Gravitational Instability and Collapse in Dusty Plasma Clouds: A Modified Jeans Analysis

The gravitational collapse of molecular clouds is fundamental to star formation, yet standard Jeans instability analyses neglect the electrostatic effects of charged dust grains that permeate these environments. We present a modified Jeans analysis incorporating charged dust dynamics into a three-component fluid framework (neutral gas, charged dust, and ion-electron plasma). Through linear perturbation analysis with strong dust-neutral coupling, we derive an effective sound speed \(c_{s,\mathrm{eff}}^2 = k_B T/\mu + \alpha n_d q_d^2\), where the second term represents electrostatic pressure from charged grains. This yields modified Jeans length \(\lambda_J = \sqrt{\pi(c_s^2 + \alpha n_d q_d^2)/G\rho_0}\) and Jeans mass \(M_J = (\pi^{5/2}/6)\sqrt{1/G^3\rho_0}\,(c_s^2 + \alpha n_d q_d^2)^{3/2}\). For typical interstellar medium conditions (\(T=10\,\mathrm{K}\), \(n_H=10^4\,\mathrm{cm}^{-3}\), \(q_d=100e\)), charged dust increases the Jeans mass by \(30\%\)--\(100\%\), suppressing low-mass star formation. Stability analysis reveals that the dust charge parameter \(\beta = \alpha n_d q_d^2/c_s^2\) reduces instability growth rates by up to \(60\%\) for \(\beta=5\), while the free-fall time \(t_{\mathrm{ff}}=\sqrt{3\pi/(32G\rho_{\mathrm{eff}})}\) remains unchanged. We construct phase diagrams showing how charged dust shifts collapse thresholds across different astrophysical environments (diffuse clouds, molecular clouds, protostellar cores, AGN tori) and suppresses brown dwarf formation. The fragmentation cascade is altered, with fewer fragmentation generations leading to more massive final cores. Observable signatures include enhanced broadband emission and anomalous microwave emission (AME) from spinning dust, with peak frequency \(\nu_{\mathrm{peak}} \propto q_d\), providing direct observational tests with Planck, ALMA, and JWST. Our results demonstrate that charged dust electrostatically stabilizes molecular clouds, offering a microphysical explanation for environmental variations in the stellar initial mass function and star formation efficiency.