## 2 Research Method
### Method 3 of 4, Regression Fits for Raeder's Tabulated Results: Integration  
The third method to estimate the temperature-dependent electron-neutral collision cross sections is introduced herein by applying linear regression modeling to published numerical results [42] in the form of tabulated data. The raw results were for 9 gaseous species, covering electron temperature values from 2000 K to 4000 K, with a step of 100 K. The linear regression functions derived herein are based on the part of the raw data corresponding to the temperature range of interest only (between 2000 K and 3000 K) for the 6 covered species in the present study. The choice of linear regression is based on visually observing the trend of the plotted discrete data points (11 points per species), which was reasonably described with a straight line. The suitability of the linear fitting functions was later confirmed by computing the R-squared (\(R^{2}\)) value for each proposed linear regression function as a non-dimensional quantitative measure of the function's ability to reproduce the discrete data points. All the R-squared values were very close to their perfect upper limit of 1 [43; 44; 45], indicating excellent goodness of fit. Each published discrete value of the electron-neutral collision cross sections (at a given electron temperature) for each species is a result of an integration performed for the product of the monoenergetic version of the electron-neutral collision cross sections and a weight function (\(g\)), as follows:  
\[\overline{Q}(T_{e})=\int\limits_{0}^{\infty}Q(v_{e})g(x)dx \tag{25}\]  
The variable (\(x\)) in the above equation is a non-dimensional version of the electron kinetic energy, after being normalized with the product of the Boltzmann constant and the absolute electron temperature (\(k_{B}T_{e}\)), which is called herein the \"thermal energy\" of the electrons. Thus, the non-dimensional variable (\(x\)) is a ratio of two energies (kinetic energy to thermal energy), or  
\[x=\frac{0.5m_{e}v_{e}^{2}}{k_{B}T_{e}}=\frac{u_{e}}{k_{B}T_{e}} \tag{26}\]  
In the Maxwell-Boltzmann probability distribution of thermal speeds, the quantity (\(2\ k_{B}T_{e}/m_{e}\)) is equal to the square of the most-probable speed (\(v_{p}\)), at which this probability distribution has its peak. Thus,  
\[v_{p}^{2}=\frac{2k_{B}T_{e}}{m_{e}} \tag{27}\]  
Or  
\[k_{B}T_{e}=0.5m_{e}v_{p}^{2} \tag{28}\]  
Using Equation (28) in Equation (26), the integration variable (\(x\)) can be viewed as the square of the ratio of the electron speed to its most-probable speed, or  
\[x=\frac{0.5m_{e}v_{e}^{2}}{0.5m_{e}v_{p}^{2}}=\left(\frac{v_{e}}{v_{p}}\right) ^{2} \tag{29}\]  
Alternatively, for a known value of (\(x\)), the corresponding value of \(v_{e}\) can be calculated, provided that the electron temperature is also specified, according to  
\[v_{e}=v_{p}(T_{e})\sqrt{x}=\sqrt{\frac{2k_{B}T_{e}}{m_{e}}}\sqrt{x} \tag{30}\]  
Thus, for a given electron absolute temperature (\(T_{e}\)) that is treated as a fixed parameter when performing the integration in Equation (25), that integration can easily be viewed as being performed with respect to the non-dimensional variable (\(x\)) instead of the dimensional monoenergetic electron speed (\(v_{e}\)), as\[\overline{Q}(T_{e})=\int\limits_{0}^{\infty}Q(x;T_{e})g(x)dx \tag{31}\]  
This means that the electron temperature implicitly affects the integral through \(v_{p}\) or \(x\). The weight function \(g(x)\) has the following form:  
\[g(x)=0.5x^{2}e^{-x} \tag{32}\]  
The value of \(g\) is 0 at \(x=0\). It has a single peak, with a maximum value of 0.27067 at \(x=2\). The value of \(g\) approaches 0 at the limit of \(x\rightarrow\infty\). This means that practically, the integration in Equation (31) can be performed over a finite range of \(x\) up to approximately \(x=10\) (where \(g\) diminishes to 0.00227).