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The effect of quantized anisotropic magnetic pressure, arising due to a strong magnetic field, is studied on the growth rate of Jeans instability of quantum electron–ion and classical dusty plasma. Two dispersion relations are formulated for the propagation of the gravitational waves that are perpendicular and parallel to the magnetic field, respectively. It is shown that for perpendicular propagation, the quantized magnetic pressure is stabilizing the Jeans instability, whereas for the parallel propagation the plasma become more unstable with the increase in the magnetic field or we can say that Jeans instabilities becomes anisotropic in this case. The corresponding Jeans wave number is calculated in the absence of tunneling effect which shows that the magnetic field anisotropy of the Galaxy is connected with Jeans effect. Further, it is shown that the Madelung term leads to the inhomogeneity of the plasma medium. Numerical results are presented to show the effect of the anisotropic magnetic pressure on the Jeans instability.

The dispersive properties of Jeans surface instability at a charged plane interface of weakly coupled, classical self-gravitating, magnetized electron-ion plasma and vacuum, are investigated. The general dispersion relation is formulated by employing  one fluid magnetohydrodynamic (MHD) model and Poisson’s equations for both electrostatic and gravitational potentials. It is shown that both the surface charge and  mass density fluctuations are coupled  together effectively, to enhance the growth rate of  surface Jeans instability, whereas an increase in magnetic field and surface tension tends to stabilize Jeans surface instability. The modified Jeans criterion of gravitational instability is obtained by formulating  a new definition of Jeans wavelength. These results may be useful to examine the gravitational collapse of self-gravitating electron-ion plasma, leading to star formation in astro-cosmic environments.

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