## 5 Discussion  
We present analytical expressions of the relationships between \(\log(T)\), \(\log(T_{\perp})\) and \(\log\Bigl{(}T_{\parallel}\Bigr{)}\). According to the analytical expressions, these relationships are linear for constant temperature anisotropy. Any anisotropy variations will affect the ordinate of the linear model. The ordinate of the linear model \(\log(T)\) vs. \(\log(T_{\perp})\) is:  
\[A=\log\biggl{(}\frac{1+2\alpha}{3\alpha}\biggr{)}, \tag{9}\]  
and we estimate its fluctuations, \(\delta A\), due to the anisotropy fluctuations \(\delta\alpha\), by using the error propagation formula:  
\[\delta_{A}=\biggl{|}\frac{\theta A}{\theta\alpha}\biggr{|}\delta_{\alpha}= \frac{\delta_{\alpha}}{\alpha(2\alpha+1)\ln(10)} \tag{10}\]  
Similarly, the ordinate of the linear model \(\log(T)\) vs. \(\log\Bigl{(}T_{\parallel}\Bigr{)}\) is:  
\[B=\log\biggl{(}\frac{1+2\alpha}{3}\biggr{)}, \tag{11}\]  
with fluctuations:  
\[\delta_{B}=\frac{2\delta_{\alpha}}{(2\alpha+1)\ln(10)}. \tag{12}\]  
Finally, the ordinate of the model \(\log(T_{\perp})\) vs. \(\log\Bigl{(}T_{\parallel}\Bigr{)}\) is:  
\[\Gamma=\log(\alpha), \tag{13}\]  
with fluctuations:  
\[\delta_{\Gamma}=\frac{\delta_{\alpha}}{\alpha\ln(10)}. \tag{14}\]  
The proton plasma parameters we analyze here have a typical temperature anisotropy \(\alpha\) -0.8 (see Figure 2). For this anisotropy value, and according to Equations (10), (12) and (14), we find that \(\delta_{A}<\delta_{B}<\delta_{\Gamma}\). This explains why the Pearson correlation coefficient is largest for \(\log(T)\) vs. \(\log(T_{\perp})\) and smallest for \(\log(T_{\perp})\) vs. \(\log\Bigl{(}T_{\parallel}\Bigr{)}\) within the same plasma population we examine. In Section 1, we show that the linear regression to \(\log(T_{\perp})\) vs. \(\log(n)\), or \(\log\Bigl{(}T_{\parallel}\Bigr{)}\) vs. \(\log(n)\) data can derive the plasma polytropic index only if the temperature anisotropy is constant within the analyzed streamlines. Notably, [30] show that there is a specific relationship between the polytropic index and the temperature anisotropy. Here, we refer to general variations in \(a\) that could be statistical variations or measurement errors. If these fluctuations are significant within a streamline, the ordinate in the relationships \(\log(T_{\perp})\) vs. \(\log(n)\), and \(\log\Bigl{(}T_{\parallel}\Bigr{)}\) vs. \(\log(n)\) exhibits significant fluctuations as well, and the relationships are not linear. According to Equations (7) and (8), and the propagation of errors, we find that the ordinate fluctuations in \(\log(T_{\perp})\) vs. \(\log(n)\) are \(\delta_{A}\), exactly as given in Equation (10). The ordinate fluctuations in \(\log\Bigl{(}T_{\parallel}\Bigr{)}\) vs. \(\log(n)\) are \(\delta_{B}\), given by Equation (12). Therefore, we expect that the linear fitting to \(\log(T_{\perp})\) vs. \(\log(n)\) derives the polytropic index more accurately than the linear fitting to \(\log\Bigl{(}T_{\parallel}\Bigr{)}\) vs. \(\log(n)\). The 2D histograms of the plasma temperature and density in Figure 4 verify this expectation, showing that the relationship between \(\log(T_{\perp})\) and \(\log(n)\) has many more similarities to the relationship between \(\log(T)\) and \(\log(n)\) than the relationship between \(\log(T_{\parallel})\) and \(\log(n)\). This is not surprising, considering also the definition of \(T\) in Equation (1) and that the typical anisotropy value is 0.8, meaning that \(T_{\perp}\sim 0.8T_{\parallel}\). All three histograms in Figure 4 show evidence that the large-scale variations of the plasma density and temperature follow a linear model \(\log(T)\propto-2\log(n)\). [PERSON] et al. [31] discuss the specific relationship which is apparent in the ambient solar wind and reflects the differences in the solar wind origins, and that it is not necessarily relevant to the solar wind expansion.