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.

Meanwhile, an analytic light scattering theory of fractal aggregates, based on a statistical distribution model of constituent particles (henceforth called monomers), has proven to be successful (Berry & Percival 1986;Rannou et al. 1997;Tazaki et al. 2016;Tazaki & Tanaka 2018). This success motivated us to develop an analytic model of geometric cross-sections of fractal aggregates based on a similar statistical approach. To our knowledge, this is the first attempt at this type of approach.
This study presents a novel analytic model of geometric crosssections of fractal aggregates by applying a statistical distribution model of monomers. We demonstrate that our analytic model successfully reproduces numerically measured cross-sections reported in earlier studies (e.g., Meakin & Donn 1988;Minato et al. 2006;OTS09). Moreover, our model is shown to be compatible with the mean-field light scattering theory .
This paper is organized as follows. In Section 2, we summarize scaling properties of geometric cross-sections of fractal aggregates based on earlier studies. In Section 3, we derive an analytic expression for geometric cross-sections and test its validity by comparison with numerical results reported in previous studies. A relationship between the analytic expression and the mean-field light scattering theory is discussed in Section 4. In Section 5, we further extend the analytic expression to treat the cross-sections of inhomogeneous aggregates and test its validity. Also, we discuss the applicability of empirical formulas that have been commonly used in previous studies. A summary is given in Section 6.

Essential parameters describing fractal aggregates
The structure of a fractal aggregate is characterized by three parameters: the fractal dimension , fractal prefactor 0 , and the number of monomers . The radius of gyration is commonly used to describe an aggregate radius and obeys a well-known fractal law: where 0 is the monomer radius. The characteristic radius of the aggregate, = √︁ 5/3 , has likewise been employed Mukai et al. 1992), instead of the radius of gyration. In Table 1, we summarize some definitions of aggregate radii used in this paper.
The values of and 0 depend on aggregation processes. A primary stage of aggregation often occurs by a process called clustercluster aggregation (CCA). CCA results in forming aggregates with fluffy structure (Figure 1). Numerical simulations suggest that a CCA cluster typically has = 1.7−2.1 (e.g., Meakin 1984a,b;Sorensen & Roberts 1997;Kempf, Pfalzner, & Henning 1999). A fractal dimension of a non-ballistic CCA cluster can be even lower, namely as low as ∼ 1.1−1.4, owing to the effect of rotation of aggregates during collisions (Blum et al. 2000;Krause & Blum 2004;Paszun & Dominik 2006). Ballistic particle-cluster aggregation (BPCA) offers the opposite limiting case to CCA, as it results in forming nearly homogeneous aggregates with ∼ 3.0 with approximately 85 per cent porosity .
In this study, we consider the following three well-known fractal aggregation models: ballistic-CCA (BCCA), BPCA, and linear chain, where each model exhibits ( = 1.9, 0 = 1.04), 3), respectively (e.g., Tazaki et al. 2016;Sorensen & Roberts 1997). The linear chain cluster may be regarded as an analogue of aggregates formed via non-ballistic CCA. The fractal dimensions and fractal prefactors of these aggregates approximately follow the linear anti-correlation: Equation (2) is linear interpolation between = 1.0 and = 3.0 and reproduces the measured 0 value of the BCCA model within the error of 5 per cent.

Scaling properties of geometric cross-sections
In general, the geometric cross-section depends on orientation of a fractal aggregate (see e.g., Figure 1). The cross-section also depends on the formation history of each aggregate. For example, BCCA clusters consisting of monomers of radius 0 exhibit various sizes and shapes. Meanwhile, model calculations of fractal grain growth often require an average cross-section of these aggregates rather than a cross-section measured at a specific orientation and a formation history. Therefore, it is reasonable to take the average of the cross-sections across various orientations for each aggregate and an ensemble of aggregates with different formation histories. Let denote this averaged geometric cross-section. Hereafter, we only focus on how the average cross-section is expressed as a function of 0 , , 0 , and . The simplest approach for estimating the cross-section is to employ the characteristic cross-section: It is useful to normalize by 2 0 , where 2 0 is the sum of the cross-sections of individual monomers, and therefore, in general, we have / Equation (4) demonstrates that the characteristic cross-section is problematic when < 2. In this case, / 2 0 increases with and eventually exceeds unity, which is clearly unphysical. Minato et al. (2006) measured geometric cross-sections of BCCA and BPCA clusters and derived fitting formulas. For < 16, the two types of clusters exhibit approximately the same cross-section 2 0 = 12.5 −0.315 exp −2.53/ 0.0920 , For BPCA, Equation (6) yields / 2 0 ∝ −0.315 for large . A similar exponent has been derived in previous studies:  derived the exponent of −0.302, and Ossenkopf (1993) found −1/3. These exponents seemingly agree with that predicted by the characteristic cross-section, −1/3 for = 3 in Equation (4).
Therefore, according to the numerical measurements, it is expected that the geometric cross-sections of fractal dust aggregates with sufficiently large scale as At this point, this scaling law is purely empirical. However, our physically motivated analytic expressions, derived in Section 3, provide a clear justification for this scaling law.

ANALYTIC EXPRESSIONS FOR GEOMETRIC CROSS-SECTIONS
We derive analytic expressions for geometric cross-sections of fractal aggregates.

Geometric cross-section of a distribution of monomers
We start with the simplest case, namely, the case of two spheres ( Figure 2a). If the two spheres overlap along a light ray direction, one sphere (the th sphere) casts a shadow onto the other one (the th sphere). By defining as the overlapping area of the two spheres in projection onto the plane perpendicular to the light direction, it is thus given by where = /2 0 , = sin , and and denote the distance and angle between the two spheres, respectively (see Figure 2). For the two spheres to overlap, 0 1 is required. Next, we derive an analytic expression for the geometric crosssection of an aggregate of monomers. Although we will derive this expression in a more general way later (Section 4), below, we first motivate the expression by deriving it for a simple case without In this case, the -th monomer overlaps with four monomers.
represents the unshaded area of the -th monomer. The shaded region indicated by represents the area of the overlapping region of the spheres and . multiple overlaps of monomers along a light ray and then proceed to demonstrate its general applicability.
If ≠ 2 0 and monomer pair distribution can be regarded as isotropic, multiple overlaps would be sub-dominant, and therefore, every overlapping region on a monomer is likely to be isolated each other ( Figure 2b). In this case, the unshaded area of the -th monomer is By using , the geometric cross-section of an aggregate may be approximated by where we have neglected a contribution of the overlapping regions to the cross-section. To evaluate ≠ in Equation (9), we consider the average overlapping area across all pairs = and assume that the overlap occurring in each pair is equal to the average value: = . We further introduce a normalized version of the average overlapping area˜= / 2 0 , which is referred to as the overlapping efficiency in this paper. With these assumptions and definitions, Equations (9 and 10) become Equation (11) is valid only when ( − 1)˜ 1, and hence, it may be rewritten as Equation (12) might be applicable even when multiple overlaps are dominant, i.e., ( − 1)˜> 1. For example, in the opposite extreme case for fully overlapping monomers in a linear chain cluster (˜= 1), Equation (12) yields a correct geometric cross-section = 2 0 regardless of a value of . Thus, this equation seems to capture the cross-section in both limiting cases. Equation (12) can also be justified in terms of a light scattering theory of aggregates. In Section 4, we obtain an identical expression to Equation (12) by considering the geometrical optics limit of the mean-field light scattering theory (Berry & Percival 1986;). Thus, we use Equation (12) as a basic equation to compute geometric cross-sections of aggregates.
To find˜, we employ a statistical distribution model of monomers in fractal aggregates, namely, the two-point correlation function (e.g., Meakin 1991;Tazaki et al. 2016;Tazaki & Tanaka 2018). If the distribution of monomer pairs is isotropic, the correlation function depends only on the relative distance of a pair . In this case, the form of the correlation function for fractal aggregates is (e.g., where ( ) yields the probability of finding a pair of monomers separated by a distance between and + . Equation (13) is normalized such that ∫ ∞ 0 4 2 ( ) = 1. Although Equation (13) is a general expression describing fractal aggregates (e.g., Meakin 1991), the form of the cut-off function (Equation 14) is non-trivial. This study adopts a model with the cutoff power of as a fiducial case, as suggested by . Tazaki & Tanaka (2018) confirmed that this cutoff model is an appropriate choice for BCCA and BPCA. Another possible choice is a model with the cut-off power of 2 (e.g., Tazaki et al. 2016), which provides results that are only slightly less accurate than those of the model of . However, a model with the cutoff power of (Berry & Percival 1986) is significantly inaccurate compared to the other models. The detailed comparison between these three cut-off models is presented in Appendix B.
Employing the two-point correlation function, the average overlapping area is given by By replacing the integration variables from ( , ) to ( , ), the overlapping efficiency becomes A further analytical reduction of Equation (16) is possible by assuming 1, i.e., the distance between two monomers is larger than the monomer diameter. In this case, the integration with respect to is approximated as where we used 1, since 1. In general, this approximation tends to be inaccurate for lower . However, even for such cases, this approximation only slightly affects accuracy of , since these aggregates tend to have lower overlapping efficiency (i.e.,˜ 1), and then the error of˜weakly affects a resultant value of (see Section 3.4).
Hence, Equation (16) is reduced tõ By using Equations (13) and (14), we obtaiñ When 2, numerical integration is necessary to evaluate˜. However, when 2 < 3, we have a more convenient expression: where Γ is the Gamma function, and is the incomplete Gamma function defined by where > 0 (Abramowitz & Stegun 1972). ( , ) can be efficiently computed without numerical integration (Press et al. 1992). This is the main practical reason for using the incomplete Gamma function rather than performing the numerical integration. Consequently, the analytic expressions for the overlapping efficiency are expressed as

Connection to small non-fractal cluster limit
The fractal scaling law (Equation 1) breaks down when is small. In such small clusters, geometric cross-sections would be almost independent of their clustering processes, and therefore the formal fractal dimension. In fact, Minato et al. (2006) employed the same fitting formula for the cross-sections of BCCA and BPCA clusters when < 16 (see also Mukai et al. 1992;Ossenkopf 1993). In contrast, our analytic expression above was derived on the premise that the fractal scaling law is valid even for a small number of monomers, and therefore, it depends on even for the small cluster limit.
To reconcile this problem, we adopt a piecewise approach. For < th , where th is a small number (e.g., below approximately ten), it is reasonable to assume that geometric cross-sections are nearly independent of aggregation processes, and hence, we may use Equation (5). Although Minato et al. (2006) practically adopted th = 16 in their fitting formulas, a slight difference between measured cross-sections of BCCA and BPCA clusters can be seen at even smaller aggregates, ∼ 8 Ossenkopf 1993;Minato et al. 2006). Thus, in this study, we adopt th = 8 for BCCA and BPCA. If we consider further lower values, th can be even smaller. For example, The cross-sections of linear chain clusters ( = 1) start to deviate from those predicted by Equation (5) at ∼ 3. Taking this into consideration, we empirically adopt Equation (25) yields th = 8.0 and th = 2.5 for 1.5 and = 1.0, respectively. th for 1.0 < < 1.5 is determined by linear interpolation.

Summary of our analytic expression
Our analytic expressions for the geometric cross-section of a fractal aggregate are summarized to be where is the numerical factor that connects two regimes continuously at = th .˜and th are given by Equations (24) with (21) and Equation (25), respectively. Our formulation does not involve any computationally challenging tasks; thus, the computational time is almost negligible regardless of values of and . The source codes are publicly available at the author's GitHub repository 1 .

Tests of our analytic expression
To test the validity of our analytic expression (Equation 26), we compare it with the fitting formulas for measured cross-sections of BCCA and BPCA clusters (Equations 5 and 6). We also compare it with the exact expression for the cross-section of a linear chain cluster (see Appendix A), which can be written by Figure 3 compares our analytic expression (Equation 26) with measured or exact cross-sections for the three types of fractal dust aggregates. Our expression successfully reproduces the geometric cross-sections for all three types of aggregates. The relative error of our expression is only less than 3 per cent (see also Figure B1). We also compared our expression with a fitting formula for the crosssections of BCCA clusters proposed by Meakin & Donn (1988). As a result, we found our expression agrees with the fitting formula within the error of 4.5 per cent for < 10 8 , where we used = 1.95 and 0 = 1.0 in the comparison (Meakin 1984a   We conducted another test to confirm the validity of the approximation made in Equation (17). Consequently, we found that this approximation causes a relative error below ∼ 1 per cent for 1 3 and 1 10 10 , where we used Equation (2) to find 0 for each value of . Therefore, the approximation is applicable for a wide parameter space.

Scaling properties of our analytic expression
One of the advantages of our analytically derived expression is its applicability to the full range of fractal dimensions. To demonstrate this, Figure 4 shows geometric cross-sections of aggregates obtained by our expression for various values of and . As a general tendency, a higher fractal dimension yields a higher overlapping efficiency, and consequently, a smaller geometric cross-section. For comparison, we plotted the cross-section of non-porous compact spheres, which obey / 2 0 = −1/3 . Although both BPCA clusters and compact spheres have = 3.0, BPCA clusters exhibit significantly larger cross-sections than compact spheres.
Depending on , the slope with respect to changes at sufficiently large . To assess whether our expression satisfies the scaling law expected in Equation (7), we investigate the asymptotic behaviour of the analytic expression. The asymptotic behaviour can be classified into three cases (i) 2 < 3, (ii) = 2, and (iii) < 2.
3.5.1 Case of 2 < 3 For sufficiently large aggregates, 1. Hence, Equation (24) is reduced tõ Thus, the geometric cross-section scales as This scaling property is in harmony with Equation (7).

Case of = 2
In this case, Equation (20) gives rise to the exponential integral. To assess its asymptotic behaviour approximately, we omit the cut-off function in the integrand in Equation (20) Therefore, the geometric cross-section scales as This result is consistent with Equation (7).

Case of < 2
Similar to the case of = 2, we omit the cut-off function to estimate its asymptotic behaviour. In this case, Equation (20) becomes Therefore, the geometric cross-section scales as 2 0 Therefore, this is also consistent with Equation (7).

SHORT-WAVELENGTH LIMIT OF MEAN-FIELD LIGHT SCATTERING SOLUTION
In Section 3, we derived an analytic expression for geometric crosssections of aggregates by considering a simple case without multiple overlaps of monomers. Another possible approach to derive the expression is to take the short-wavelength limit of the mean-field light scattering theory . The meanfield theory successfully reproduces the extinction cross-sections of BCCA and BPCA clusters (Tazaki & Tanaka 2018). Since the extinction cross-section approaches twice the geometric cross-section at a sufficiently short wavelength (e.g., Bohren & Huffman 1983), it can be used to measure the geometric cross-section. In this section, we re-derive Equation (12) by considering the short-wavelength limit of the mean-field theory.

Summary of mean-field equations
Here, we summarize basic equations of the mean-field theory ). The extinction cross-section of an aggregate is given by where is the wavenumber, and (¯( 1, ) are the mean-field scattering coefficients obtained by solving a set of linear equations  (1) 1, (2) 1, 1, 1, = 2 2 + 1 ( + 1) ( + 1) where ( , ) are the scattering coefficients of a spherical monomer obtained by the Lorenz-Mie theory (Bohren & Huffman 1983), stop is the truncation order of the scattering coefficients, is the Legendre polynomial function, and is the associated Legendre function,

+1/2 and
(1) +1/2 are the Bessel and Hankel functions, respectively. We have changed the lower bound of the integration in Equation (41) from zero to 2 0 to make the formulation consistent with that in Section 3.

Solution to mean-field equations in short-wavelength limit
In general, the mean-field scattering coefficients (¯( 1) 1, ,¯( 2) 1, ) cannot be obtained analytically. However, there is an exceptional case that yields an analytic solution. First of all, we impose a wavelength much shorter than both monomer and aggregate radii so that 0 = 0 1 and = 1. Also, since we aim to evaluate geometric crosssections, it is useful to assume perfectly absorbing monomers. With these assumptions, we derive an analytic solution to the extinction cross-section.
For sufficiently large dust aggregates ( 1), the asymptotic forms of the Bessel and Hankel functions yield +1/2 ( ) (1) +1/2 ( ) ∼ ( ) −1 . Thus, Equation (41) where we used Equation (19). Because no longer depends on index ,¯1 , 1, and¯1 , 1, are reduced to a rather simple form For 0 1, the Lorenz-Mie scattering coefficients are decomposed into two components: where the first term represents Fraunhofer diffraction, and the second term represents the sum with respect to Fresnel reflection ( = 0: external reflection, = 1: transmitted light, 2: internally reflected light) (e.g., van de Hulst 1957;Tazaki et al. 2021). From the assumption of perfectly absorbing monomers ( ( ) = ( ) = 0), the scattering coefficients become To highlight the meaning of Equation (47), we consider the extinction cross-section of an isolated monomer: where we used stop ∼ 0 , as guaranteed by the localization principle (van de Hulst 1957). Likewise, it is straightforward to show that the absorption and scattering cross-sections are 2 0 , respectively (e.g., Bohren & Huffman 1983). As a result, the extinction cross-section of an isolated monomer is twice its geometric cross-section, and the half comes from Fraunhofer diffraction and the other half from absorption.
Equation (47) makes the mean-field equations exactly symmetric for the two modes, leading to a solution of¯ (   1) 1, =¯ (   2) 1, . Therefore, where = ( − 1) . Equation (51) can be solved analytically, and we obtain (1) Using stop =1 2 0 and Equations (43) and (52), Equation (34) Equation (53) is independent of the wavenumber , indicating that the expression is in the geometrical optics limit. Since we assumed a perfectly absorbing aggregate, we can anticipate ext = 2 as a consequence of diffraction and absorption. Therefore, the geometric cross-section is given by Equation (54) is identical to Equation (12). Consequently, our analytic expression for geometric cross-sections is identical to the half of the extinction cross-section in the mean-field theory in the geometrical optics limit.

Application to geometric cross-sections of QBCCA clusters
OTS09 proposed another type of aggregation called quasi-BCCA (QBCCA). Here, we calculate geometric cross-sections of QBCCA clusters by extending our analytic expression to treat its inhomogeneous structure, and then, test its validity by comparing the measured cross-sections presented in OTS09.
The structure of a QBCCA cluster is specified by a parameter (0 < 1), which determines a mass ratio of the two aggregates in each collision. QBCCA of = 1 corresponds to BCCA. For a given value of , QBCCA is identical to BPCA when 1.5/ . On the other hand, when > 1.5/ , a QBCCA cluster exhibits a fractal dimension of ( ) (QBCCA regime). The fractal dimension increases with decreasing , i.e., ( ) 1.92 and 2.05 for = 0.325 and 0.05, respectively. Thus, a QBCCA cluster has inhomogeneous structure: the small scale structure is relatively dense, whereas the large scale structure is relatively fluffy.
Our analytic expression (Equation 26) is based on a homogeneous fractal cluster, and hence, it is not directly applicable to QBCCA clusters. To mimic the inhomogeneous structure, we adopt the following expressions where 1 and 2 are numerical factors to connect each regime continuously, and˜( BPCA) is the overlapping efficiency of BPCA. The first and second expressions are identical to Equation (26) for BPCA. The third expression represents the cross-section at the QBCCA regime. If 1.5/ th , we directly connect the first and third expressions at = th . We consider four QBCCA models: = 0.325, 0.1, 0.05, 0.01. We first determine the fractal dimension ( ) by fitting the radius of gyration of each model using ∝ 1/ ( ) at the QBCCA regime, and then, the fractal prefactor by using Equation (2) for a given value of ( ). We do not use a value of 0 directly measured from Equation (1) because this value is affected by the small-scale dense structure and is not a good indicator of the overlapping efficiency of the QBCCA regime. Figure 5 compares geometric cross-sections of the QBCCA clusters obtained by Equation (55) and numerically measured by OTS09. We find excellent agreement between the two results. For 0.05, the relative error is less than 2.1 per cent. For = 0.01, the measured cross-sections show oscillatory behaviour; nevertheless, our expression reproduces these cross-sections within the error of only 5.7 per cent.
The excellent agreement in Figure 5 demonstrates that our formulation is valid for a fractal dimension between 1.9 (BCCA) and 3.0 (BPCA). Also, our analytic expression can be successfully extended to the case of an inhomogeneous fractal cluster by employing a simple prescription, as we employed Equation (55) for QBCCA clusters.

Comparison with commonly used estimates of cross-sections
Model calculations of fractal grain growth often adopt various estimates of geometric cross-sections of fractal aggregates (e.g., Cabane et al. 1993;Wolf & Toon 2010;Okuzumi et al. 2012;Krĳt et al. 2015;). Here, we discuss how our analytically derived expression (Equation 26) differs from various empirical formulas in the literature.

Characteristic cross-sections
Characteristic cross-sections (Equation 3), or the almost similar expression 2 , have been frequently used in the literature to estimate geometric cross-sections (e.g., Cabane et al. 1993;Wolf & Toon 2010;. As mentioned in Section 2.2, Equation (3) becomes unphysical when < 2. Thus, we adopt a simple prescription: Figure 6 compares our analytic expression (Equation 26) with Equation (56), and Figure 7 shows the relative error between them. Equation (56) shows moderate agreement with our analytic expression when = 3.0. To compare the two results, we introduce the area-equivalent radius: = √︁ / . At = 10 3 , the area-equivalent radius differs from the characteristic radius by only 4 per cent, which is quantitatively consistent with measurements reported in OTS09. However, the difference increases with increasing . For sufficiently large , using Equation (28), we have 2 1+ 3/5 where we substituted 0.917 for BPCA. Therefore, the areaequivalent radius of BPCA is approximately 14 per cent larger than the characteristic radius at large , and then a relative error of becomes ∼ 23 per cent.
The relative error of Equation (56) increases with decreasing for 2 < 3. In this range of fractal dimensions, both our expression and Equation (56) fulfill the scaling law (Equation 7); however, Equation (56) significantly overestimates the cross-sections as approaches 2.0. For = 2.0, the relative error of the characteristic cross-section exceeds 100 per cent at ∼ 10 3 , and the error monotonically increases with rising . This is because Equation (56) has a constant value, whereas our expression has logarithmic dependence (Section 3.5.2). For < 2, the relative error decreases with decreasing at large . For = 1.0, the relative error is approximately 18 per cent at large . Therefore, the characteristic cross-section contains the relative error of ∼ 20 per cent when = 1.0 and = 3.0, whereas the error is particularly pronounced at ∼ 2 at large .  realized that fluffy aggregates satisfy the following empirical relation:

Paszun & Dominik (2009)
where out is the outermost radii of aggregates. This relation is valid when out / > 1.2, indicating that it is not applicable to compact aggregates, i.e., higher values of .

OTS09
OTS09 proposed another empirical formula, which provides a better fit to measured cross-sections of QBCCA clusters than Equation (58). This formula has been widely used in model calculations of fractal grain growth in protoplanetary discs and planetary atmospheres (e.g., Okuzumi et al. 2012;Krĳt et al. 2016;. The empirical formula is given by where BCCA is the cross-section of a BCCA cluster, obtained from the fitting formulas in Minato et al. (2006) (Equations 5 and 6), and ,BCCA is the characteristic radius of the BCCA cluster. At sufficiently large , Equation (59) yields BCCA in the BCCA limit and 2 in the BPCA limit. Because Equation (59) sometimes yields > 2 0 at small , we set = 2 0 in such cases. We compare our analytic expression with the empirical formula (Equation 59) in Figure 6. We also show the relative error between them in Figure 7. As an overall tendency, the empirical formula agrees with our analytic expression within the error of 40 per cent for small aggregates ( < 10 6 ) of an arbitrary fractal dimension. In contrast, the error tends to increase for larger aggregates ( > 10 6 ) particularly at ∼ 1 and ∼ 2.2. The two methods are in good agreement for = 1.9, since both of them can reproduce the crosssections of BCCA clusters.
Main differences between the empirical formula (Equation 59) and our analytic expression are as follows.

•
= 1.1: The empirical formula underestimates a cross-section when 3 and 500. At large , the relative error gradually increases with rising and exceeds 20 per cent for 10 4 , as the term BCCA tends to govern the reciprocal sum.
• = 2.2: The empirical formula underestimates a cross-section by at most about 11 per cent when 10 4 . A similar underesti-mation has also been reported in OTS09 for QBCCA clusters. When 10 4 , the empirical formula overestimates a cross-section because it does not follow the scaling law (Equation 7).
• = 2.5: The empirical formula underestimates a cross-section by at most about 17 per cent when ∼ 10 2 −10 3 . A similar difference has been reported in Suyama et al. (2012) for compressed aggregates ( 2.5) at ∼ 10 2 − 10 3 . • = 3.0: Since the empirical formula approaches 2 at large , the relative error is almost the same as that of the characteristic cross-section (Section 5.2.1). However, the empirical formula is slightly less accurate than 2 at small aggregates ( ∼ 10 3 ), i.e., the relative error is approximately 28 per cent at ∼ 5 × 10 3 . This is because the reciprocal sum in the formula does not accurately yield ∼ 2 for such a small , which causes larger errors than 2 .
In summary, Equation (59) tends to produce better results among other empirical formulas; however, it leads to significant errors for very large aggregates ( 10 6 ). Also, Equation (59) fails to reproduce the scaling law at ∼ 2.2.

SUMMARY
We derived an analytic expression for geometric cross-sections of fractal dust aggregates by applying a statistical distribution model of monomers. Our main results are as follows.
• Our analytic expression (Equation 26) successfully reproduces the cross-sections of three types of fractal aggregates (the linear chain, BCCA, and BPCA clusters) with a relative error below 3 per cent. Furthermore, our expression naturally reproduces the scaling law (Equation 7) at sufficiently large .
• The analytic expression is shown to be identical to an expression predicted by the mean-field light scattering theory in the shortwavelength limit. Therefore, our formulation for geometric crosssections is compatible with the mean-field theory.
• We extended our analytic expression to calculate the crosssections of QBCCA clusters, which exhibit inhomogeneous structure. The cross-sections obtained by the extended expression show 10 0 10 2 10 4 10 6 10 8 10 10 10 12 N  excellent agreement with the measured cross-sections, where the error is only less than 5.7 per cent. This agreement suggests that our formulation is valid even for a case of a fractal dimension between 1.9 (BCCA) and 3.0 (BPCA).
• While the empirical formula in OTS09 is the best among various fitting formulas in the literature, it leads to significant errors for very large aggregates ( 10 6 ) and fails to reproduce the scaling law (Equation 7).
Our analytic expression comes at a low computational cost and yields better accuracy than the empirical formulas proposed in previous studies. Therefore, it is useful in model calculations of fractal grain growth in protoplanetary discs and planetary atmospheres.

APPENDIX A: GEOMETRIC CROSS-SECTIONS OF LINEAR CHAIN CLUSTERS ( = 1)
Assuming a linear chain cluster consisting of monomers and denoting the angle between a light ray and the linear chain cluster by Θ, we calculate the shadow area cast by this cluster onto the plane perpendicular to the light ray direction. To calculate the shadow area, we consider a rectangle circumscribing the shadow in the projection plane. Each length of the rectangle will be 2 0 and 2 0 [1 + ( − 1) sin Θ], and hence, the area of the rectangle is given by 4 2 0 [1 + ( − 1) sin Θ]. The shadow area is obtained by subtracting the area of the marginal regions between the rectangle and the shadow from that of the rectangle. Thus, the shadow area (Θ) is where we employed Equation (8)  (A3)

APPENDIX B: EFFECT OF CUT-OFF FUNCTIONS OF CORRELATION FUNCTION
In Section 3.1, we derived geometric cross-sections of aggregates by assuming a form of the two-point correlation function. Because an aggregate has a finite radius, the two-point correlation function must have a cut-off function at a length scale comparable to the aggregate radius. The cut-off function has been commonly assumed to have the following form ( / ) ∝ exp[−( / ) ], where represents the cut-off power. In our fiducial model, we adopted = (see Section 3.1) in accordance with previous studies Tazaki & Tanaka 2018). Here, we investigate how different values of affect calculations of geometric cross-sections. Berry & Percival (1986) proposed the exponential cut-off model ( = 1), given by A similar analysis given in Section 3.1 yields the following analytic expressions for the overlapping efficiency: where 1 = 2( 0 / ) [ ( + 1)/2] 1/2 . Another cut-off model is the Gaussian cut-off model ( = 2) (e.g., Sorensen, Cai, & Lu 1992;Tazaki et al. 2016), which has the form of In this case, the overlapping efficiency can be expressed as where 2 = ( 0 / ) 2 .  Figure B1. Comparison between three cut-off models: geometric cross-sections of fractal aggregates (left) and relative errors of our analytic expression (right). Dotted, dashed, and solid lines represent the results for =1, 2, and , respectively. The red, blue, and green lines represent the results for the linear chain, BCCA, and BPCA clusters, respectively. The grey solid lines in the left panel represent numerically measured or exact cross-sections of BCCA, BPCA, and linear chain clusters (Equations 5, 6, 27). Figure B1 compares geometric cross-sections obtained by three different cut-off models. For our fiducial model ( = ), the geometric cross-sections agree with the numerically measured and exact cross-sections (Equations 5, 6, 27) within the error of 3 per cent. For the Gaussian cut-off model ( = 2), the relative error is below ∼ 5 per cent. However, for the exponential cut-off model ( = 1), the relative error is above 30 per cent. Therefore, we do not recommend to use = 1. These findings are consistent with the conclusion derived in Tazaki & Tanaka (2018), where the authors compared extinction cross-sections of BCCA and BPCA clusters obtained by a rigorous light scattering simulation with those obtained by the mean-field light scattering theory. This paper has been typeset from a T E X/L A T E X file prepared by the author.