Question on using r_optical for tabulated BSSRDF

[KNiGHT]

I am not able to understand the logic behind this snippet from third edition in section 11.4.2 (Tabulated BSSRDF) discussing the sampling of the BSSRDF:

"

When a lookup for a given extinction coefficient and radius occurs at run time we find the corresponding unitless optical radius and evaluate the lower-dimensional tabulation as follows:

Sr(eta, g,rho,sigma_t,r) = sigma_t^2 * Sr(eta, g,rho,1,r_optical)

"

The book goes on to describe that this is related to Section 13.5.2, which describes the change of variables from cartesian to polar coordinates (which doesn't seem to be related to the discussion unless I'm missing something).

My gaps in understanding are as follows:

lets say sigma_t is 0.1, and we need to find Sr at r = 1, then we get the value tabulated for  r=0.1, and given that we stretched the interval to 10 times of the original one, we need to multiply by sigma_t [the Jacobian] to retain the normalized distribution. so I get why we multiply by sigma_t, but I don't get why we multiply by sigma_t^2 instead?

Many thanks for your help.