Jeans analysis of self-gravitating systems in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>R</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>gravity

Dynamics and collapse of collisionless self-gravitating systems is described by the coupled collisionless Boltzmann and Poisson equations derived from $f(R)$ gravity in the weak field approximation. Specifically, we describe a system at equilibrium by a time-independent distribution function ${f}_{0}(x,v)$ and two potentials ${\ensuremath{\Phi}}_{0}(x)$ and ${\ensuremath{\Psi}}_{0}(x)$ solutions of the modified Poisson and collisionless Boltzmann equations. Considering a small perturbation from the equilibrium and linearizing the field equations, it can be obtained a dispersion relation. A dispersion equation is achieved for neutral dust-particle systems where a generalized Jeans wave number is obtained. This analysis gives rise to unstable modes not present in the standard Jeans analysis (derived assuming Newtonian gravity as weak filed limit of $f(R)=R$). In this perspective, we discuss several self-gravitating astrophysical systems whose dynamics could be fully addressed in the framework of $f(R)$ gravity.

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