of Motion
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Following its collapse into the potential of the galactic dark matter halo, the RHG is shock-heated to the virial temperature of the potential, and rapidly attains a quasi hydrostatic equilibrium. For computational simplicity, we adopt a Cartesian coordinate system in which the galactic potential $g$ is imposed in the $y$ direction. For a spherically symmetric potential, $y$ corresponds to the radial direction. The equation of motion of the RHG becomes $${d V_{h x} \over d x} = - {1 \over \rho_h} {d P_t \over d x}$$ $${d V_{h y} \over d y} = - {1 \over \rho_h} {d P_t \over d y} -g$$ where $\rho_h$, $P_h$, $V_{h, x}$, and $V_{h,y}$ are respectively the density, pressure, two-velocity components of the RHG, $P_t = P_h +
P_w$ is the total pressure, $\rho_w$, $P_w$, $V_{w, x}$, and $V_{w,y}$ are respectively the density, pressure and two velocity components of WFC’s. The equation of motion for the WFC’s is similar, $${d V_{w x} \over d x} = - {1 \over \rho_w} {d P_t \over d x}$$ $${d V_{w y} \over d y} = - {1 \over \rho_w} {d P_t \over d y} -g+F_D,$$ where $F_D$ is a drag force term, which is 