## 1 Introduction  
The solar wind is a very dynamic system, with plasma and electromagnetic field fluctuations occurring in a wide range of scales (e.g., [1], and references therein). Plasma and field measurements should be analyzed properly in order to understand the mechanisms that govern solar wind and space plasmas in general. For instance, knowing the velocity distribution functions of the plasma species makes it possible to derive the plasma bulk parameters, which are crucial in investigating mechanisms that involve momentum, heat, and energy transfer. Certain dynamical processes need to be investigated along specific streamlines. If plasma observations are obtained in different streamlines, plasmas with entirely different properties may end up being analyzed, regardless of the mechanisms acting on them. Different streamlines in the solar wind may contain plasmas that originated in vastly different structures of the Sun. Therefore, if we do not analyze specific streamlines separately, the observed plasma variations may reflect the different plasma sources and not the effects of interesting dynamical processes. For example, empirical determination of the plasma polytropic index from the plasma density and temperature fluctuations has meaning only for observations obtained within the same plasma streamline (e.g., [2, 3]). The Bernoulli integral, which is expressed in terms of energy per mass, is conserved along specific streamlines. Therefore, typical analyses can identify plasma streamlines by detecting time intervals for which the standard deviation of the Bernoulli integral is below a predefined threshold value (e.g., [4, 5, 6, 7, 8, 9]). The Bernoulli integral is given by  
\[\frac{1}{2}v^{2}+\frac{\gamma}{\gamma-1}\frac{P}{\rho}+\frac{B^{2}}{2\rho\mu_ {0}}=C\text{ for }\gamma\
eq 1, \tag{1}\]and  
\[\frac{1}{2}v^{2}+\ln\rho\frac{P}{\rho}+\frac{B^{2}}{2\rho\mu_{0}}=C\text{ for }\gamma \
eq 1, \tag{2}\]  
where \(v\) is the plasma bulk speed, \(P\) is the thermal pressure, \(\rho\) is the mass density, \(B\) is the magnetic field, \(\gamma\) is the polytropic index, \(\mu_{0}\) is the magnetic permeability constant, and \(C\) is a constant characteristic for a streamline. The first term of the integral is the dynamic pressure over \(\rho\). The second term includes the thermal pressure normalized to \(\rho\), while the third term is the magnetic field pressure over \(\rho\). Note that the fraction \(1/2\) in the third term, comes from averaging over all the possible angles between \(B\) and \(v\) vectors (e.g., [10; 11]). The Bernoulli integral, in terms of the bulk flow speed \(v\), thermal speed \(v_{\text{th}}\), and Alfven speed \(v_{\text{A}}\), is:  
\[\frac{1}{2}v^{2}+\frac{1}{2}v_{\text{th}}^{2}\left(\frac{\gamma}{\gamma-1} \right)+\frac{1}{2}v_{A}^{2}=C\text{ for }\gamma\
eq 1, \tag{3}\]  
and  
\[\frac{1}{2}v^{2}+\frac{1}{2}v_{\text{th}}^{2}\ln\rho+\frac{1}{2}v_{A}^{2}=C \text{ for }\gamma\
eq 1. \tag{4}\]  
In general, the first (dynamic) term is expected to be the dominant term in solar wind protons at \(\sim\)1 au, where the bulk flow speed is typically larger than the thermal and the Alfven speed. However, the second term may become significant as \(\gamma\to 1\). In this study, we investigate the significance of the Bernoulli integral terms in solar wind proton plasma. We analyze proton observations obtained at 1 au in order to quantify the contribution of each term and examine the accuracy of specific simplifications. We firstly investigate the integral terms for theoretical values of \(\gamma\) in order to cover a wide range of possible values. We then investigate the integral terms by estimating the actual \(\gamma\) of solar wind protons from the density and temperature fluctuations. The specific investigation reveals the occurrence of solar wind proton intervals in which the thermal term dominates over the other two terms. Based on this result we decide whether or not is appropriate to simplify the calculations in future applications.