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 R0, its geometric cross-section G is expressed as $G=AN\pi R_0^2/[1+(N-1)\tilde{\sigma }]$, 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.

中文翻译：

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分形尘埃聚集体几何截面的解析表达式

在原行星盘和行星大气中，尘埃颗粒凝结形成分形尘埃聚集体。这些聚集体的几何横截面是表征空气动力摩擦、碰撞率和不透明度的关键参数。然而，横截面的数值测量通常很耗时，因为骨料呈现出复杂的形状。在这项研究中，我们推导出了分形聚集体几何横截面的新解析表达式。如果一个聚合体由 N 个半径为 R0 的单体组成，其几何截面 G 表示为 $G=AN\pi R_0^2/[1+(N-1)\tilde{\sigma }]$，其中 $\波浪号{\sigma }$ 是重叠效率，A 是将解析表达式连接到小的非分形簇限制的数值因子。重叠效率取决于分形维数、分形预因子、和N，并推导出其解析表达式。解析表达式成功地再现了骨料的数值测量横截面。我们还发现我们的表达式与几何光学极限中聚集体的平均场光散射理论兼容。解析表达式极大地简化了繁琐的计算，并将在原行星盘和行星大气中的分形晶粒生长的模型计算中有用。