We can also use the principle of hydrostatic equilibrium to estimate the velocity dispersion of dark matter in clusters of galaxies. Only baryonic matter (or, rather, the collisions thereof) emits X-ray radiation. The absolute X-ray luminosity per unit volume takes the form L X = Λ Λ ( T B ) ρ ρ B 2 where T B and ρ ρ B are the temperature and density of the baryonic matter, and Λ Λ ( T ) is some function of temperature and fundamental constants. The baryonic density satisfies the above equation d P = − − ρ ρ g d r : p B ( r + d r ) − − p B ( r ) = − − d r ρ ρ B ( r ) G r 2 ∫ ∫ 0 r 4 π π r 2 ρ ρ M ( r ) d r . The integral is a measure of the total mass of the cluster, with r being the proper distance to the center of the cluster. Using the ideal gas law p B = k T B ρ ρ B / m B ( k is the Boltzmann constant and m B is a characteristic mass of the baryonic gas particles) and rearranging, we arrive at d d r ( k T B ( r ) ρ ρ B ( r ) m B ) = − − ρ ρ B ( r ) G r 2 ∫ ∫ 0 r 4 π π r 2 ρ ρ M ( r ) d r . Multiplying by r 2 / ρ ρ B ( r ) and differentiating with respect to r yields d d r [ r 2 ρ ρ B ( r ) d d r ( k T B ( r ) ρ ρ B ( r ) m B ) ] = − − 4 π π G r 2 ρ ρ M ( r ) . If we make the assumption that cold dark matter particles have an isotropic velocity distribution, then the same derivation applies to these particles, and their density ρ ρ D = ρ ρ M − − ρ ρ B satisfies the non-linear differential equation d d r [ r 2 ρ ρ D ( r ) d d r ( k T D ( r ) ρ ρ D ( r ) m D ) ] = − − 4 π π G r 2 ρ ρ M ( r ) . With perfect X-ray and distance data, we could calculate the baryon density at each point in the cluster and thus the dark matter density. We could then calculate the velocity dispersion σ σ D 2 of the dark matter, which is given by σ σ D 2 = k T D m D . The central density ratio ρ ρ B ( 0 ) / ρ ρ M ( 0 ) is dependent on the redshift z of the cluster and is given by ρ ρ B ( 0 ) / ρ ρ M ( 0 ) ∝ ∝ ( 1 + z ) 2 ( θ θ s ) 3 / 2 where θ θ is the angular width of the cluster and s the proper distance to the cluster. Values for the ratio range from 0.11 to 0.14 for various surveys.