Finite-Resistivity Effects on Jeans Instability in Quantum Dusty Plasmas

This study examines Jeans instability in quantum dusty plasma, incorporating resistive effects, using the quantum magnetohydrodynamic (QMHD) model. Linearized perturbation equations are derived via normal mode analysis to obtain the general dispersion relation, enabling analysis of the instability growth rate and criteria. The dispersion relation is specialized for both longitudinal and transverse propagation modes to assess the impacts of resistivity and quantum effects. For transverse propagation, the gravitating mode's growth rate is influenced by resistivity and magnetic field strength, while the Jeans instability criterion is modified solely by the quantum term. In the longitudinal direction, the gravitating mode is altered by quantum corrections, and the non-gravitating Alfvén mode is affected by both finite electrical resistivity and quantum effects. Numerical calculations demonstrate that resistivity exerts a destabilizing influence on the Jeans instability growth rate, with quantum parameters further modulating it. These findings apply to astrophysical dusty structures such as white dwarf interiors and molecular clouds, as well as low-temperature laboratory plasmas. Typical values include a Jeans length of $\sim 10^5$ m and a gravitational collapse timescale of $\sim 2.5$ s.