From Isopoly to Bipoly: Refining Solar Wind Thermal Modeling with Solar Orbiter

(Solar Orbiter Nugget #74 by J-B. Dakeyo¹, P. Démoulin^2,4, A.P. Rouillard³, M. Maksimovic², A. Chapiron³, S.D.Bale³)

Introduction

The observed non-adiabatic evolution of the solar wind leaving the solar atmosphere reveals the presence of some heating mechanism. Regarding the inner heliosphere evolution, observational studies [1,2,3] have shown that the temperature of protons and electrons appears to decrease slower with radial distance than as theoretically expected.

The thermal evolution can be characterized by the polytropic index γ, which takes a value of γ=5/3 for and adiabatic expansion (no heating), but in practice exhibits values of γ_p≈1.45 and γ_e≈1.25, for protons and electrons respectively, within radial distance from 0.3 au to 1 au [2,3]. A recent study has tried to include such a thermal behavior influence into the so-called “isopoly” solar wind model, assuming an isothermal evolution of the wind within the corona followed by a polytropic one based on the observed γ_p and γ_e, to study in what extent the thermal pressure could be responsible of the solar wind acceleration [4]. 

In addition to the interplanetary thermal constraints, the coronal effect has been modeled to account for typical expansion factor profiles of the magnetic flux tube (due to the balance of magnetic pressure in the corona) deduced from PFSS modeling based on the photospheric magnetic connectivity of Solar Orbiter’s observations. To capture each wind speed specificities, observations have been split into 5 wind speed populations based on a quantile classification of the wind. 
Nevertheless, the isopoly approach has faced one of the most common modeling issues, which is that the interplanetary fast solar wind speed can only be modeled assuming a higher proton coronal temperature (∼3MK) compared to the order of what is actually observed ( ∼1MK from spectroscopic observations [5]). 

In this study, we adapt the actual existing isopoly modeling, with the bipoly modeling that can account for more realistic coronal temperatures. These are deduced from the ion charge state ratio O⁷⁺/O⁶⁺ observations of Solar Orbiter (SO). This provides a proxy to the solar wind coronal temperature. 

Figure 1. Relationship between the charge-state ratio and the estimated “freeze-in” temperature based on the equations of [6]. The highlighted part of each curve represents the range of charge-state ratio values found in the SO observations (HIS instrument from 17/01/2022 to 27/04/2023).

Results

Electron coronal temperature estimation from charge state ratio

We first aim to define coronal constrains for the modeling based on in-situ observations of charge state ratios observed by the Heavy Ions Sensor (HIS) onboard SO.

Based on the work of [6], the densities of ions measured in-situ can be used to determine the wind source electron temperatures, i.e., in regions where the coronal plasma is still collisional. This is based on the property that in the collisional corona, the heavy ions become increasingly ionized by electrons as the ambient temperature and therefore the electron kinetic energy rise. The hotter, the denser, and the more time the electrons remain in the collisional corona, the higher the ionization level of the atoms. Using the known ionization and recombination rates of different ion species, it is possible to determine T_e in the corona as a function of the ratio of ionic fractions. This way, T_e can be determined in the corona around a “freezing height” located at the radius r_f, where collisional frequencies are too low to change ionization levels. The relevant r_f values are typically below ∼1.3 r_⊙, so in our wind modeling study, we set the low coronal temperatures to be the same as the estimated freeze-in temperatures (set in the model to  r ≈ r_⊙).

The carbon and oxygen ions measured routinely in the solar wind, are typically used to infer T_e in the corona. Indeed, their charge-state ratios are sensitive to electron temperatures that are encountered in this medium, i.e., from ∼0.5 to ∼5 MK for carbon, and ∼1 to ∼10 MK for oxygen [6]. Figure 1 shows the relationship between the charge state ratio C⁶⁺/C⁵⁺ and O⁷⁺/O⁶⁺ as a function of the freeze-in temperature. Since the charge state ratios of O⁷⁺/O⁶⁺ and H⁺/H have similar ionisation thresholds, the freeze-in temperatures of both populations are expected to be close. Therefore, we assume that the freeze-in temperature of O⁷⁺/O⁶⁺ represents both the electron and proton coronal temperatures well  (i.e. T_e0≈T_p0).   

Charge state ratio and electron coronal temperature relation with bulk speed

Figure 2 presents the relation between the observed charge state of the ionic ratio O⁷⁺/O⁶⁺, and its corresponding coronal temperature, as a function of the observed proton bulk speed of the wind measured by the Proton Alpha Sensor (PAS) onboard SO. Both panels reveal a clear anti-correlation with the wind speed, providing quantitative constraints in function of the interplanetary wind speed, for the solar wind modeling. The coronal temperatures T_e0 for each of the 5 wind speed populations are ranging between 1.6 MK for slow winds (∼350 km/s at 1 au) down to 1.2 MK for fast wind (∼650 km/s at 1 au).

Figure 2. Charge-state ratios and equivalent electron temperature from SO measurements (from PAS and HIS instruments) between 17/01/2022 and 27/04/2023. Panel 'c' : Observed wind speed as a function of charge-state ratio O⁷⁺/O⁶⁺. Panel 'd': Observed wind speed as a function of the equivalent "freeze-in" temperature from panel (c). The black dots represent the median coronal temperature estimated from a quantile classification applied on SO, following the method of Dakeyo et al 2022 [3]. 

Generalization of the two-thermal regime approach: Bipoly modeling 

To overcome the difficulties faced by the isopoly approach to model fast wind with low coronal temperature, we relax the isothermal condition in the corona to a polytropic evolution. This leads to two thermal regimes, both being polytropic. The temperature can either increase or decrease (respectively γ<1 and γ>1), giving a more realistic physical description with regards to the main candidate for coronal heating (Alfvén waves dissipation or ion-cyclotron heating [8]). It also provides more flexibility for the model to fit coronal temperatures, derived from spectroscopic and charge state ratio observations.

We compare the bipoly model with in-situ solar wind observations, i.e. bulk speed, proton and electron temperature, and electron density. We distinguish two types of wind solution: “f-subsonic” and “f-supersonic”, which are subsonic and supersonic, respectively, in the super-expansion region below r_ss. Finally, we fit bipoly solutions to the set of pre-derived coronal constrains (coronal temperatures from O⁷⁺/O⁶⁺), and interplanetary observations from the Parker Solar Probe (PSP) and Helios missions. To better see any wind speed dependence in the wind behavior, the PSP and Helios dataset have been classified into 5 wind speed population. Observations have been cut into radial bins from 0.05 au to 1au, and sorted in quantiles on each radial bin (see [3] for more details).

The results are given in Figure 3: the relaxation of the first isothermal thermal regime into a polytropic one allows us to model the slow and fast wind interplanetary medium bulk speeds (panel (a)), proton and electron temperatures (panel (b) and (d) respectively), and plasma density (panel (c)), as well as the coronal temperatures deduced from charge state ratio observations.  Slow wind shows  a decrease of the proton temperature in the super-expansion region (grey shaded region), while the fast wind presents a temperature increase, more realistic with remote-sensing spectroscopic observations of solar sources [5,6]. All the different modeled quantities range within values expected to be observed, in the corona as well as further away in the inner heliosphere.
 

Figure 4. Bipoly solutions fitted manually to the in-situ data set used by [3]. Panel (a): Velocity profiles; Panel (b): Proton temperature profiles; Panel (c): Density profiles; Panel (d): Electron temperature profiles. Models include the flux tube expansion factor profiles computed from PFSS, and ( T_e0,T_p0 ) deduced from charge-state ratio. The line colors correspond to the wind populations A to E such as labeled on the right side of panel (a). Each of the five bipoly models is computed by taking into account a median expansion profile for its observed speed. This is based on the PFSS profiles from [4] for the two range of values: ( f_ss  < 7) and (100 < f_ss< 250), where f_ss is the expansion factor at r_ss from PFSS models. The data used for fitting are added in all panels as dots linked with straight segments of the same color of its corresponding wind population. The region of super radial expansion (up to r_ss=2.5 r_⊙) is delineated by the gray shaded area. The critical radius location r_c, of each solution is annotated on each profile by a cross. The f-subsonic (r_c>r_ss) and the f-supersonic (r_c<r_ss) solutions are plotted in solid and dashed lines respectively (population E has only f-supersonic solutions).

Conclusions

Our results indicate that the fast wind can actually also be described by a low complexity solar wind modeling, and that the bipoly approach is the first “simple” model capable of modeling global and detailed features with such a precision regarding a variety of observations. These solutions offer solid constrains to disentangle the major heating mechanisms influence, by providing detailed proton and electron heating rates, and their radial evolution. Indeed, heating rates are crucial for the understanding of the solar wind radial evolution, thus their quantitative description gives a new framework for more sophisticated theoretical studies.
Possible future improvements to solar wind forecasting could also be achieved by considering these physically derived fast computation solutions, regarding time travel, wind interactions with other wind streams or coronal mass ejections. 

Affiliations

(1) Space Sciences Laboratory, University of California, Berkeley, CA, USA
(2) LIRA, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université, Université de Paris, 5 place Jules Janssen, 92195 Meudon, France
(3) IRAP, Observatoire Midi-Pyrénées, Université Toulouse III – Paul Sabatier, CNRS, 9 avenue du Colonel Roche, 31400 Toulouse, France 
(4) Laboratoire Cogitamus, 75005 Paris, France  
 

References

[1] Schwartz & Marsch 1983 « The radial evolution of a single solar wind plasma parcel”, DOI : 10.1029/JA088iA12p09919

[2] Maksimovic et al. 2020 « Anticorrelation between the Bulk Speed and the Electron Temperature in the Pristine Solar Wind: First Results from the Parker Solar Probe and Comparison with Helios “, DOI : 10.3847/1538-4365/ab61fc

[3] Dakeyo et al. 2022 « Statistical Analysis of the Radial Evolution of the Solar Winds between 0.1 and 1 au and Their Semiempirical Isopoly Fluid Modeling”, DOI : 10.3847/1538-4357/ac9b14

[4] Dakeyo et al. 2024b Testing the flux tube expansion factor -- solar wind speed relation with Solar Orbiter data”, DOI: 10.1051/0004-6361/202451272

[5] Cranmer 2022 “Coronal Holes and the High-Speed Solar Wind” , DOI : 10.1023/A:1020840004535

[6] Cranmer 1999 “Spectroscopic Constraints on Models of Ion Cyclotron Resonance Heating in the Polar Solar Corona and High-Speed Solar Wind”, DOI : 10.1086/307330

[7] Ko et al. 1997. “An Empirical Study of the Electron Temperature and Heavy Ion Velocities in the South Polar Coronal Hole”, DOI : 10.1023/A:1004943213433

[8] Wang 1993, “Flux-Tube Divergence, Coronal Heating, and the Solar Wind” , DOI : 10.1086/186895

[9] Parker 1958, « Dynamics of the Interplanetary Gas and Magnetic Fields“, DOI : 10.1086/146579
 

Acknowledgements

We acknowledge the NASA Parker Solar Probe Mission and the SWEAP team led by J. Kasper for the use of data. We thank the instrumental Solar Wind Analyser team (SWA) for valuable discussions. This research was funded by the European Research Council ERC SLOW_SOURCE (DLV- 819189) project. This work was supported by CNRS Occitanie Ouest and LESIA. We recognize the collaborative and open nature of knowledge creation and dissemination, under the control of the academic community as expressed by Camille Noûs.