 Local binding-energy allocation in galactic feedback

Self-gravitating energy is nonlocal. Shell assembly assigns $-GM(<r)dM/r$ and counts pairs once; the density $\rho \Phi /2$ is another total-energy allocation. Removing a test parcel from a fixed potential instead requires $-\Phi dM$. These prescriptions need not give the same radial feedback threshold, even though their units agree.