= Solution
For the quadratic <Planck function> profile,
$$
I_0=\pi B_0+2\pi\beta\int_0^{\tau_s}\tau e^{-\tau}\,d\tau
=\pi B_0+2\pi\beta[1-(1+\tau_s)e^{-\tau_s}].
$$
The optically thick limit and the instantaneous doubled-abundance result are therefore
$$
\boxed{I_0=\pi(B_0+2\beta),\qquad
\widetilde I_0=\pi(B_0+\beta/2),\qquad
\Delta I_0=-\frac32\pi\beta.}
$$
The new coefficient is $\beta/4$, since $B(\widetilde\tau)=B_0+\beta(\widetilde\tau/2)^2$ at fixed physical <temperature>. For positive $\beta$, the outgoing <spectral irradiance> decreases. The factor differs from the linear case because the temperature-dependent source function curves with depth; doubling abundance does not mean holding its old functional coefficients fixed in the new optical-depth coordinate.
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