Strip density of a spherical system ($S(x)=2\pi {\int }_x^{\infty }r\rho (r)dr$)

The strip density is the mass per unit projected Cartesian coordinate $x$: $S(x)={\int }_{-\infty }^{\infty }\Sigma (\sqrt{x^2+y^2})dy$. It is even in $x$. For $x\ge 0$, changing variables to projected radius gives $S(x)=2{\int }_x^{\infty }\Sigma (R)R/\sqrt{R^2-x^2}dR$. Alternatively integrate the three-dimensional mass density over the plane of coordinates $y,z$ to obtain the displayed expression. Consequently $S^{\prime }(x)=-2\pi x\rho (x)$ for $x>0$.

For an isolated spherical distribution, shell assembly gives gravitational potential energy $W=-{\int }_0^{\infty }GM(r)r^{-1}dM(r)$. Use $S^{\prime }=-2\pi r\rho$ and $M^{\prime }=-2r{S}^{\prime }$ to write $W=2G\int M{S}^{\prime }dr=-2G\int S{M}^{\prime }dr=4G\int rS{S}^{\prime }dr$. A second integration by parts gives the displayed negative quadratic functional. The boundary terms $MS$ and $r{S}^2$ must vanish; finite-mass regular profiles with finite binding energy satisfy the usual sufficient decay conditions.