Instability occurs in the space plasma in general way due to the redistribution of free energy which has accumulated in a non-equilibrium state that is outcome of variation in characteristics parameter of a plasma i.e. temperature, pressure, density etc. Instabilities are normal modes of a system that grow in space or time (Treumann et al., 2001) [1].
The word 'instability' implies a well-defined relationship between wave vector k and frequency ω. The source of development of a micro instability in plasma is roughly called free energy or unconfined energy, and anisotropy or inhomogeneity in the zeroth-order velocity distribution function. The space plasma of different astrophysical region is inhomogeneous to some degree, and the related plasma gradients are origin of free energy that can propel plasma instabilities (Gary, 1993) [2].
In the space plasma, how the firehose instability excited by the anisotropy, and how the anisotropy is formed due to the super thermal particles, first we understand this phenomenon and it's general mechanism in brief ,in the light of study done by many researchers in the past.
In the astrophysical region where plasma density is low, and binary collisions are almost absent, plasma particles gains energy through wave particle interaction, not by particle-particle interaction, which makes reason for super thermal character of the particles. In the regime of linear waves, population of plasma, gets energy through cyclotron resonance and Landu damping (Fisk, 1976) [3]. These particles get accelerated, parallel to the ambient magnetic field, than to perpendicular magnetic field, due to many different reasons, it is generally observed in the solar flares and solar wind (Peasold et al., 1999) [4].Thus the particle distribution are not in form of maxwellian rather in form of quasi-maxwellian distribution (Pilipp et al., 1990) [5] and called anisotropic as it is now governed by direction in velocity space (Salem et al., 2003) [6].These anisotropic distribution of super thermal particles have considerable quota of unconfined energy, pervaded in the direction of the magnetic field that arouse the kinetic instabilities (Paesold et al.,1999) [4]; (Fahr et al., 2007) [7] like firehose. The Firehose instability can be compare to an usual garden or fire hose with a swift water flow, where slight disturbance can upshot a fierce movement of hose. The non-resonant nature of firehose instability is due to it's bulk features (Parker, 1958) [8]; (Kennel et al.,1967) [9]; (Hollweg et al., 1970) [10].
Here are some important analytical and numerical analysis and their conclusion, studied by some authors in different astrophysical plasma regions with evidence of observations and computer simulation ,are presented in the following script.
The link between the energy faded particles and super thermal population is root cause of stimulation and further enhancement of the functions of waves and instabilities in magnetized plasma as it is observed in different space plasma regions. The assembly of Kappa functions is highly applicable treatment for such population with super thermal high energy tails (Vasyliunas 1968) [11]; (Summers et al.,1991) [12], and shows unalike dispersion and stability properties of waves, from maxwellian. A concise and clear explanation of these procedure of acceleration is described by Pierrard at el., (2010) [13].
At 1 AU distance, observation recommends that both proton and electron constituent in astrophysical plasma, habitually thermally anisotropic (Hundhausen et al.,1967a, b) [14]; (Montgomery et al.,1968) [15]; (Formisano 1969) [16] which creates the condition for electron firehose instability. Similarly in the solar wind, existence of electron and proton thermal anisotropy also described by Hollweg et al., (1970) [10].The occurrence of firehose instability in the planetary magnetosphere due to the temperature and pressure anisotropy is confirmed by Li et al., (1995) [17], Kaufinann et al., (2002) [18].
When kinetic energy is surplus, along the background planetary magnetic field, i.e.\(\frac{{T}_{\parallel ,a}}{{T}_{\perp ,a}}>1,\) it triggers firehose instability (where a is labeled for different charged species, electrons, protons, and \(\parallel\) and ⊥ denotes directions relative to the background magnetic field)( Lazor et al., 2009) [19]; (Lazor et al., 2011) [20].If \({T}_{\parallel ,e}>{T}_{\perp e}\),the left hand circularly polarized electron firehose instability (EFHI) starts, and for condition \({T}_{\parallel ,p}>{T}_{\perp p}\), right hand circularly polarized proton firehose instability.(PFHI) (Hollweg et al.,1970) [10]; (Pilipp et al.,1971) [20].PFHI is studied in linear regime by Lazor et al., 2011 [21].
Both of these branches (PFHI, EFHI) have been established by computer simulation as well as spacecraft observation. Two dimensional Particle-In-Cell (PIC) simulations with a physical mass-ratio for the fiirehose instability has been studied by Camporeale et al., (2008) [22] and reviewed by Lazar et al., (2009) [19]. The observations made by spacecrafts such as Helios, Ulysses, and Cluster exhibit the anisotropic behavior of ions, and their thermal gradients traits reported by Marsch et al.,(1982) [23], Marsch et al. (2004) [24]; Marsch et al.(2006) [25], and for electrons, studied by Stverak et. al.(2009) [26], at radial distances up to 5AU (Stverak et al.2008) [27]. The significance of kinetic process to constrain the anisotropies observed at heliocentric distances is reported by Hellinger et al., 2006) [28].
The importance of this mode is for the transportation of the energy is explained by Gary (1993) [2]; Marsch (2006) [25]. In the solar flares and solar wind, the role of electron firehose instability and its possible applications has been discussed by Paseold et al.,(1999) [4] and Li et al., (2000) [29].
Shaaban et al., (2018) [30] have given the full gamut of electron firehose instabilities in the vicinity of super thermal electron population in positron-electron plasma based on particle simulations to deduce instability criteria and growth rate by applying by linear fluid theory. They solved dispersion relation employing an advanced numerical solver, DSHARK, designed for such plasmas, deviated from maxwellian distribution, the same is also studied by Astfalk et al., (2015) [31].
The environment near the earth provides a plausible laboratory for plasma research. In the magnetosphere variety of waves and instability associated with superthermal population arouse due to plasma instabilities. Observational data from space craft (CRRES, DEL & 2, ISEEL & 2, DMSP, IMP8) numerous ground stations in Australia, New Zealand ,Antarctica are used to study such instability and waves in magnetosphere and ionosphere (Ahirwar 2012) [32]. Direct In situ evidence of firehose instability in multiple reconnection, is stated by Alexandrova et al., (2020) [33] in earth's magnetotail.
In the solar wind and terrestrial magnetosphere direct in situ measurements implies that the velocity distribution functions of space plasmas particles, quasi maxwellian up to the average thermal velocities while they show non maxwellian superthermal tails at higher energies (Marsch et al., 2006) [25]; (Pierrard et al., 2010) [13].
As the instability takes place, it limits the growth of electron temperature anisotropy, and this incident is confirmed by observation and through simulations by Messmer (2002) [34]; Gary and Nishimura (2003) [35]; Paesold et al., (2003) [36]; Camporeale et al., (2008) [22] .
A variety of activities of space plasma occur in aurora region from solitary waves to different types of instabilities and also large scale MHD phenomena. Ion heating/acceleration transverse to the geomagnetic field line in the auroral region is main cause of flowing of ions from the ionosphere to the magnetosphere. (Agrwal et al., 2011) [37].
Matteini et al., (2015) [38] reported that parallel fire hose instability is generated by anisotropic alpha particles in the same way as that generated by protons., In homogeneous and collisionless solar wind plasma, quasi-linear kinetic theory is applied (Safraraj et.al. 2017, 2019) [39, 40] to study the effect of change in time parameter when the norms of temperature is \({T}_{\parallel }\)>\({T}_{\perp }\), also described the physical aspect of EFI which play major role in electron temperature anisotropy under dilute space plasma condition. They have also studied the role of PFI in determining parallel temperatures of protons in solar wind. On the basis of particle simulations and linear fluid theory FHI has been studied by Jao et al.,2020 [41] in electron -positron plasma. Firehose Instability and adaptive critical balance in an expanding, turbulent, collisionless Plasma is investigated by Bott et al.,(2021) [42]. Recently in Kappa-Maxwellian distributed space plasma, Nazeer et al., (2021) [43] have studied the firehose instability. Electron-Driven propagating FH instability in the solar wind have been studied by Verscharen et al., et al. (2022) [44].
In the present paper the work nearly related to the paper presented by Lazor et al., 2009 [19]. They found out dispersion relation of FHI, triggered by kappa type distribution function of electrons, in solar flares and solar wind with kinetic approach.
So far the firehose instability with kappa distribution function, has been analyzed either for electrons, or for ions, or for both ,but to our knowledge no one has included yet dust particles. Most of the research work have been done about firehose instability in many different space plasma regions but no one has yet analyzed it in plasma of auroral accelration region of earth's magnetosphere.
The auroral acceleration is a near earth process, often less than 8000 km in altitude reported by S3-3 satellite mission and below 2 RE established by polar observation (Ahirwar, 2011) [45]. The auroral acceleration region provides a plausible mechanism, to excite FHI. The heating and acceleration of ions along and perpendicular magnetic field, energy exchange of the ions with background magnetic field, are common features (Ahirwar, 2012) [32] which create basics to frame FHI in auroral acceleration region for our present work.