## 5 Discussion  
We examine the terms of the Bernoulli integral in solar wind protons at \(\sim\)1 au. First, we show that \(\frac{1}{2}v^{2}\) is typically by \(\sim\)2 orders of magnitude larger than \(\frac{1}{2}v_{\text{th}}^{2}\) and \(\frac{1}{2}v_{\text{A}}^{2}\). We then verify that the second term \(\frac{1}{2}v_{\text{th}}^{2}\left(\frac{\gamma}{\gamma-1}\right)\) of the integral for non-isothermal plasmas, may become significant for \(\left|\frac{\gamma}{\gamma-1}\right|>5\), which corresponds to \(\frac{5}{6}<\gamma<\frac{5}{4}\). Moreover, we quantify the relative standard deviation of the Bernoulli integral within short intervals, assuming different \(\gamma\) values. Our results show that the second and third terms of the integral affect the overall deviation of the integral by just a few percent, even for \(\gamma\sim 5/6\) and \(\gamma\sim 5/4\).  
We further quantify the error of the second term as a function of the polytropic index uncertainty. We show that for the range of \(\gamma\) values for which the term becomes significant (\(\left|\frac{\gamma}{\gamma-1}\right|>5\)), the absolute value of its relative uncertainty exceeds 50%, even for small \(\gamma\) uncertainties (e.g., \(\frac{\delta_{\gamma}}{\gamma}=0.1\)). Therefore, in order to maintain a high accuracy in the calculations of the Bernoulli integral, we need to ensure higher accuracy of \(\gamma\) as \(\gamma\to 1\).  
Based on our results, we could potentially drop the less significant terms of the Bernoulli integral without reducing the accuracy of the calculations. In other words, we could ignore the second and the third terms of the integral, or just the second term for a reasonable range of \(\gamma\) close to unity. Note that by excluding the second term, we also reduce the large uncertainties of the integral due to the uncertainty \(\delta_{\gamma}\). Such an approach may be useful when accurate measurements of the plasma density, temperature, or magnetic field are not available.  
The study by [5] derives \(\gamma\) of solar wind protons at 1 au by applying linear fittings to \(\ln(n)\) vs. \(\ln(T)\) observations. They exclude all the intervals for which the linear correlation between \(\ln(n)\) and \(\ln(T)\) is smaller than a threshold value. The correlation filter excludes all cases with \(\gamma\to 1\) (see Figure 3 of [5] and the extensive discussion in [21]). The authors simplify their calculations further, by neglecting the internal energy term and the magnetic term, exactly as in [4]. This simplification is equivalent to using the Bernoulli integral expression for incompressible, non-magnetized plasmas, which is independent of \(\gamma\). Although [22] argue that this approach is not appropriate, we support that when analyzing solar wind protons at 1 au, there is no apparent impact on the results if the cases with \(\gamma\to 1\) are properly treated or excluded from a large statistical sample.  
We also investigate the complete Bernoulli integral, which is a continuous function of \(\gamma\). We examine the significance of the thermal term against the dynamic term as calculated for the solar wind protons at \(\sim\)1 au assuming \(\gamma=5/6\) and \(\gamma=5/4\), respectively. We show that the second term is even less significant in this consideration than it is when we use the non-isothermal expression in Equations (1) and (3). However, we need to emphasize that the significance of the second term can be very large for a range of \(\gamma\) and \(\frac{\rho}{\rho_{*}}\) values. For example, for \(\frac{\rho}{\rho_{*}}=0.1\), the thermal term is \(<-5v_{\rm th}^{2}\) as \(\gamma\to 2\). For completeness, we repeat our calculations in the solar wind protons using \(\gamma\) values determined from linear fittings to \(\ln(n)\) vs. \(\ln(T)\) data points within short subintervals. Our calculations verify that the second term of the integral is insignificant in solar wind protons at \(\sim\)1 au, especially when we exclude intervals with high uncertainty of the derived polytropic index.