−

1− p



2 β

+

3 Vm

−1 ( p σ + 1)

Vm

1− p

1+α

Vm

2 β



p



1

1+α

+



1 1− p

3 Vm

Vm 1+α

−

2 β



3 Vm



σ

−

2 β



3 Vm

Here we have defined the quantum parameter as H = H e = H p since positron and electron have equal mass. In order to find the analytical solution of the KdVB equation given in (51), we define the transformed coordinate ζ such that ζ = χ (ξ − U τ ) in the comoving frame with speed U of the nonlinear structure, where χ is a dimensionless nonlinear wave number to be determined later on. We have used the boundary conditions (1) (1) (1) v px → 0 and ∂ζ v px , ∂ζ2 v px , for ζ → ∞ for localized solution. The shock solution of KdVB equation (51), obtained using tanh method [21], is given by: (1 )

v px (ζ ) =

3C 2 25B

+

 sech2

6C

2

25B



C 10B

 (ξ − U τ ) 

1 − tanh

C 10B

 (ξ − U τ )

The speed of the comoving frame U = (1)

The next higher order terms (∼ 5/2 ) of y component of Ampere’s laws is written as:

−



1− p

Vm

6C 2 25 A B

(52)

is obtained using the (1)

(1)

boundary conditions ζ → ∞, then v px (ζ ) → 0, ∂ v px (ζ ), ∂ 2 v px (ζ )

(1 )

∂ Ex − V mα =0 ∂ξ

(2)

3V m

1− p

2

(2 )

+

1+α p

(48)

The next higher order terms (∼ ) of x component of Ampere’s law gives:

4 Vm α

C = V m η0 3V m −

The next higher order terms ( 5/2 ) of electron continuity and x and y components of positron momentum equation give the following set of equations: (1 )

(2)

(51)

 −1  6 + 4( p σ + 1− p ( p σ + 1)



(44)

∂np ∂np ∂  (1) (1)  ∂ v px − Vm n p v px + + =0 ∂τ ∂ξ ∂ξ ∂ξ

Vm 1+α

Vm

−

(1 ) (2 ) (1 ) + v ey + ne(1) v ey + B (1) v ey



1− p

1

−

(42)

=0

(2)

(1 )



p

−

1+α

1

(1 )

−

(1 )

Vm

A = 3V m −

The next higher order terms (∼ 5/2 ) of electron continuity and x and y components of electron momentum equation give the following set of equations:

(2)

where the coefficients A, B and C are defined as:

(1 )

(2 )

(2)

(2)

(1 )

∂2v − η0 2 1x = 0 ∂ ξ

(2)

∂ v px ∂ 3 v px ∂ 2 v px (1) ∂ v px + v px − C =0 +B ∂τ ∂ξ ∂ξ 3 ∂ξ 2

(2 )

  ∂ v ix ∂v ∂v + v (ix1) ix − V m ix − E (x2) + v (iy2) − B (1) v (iy1) ∂τ ∂ξ ∂ξ

(2)

E x , E y and B (2) , from Eqs. (39)–(50) and using relation given in Eq. (31), we obtain the KdVB equation for magnetosonic wave in dense e–p–i plasma described as follows:

A

∂ n(i 1) ∂ n (2 ) ∂ v (2 ) ∂ − V m i + n(i 1) v (ix1) + ix = 0 ∂τ ∂ξ ∂ξ ∂ξ

(2)

Finally, eliminating ni , ne , n p , v ix , v ex , v px , v iy , v ey , v py ,

∂ E (y2) ∂ E (y1) =0 +α ∂ξ ∂τ (50)

2

C ± 10B ,

and tanh ζ → 1 and χ = which is related with the weak dispersive and dissipative coefficients in the KdVB equation. 4. Results and discussion In this section, we investigate the effects of positron concentration and ion dissipation on shocks in e–p–i dense magnetoplasmas. For numerical studies of KdVB equation (51), we have used the plasma parameters such as plasma density and external magnetic field lying in the ranges ne0 = 1026 –1028 cm−3 , B 0 = 109 –1011 G. Moreover, we know there is a temperature in

S. Hussain et al. / Physics Letters A 377 (2013) 2105–2110

Fig. 1. The plot of normalized phase velocity of magnetosonic wave vs positron density in dense e–p–i plasmas is shown for dense astrophysical plasma parameters i.e., ne0 = 1027 cm−3 and B 0 = 1010 G.

the range T = 8000–40,000 K in the outer layers of dense astrophysical objects such as neutron star/pulsar [22]. In order to check the validity of our plasma model for the consideration of quantum mechanical effects, we consider the following densities of each species: ne0 = 1027 cm−3 , ni0 = 0.8 × 1027 cm−3 , n p0 = 0.2 × 1027 cm−3 . Moreover, we consider a magnetic field of B 0 = 1010 G for dense plasma situations. The Fermi temperatures of the electron and positron for these parameters are turned out to be, respectively, T Fe = 4.2 × 107 K and T Fp = 1.4 × 107 K which are much larger than the system temperature. These values of the Fermi temperatures validate the inclusion of their quantum mechanical effects in their momentum equations. The ion Fermi temperature is calculated as T F i = 1989 K which is much smaller than the system temperature and validate their classical behavior in our dense e–p–i magneto-plasma model. We have neglected the spin effects of electrons and positrons since the condition μ B B 0 (magnetization energy)  k B T Fe (Fermi energy) (where μ B is the e Bohr magneton for electron and defined as μ B = 2m ) is valid ec and it holds in our numerical calculations as well. We have also calculated the numerical values of the these parameters μ B B 0 = 9.274 × 10−11 ergs and Fermi energies of electrons and positrons are k B T Fe = 5.842 × 10−11 ergs, k B T Fp = 1.99 × 10−11 ergs respectively. We have treated our model in the nonrelativistic limit because Fermi energies of electron and positron are turned out to be much less than their rest mass energies. The quantum parameter for electron quantum fluid turns out to be H e = 4.35 × 10−4 , the ion gyroradius is ρs = 6.165 × 10−7 cm and the plasma β is 0.5873. The effect of positron concentration on the magnetosonic phase speed is shown in Fig. 1. It can be seen that phase speed of the magnetosonic wave increases with the rise in the concentration of positrons in dense e–p–i plasmas. This evidence is due the addition of positrons in dense electron–ion plasmas, the Fermi pressure due to positrons added with electrons, so the total Fermi pressure is enhanced due to which the phase speed of the magnetosonic wave is increased in dense e–p–i plasma. Moreover, in presence of positrons the ion density is reduced due to which the Alfvén phase velocity is increased which increases the phase velocity of magnetosonic wave in dense e–p–i plasmas. It is clear form Fig. 1, that when the number of ions are larger in comparison to number of positrons for the fixed value of the electrons in dense e–p–i plasmas, then phase velocity of the magnetosonic wave increases slowly. However, in the reverse situation, when the positron concentration is larger than ion concentration in dense e–p–i plasmas the phase velocity of magnetosonic waves increases rapidly. The magnetosonic shock structures with variation in positron density and ion kinematic viscosity are shown, respectively, in Figs. 2 and 3. The decrease in the strength of magnetosonic shocks with the increase in positron density in dense

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Fig. 2. The decrease in the shock strength with the increase in positron concentration in dense e–p–i plasmas is shown for n p0 = 0 (bold curve), 0.2 × 1027 cm−3 (dashed thick curve) 0.4 × 1027 cm−3 (dashed thin curve) and the corresponding plasma β values are β = 0.73, 0.58 and 0.44 respectively, where the external magnetic field is B 0 = 1010 G and ion kinematic viscosity is η = 0.001.

Fig. 3. The increase in the magnetosonic shock strength with variation in the kinematic viscosity of ions i.e., η = 0.001 (dashed curve) and η = 0.0015 (bold curve) is shown for ne0 = 1027 cm−3 , n p0 = 0.2 × 1027 cm−3 and B 0 = 1010 G.

e–p–i plasmas is shown in Fig. 2. The shock strength decreases with the increase in the positron concentration because the ion density decreases which reduces the inertial force to derive the magnetosonic waves in dense e–p–i plasmas. The increase in the positron density i.e., n p0 = 0.2 × 1027 cm−3 to 0.4 × 1027 cm−3 results increase in the value of phase velocity of magnetosonic wave i.e., V m = 7.71 × 107 cm/s to V m = 8.91 × 107 cm/s respectively in dense e–p–i plasmas for ne0 = 1027 cm−3 and B 0 = 1010 G. The value of the plasma β is found to be decreased from β = 0.58 to 0.44 by increasing the value of positron concentration. The increase in the ion kinematic viscosity from η = 0.001 to η = 0.0015 enhances the strength of the magnetosonic shocks in dense e–p–i plasma system as shown in Fig. 3. 5. Conclusion To conclude, we have investigated the magnetosonic shock waves in homogeneous collisionless, magnetized dissipative, dense e–p–i plasmas. The ions are taken to be non-degenerate and their dissipation is assumed through kinematic viscosity due to Coulomb interaction in dense plasmas. Electrons and positrons are taken to be degenerate fluid and quantum mechanical effects are considered through their fermionic nature (Fermi statistics) and Bohm potential (quantum tunneling effects) in dense e–p–i plasmas. Multifluid QMHD model is employed to derive the KdVB equation for magnetosonic waves in dense e–p–i plasmas by using well-known reductive perturbation method. The tanh method is employed for the solution of the KdVB equation, which gives a shock wave solution. The phase velocity of the magnetosonic wave in dense e–p–i plasma is found to be increased with the rise in the positron density. The effects of positron concentration and ion kinematic viscosity on the profile of nonlinear magnetosonic shocks are also

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studied numerically by taking the dense astrophysical parameters exist in the literature. It is found that by increasing the positron concentration in dense e–p–i plasmas the strength of the shock is found to be decreased. However, the magnetosonic shock strength is found to be increased with the increase in the ion kinematic viscosity. The present study is useful to explain the charge particle acceleration observed in the colliosionless dense astrophysical plasma situations such as in white dwarfs and neutron stars/pulsars. References [1] [2] [3] [4] [5]

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