Coexistence of magneto-rotational and Jeans instabilities in an axisymmetric nebula

Aims. We analyze the magneto-rotational instability (MRI) effects on gravitational collapse and its influence on the instability critical scale. Methods. In particular, we study an axisymmetric nonstratified differentially rotating cloud, embedded in a small magnetic field, and we perform a local linear stability analysis, including the self gravity of the system. Results. We demonstrate that the linear evolution of the perturbations is characterized by the emergence of an anisotropy degree of the perturbed mass densities. Starting with spherical growing overdensities, we see that they naturally acquire an anisotropy of order unity in their shape. Despite the linear character of our analysis, we infer that such a seed of anisotropy can rapidly grow in a nonlinear regime, leading to the formation of filament-like structures. However, we show how such an anisotropy is essentially an intrinsic feature of the Jean instability, and how MRI only plays a significant role in fixing the critical scale of the mode spectrum. We then provide a characterization of the present analysis in terms of the cosmological setting, in order to provide an outlook of how the present results could concern the formation of large-scale structures across the Universe