The Polytropic Behavior of Solar Wind Protons as Observed by the Ulysses Spacecraft during Solar Minimum

Abstract

We analyze proton bulk parameters derived from Ulysses observations and investigate the polytropic behavior of solar wind protons over a wide range of heliocentric distances and latitudes. The large-scale variations of the proton density and temperature over heliocentric distance indicate that plasma protons are governed by subadiabatic processes (polytropic index γ < 5/3), if we assume protons with three effective kinetic degrees of freedom. From the correlation between the small-scale variations of the plasma density and temperature in selected subintervals, we derive a polytropic index γ ∼ 1.4 on average. Further examination shows that the polytropic index does not have an apparent dependence on the solar wind speed. This agrees with the results of previous analyses of solar wind protons at ∼1 au. We find that the polytropic index varies slightly over the range of the heliocentric distances and heliographic latitudes explored by Ulysses. We also show that the homogeneity of the plasma and the accuracy of the polytropic model applied to the data points vary over Ulysses' orbit. We compare our results with the results of previous studies that derive the polytropic index of solar wind ions within the heliosphere using observations from various spacecraft. We finally discuss the implications of our findings in terms of heating mechanisms and the effective degrees of freedom of the plasma protons.

Transport Equation of Kappa Distributions in the Heliosphere

“…We derive the radial transportation of the value of kappa of the solar wind plasma protons in the heliosphere from R = 1 up to R = 100 au. We consider two radial profiles for the polytropic index, (a) a constantly adiabatic expansion of plasma protons that corresponds to a fixed polytropic index of γ = 5/3 (e.g., Kartalev et al 2006;Nicolaou et al 2014;Livadiotis & Desai 2016;Nicolaou et al 2023), and (b) the sub-adiabatic expansion with a polytropic index γ(R) that follows the model in Equation ( 22). Figure 3 (upper panels) shows these two cases of polytropic index radial profile throughout the heliosphere.…”

“…The solar wind flows radially outward from the Sun and expands in the inner heliosphere (heliocentric distance R less than ∼ 10 au) in an almostadiabatic cooling (γ ∼ 1.4-1.6, e.g., see Livadiotis & Desai 2016;Nicolaou et al 2023), namely, the flowing solar wind plasma expands in the heliosphere without significant energy exchange (negligible heating rate). However, both observations and existing models agree that the polytropic index further out from the heliosphere differs significantly from being adiabatic.…”

Connection between Polytropic Index and Heating

“…Equation 47b is fitted to the three observations shown in Figure 3(b), that is, γ ∼ 1.6 near R ∼ 1 au, using Wind data (Livadiotis & Desai 2016), γ ∼ 1.4 near R ∼ 5 au, using Ulysses data (Nicolaou et al 2023), and γ ∼ 1 near R ∼ 30 au, using New Horizons data (Elliott et al 2019). The above example is simplified and only included solar wind protons, excluding both pickup ions and electrons, or in general, the multi-particle description.…”

“…The heating of the solar wind increases with heliocentric distance, becoming more significant in the outer heliosphere, and thus solar wind thermodynamics is described by nonadiabatic processes, especially in the outer heliosphere. Indeed, while near adiabatic processes are observed within a few au of the Sun (e.g., Livadiotis & Desai 2016, near 1 au; Nicolaou et al 2023, up to 5 au), nonadiabatic processes are increasingly observed in the outer heliosphere (Elliott et al 2019). If the plasma expansion was exactly adiabatic, the temperature would be much lower than the observations revealed; this is because nonadiabatic temperatures have higher values than they would have been under an adiabatic cooling, indicating the presence of significant heating.…”

“…In the paper, we consider both the precise analytical formulations with variable polytropic indices, and also provide their approximations, where the indices are treated as constants along an appropriately small distance of the flow. For instance, the polytropic index is nearly adiabatic at R ∼ 1 au, e.g., γ ∼ 1.6 (Livadiotis & Desai 2016), and drops only a little, to γ ∼ 1.4, within a few au from the Sun (Nicolaou et al 2023).…”