### Polytropic Process  
The polytropic process describes the transition of a fluid from one thermodynamic state to another through the relationship between the fluid density and temperature (or pressure [10,11]). Through the polytropic equation, we achieve a closure between the higher order moments and the plasma density. In terms of plasma density \(n\) and temperature \(T\), the polytropic equation is:  
\[T=Cn^{\gamma-1}, \tag{5}\]  
where C is a constant characterizing specific streamlines, and \(\gamma\) is the polytropic index. The value of \(\gamma\) is indicative for the energy transfer during the process. The determination of \(\gamma\) in several applications is useful in understanding the physical mechanisms in the system. We take the logarithm of Equation (5):  
\[\log(T)=(\gamma-1)\log(n)+\log C, \tag{6}\]  
so that \(\gamma\) is determined by the slope of the linear relationship between \(\log(T)\) and \(\log(n)\). The polytropic index is a useful tool to study dynamic plasma fluctuations [12] and describes the heating/cooling of the plasma without the necessity to solve the energy equation, which can be very complicated [13]. Several studies analyze plasma observations and determine the polytropic index of different species in several regions through the heliosphere. For instance, [14] determine the polytropic index of plasma electrons in the Saturnian magnetosphere from Cassini observations, while [15] derive the polytropic index of energetic plasma ions in the magnetosphere of Saturn, observed by the same spacecraft. The studies by [16; 17] use a polytropic model to characterize the temperature and density fluctuations in the distant Jovian magnetosheath and boundary layer. In other examples, [18; 19] determine the polytropic expansion of the solar wind protons in the inner heliosphere, while [20; 21; 22] calculate \(\gamma\) of protons at \(\sim\)1 au. [PERSON] et al. [23] calculate \(\gamma\) of solar wind protons through the outer heliosphere, and [24] determines the polytropic process in the inner heliosheath. Finally, [25] discuss the radial profile of the polytropic index within the heliosphere. As mentioned in the previous subsection, in situ plasma measurements cannot always determine the full temperature tensor. Therefore, we would like to determine the polytropic relationship as a function of the temperature in a single direction. For simplicity, we investigate the polytropic relationship with \(T_{\perp}\) and \(T_{\parallel}\), although we could generalize for any direction. By combining Equations (3), (4) and (6), we can express the polytropic equation in terms of \(T_{\perp}\) and \(\alpha\):  
\[T_{\perp}=C\bigg{(}\frac{3\alpha}{2\alpha+1}\bigg{)}n^{\gamma-1}\Rightarrow \log(T_{\perp})=\log C+\log\bigg{(}\frac{3\alpha}{2\alpha+1}\bigg{)}+(\gamma -1)\log(n), \tag{7}\]  
or in terms of \(T_{\parallel}\) and \(\alpha\):  
\[T_{\parallel}=C\bigg{(}\frac{3}{2\alpha+1}\bigg{)}n^{\gamma-1}\Rightarrow \log\Big{(}T_{\parallel}\Big{)}=\log C+\log\bigg{(}\frac{3}{2\alpha+1}\bigg{)} +(\gamma-1)\log(n). \tag{8}\]  
According to Equations (7) and (8), we can determine a linear relationship between either \(\log(T_{\perp})\) and \(\log(n)\), or \(\log\Big{(}T_{\parallel}\Big{)}\) and \(\log(n)\), only if \(\alpha\) does not vary within the examined interval. In both cases, the slope of the linear model defines \(\gamma\). In other words, if we use a temperature element along one direction only and we do not know the temperature anisotropy, we can derive the polytropic index only in intervals with nearly constant temperature anisotropy. This paper examines the polytropic behavior of solar wind protons in an annual interval, while a future study will expand the analysis in a dataset covering more than 25 years of observations. Our selected time period shows clear evidence of anisotropic heating within stream-interaction regions.