Theory for Optically Observing Cloud Aerosol Critical Activation Diameter

Abstract. Aerosol activation is the fundamental microphysical process governing cloud droplet formation. The critical activation diameter, 𝐷𝑎, is a key parameter that characterizes cloud supersaturation and aerosol activation and serves as a fundamental variable in simulations of aerosol–cloud interactions. Kuang et al. (2025) proposed a machine-learning framework for the instantaneous retrieval of 𝐷𝑎 from spectral scattering measurements of interstitial and activated aerosols. However, the lack of a rigorous theoretical foundation has limited the understanding of the method's physical basis, applicability, and limitations in field observations. Here, we establish the missing theoretical foundation by integrating aerosol activation theory with Mie scattering theory to derive a unified closed-form solution linking optical scattering to the 𝐷𝑎. The resulting theory provides the first analytical explanation for the approximately linear relationship between 𝐷𝑎 and the scattering fraction of interstitial aerosols reported by Kuang et al. (2025), while also revealing the strongly nonlinear behavior near the limiting scattering regimes. Furthermore, the theoretical detection limit of the optical approach is quantified, and its potential applicability to different cloud types is evaluated. Overall, this work provides a rigorous theoretical foundation for optical observations of cloud aerosol activation. The unified analytical solution bridges all limiting regimes, clarifies the physical basis of optical retrievals of the critical activation diameter, and offers theoretical guidance for the design and optimization of future optical instruments for cloud activation measurements.