= Solution
Denote <aerosol optical depth> by $\tau_a$ to distinguish it from <Optical depth>[cloud optical depth]. Retrieve $N$ as in <2/d>, collocate it with $\tau_a$, and estimate an <Aerosol-cloud interaction>[aerosol-to-droplet susceptibility]
$$
s_a=\left.\frac{d\log N}{d\log\tau_a}\right|_{\text{matched meteorology and cloud-water conditions}}.
$$
A <regression> should control for <Liquid-water content>[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:
$$
d\log\frac A{1-A}=\tfrac13d\log N,
\qquad
\boxed{A_2=\frac{2^{s_a/3}A_1}{1-A_1+2^{s_a/3}A_1}.}
$$
For a small response, $\Delta A\simeq A_1(1-A_1)s_a\log2/3$. If $F_{\rm sw}$ is the appropriate mean incoming shortwave flux and $f_c$ the fraction of the scene occupied by these <Cloud>[clouds], the leading top-of-atmosphere perturbation, positive downward, is
$$
\boxed{\Delta F_{\rm TOA}\simeq-f_cF_{\rm sw}\,\mathcal T_{\rm above}^{\downarrow}\mathcal T_{\rm above}^{\uparrow}\,(A_2-A_1).}
$$
The two <Transmittance>[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 $F_{\rm sw}=L_{\circledast}/4$. Positive $s_a$ gives increased reflection and negative <Radiative forcing>[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, <Liquid-water path>[water path] or lifetime are additional adjustments, not the fixed-water Twomey term derived here.
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