Theorem: at tau = ln(2), I(tau) = (I_0 + B)/2 exactly (half-intensity optical depth)

Theorem (Schwarzschild half-intensity canonical): the formal solution I(tau) = I_0 exp(-tau) + B(1 - exp(-tau)) at tau = ln(2) gives I(ln 2) = I_0/2 + B/2 = (I_0 + B)/2 exactly, because exp(-ln 2) = 1/2. This is the optical depth at which…