## 5 Properties of Shape Groups
### Relationships between Microphysical Properties
#### 5.2.2 Area Ratio and Aspect Ratio  
Area ratio versus aspect ratio (\(A_{\rm r}\) vs. \(A_{\rm s}\)) data can be fitted to the linear relationship:  
\[A_{\rm r}=C_{5}+C_{6}\cdot A_{\rm s}, \tag{5}\]  
where \(C_{5}\) and \(C_{6}\) are the fit parameters.  
Figure 6 shows the \(A_{\rm r}\) vs. \(A_{\rm s}\) relationship for each shape group. For comparison, the equality \(A_{\rm r}=A_{\rm s}\) is displayed as a gray dashed line. We note that particles in shape groups (_3_) Thick columns and bullets, (_5_) Plates, (_13_) Ice and melting or sublimating particles,_ and (_15_) _Spherical particles_ are closest to this line. As can be seen in Table 9, these groups, together with group (_1_) _Needles and thin or long columns_, have a good correlation between area and aspect ratio (\(R_{b}^{2}\geq 0.7\)). On the other hand, shape groups (_2_) _Crossed needles and crossed columns, (_6_) _Stellar crystals, (_7_) Bullet rosettes,_ and (_9_) _Side planes_ have a low correlation (\(R_{b}^{2}<0.5\)). For instance, in the case of the shape group (_6_) _Stellar crystals_, this might be due to several factors: the empty space between their branches resulting in low \(A_{\rm r}\) values and/or significant variations in \(A_{\rm s}\) due to particle orientation. Shape groups (_1_) _Needles and thin or long columns, (_3_) _Thick columns and bullets, (_5_) _Plates_, and (_15_) _Spherical particles_ have the highest correlation (\(R_{b}^{2}>0.8\)).  
\begin{table}
\begin{tabular}{l c c c c c c} \hline \hline \multirow{2}{*}{**Shape Groups (1–15)**} & \multicolumn{3}{c}{\(A\) **vs. \(D_{\rm max}\)**} & \multicolumn{3}{c}{\(A_{\rm r}\) **vs. \(A_{\rm s}\)**} \\ \cline{2-7}  & \(C_{1}/10^{-12}\) m\({}^{2}\) & \(C_{2}\) & \(R_{a}^{2}\) & \(C_{5}\) & \(C_{6}\) & \(R_{b}^{2}\) \\ \hline _(1) Needles and thin or long columns_ & 108 & 1.05 & 0.76 & 0.0183 & 0.698 & 0.90 \\ _(2) Crossed needles and crossed columns_ & 128 & 1.05 & 0.73 & 0.0591 & 0.252 & 0.32 \\ _(3) Thick columns and bullets_ & 31.6 & 1.25 & 0.85 & 0.0415 & 0.756 & 0.89 \\ _(4) Capped columns and capped bullets_ & 5.31 & 1.59 & 0.79 & 0.0120 & 0.654 & 0.46 \\ _(5) Plates_ & 3.17 & 1.72 & 0.94 & \(-\)0.00869 & 0.894 & 0.80 \\ _(6) Stellar crystals_ & 0.531 & 1.93 & 0.86 & 0.123 & 0.395 & 0.25 \\ _(7) Bullet rosettes_ & 5.35 & 1.60 & 0.82 & 0.0991 & 0.450 & 0.25 \\ _(8) Branches_ & 2.11 & 1.73 & 0.84 & 0.0845 & 0.483 & 0.47 \\ _(9) Side planes_ & 1.88 & 1.76 & 0.88 & 0.124 & 0.457 & 0.32 \\ _(10) Spatial plates_ & 4.64 & 1.66 & 0.86 & 0.0718 & 0.698 & 0.61 \\ _(11) Spatial stellar crystals_ & 1.88 & 1.78 & 0.95 & 0.0736 & 0.556 & 0.55 \\ _(12) Graupel_ & 0.324 & 1.97 & 0.98 & 0.201 & 0.518 & 0.45 \\ _(13) Ice and melting or sublimating particles_ & 5.87 & 1.54 & 0.91 & 0.0351 & 0.769 & 0.68 \\ _(14) Irregulars and aggregates_ & 3.54 & 1.66 & 0.90 & 0.0158 & 0.648 & 0.52 \\ _(15) Spherical particles_ & 0.795 & 1.99 & 0.99 & \(-\)0.168 & 1.14 & 0.93 \\ \hline \multicolumn{7}{c}{**All shape groups together**} \\ \multicolumn{7}{c}{**(regardless of shape)**} \\ \hline \hline \end{tabular}
\end{table}
Table 9: Area versus particle size (\(A\) vs. \(D_{\rm max}\)) and area ratio versus aspect ratio (\(A_{\rm r}\) vs. \(A_{\rm s}\)) relationships fitted to Equations (3) and (4) for each shape group and for all data, i.e., for all the shape groups regardless of shape. The parameters \(C_{1}\), \(C_{2}\), \(C_{5}\), and \(C_{6}\) and the correlation coefficients \(R_{a}^{2}\) and \(R_{b}^{2}\) are shown.