In this paper, we analyze the stability of a homogeneous self-gravitating plasma, having a non-zero resistivity. This study provides a generalization of the Jeans paradigm for determining the critical scale above which gravitational collapse is allowed. We start by discussing the stability of an ideal self-gravitating plasma embedded in a constant magnetic field. We outline the existence of an anisotropic feature of the gravitational collapse. In fact, while in the plane orthogonal to the magnetic field the Jeans length is enhanced by the contribution of the magnetic pressure, outside this plane perturbations are governed by the usual Jeans criterium. The anisotropic collapse of a density contrast is sketched in details, suggesting that the linear evolution provides anisotropic initial conditions for the non-linear stage, where this effect could be strongly enforced. The same problem is then faced in the presence of non-zero resistivity and the conditions for the gravitational collapse are correspondingly extended. The relevant feature emerging in this resistive scenario is the cancellation of the collapse anisotropy in weakly conducting plasmas. In this case, the instability of a self-gravitating resistive plasma is characterized by the standard isotropic Jeans length in any directions. The limit of very small resistivity coefficient is finally addressed, elucidating how reminiscence of the collapse anisotropy can be found in the different value of the perturbation frequency inside and outside the plane orthogonal to the magnetic field.
In many astrophysical and cosmological systems the presence of a plasma component has a very important role in determining the shape and the behavior of their equilibrium configurations. The peculiarity of these plasma configurations with respect to those ones observed in laboratory, relies on the dominant character of the gravitational interaction in determining the stability properties. Indeed, as firstly suggested by Jeans [1,2], the gravitational interaction is able to induce the collapse, as long as a critical scale of the configuration is reached. Such a scale, depending on the sound speed and on the mass density of the medium is commonly known as the Jeans length. On the other hand, dealing with a self-gravitating plasma instead of a fluid brings in the equilibrium all the typical features observed in magnetically confined and highly ionized gases, like the emergence of Alfvén and magnetosonic waves. Moreover, we stress that the possibility to postulate the presence of a magnetic field, is allowed since it is observationally demonstrated by the direct observation of astrophysical systems (see e.g. Refs. [3,4,5,6,7,8,9,10] and, more recently, Refs. [11,12,13,14,15,16,17,18]; see also Refs. [19,20,21,22,23] for review works).
The fact that, in our analysis, both the background mass density and magnetic field of the configuration are taken homogeneous is justified by the often slow variation of these quantities in real astrophysical systems, even over scales for which the self-gravity is already relevant (for instance, the primordial cosmological plasma and the ionized intergalactic baryonic component [24]). Indeed, our study concerns the linear stability of a homogeneous magnetized and self-gravitating plasma, endowed with a finite value of the resistivity coefficient. This latter dissipative feature is here introduced to account for the non-ideal nature of the most commonly observed space plasma. Significant reconnection processes of the magnetic profile are often observed or argued via the interpretation of data from astrophysical configurations. Despite the effects of a finite resistivity coefficients are particularly important in the non-linear regimes, where the establishment of a turbulent profile of the plasma can phenomenologically enforces the resistivity (see for instance the question concerning the so-called anomalous resistivity in the configurations of stellar accretion disks [25] from the plasma instabilities raised from the streaming of electrons, and also [26,27,28]), nonetheless, we will show how its presence is crucial already in the linear case, when dealing with the stability properties.
As a first step, we analyze the linear stability for the ideal case, when the resistivity coefficient of the plasma vanishes. In this limit, we essentially reconstruct the Jeans paradigm for the gravitational stability of the plasma structure. The stability out of the plane orthogonal to the constant magnetic field remains still characterized by the same Jeans length obtained originally for the fluid scheme. A relevant new feature emerges, however, in the plane perpendicular to the magnetic force straight lines, where the contribution due to the magnetic pressure affects the equilibrium enhancing the value of the Jeans length by a term corresponding to the square of the Alfvén velocity in the plasma. Such additional contribution enters the Jeans length expression on the same footing as the sound speed contribution and therefore its relevance strictly depends on the ratio of the sound speed to the Alfvén one. In particular, the greater the Alfvén speed is, the larger is the anisotropy in the gravitational collapse, inside and outside the orthogonal plane. We properly describe this effect by following the behavior of a over-dense region during the linear evolution, which is accordingly squeezed on the orthogonal plane. Indeed, in the linear regime, the density contrasts grows without a real gravitational collapse (that takes place essentially in the non-linear stage of the evolution), but the growth is slower on the plane where the magnetic pressure affects the Jeans scale. It is worth noting that, in the perturbation scheme, the magnetic pressure is, despite its name, anisotropic, being provided by the scalar product between the background and the perturbed magnetic field. It is just this feature that introduces the anisotropy in the perturbation evolution. The mode that becomes unstable on the orthogonal plane corresponds to the quasi-stationary one (typical of the slow magnetosonic configuration of a non-gravitating plasma). The presence of gravity alters the nature of this mode, making the system unstable, but with a greater Jeans length with respect to the directions out of this plane, for which the quasi-stationary mode would be absent in the non-gravitational case too.
The gravitational stability analysis is then faced taking into account a non-vanishing resistivity coefficient. In this case
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