## 2 Material and Experimental Methods
### Two-Color Pyrometry Method  
To calculate the temperature distribution on the surface inside the keyhole, we used Thermera-HS software (Phantom Co., Ltd., Tokyo, Japan) with the two-color pyrometry method. The spectral sensitivity of each color sensor corresponding to wavelengths is shown in Figure 4. The spectral sensitivities of the \(R\), \(G\), and \(B\) color sensors were \(Q_{R}(\lambda)\), \(Q_{G}(\lambda)\), and \(Q_{B}(\lambda)\), where \(\lambda\) (m) is wavelength, respectively. Using this camera, we measured three experimental radiation intensities \(I_{R}^{meas}\), \(I_{G}^{meas}\), and \(I_{B}^{meas}\) simultaneously in each experiment. If we assume that the surface along the keyhole wall is a black body, then theoretical radiation intensity ratios of \(I_{R}^{theo}/I_{B}^{theo}\), \(I_{G}^{theo}/I_{B}^{theo}\), and \(I_{R}^{theo}/I_{G}^{theo}\), where each \(I^{theo}\) is a function of temperature \(T\) (K) can be computed as:  
\[I_{R}^{theo}(T)=\sum_{\lambda=0}^{\infty}Q_{R}(\lambda)\times B(\lambda,T) \tag{1}\]  
Figure 4: Experimental setup of temperature distribution measurement.  
Figure 5: Schematic illustration of temperature measurement on the surface along the keyhole wall.  
\[I_{G}^{thno}(T)=\sum_{\lambda=0}^{\infty}Q_{G}(\lambda)\times B(\lambda,T) \tag{2}\]  
\[I_{B}^{thno}(T)=\sum_{\lambda=0}^{\infty}Q_{B}(\lambda)\times B(\lambda,T) \tag{3}\]  
where:  
\[B(\lambda,T)=\frac{2 hc^{2}}{\lambda^{5}}\times\frac{1}{e^{\frac{hc}{4 hg^{7}}}-1} \tag{4}\]  
where \(B(\lambda,T)\) (Ws r\({}^{-1}\) m\({}^{3}\)) is the spectral radiation of the weld pool, \(h=6.63\times 10^{-34}\) m\({}^{2}\)kg\({}^{-1}\)s\({}^{-1}\) is the Planck constant, \(C=3\times 10^{8}\) m s\({}^{-1}\) is the speed of light, \(k_{B}=1.38\times 10^{-23}\) J K\({}^{-1}\) is the Boltzmann constant, and \(T\) (K) is the temperature of the surface along the keyhole wall. From Equations (1)-(3), we obtained the theoretical ratios \(I_{R}^{thno}/I_{B}^{thno},I_{G}^{thno}/I_{B}^{thno}\), and \(I_{R}^{thno}/I_{G}^{thno}\) corresponding to each temperature as described in Figure 6. Comparing the radiation intensity ratios \((I_{R}^{trans}/I_{B}^{trans},I_{G}^{trans}/I_{B}^{trans},\) and \(I_{R}^{trans}/I_{G}^{trans})\) measured the in experiments with the theoretically calculated radiation intensity ratios \((I_{R}^{thno}(T)/I_{B}^{thno}(T),\)\(I_{G}^{thno}(T)/I_{B}^{thno}(T),\) and \(I_{R}^{thno}(T)/I_{G}^{thno}(T))\), the temperatures (\(T_{i}\)) on the surface along keyhole wall were determined. As indicated in Figure 7, only one temperature value exists for each intensity ratio in the case of the \(I_{R}^{thno}(T)/I_{G}^{thno}(T)\) ratio, this ratio was chosen to calculate the temperature in this study.  
A recent paper reported that the two-color pyrometry method, as explained in this paper, can be used to detect the temperature of the weld pool surface for the tungsten inner gas (TIG) welding process with acceptable accuracy [24]. Other papers confirmed that the maximal temperatures on the weld pool surface in KPAW determined using experiments and simulation are nearly equal, with less than 5% deviation (about 1875 K per experiments and 1945 K per simulation) [18; 25]. We used many experiments to confirm the accuracy of this method. The results showed that the average maximal temperature on the weld pool surface in KPAW is about 1875 K, with a deviation of \(\pm 50\) K. From all of above, we found that the uncertainly in the measured temperatures and the possible wavelength variation were slight. Therefore, the measured temperature is reliable for considering the action of Marangoni force.  
Figure 6: Relationship between spectral sensitivities and wavelength.