###### Abstract  
We examine the relationships between temperature tensor elements and their connection to the polytropic equation, which describes the relationship between the plasma scalar temperature and density. We investigate the possibility to determine the plasma polytropic index by fitting the fluctuations of temperature either perpendicular or parallel to the magnetic field. Such an application is particularly useful when the full temperature tensor is not available from the observations. We use solar wind proton observations at ~1 au to calculate the correlations between the temperature tensor elements and the scalar temperature. Our analysis also derives the polytropic equation in selected streamlines of solar wind plasma proton observations that exhibit temperature anisotropies related to stream-interaction regions. We compare the polytropic indices derived by fitting fluctuations of the scalar, perpendicular, and parallel temperatures, respectively. We show that the use of the parallel or perpendicular temperature, instead of the scalar temperature, still accurately derives the true, average polytropic index value, but only for a certain level of temperature anisotropy variability within the analyzed streamlines. The use of the perpendicular temperature leads to more accurate calculations, because its correlation with the scalar temperature is less affected by the anisotropy fluctuations.  
space plasmas; solar wind; thermodynamics 202122 April 2021  
the shocked solar wind interacts with the planet's ionosphere. In another example, [8] examine ARTEMIS ion observations of a long-duration magnetotail exhaust. The authors show that the parallel temperature dominates far away from the x-line, indicating strong parallel ion heating. Near the neutral plane, however, the perpendicular temperature dominates, indicating that ions undergo a Speiser-like motion.  
In gyro-tropic plasmas, the scalar temperature \(T\) is the trace of the temperature tensor:  
\[T=\frac{1}{3}T_{\parallel}+\frac{2}{3}T_{\perp}, \tag{1}\]  
and the temperature anisotropy is the ratio:  
\[\alpha=\frac{T_{\perp}}{T_{\parallel}}. \tag{2}\]  
Unfortunately, in situ observations are not always optimal to construct 3D VDFs or full pitch-angle distributions from which we can determine the full temperature tensor. For instance, in some occasions, Faraday cup observations obtain 1D VDFs from which we can derive the temperature along one direction only [9]. Therefore, here, we would like to explore the correlations between the temperature tensor elements, the scalar temperature, and the temperature anisotropy. These correlations are useful to understand what plasma dynamics information we lose when accurate temperature measurements occur in only one direction. By combining Equations (1) and (2), we express the scalar temperature as a function of \(\alpha\) and \(T_{\perp}\):  
\[T=\frac{1}{3\alpha}T_{\perp}+\frac{2}{3}T_{\perp}=T_{\perp}\left[\frac{1}{3} \bigg{(}\frac{1}{\alpha}+2\bigg{)}\right]\Rightarrow\log T=\log T_{\perp}+ \log\bigg{(}\frac{1+2\alpha}{3\alpha}\bigg{)}, \tag{3}\]  
and as a function of \(\alpha\) and \(T_{\parallel}\):  
\[T=\frac{1}{3}T_{\parallel}+\frac{2}{3}\alpha T_{\parallel}=T_{\parallel} \bigg{(}\frac{1+2\alpha}{3}\bigg{)}\Rightarrow\log T=\log T_{\parallel}+ \log\bigg{(}\frac{1+2\alpha}{3}\bigg{)}. \tag{4}\]  
We would like to examine how the above relationships apply in solar wind proton observations, and how well \(T_{\perp}\) or \(T_{\parallel}\) can be used to study a specific thermodynamic process; the polytropic process.