## 2 Error Analysis of the Center of Mass Method
### Sampling Window Truncation Error  
From Equations (21) and (22), we can see that the systematic error in the CoM method will be reduced by increasing the Gaussian width of the star spot when only considering the sampling frequency limitation. Nevertheless, the received starlight energy becomes weaker for a pixel as the Gaussian width increases, which results in deterioration of the SNR of the star image, resulting in greater random error. Therefore, a sampling window is utilized to limit the number of pixels around the actual star spot. When the sampling window size is relatively small, starlight energy falls out the window, and the signal of the detected star may be truncated asymmetrically; this causes a truncation error. Figure 3 illustrates how window truncation leads to error in the centroid calculation process.  
In Figure 3a, it can be seen that the discrete star spot signal function \(e(k)\) detected by CCD pixels occupies about four pixels in the image plane, while the width of the sampling window is only three pixels (Figure 3b); as a result, the discrete signal function \(e(k)\) is truncated asymmetrically when the pixel detected signal is multiplied by the window response function. As shown in Figure 3c, the partial pixel detected signal in \(e(k)\) located outside the sampling window \(r(x)\) will be excluded in calculating the star centroid position. The nonsymmetrical truncation causes the estimated centroid \(\hat{x}_{g}\) to be smaller than the ideal centroid \(x_{0}\), that is, a truncation error is introduced into the final estimated results of the CoM method.  
Here, we analyze the numerical simulation results under Gaussian widths \(\sigma_{PSF}\) of 0.3 and 0.671 pixels to present the influence of the truncation error on the relationship curve of \(\sigma_{\hat{x}_{g}}\) and \(x_{0}\).  
Figure 4 shows the respective pixel detected signals in the three-pixel sampling window when the ideal star centroid positions are \(x_{0}=\) 2, 2.25, 2.5 under Gaussian widths \(\sigma_{PSF}\) = 0.3, 0.671 pixels. We can see that the complete pixel detected signal is included in the sampling window with \(\sigma_{PSF}\) = 0.3 pixels during the process of moving the star centroid from the pixel center to the edge (Figure 4a-c), since the width of the effective pixel detected signal is less than three pixels when \(\sigma_{PSF}\) is at a small value. In this case, the pixel detected signal will not be truncated asymmetrically by the sampling window; thus, the systematic error is associated with the discrete approximation error, and the relationship curve between \(\sigma_{\hat{x}_{g}}\) and \(x_{0}\) is sinusoid (Figure 5a). However, for the pixel detected signal with \(\sigma_{PSF}\) = 0.671 pixels, the width is relatively lager than three pixels. As a result, the detected signal is truncated asymmetrically when it locates at a non-central position of the pixel, as shown in Figure 4e,f. Under this condition, the systematic error is mainly determined by window truncation error, the relationship curve becomes linear (Figure 5b), and the maximum error value appears when the star centroid is located at the pixel edge.  
Figure 3: A truncated pixel detected signal in the centroid estimation process. (**a**) Signal intensity profiles and pixel detected signal (line: signal intensity profile \(e(x)\), bar: pixel detected signal \(e(k)\)). (**b**) The three-pixel sampling window \(r(x)\). (**c**) The windowed pixel signal (dark bar: pixel detected signal in \(r(x)\), light bar: pixel detected signal outside \(r(x)\)).