Surface density of a disk ($\Sigma$)

The surface density of a disk is the mass density integrated through its vertical column: $\Sigma (r)={\int }_{-\infty }^{\infty }\rho (r,z)dz$. Its dimensions are mass per area. An axisymmetric annulus of width $dr$ therefore contains $dM=2\pi r\Sigma (r)dr$. This reduces a three-dimensional mass inventory to a radial one, which is useful in viscous evolution of an accretion disk. Distinct vertical profiles can have the same column mass, so $\Sigma$ alone does not determine the vertical structure of an astrophysical disk.