Axial gravitational field of a uniform ring ($g_z=-GMz/(a^2+z^2{)}^{3/2}$)

A uniform thin ring of radius $a$ and mass $M$ has axial gravitational field $g_z=-GMz/(a^2+z^2{)}^{3/2}$. Every ring element is the same distance from the axial point; transverse contributions cancel and axial projection supplies the factor $z/\sqrt{a^2+z^2}$. This also follows by differentiating the axial potential $\Phi =-GM/\sqrt{a^2+z^2}$. Near the centre $g_z\simeq -GMz/a^3$, whereas far away it approaches the point-mass field.