Analytic expressions for geometric cross-sections of fractal dust aggregates

In protoplanetary discs and planetary atmospheres, dust grains coagulate to form fractal dust aggregates. The geometric cross-section of these aggregates is a crucial parameter characterizing aerodynamical friction, collision rates, and opacities. However, numerical measurements of the cross-section are often time-consuming as aggregates exhibit complex shapes. In this study, we derive a novel analytic expression for geometric cross-sections of fractal aggregates. If an aggregate consists of $N$ monomers of radius $R_0$, its geometric cross-section $G$ is expressed as \begin{equation} \frac{G}{N\pi R_0^2}=\frac{A}{1+(N-1)\tilde{\sigma}}, \nonumber \end{equation} where $\tilde{\sigma}$ is an overlapping efficiency, and $A$ is a numerical factor connecting the analytic expression to the small non-fractal cluster limit. The overlapping efficiency depends on the fractal dimension, fractal prefactor, and $N$ of the aggregate, and its analytic expression is derived as well. The analytic expressions successfully reproduce numerically measured cross-sections of aggregates. We also find that our expressions are compatible with the mean-field light scattering theory of aggregates in the geometrical optics limit. The analytic expressions greatly simplify an otherwise tedious calculation and will be useful in model calculations of fractal grain growth in protoplanetary discs and planetary atmospheres.