Dust Acoustic Solitons in Saturn’s Dust-Filled Magnetosphere

The possibility is considered of propagation of localized wave structures, such as dust acoustic solitons, in the plasma of the dust-filled Saturn’s magnetosphere, which contains electrons of two sorts (hot and cold) subject to kappa distribution, ions, and charged dust grains. The ranges of possible velocities and amplitudes of the solitons are determined. Soliton solutions for different sizes and concentrations of dust grains in the dust-filled Saturn’s magnetosphere are found.


INTRODUCTION
It is difficult to imagine a plasma-filled region of the Solar system that would be free of finely dispersed dust grains [1]. Nano-and microsized dust grains are found in interplanetary space, in ionospheric and magnetospheric plasma of the planets of the Solar system, in planetary rings, in the vicinity of space bodies devoid of their own atmosphere, and so on. An important object from the viewpoint of dusty plasma studies is Saturn's magnetosphere [2,3]. Its plasma parameters were measured as early as in the 1980s by the Voyager 1 and 2 space missions [4]. Also based on the data collected by the Voyager 1 spacecraft [5], it was shown that waves exist in the magnetospheric plasma of Saturn. Theoretical studies of ion acoustic waves in Saturn's magnetosphere were carried out in [6]. The dusty plasma in the vicinity of Saturn's satellite Enceladus was discovered during the Cassini mission [2,3].
The plasma in Saturn's magnetosphere has a number of specificities compared with other space systems that are now actively studied (studies of the dusty plasma on Moon and Mars see, e.g., in [7,8]). For example, measurements of the parameters of electrons in Saturn's magnetosphere obtained by the Voyager [4,5] and Cassini [9] missions showed the existence of two types (hot and cold) of electrons. It turned out [9] that the velocities of these electron populations are subject to the so-called kappa distribution with independent low values of κ.
All of this indicates the importance of studying the nonlinear wave structures in the dusty plasma under conditions characteristic of Saturn's magnetosphere, for which, dust acoustic waves are typical. An import-ant type of nonlinear structures observed in space [10,11] are solitons. In this work, we consider nonlinear sound structures in the dust-filled Saturn's magnetosphere characteristic of dusty plasma, namely, dust acoustic solitons [12]. We take into account the two types of electrons (hot and cold) that coexist in Saturn's magnetosphere and are subject to two different κ distributions. The study is carried out for arbitrary (not small) soliton amplitudes, which appears important for interpreting future space observations.

BASIC EQUATIONS
To describe dust acoustic solitons (see, e.g., [13][14][15]), the following set of equations can be used, which includes the continuity equation, the (Euler) motion equation for the dust grains, and Poisson's equation for the self-consistent electrostatic potential in the dusty plasma of Saturn's magnetosphere: (1) where the space variable x corresponds to the propagation direction of the wave perturbation, n d is the concentration of dust grains, is their velocity, m d and q d = Z d e is the mass and charge of dust grains, is the electron charge, n e,c(h) is the concentration of hot (cold) electrons, and n i is the concentration of ions.

DUSTY PLASMA
Additionally, it is necessary to take into account the ion and electron distributions that have time to get established over the dust acoustic time scale. It was already noted that, in Saturn's magnetosphere, the electrons have a κ velocity distribution [16] (2) where T e,c(h) is the temperature of cold (hot) electrons in energy units, index 0 corresponds to unperturbed states, and κ c and κ h are the parameters of kappa distributions of the cold and hot electrons, respectively. Note that κ c , κ h > 3/2.
The ions are subject to Boltzmann's distribution (3) where T i is the ion temperature in energy units.
Taking into account that dust acoustic solitons are relatively slow, and consequently, their characteristic time scale significantly exceeds the characteristic time scale of the change of the charge of dust grains (see, e.g., [17]), the dust grain charges have time to adjust to the plasma parameters and can be determined by the equation (4) where the macroscopic currents of cold (hot) electrons on the surface of the dust grain are described by the expression [18] (5) and the microscopic ion current has the form [18] Here, a is the dust grain size, m e(i) is the electron (ion) mass, and Γ(κ c(h) ) is the gamma function. Note that the values Z d in the soliton obtained from Eq. (4), generally speaking, depend on the electrostatic potential ϕ. However, according to the conducted calculations that take into account the changes of plasma parameters in the soliton, this dependence is not very strong. For example, even in the maximum of the soliton, where the deviation of its charge from its nonper-turbed state is maximum, the charge differs less than 5% from its value at the base of the soliton, including in solitons with a high amplitude (see [17]). This fact allows one to consider the charges of dust grains in the soliton approximately constant, which will be used later.
The nonperturbed concentrations are connected by the quasi-neutrality condition (7) It is convenient to introduce the total concentration of nonperturbed cold and hot electrons (8) and the ratio α between the concentrations of hot and cold electrons ( ). Then we have

SOLITON SOLUTIONS
The system of equations (1)- (7) can be solved by the standard method of Sagdeev potential [19]. To determine the localized wave solution that propagates at some constant velocity M, we can transfer to a new system of coordinates connected to the wave. In this case, all the parameters of the problem depend only on the new variable . The Sagdeev potential in dimensionless variables , , , where and has the form where , , .
The soliton solution is found from the relation (12) where ϕ ξ is the first derivative of function ϕ with respect to ξ.
Below, results are given for calculations that were carried out for the following parameters of Saturn's magnetosphere (see [4,6,9]): n i0 = 10 cm -3 , T i = 100 K, T ec = 10 eV, T eh = 700 eV, , and κ c = κ h = 2. Figure 1 shows the dependences of charge numbers Z d of the dust grains and total electron concentrations n e0 on dust grain size for different concentrations of dust grains. The values Z d and n e0 were obtained at the given value of n i0 based on the self-consistent solution of Eqs. (4)- (7).

CONFLICT OF INTEREST
The authors declare no conflict of interest.

OPEN ACCESS
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