Appendix A: Definitions

Radius definitions

Because of its complex shapes, there are several ways to define the size of an aggregate. We will use two definitions: (i) the volume-equivalent radius \(a_\mathrm{V}\) and (ii) the characteristic radius \(a_\mathrm{c}\). The former and latter definitions are, so-to-speak, measuring the mass and the apparent size of an aggregate, respectively.

The volume-equivalent radius is the radius of a sphere formed by squashing an aggregate while retaining the material volume. Since the volume of the dust particle is the same before and after the squashing process, we have

\[\frac{4}{3}\pi a_\mathrm{V}^3=N \frac{4}{3}\pi a_\mathrm{mon}^3,\]

where \(a_\mathrm{mon}\) is the monomer radius and \(N\) is the number of monomers. We have also assumed all monomers in the aggregate are identical and spherical. From this, we obtain

\[a_\mathrm{V}=a_\mathrm{mon}N^{1/3}.\]

It is obvious however that an apparent size of an aggregate could be much larger than the volume-equivalent radius, particularly when the porosity is high. We thereby introduce the second definition for the aggregate radius: the characteristic radius \(a_\mathrm{c}\) (Mukai et al. 1992, Kozasa et al. 1992):

\[a_\mathrm{c} = \sqrt{\frac{5}{3}}a_\mathrm{g},\]

where \(a_\mathrm{g}\) is the radius of gyration defined by

\[\begin{split}a_\mathrm{g} = \sqrt{\sum_{i=1}^{N} \frac{ \\|\vec{r_i}-\vec{r_\mathrm{M}}\\|^2 }{N}},\end{split}\]

where \(\vec{r_i}\) is the position vector of \(i\)-th monomer, and \(\vec{r_\mathrm{M}}\) is the position vector of the center of mass of the aggregate. The radius of gyration therefore represents a standard deviation of the distance of monomers with respect to the center of mass.

Fractal dimension, prefactor, and porosity

By using the above two radius definitions, we define the porosity of an aggregate by

\[\mathcal{P} = 1 - \left(\frac{a_\mathrm{V}}{a_\mathrm{c}}\right)^3.\]

Since both \(a_\mathrm{c}\) and \(a_\mathrm{V}\) are proportional to the monomer radius, the porosity is independent of the monomer radius.

It is also useful to introduce the fractal dimension \(D_\mathrm{f}\) defined as follows:

\[N = k_\mathrm{f}\left(\frac{a_\mathrm{g}}{a_\mathrm{mon}}\right)^{D_\mathrm{f}},\]

where \(k_\mathrm{f}\) is the fractal prefactor.

The fractal dimension is a quantity that characterizes the geometric shape of an aggregate. For example, an aggregate with a fractal dimension of 3 has monomers distributed in such a way that the aggregate’s mass grows with the cube of its radius of gyration, similar to an elementary scaling for a single spherical particle. In this case, the aggregate tends to have a compact structure. A fractal dimension may also be less than 3. The lower the fractal dimension is, the more open the aggregate structure becomes. For example, an aggregate consisting of monomers lined up in a straight chain would have a volume proportional to its radius, resulting in a fractal dimension of 1.

The fractal prefactor \(k_\mathrm{f}\) may be regarded as a packing fraction in \(D_\mathrm{f}\) dimensional space. Its meaning becomes intuitive if we consider the case of \(D_\mathrm{f}=3\). In this case, the above fractal scaling law gives

\[\left(\frac{a_\mathrm{V}}{a_\mathrm{c}}\right)^3=\left(\frac{3}{5}\right)^{3/2}k_\mathrm{f}\approx0.465 k_\mathrm{f} ~~(D_\mathrm{f}=3)\]

By substituting this expression into the definition of porosity, we obtain

\[\mathcal{P}=1-0.465 k_\mathrm{f}~~(D_\mathrm{f}=3)\]

In the three dimensional space, the lowest porosity of a cluster of (hard) monodisperse spheres is known to be about 0.26, therefore, we expect \(k_\mathrm{f}\lesssim1.59\) for \(D_\mathrm{f}=3.0\).

Definitions of scattering properties

Notations

The definition of the optical properties written in a file dustkapscatmat_XXX.inp follows the definitions of the ones used in RADMC-3D (see also the manual of RADMC-3D).

Suppose we observe scattered light from a collection of identical dust particles and the dust particles are randomly orientated with equal probability, then the scattering matrix, which relates the incident and scattered Stokes parameters, will have 6 independent elements:

\[\begin{split}\begin{pmatrix} I_\mathrm{sca} \\ Q_\mathrm{sca} \\ U_\mathrm{sca} \\ V_\mathrm{sca} \\ \end{pmatrix} = \frac{m_\mathrm{dust}}{d^2} \begin{pmatrix} Z_{11} & Z_{12} & 0 & 0 \\ Z_{12} & Z_{22} & 0 & 0 \\ 0 & 0 & Z_{33} & Z_{34} \\ 0 & 0 & -Z_{34} & Z_{44} \\ \end{pmatrix} \begin{pmatrix} I_\mathrm{inc} \\ Q_\mathrm{inc} \\ U_\mathrm{inc} \\ V_\mathrm{inc} \\ \end{pmatrix},\end{split}\]

where \(d\) is the distance from a dust particle to the observer, \(m_\mathrm{dust}\) is the mass of a dust particle (\(\mathrm{g}\)), \(Z_{ij}\) is scattering matrix elements. If each dust particle is not spherically symmetric, we have \(Z_{11}\neq Z_{22}\) and \(Z_{33}\neq Z_{44}\).

The scattering matrix elements \(Z_{ij}\) (\(\mathrm{cm}^2/\mathrm{g}/\mathrm{str}\)) are related to the traditional notation via:

\[Z_{ij}=\frac{S_{ij}^\mathrm{BohrenHuffman}}{k^2m_\mathrm{dust}},\]

where \(k\) is the wave number (\(\mathrm{cm}^{-1}\)), and \(S_{ij}\) are the scattering matrix elements defined by Bohren & Huffman (1983). Note that RADMC-3D adopts the IAU 1974 definition for the Stokes vectors, and it is different from what Bohren & Huffman’s book follows. As a result, we need a minus sign for the \(Z_{13}\), \(Z_{14}\), \(Z_{23}\), \(Z_{24}\), \(Z_{31}\), \(Z_{41}\), \(Z_{32}\), \(Z_{42}\) elements because of the opposite \(U\) and \(V\) sign conventions (But, these elements are zero for our cases).

The integration of \(Z_{11}\) over all solid angles gives the scattering opacity (\(\mathrm{cm}^2/\mathrm{g}\))

\[\kappa_\mathrm{sca}(\lambda)=\int_{4\pi} Z_{11}(\theta,\lambda)d\Omega.\]

The definition of the phase function is the same as Bohren & Huffman’s one (see Eq. 13.3 in Chapter 13):

\[p(\theta)\equiv\frac{1}{C_\mathrm{sca}}\frac{dC_\mathrm{sca}}{d\Omega}=\frac{1}{C_\mathrm{sca}/m_\mathrm{dust}}\frac{S_{11}}{k^2m_\mathrm{dust}}=\frac{Z_{11}(\theta)}{\kappa_\mathrm{sca}},\]

where \(C_\mathrm{sca}\) is the scattering cross section, and \(dC_\mathrm{sca}/d\Omega\) is the differential scattering cross section. The phase function is therefore normalized such that \(\int_{4\pi} p(\theta)d\Omega=1\). The asymmetry parameter is defined by (p. 72, Section 3.4 in Bohren & Huffman’s book):

\[g \equiv \int_{4\pi} p(\theta) \cos\theta d\Omega=\frac{2\pi}{\kappa_\mathrm{sca}}\int_{-1}^{+1} Z_{11}(\mu)\mu d\mu,\]

where \(\mu=\cos\theta\); \(\theta\) is the scattering angle (\(\theta=0^\circ\) is the forward scattering direction). \(g\) is a non-dimensional quantity.

Size distribution averaging

This package provides the optical properties averaged over a particle-size distribution. The size distribution is assumed to obey a power-law function:

\[n(a_\mathrm{V})da_\mathrm{V}\propto a_\mathrm{V}^{-q}da_\mathrm{V}~(a_\mathrm{min}\le a_\mathrm{V}\le a_\mathrm{max}),\]

where \(a_\mathrm{V}\) is the volume-equivalent radius of a dust particle, and \(n(a_\mathrm{V})da_\mathrm{V}\) is the number density of particles within [\(a_\mathrm{V},a_\mathrm{V}+da_\mathrm{V}\)]. By default, we adopt \(q=3.5\).

The average scattering matrix elements for a collection of various particle sizes can be expressed as the following integration:

\[Z_{ij,\mathrm{ave}}(\theta)=\frac{\int_{a_\mathrm{min}}^{a_\mathrm{max}}a_\mathrm{V}^{3-q}Z_{ij}(a_\mathrm{V},\theta) da_\mathrm{V}}{\int_{a_\mathrm{min}}^{a_\mathrm{max}} a_\mathrm{V}^{3-q}da_\mathrm{V}},\]

where \(Z_{ij,\mathrm{ave}}\) are the scattering matrix elements averaged over the size distribution (Remember that \(Z_{ij}\) are quantities per unit mass). Similarly, the distribution-averaged absorption opacity is calculated by

\[\kappa_\mathrm{abs,ave} = \frac{\int_{a_\mathrm{min}}^{a_\mathrm{max}}a_\mathrm{V}^{3-q}\kappa_\mathrm{abs}(a_\mathrm{V}) da_\mathrm{V}}{\int_{a_\mathrm{min}}^{a_\mathrm{max}} a_\mathrm{V}^{3-q}da_\mathrm{V}}.\]

The averaged scattering opacity and asymmetry parameter are calculated from the averaged value of \(Z_{11,\mathrm{ave}}\):

\[\begin{split}\kappa_\mathrm{sca,ave}=2\pi \int_{-1}^{+1} Z_{11,\mathrm{ave}}(\mu)d\mu,\\ g_\mathrm{ave}=\frac{2\pi}{\kappa_\mathrm{sca,ave}}\int_{-1}^{+1} Z_{11,\mathrm{ave}}(\mu)\mu d\mu.\end{split}\]