Analysis of instability of systems composed by dark and baryonic matter (Q2809400)

The authors analyze the dynamics and the collapse of a collisionless selfgravitating system composed by dark and baryonic matter. This system is described by two Boltzmann equations, one for each constituent, and the Poisson equation for the gravitational field. The evolution equations of the distribution functions of baryonic matter \(f_b\equiv f(x,v_b,t)\) and dark matter \(f_d\equiv f(x,v_d,t)\) are the collisionless Boltzmann equations NEWLINE\[NEWLINE \frac{\partial f_b}{\partial t}+v_b\cdot\frac{\partial f_b}{\partial x} -\nabla\Phi\cdot\frac{\partial f_b}{\partial v_b}=0,NEWLINE\]NEWLINE NEWLINE\[NEWLINE \frac{\partial f_d}{\partial t}+v_d\cdot\frac{\partial f_b}{\partial x} -\nabla\Phi\cdot\frac{\partial f_d}{\partial v_d}=0.NEWLINE\]NEWLINE The gravitational field must fulfill the Poisson equation NEWLINE\[NEWLINE \nabla^2\Phi=4\pi G\bigg(\int f_bdv_b +\int f_d dv_d\bigg)=4\pi G(\rho_b+\rho_d), NEWLINE\]NEWLINE where \(\rho_b\) and \(\rho_d\) are the mass densities of the baryonic and dark matter, respectively.