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Shocklets in the Comet Halley Plasma

Dust acoustic (DA) waves evolving into shocklets are investigated in the Comet Halley plasma system relaxing to Maxwellian, Kappa and Cairns distributions. Here dynamics of dust is described by the fully nonlinear continuity and momentum equations. A set of two characteristic wave nonlinear equations is obtained and numerically solved to examine the DA solitary pulse which develops into oscillatory shocklets with the course of time such as at time {\tau}=0, symmetric solitary pulses are formed, which develop into oscillatory shocklets. It has been observed that variation in superthermality strongly affects the profiles of nonlinear DA structures in terms of negative potential, dust velocity and density.


I. INTRODUCTION
In our solar system, billions of comets can be found circumnavigating the Sun. These are snowballs of frozen gases, rocks and cosmic dust particles. It is widely believed that like nucleus in an atom, nucleus of the comet is discrete and cohesive made of volatile solids containing all the mass of the comet. While orbiting around the Sun, when a comet approaches closer to the Sun, it expels dust particles and ionized gases in huge quantity into a behemoth like brightening head bigger than most planets. This dust and gases form a dusty plasma and is in the form of a tail that expands outwardly away from the Sun for millions of miles. These dust grains are electrically charged by virtue of plasma which permeates the solar system. From the observation of dust impact analyzer installed on Vega spacecraft supported that much of the carbon is trapped within dust grains [1][2][3].
Due to low geometric albedo (∽ 0.04), comets are considered darkest in the solar system. Whereas Halley's comet is the most luminous one which appears after an interval of 76 years.
Surface of the comet is covered with a dust mantle and has temperature of ∽ 300 − 400 K where the latter was determined from the infrared spectrometer (IKS) installed on the Vega 2 spacecraft [4].
Whipple revitalize the discrete nucleus concept and developed a seminal model [5] which could potentially explain the dynamic affects on the orbits of comets (such as nongravitational recoil effects) as well as other vital features of cometary observations. Despite large volume (∽ 500 km 3 ), the small bulk density (0.1 ∽ 0.2gcm −3 ) can be due to the small mass of Halley's comet (∽ 5×10 16 −10 17 g) which has been estimated independent of gravitational effects of its orbit [6].
H 2 O is the main constituent of the volatile nucleus, usually volatiles like CO would be caged within the water -ice lattice and form clathrate hydrate. It is also anticipated that when water-ice sublimation starts, significant coma activity initiates and this happens at about distance of 2 − 3 AU [7] . In a new comet water ice is of amorphous nature and transforms into crystalline structure as the kinetic energy of the cometary nucleus increases and reaches to a critical value. At this point usually an unanticipated nucleus burst out and this happens between about 3 and 6 AU [8] .
Mendis in 1986 calculated the flights of the dust grains of different sizes (micronsubmicron) that are anticipated to be expelled out of the cometary nucleus. Observational data from Vega 1 & 2 and Giotto spaceports shows that plasma of comae of comet Halley contains ions of both positive and negative polarity other than cosmic dust and electrons.
This kind of plasma with additional negative ions known as electronegative dusty plasma is chemically very reactive. Electronegative plasma is also found in other astrophysical environments such as D-region of the ionosphere, themesosphere, photosphere of Sun etc. and plays a vital role in the semiconductor technology. In most of these astrophysical and laboratory plasmas, electrons can stick to the surface of dust and so their density reduces and such system can also be treated as electron depleted plasma. Therefore, in this setting.
role of negative ions becomes more crucial [10] .

Observations reveal excitation of foreshock and bow shock like nonlinear structures at
Halley's comet. Shock waves are quite common nonlinear structures in astrophysical settings for instance are triggered when interaction of solar wind with the comets or magnetic field of different planets takes place, these waves are also anticipated to appear in galactic dust clouds at the beginning stage of star creation. Waves of compressional nature are generated when dust particles impact with solid surface of a spacecraft, in both the dust grain and surface of the spacecraft.
Shocks are outcome of discontinuous disturbances traveling in a medium where an abrupt jump in physical quantities like density, pressure and temperature across some narrow region leads to such structures. Dissipation structures like kinematic viscosity (also known as momentum diffusivity) feature of collisions between dust and ions and the fluctuating charge of dust particles lead to the formation of dust acoustic (dust ion) DA (DIA) shock waves in dusty plasmas with Γ c << 1 (coupling parameter). Whereas in opposite case when plasma is strongly coupled (Γ c >> 1), shear and bulk viscosities are considered crucial in the excitation of DA or DIA shock waves of monotone and oscillatory nature [11] . Trigger of shock waves is an irreversible phenomena and contrary to nonlinear solitary waves, there is a sharp dissipation of energy and speed with the distance of the shock wave alone.
For the better understating of this important class of nonlinear waves in dusty plasma which has enormous significance in the astrophysical context, many experimental investigations are worth mentioning. For example first observation of shock waves in radio frequency 3D dusty (complex) plasma discharge under minute gravity condition was carried in the gered with the aid of impulsive axial magnetic field was examined [13]. In another interesting study, excitation of a bow shock in a 2D dusty plasma was examined where a super-sonic flow of micro sized negatively dust particles around a stationary object was used. this exactly mimics the aforementioned situation when shock waves are generated due to the impact of There has been extensive theoretical investigation of DA and DIA shock waves [14][15][16][17][18][19]22], in one of earlier works by Ghosh, Ehsan, and Murtaza [11], authors investigated shock waves deriving KdV-Burger equation for the Comet Halley plasma where nonsteady dust charge variation produces anomalous dissipation that leads to the shock wave. There authors transformed KdV burger equation to another one analogous to damped anharmonic oscillator and using fifth order Runge-Kutta technique, oscillatory nature shock amplitudes were studied.
It was reported that shock wave is more dispersive in nature for a the higher strength of magnetic field. whereas for the parallel propagation a monotonic shock was observed.
In the present manuscript we aim to present a fully nonlinear theory to study largeamplitude DA waves developing into the DA shocklet dust acoustic in the Cometary plasma using diagonalization method [23] . For the readers it is worth mentioning the term shocklet was introduced to represent sporadic steep fronts observed in high-speed turbulent compressible fluid and in space as flank formations associated with planetary-scale shocks [20,21] . When solar wind passes through a comet makes it a complex plasma system which is far away from the state of equilibrium. In this scenario, cometary electrons and ions will asymptotically approach to a nonequilibrium steady state with power-law distributions, e.g., often fitted with the Cairns, Kappa or r, q distribution [24][25][26] .
To the best of authors' knowledge this has not been studied earlier and results of the present investigation are useful for understanding the physics of formation of shocklets at Halley's comet.
The paper is organized in the following manner: In Sec. II, the basic set of equations for the dust acoustic shock wave of comet Halley plasma are given. Section III deals with the derivation nonlinear equation. In Sec. IV and V provide quantitative analysis and conclusions, respectively.

II. BASIC SET OF EQUATIONS
Consider a one dimensional, collisionless unmagnetized plasma, whose components are the four species consisting of electrons, singly ionized negatively and positively charged ions with dust grains that are negatively charged. We assumed that spherically shaped dust grains are carrying a constant charge because the time for the dust acoustic shocklets to be discussed is much smaller than that required for further significant change in dust charge.
The condition of charge neutrality at equilibrium gives n i + 0 = n e0 − εZ d0 n d0 + n i − 0 , where n p0 denotes the equilibrium number density of the p th species (viz., p = i + for positively charged ions, p = i − for negatively charged ions, p = e for electrons, p = d for dust particles and Z d0 is the equilibrium dust charge state. The following set of one dimensional nonlinear fluid equations are used, which describe the dynamics of DA solitary and shock waves: and ε 0 e where , v d and φ represents the dust thermal speed, dust fluid velocity and the electrostatic potential. As the singly charged negative, positive ions and electrons are assumed to be in thermal equilibrium, so we can described their densities by the following expressions: for the case when electrons, singly charged negative and positive ions are taken as nonthermal, we can expressed the number densities for Kappa distributed electrons and singly charged negative and positive ions by the following expressions [23,25]: and for the case of Cairns distributed electrons and singly charged negative and positive ions densities are given by [23,26] where the subscript s equals e for electrons and i − for negatively charged ions. Here, Γ = 4α/(1 + 3α) comprise of Cairns parameter α that determines the population of nonthermal particles. It is important to mention here that if we consider κ → ∞ and Γ → 0 in Eqs. (6 − 9), respectively, then we can directly replicate the Maxwell-Boltzmann density distribution for electrons, positively and negatively charged ions.

III. NONLINEAR EVOLUTION EQUATION
In order to normalized the above set of Eqs. (1 − 9) we apply the scaled parameters [23] as and Mostly, the dust temperature effect is neglected because of low values of dust temperature T d and also because of the large dust charge state so we can neglect the last term in Eq.
(10) which gives the following form In the current work, we are intended to study nonstationary, one dimensional nondispersive fully nonlinear DAW's. Thus, by neglecting the dispersive term viz., ∂ 2 x φ = 0 in Eq. (3) and by utilizing Eqs. (4) and (5) for thermal distributed particles and Eqs. (6 − 9) for the case of non thermal distributed particles, yields the following results where for Maxwellian distributed pair ions and electrons for Kappa distributed pair ions and electrons Cairns distributed pair ions and electrons (15) here is the parameter that represent hot electron superthermality effects and for κ → ∞, c k → 1 displaying the reduction of Kappa distribution to standard Maxwellian distribution. Equations (12) and (14) are the normalized nonlinear coupled equations, which describes the large-amplitude nonstationary DA shock waves.
We can express Eqs. (12) and (14) in the matrix form: where the parameter χ b (U) for each distributions is expressed as given in Eq. (15) The square matrix on the left hand side of Eq. (16) can be diagonalized by means of a diagonalizing matrix technique. The two eigen values of the square matrix can be resolved by det(A − λI) = 0, given by where square matrix and unit matrix are given by The square matrix in Eq. (16) taken as A, can be diagonalized by means of a diagonalizing matrix C whose columns are the eigenvectors of A, so that Multiplying Eq. (16) by C −1 from the left gives the diagonalized system of equations Here the new variables are wave solution is found by taking either Ψ + or Ψ − to zero. Setting Ψ − to zero, we obtain V d = −F (U) and Ψ + = 2U b . Since Ψ + can be written as function of V d , so we can write Eq.