## 3 Results and Discussion
This and other approximations of the standard drag curve [31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42] overestimate subsonic \(C_{D}\) for the \(P=76\) Torr condition because they are purely functions of \(Re\) and do not account for \(Kn\), where \(Kn\) describes the degree of rarefaction by the ratio of the molecular mean free path of the fluid to the diameter of the sphere [1; 23; 30; 43; 44; 45; 46]. The \(Kn\) value for the \(P=76\) Torr condition was 0.18, corresponding to the transitional-flow regime between slip-flow and free-molecule flow [30]. A slip correction factor [30] of \(C=1.45\), given by Equation (6), may be implemented to the standard drag curve approximation in the case of \(P=76\) Torr, where continuum flow is assumed despite the \(Kn\) value indicating rarefied transitional-flow. Slip correction factors are negligible for the \(P=760\) Torr cases, where flow conditions are nearer to continuum flow. By dividing the standard drag curve by 1.45, the approximation is shown in Figure 3 to agree with the empirical results for the \(P=76\) Torr condition at \(M<0.8\). Standard drag curve approximation:  
\[C_{D}=\frac{24}{Re}\Big{[}1+0.1315 Re^{(0.82-0.05\log_{10}Re)}\Big{]}\,\ \ 0.01<Re\leq 20 \tag{4}\]  
\[C_{D}=\frac{24}{Re}\Big{[}1+0.1935 Re^{(0.6305)}\Big{]}\,\ \ 20\leq Re\leq 260 \tag{5}\]  
Figure 3: (**a**) Drag coefficient, \(C_{D}\), vs. Reynolds number, \(Re\), for the three experimental conditions. Data points are colored according to \(M\). Previously published models are provided for comparison. Error bars correspond to a 95% confidence interval (\(\pm 2\) standard deviations). (**b**) Approximate fluid flow characteristics at supersonic, high subsonic, and low subsonic microsphere velocities. Black lines represent the extents of the recirculation region, while blue lines represent streamlines. A detached bow shock is present in the supersonic condition. Steady flow features are depicted because \(Re\) values for nearly all microsphere flights studied in this work fall below the 270–300 range when the onset of unsteady vortex shedding occurs for spheres. Slip correction factor:  
\[C=1+Kn\left[2.514+0.8e^{-0.55/Kn}\right] \tag{6}\]  
Recognizing the difficulty of accurately measuring aerodynamic force empirically under high-\(M\), low-\(Re\) conditions, [PERSON] et al. [23] used direct numerical simulation to model \(C_{D}\) vs. \(Re\) within the ranges 0.3 \(<M<\) 2.0 and 50 \(<Re<\) 300. Of the \(C_{D}\) models considered by the authors, [PERSON]'s (Equation (7)) showed the best agreement with the simulation results in both subsonic and supersonic conditions. Thus, in Figure 3, the Carlson-Hoglund model is provided for the three experimental conditions in the present work. The model was developed to be valid across the subsonic, transonic, and supersonic regimes, although no specific range of validity is given by the authors [47]. It does not require implementation of the slip correction factor, as it accounts for \(Kn\) through the \(M/Re\) proportionality [47]. The model shows reasonable agreement with the \(M<\) 0.8 results from the present work. Carlson-Hoglund model:  
\[C_{D}=\left(\frac{24}{Re}\right)\left[\frac{\left(1+0.15 Re^{0.687}\right)\left( 1+e^{\left(0.427/M^{4.63}\right)-\left(3.0/Re^{0.88}\right)}\right)}{1+\left( M/Re\right)\left(3.82+1.28e^{-1.25 Re/M}\right)}\right] \tag{7}\]  
To assess the model's validity in the supersonic regime, the \(M>\) 1.2 results of the present work are provided in Figure 4 alongside the combined empirical data of [PERSON] [12], [PERSON] and [PERSON] [13], [PERSON] [14], and [PERSON] and [PERSON] [15] within the ranges 1.2 \(<M<\) 4 and \(Re<\) 300. These data, derived from wind tunnel and ballistic experiments, accessed \(Re<\) 300 by testing macro-scale spheres (0.8 \(<D_{\text{P}}<\) 25 mm) in low air pressure environments (\(P<\) 5 Torr) [12; 13; 14; 15]. Using the \(M/\sqrt{Re}\) flow regime metric [30; 48], the data were found to fall within the transitional-flow regime.