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Denote aerosol optical depth by ${\tau }_a$ to distinguish it from cloud optical depth. Retrieve $N$ as in d, collocate it with ${\tau }_a$, and estimate an aerosol-to-droplet susceptibilityA regression should control for water content, updraught and cloud regime; aerosol optical depth is not itself a direct measurement of activated nuclei. For an approximately constant $s_a$ over a factor two in aerosol, $N_2/N_1=2^{s_a}$.

Integrate the supplied cloud albedo susceptibility rather than treating a finite doubling as infinitesimal:For a small response, $\Delta A\simeq A_1(1-A_1)s_a\log 2/3$. If $F_{\mathrm{sw}}$ is the appropriate mean incoming shortwave flux and $f_c$ the fraction of the scene occupied by these clouds, the leading top-of-atmosphere perturbation, positive downward, isThe two transmissions account approximately for the atmosphere above the cloud and may be absorbed into an effective illuminating flux. In the simplest global transparent-overburden estimate $F_{\mathrm{sw}}=L_{⊛}/4$. Positive $s_a$ gives increased reflection and negative shortwave forcing, the Twomey effect. Numerical forcing cannot be determined from the given susceptibility alone: it needs baseline albedo, cloud fraction, illumination and an empirically constrained $s_a$. Humidity swelling, cloud contamination of aerosol retrievals and common meteorological influences can create spurious correlations. Changes in cloud amount, water path or lifetime are additional adjustments, not the fixed-water Twomey term derived here.