Vertical Jeans equation (${\partial }_z(n{\sigma }_z^2)+R^{-1}{\partial }_R(Rn\overline{v_R{v}_z})=-n{\partial }_z\Phi$)

For a steady axisymmetric stellar population with no mean vertical motion, the vertical Jeans equation balances the gradient of the vertical random-motion stress and the radial gradient of the tilted velocity ellipsoid against the vertical Newtonian gravitational field. Neglecting the mixed moment gives ${\partial }_z(n{\sigma }_z^2)=-n{\partial }_z\Phi$. A tracer density $n$ need not equal the density sourcing the gravitational potential.

The vertical Jeans equation determines $K_z={\partial }_z\Phi$ from a tracer's density and velocity dispersion, with a correction for the tilted velocity ellipsoid. Integrating the axisymmetric Poisson equation for Newtonian gravity givesfor $z>0$ and reflection symmetry. Subtracting independently estimated stellar and gas surface densities constrains any additional gravitating component; a dark matter halo also contributes, so an excess is not automatically a dark disk.