## 4 Performance Evaluation of the Active Support System for the 4m SiC Primary Mirror
### Correction of the Primary Mirror Surface Shape Errors Caused by Thermal and Wind Loads
#### 4.4.1 Thermal Load  
SiC has a high specific stiffness and good thermal stability, making it an ideal mirror material. The main indexes of RB SiC (reaction-bonded silicon carbide) are as follows: density 3050 kg/m\({}^{3}\), [PERSON]'s modulus 392,000 MPa, Poisson's ratio 0.25, thermal expansion coefficient 2.5 \(\times\) 10\({}^{-6}\)/K, and thermal conductivity 185 W/(m\(\cdot\)K) [30]. Due to SiC having a large thermal expansion coefficient, a primary mirror composed of SiC is sensitive to changes in the environment's temperature, and analyzing this thermal deformation is necessary [31; 32]. The designed active support system could actively correct the thermal deformation of the 4m SiC primary mirror to reduce the deformation of the mirror surface and improve the imaging quality of the telescope. Regarding the thermal load of the 4m SiC primary mirror, three representative temperature cases were considered: the axial, radial, and transverse temperature differences of the primary mirror. Figure 18 shows the temperature fields in the three situations for a temperature difference of 0.2 \({}^{\circ}\)C. The three temperature fields were applied to a 4m SiC primary mirror; the deformation of the mirror surface is shown in Figure 19.  
Figure 17: Zernike distributions of surface before and after correction (elevation angle of 90\({}^{\circ}\)). Figure 18: Three temperature fields corresponding to three types of temperature differences. (**a**) Temperature field of 0.2 \({}^{\circ}\)C axial temperature difference; (**b**) Temperature field of 0.2 \({}^{\circ}\)C radial temperature difference; (**c**) Temperature field of 0.2 \({}^{\circ}\)C transverse temperature difference. The first 30 orders of bending modes were used to perform an active optics correction on the above surface shape, and the results are shown in Figure 20. The rms values of the primary mirror surface shape before and after the correction are listed in Table 4. Under the axial temperature difference of 0.2 \({}^{\circ}\)C, the rms decreased from 1026.7 to 14.9 nm, indicating a significant correction effect. The rms of the other two cases also decreased significantly after the correction. The designed support system had a good correction effect on the thermal deformation of the primary mirror. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Temperature Field**} & \multicolumn{2}{c}{**Primary Mirror Surface Shape rms/nm**} & \multicolumn{2}{c}{**Amplitude of the**} & \multicolumn{2}{c}{**Maximum Von Mises Stress**} \\ \cline{2-5}  & **Before Calibration** & **After Correction** & **Correction Force/N** & **of the Primary Mirror/MPa** \\ \hline Axial temperature & 1026.7 & 14.9 & 303.8 & 0.34 \\ difference of 0.2 \({}^{\circ}\)C & 196.2 & 16.4 & 60.7 & 0.15 \\ Transverse temperature & 58.7 & 17.7 & 125.3 & 0.15 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Relevant data of the primary mirror surface shape before and after correction for thermal load deformation. Figure 19: The 4m SiC primary mirror surface shape caused by three types of temperature differences. (**a**) Axial temperature difference of 0.2 \({}^{\circ}\)C; (**b**) Radial temperature difference of 0.2 \({}^{\circ}\)C; (**c**) Transverse temperature difference of 0.2 \({}^{\circ}\)C.  
Figure 20: Results of correcting the thermal load deformation of the primary mirror using the first 30 orders of bending modes. (**a**) Axial temperature difference of 0.2 \({}^{\circ}\)C; (**b**) Radial temperature difference of 0.2 \({}^{\circ}\)C; (**c**) Transverse temperature difference of 0.2 \({}^{\circ}\)C.  
The Zernike polynomial was used to fit the surface shape (axial temperature difference of 0.2 \({}^{\circ}\)C) before and after the correction, as shown in Figure 21. The surface correction eliminated the defocus and spherical aberration. In addition, we performed surface corrections for the other temperature differences.