Spherical density projection
= Spherical density projection
{title2=$\Sigma(R)=2\int_R^\infty\rho(r)r(r^2-R^2)^{-1/2}\,dr$}
A spherical <mass density> $\rho$ gives projected <surface density> $\Sigma(R)=\int_{-\infty}^{\infty}\rho(\sqrt{R^2+z^2})\,dz$. On the positive half of the <line of sight>, substitute $r=\sqrt{R^2+z^2}$ to obtain the displayed <Abel transform>. Spherical symmetry is essential: a general three-dimensional density cannot be reconstructed uniquely from one projection.