We analyze measured proton and electron temperatures in the high-speed solar wind in order to calculate the separate rates of heat deposition for protons and electrons. When comparing with other regions of the heliosphere, the fast solar wind has the lowest density and the least frequent Coulomb collisions. This makes the fast wind an optimal testing ground for studies of collisionless kinetic processes associated with the dissipation of plasma turbulence. Data from the Helios and Ulysses plasma instruments were collected to determine mean radial trends in the temperatures and the electron heat conduction flux between 0.29 and 5.4 AU. The derived heating rates apply specifically for these mean plasma properties and not for the full range of measured values around the mean. We found that the protons receive about 60% of the total plasma heating in the inner heliosphere, and that this fraction increases to approximately 80% by the orbit of Jupiter. A major factor affecting the uncertainty in this fraction is the uncertainty in the measured radial gradient of the electron heat conduction flux. The empirically derived partitioning of heat between protons and electrons is in rough agreement with theoretical predictions from a model of linear Vlasov wave damping. For a modeled power spectrum consisting only of Alfvenic fluctuations, the best agreement was found for a distribution of wavenumber vectors that evolves toward isotropy as distance increases.
Using one-dimensional hybrid expanding box model we investigate properties of the solar wind in the outer heliosphere. We assume a proton-electron plasma with a strictly transverse ambient magnetic field and, beside the expansion, we take into account influence of a continuous injection of cold pick-up protons through the charge-exchange process between the solar wind protons and hydrogen of interstellar origin. The injected cold pick-up protons form a ring distribution function that rapidly becomes unstable and generate Alfvén cyclotron waves. The Alfvén cyclotron waves scatter pick-up proton…
Previous formulations of heating and transport associated with strong magnetohydrodynamic (MHD) turbulence are generalized to incorporate separate internal energy equations for electrons and protons. Electron heat conduction is included. Energy is supplied by turbulent heating that affects both electrons and protons, and is exchanged between them via collisions. Comparison to available Ulysses data shows that a reasonable accounting for the data is provided when (i) the energy exchange timescale is very long and (ii) the deposition of heat due to turbulence is divided, with 60% going to proton…
The solar wind undergoes significant heating as it propagates away from the Sun; the exact mechanisms responsible for this heating remain unclear. Using data from the first perihelion of the Parker Solar Probe mission, we examine the properties of proton and electron heating occurring within magnetic coherent structures identified by means of the Partial Variance of Increments (PVI) method. Statistically, regions of space with strong gradients in the magnetic field, $PVI \geq 1$, are associated with strongly enhanced proton but only slightly elevated electron temperatures. Our analysis indicat…
According to \emph{Wind} observations between June 2004 and May 2019, this Letter investigates the proton and alpha particle temperatures in the space of ($θ_d$, $V_d/V_A$) for the first time, where $θ_d$ and $V_d$ are the radial angle and magnitude of alpha$-$proton differential flow vector respectively, $V_A$ is the local Alfvén speed. Results show that the temperatures significantly depend on $θ_d$ as well as $V_d/V_A$. In case of low proton parallel beta ($β_{p{\parallel}} < 1$), it is found that the proton perpendicular temperature is clearly enhanced when $θ_d$ is small ($\lesssim 45^\ci…
The solar wind undergoes significant heating as it propagates away from the Sun; the exact mechanisms responsible for this heating are not yet fully understood. We present for the first time a statistical test for one of the proposed mechanisms, stochastic ion heating. We use the amplitude of magnetic field fluctuations near the proton gyroscale as a proxy for the ratio of gyroscale velocity fluctuations to perpendicular (with respect to the magnetic field) proton thermal speed, defined as $ε_p$. Enhanced proton temperatures are observed when $ε_p$ is larger than a critical value ($\sim 0.019 …
We extend previous theories of stochastic ion heating to account for the motion of ions along the magnetic field. We derive an analytic expression for the ion-to-proton perpendicular temperature ratio in the solar wind for any ion species, assuming that stochastic heating is the dominant ion heating mechanism. This expression describes how this temperature ratio depends upon the average velocity of the ions along the magnetic field direction and the ratio of the parallel proton pressure to the magnetic pressure. We compare our model with previously published measurements of alpha particles and…
This paper employs a recent turbulent heating prescription to predict the ratio of proton-to-total heating due to the kinetic dissipation of Alfvenic turbulence as a function of heliocentric distance. Comparing to a recent empirical estimate for this turbulent heating ratio in the high-speed solar wind, the prediction shows good agreement with the empirical estimate for R >~ 0.8 AU, but predicts less ion heating than the empirical estimate at smaller heliocentric radii. At these smaller radii, the turbulent heating prescription, calculated in the gyrokinetic limit, fails because the turbulent …
Stochastic heating refers to an increase in the average magnetic moment of charged particles interacting with electromagnetic fluctuations whose frequencies are smaller than the particles' cyclotron frequencies. This type of heating arises when the amplitude of the gyroscale fluctuations exceeds a certain threshold, causing particle orbits in the plane perpendicular to the magnetic field to become stochastic rather than nearly periodic. We consider the stochastic heating of protons by Alfvén-wave (AW) and kinetic-Alfvén-wave (KAW) turbulence, which may make an important contribution to the hea…
We develop a 1D solar-wind model that includes separate energy equations for the electrons and protons, proton temperature anisotropy, collisional and collisionless heat flux, and an analytical treatment of low-frequency, reflection-driven, Alfven-wave turbulence. To partition the turbulent heating between electron heating, parallel proton heating, and perpendicular proton heating, we employ results from the theories of linear wave damping and nonlinear stochastic heating. We account for mirror and oblique firehose instabilities by increasing the proton pitch-angle scattering rate when the pro…
How ions are energized and heated is a fundamental problem in the study of energy dissipation in magnetized plasmas. In particular, the heating of heavy ions (including ${}^{4}\mathrm{He}^{2+}$, ${}^{3}\mathrm{He}^{2+}$ and others) has been a constant concern for understanding the microphysics of impulsive solar flares. In this article, via two-dimensional hybrid-kinetic Particle-in-Cell simulations, we study the heating of Helium ions (${}^{4}\mathrm{He}^{2+}$) by turbulence driven by cascading waves launched at large scales from the left-handed polarized Helium ion cyclotron wave branch of a…
Magnetohydronamic turbulence is believed to play a crucial role in heating the laboratorial, space, and astrophysical plasmas. However, the precise connection between the turbulent fluctuations and the particle kinetics has not yet been established. Here we present clear evidence of plasma turbulence heating based on diagnosed wave features and proton velocity distributions from solar wind measurements by the Wind spacecraft. For the first time, we can report the simultaneous observation of counter-propagating magnetohydrodynamic waves in the solar wind turbulence. Different from the tradition…
We consider the radial evolution of perpendicular ion heating due to the violation of magnetic moment conservation caused by Alfven and kinetic Alfven wave turbulence. This process, referred to as stochastic heating, can be quantified by the ratio between the average velocity fluctuations at the ion gyroradius and the perpendicular ion thermal speed $ε\equiv δv / v_{t\perp}$. Using 17 years of Helios observations, we constrain how much energy could be dissipated by this mechanism between 0.29 and 0.98 au. We find that stochastic heating likely operates throughout the entire inner heliosphere, …
Based on in-situ measurements by Wind spacecraft from 2005 to 2015, this letter reports for the first time a clearly scale-dependent connection between proton temperatures and the turbulence in the solar wind. A statistical analysis of proton-scale turbulence shows that increasing helicity magnitudes correspond to steeper magnetic energy spectra. In particular, there exists a positive power-law correlation (with a slope $\sim 0.4$) between the proton perpendicular temperature and the turbulent magnetic energy at scales $0.3 \lesssim kρ_p \lesssim 1$, with $k$ being the wavenumber and $ρ_p$ bei…
Various remote sensing observations have been used so far to probe the turbulent properties of the solar wind. Using the recently reported density modulation indices that are derived using angular broadening observations of Crab Nebula during 1952 - 2013, we measured the solar wind proton heating using the kinetic $\rm Alfv\acute{e}n$ wave dispersion equation. The estimated heating rates vary from $\approx 1.58 \times 10^{-14}$ to $1.01 \times 10^{-8} ~\rm erg~ cm^{-3}~ s^{-1}$ in the heliocentric distance range 5 - 45 $\rm R_{\odot}$. Further, we found that heating rates vary with the solar c…
We investigate how the proton distribution function evolves when the protons undergo stochastic heating by strong, low-frequency, Alfvén-wave turbulence under the assumption that $β$ is small. We apply our analysis to protons undergoing stochastic heating in the supersonic fast solar wind and obtain proton distributions at heliocentric distances ranging from 4 to 30 solar radii. We find that the proton distribution develops non-Gaussian structure with a flat core and steep tail. For $r >5 \ R_{\rm S}$, the proton distribution is well approximated by a modified Moyal distribution. Comparisons w…
Spacecraft measurements show that protons undergo substantial perpendicular heating during their transit from the Sun to the outer heliosphere. In this paper, we use {\em Helios~2} measurements to investigate whether stochastic heating by low-frequency turbulence is capable of explaining this perpendicular heating. We analyze {\em Helios~2} magnetic-field measurements in low-$β$ fast-solar-wind streams between heliocentric distances $r=0.29$ AU and $r=0.64$ AU to determine the rms amplitude of the fluctuating magnetic field, $δB_{\rm p}$, near the proton gyroradius scale $ρ_{\rm p}$. We then e…
The properties of a solar wind stream are determined by its source region and by transport effects. Independently of the solar wind type, the solar wind measured in situ is always affected by both. We consider the proton-proton collisional age as an ordering parameter for the solar wind at 1AU and explore its relation to the solar wind classification scheme developed by Xu & Borovsky (2015). We use this to show that explicit magnetic field information is not required for this solar wind classification. Based on the observation that the three basic solar wind types from this categorization cove…
Evidence for inhomogeneous heating in the interplanetary plasma near current sheets dynamically generated by magnetohydrodynamic (MHD) turbulence is obtained using measurements from the ACE spacecraft. These coherent structures only constitute 19% of the data, but contribute 50% of the total plasma internal energy. Intermittent heating manifests as elevations in proton temperature near current sheets, resulting in regional heating and temperature enhancements extending over several hours. The number density of non-Gaussian structures is found to be proportional to the mean proton temperature a…
Protons in the solar corona and heliosphere exhibit anisotropic velocity distributions, violation of magnetic moment conservation, and a general lack of thermal equilibrium with the other particle species. There is no agreement about the identity of the physical processes that energize non-Maxwellian protons in the solar wind, but a traditional favorite has been the dissipation of ion cyclotron resonant Alfven waves. This paper presents kinetic models of how ion cyclotron waves heat protons on their journey from the corona to interplanetary space. It also derives a wide range of new solutions …
Parker Solar Probe Observations of Preferential Heating of Protons over Alpha Particles near Turbulent Coherent Structures
Yunnan University（云南大学）；University College London（伦敦大学学院）；Mullard Space Science Laboratory, University College London（穆拉德空间科学实验室，伦敦大学学院）；University of Delaware（特拉华大学）；CNR, Institute for Plasma Science and Technology (ISTP)（意大利国家研究委员会等离子科学与技术研究所（ISTP））；KTH - Royal Institute of Technology（瑞典皇家理工学院）；China University of Petroleum (East China)（中国石油大学（华东））；State Key Laboratory of Deep Oil and Gas, China University of Petroleum (East China)（中国石油大学（华东）深油气资源与地质国家重点实验室）；Beihang University（北京航空航天大学）；National Institute of Natural Hazards, Ministry of Emergency Management of China（中国应急管理部自然灾害防治研究院）；National University of Defense Technology（国防科技大学）
Remote sensing observations of coronal holes show that heavy ions are hotter than protons and their temperature is anisotropic. In-situ observations of fast solar wind streams provide direct evidence for turbulent Alfvén wave spectrum, left-hand polarized ion-cyclotron waves, and He$^{++}$ -- proton drift in the solar wind plasma, which can produce temperature anisotropies by resonant absorption and perpendicular heating of the ions. Furthermore, the solar wind is expected to be inhomogeneous on decreasing scales approaching the Sun. We study the heating of solar wind ions in inhomogeneous pla…
The fast solar wind's high speeds and nonthermal features require that significant heating occurs well above the Sun's surface. Two leading theories seem incompatible: low-frequency "Alfvénic" turbulence, which transports energy outwards and is observed ubiquitously by spacecraft but struggles to explain the observed dominance of ion over electron heating; and high-frequency ion-cyclotron waves (ICWs), which explain the nonthermal heating of ions but lack an obvious source. Here, we argue that the recently proposed "helicity barrier" effect, which limits electron heating by inhibiting the turb…
Solar wind heating rates have often been calculated by fitting plasma and magnetic field data with a set of model functions. In this letter, we show that the rates obtained by such an approach strongly depend on the rather arbitrary choice one makes for these model functions. An alternative approach, consisting in monitoring the radial evolution of the adiabatic invariants, based on locally and consistently measured plasma and magnetic field parameters, is free from such a flaw. We apply this technique to a recently released Helios proton dataset, and confirm the existence of a clear perpendic…
Several studies in the near-Earth environment show that intermittent structures are important sites of energy dissipation and particle energization. Recent Parker Solar Probe (PSP) data, sampled in the near-Sun environment, have shown that proton heating is concentrated near coherent structures, suggesting local heating of protons by turbulent cascade in this region. However, whether electrons exhibit similar behavior in the near-Sun environment is not clear. Here, we address this question using PSP data collected near the Sun during the first seven orbits. We use the partial variance of incre…
The velocity of alpha particles relative to protons can vary depending on the solar wind type and distance from the Sun (Marsch 2012). Measurements from the previous spacecraft provided the alpha-proton's differential velocities down to 0.3 au. Parker Solar Probe (PSP) now enables insights into differential flows of newly accelerated solar wind closer to the Sun for the first time. Here, we study the difference between proton and alpha bulk velocities near PSP perihelia of Encounters 3-7 when the core solar wind is in the field of view of the Solar Probe Analyzer for Ions (SPAN-I) instrument. …
We propose that coronal heating in EUV coronal plumes is weaker, not stronger, than in adjacent non-plume coronal magnetic funnels. This expectation stems from (i) the observation that an EUV plume is born as the magnetic flux at the foot of the plume's magnetic funnel becomes tightly packed together, and (ii) the observation that coronal heating in quiet regions increases in proportion to the coast-line length of the underlying magnetic network. We do not rule out the possibility that coronal heating in EUV plumes might be stronger, not weaker, but we point out how the opposite is plausible. …
We use data from the first six encounters of Parker Solar Probe and employ the Partial Variance of Increments ($PVI$) method to study the statistical properties of coherent structures in the inner heliosphere with the aim of exploring physical connections between magnetic field intermittency and observable consequences such as plasma heating and turbulence dissipation. Our results support proton heating localized in the vicinity of, and strongly correlated with, magnetic structures characterized by $PVI \geq 1$. We show that on average, such events constitute $\approx 19\%$ of the dataset, tho…
A model is presented for generation of fast solar wind in coronal holes, relying on heating that is dominated by turbulent dissipation of MHD fluctuations transported upwards in the solar atmosphere. Scale-separated transport equations include large-scale fields, transverse Alfvenic fluctuations, and a small compressive dissipation due to parallel shears near the transition region. The model accounts for proton temperature, density, wind speed, and fluctuation amplitude as observed in remote sensing and in situ satellite data.
Polytropic stellar wind models with strongly localized heating
Ruhr-Universität Bochum（波鸿鲁尔大学）；Ilia State University（伊利亚国立大学）；Evgeni Kharadze Georgian National Astrophysical Observatory（叶夫根尼·哈拉泽格鲁吉亚国家天文台）；Institut für Weltraumforschung, Österreichische Akademie der Wissenschaften（奥地利科学院空间研究所）；Institute of Radio Astronomy, National Academy of Sciences of Ukraine（乌克兰国家科学院无线电天体物理研究所）
We propose a novel one-dimensional model that includes both shock and turbulence heating and qualify how these processes contribute to heating the corona and driving the solar wind. Compressible MHD simulations allow us to automatically consider shock formation and dissipation, while turbulent dissipation is modeled via a one-point closure based on Alfvén wave turbulence. Numerical simulations were conducted with different photospheric perpendicular correlation lengths $λ_0$, which is a critical parameter of Alfvén wave turbulence, and different root-mean-square photospheric transverse-wave am…
In fast wind or when the local Coulomb collision frequency is low, observations show that solar wind minor ions and ion sub-populations flow with different bulk velocities. Measurements indicate that the drift speed of both alpha particles and proton beams with respect to the bulk or core protons rarely exceeds the local Alfvén speed, suggesting that a magnetic instability or other wave-particle process limits their maximum drift. We compare simultaneous alpha particle, proton beam, and proton core observations from instruments on the Wind spacecraft spanning over 20 years. In nearly collision…
One of the important discoveries made by Voyager-2 is the nonadiabatic radial profile of the solar wind proton temperature. This phenomenon has been studied for several decades. The dissipation of turbulence energy has been proposed as the main physical process responsible for the temperature profile. The turbulence is both convected with the solar wind and originated in the solar wind by the compressions and shears in the flows and by pick-up ions. The compression source of the solar wind heating in the outer heliosphere appears due to shock waves, which originated either in the solar corona …
We obtain scatter broadened images of the Crab Nebula at 80 MHz as it transits through the inner solar wind in June 2016 and 2017. These images are anisotropic, with the major axis oriented perpendicular to the radially outward coronal magnetic field. Using these data, we deduce that the density modulation index (δN_e/N_e) caused by turbulent density fluctuations in the solar wind ranges from 1.9 \times 10^{-3} to 7.7 \times 10^{-3} between 9-20 R_{\odot}. We also find that the heating rate of solar wind protons at these distances ranges from 2.2 \times 10^{-13} to 1.0 \times 10^{-11} ~erg~cm^…
We generate a model description of the solar wind based on an explicit wave-turbulence-driven heating mechanism, and constrain our model with observational data. We included an explicit coronal heating source term in the general 3D magnetohydrodynamic code NIRVANA to simulate the properties of the solar wind. The adapted heating mechanism is based on the interaction and subsequent dissipation of counter-propagating Alfvén waves in the solar corona, accounting for a turbulent heating rate Q_p. The solar magnetic field is assumed to be an axisymmetric dipole with a field strength of 1 G. Our mod…
The SWICS instrument aboard the ACE satellite has detected frequent intervals in the slow solar wind and interplanetary coronal mass ejections (ICMEs) in which C6+ and other fully stripped ions are strongly depleted, though the ionization states of elements such as Si and Fe indicate that those ions should be present. It has been suggested that these outlier or dropout events can be explained by the resonant cyclotron heating process, because these ions all have the same cyclotron frequency as He2+. We investigate the region in the corona where these outlier events form. It must be above the i…
We have performed a 2.5 dimensional magnetohydrodynamic simulation that resolves the propagation and dissipation of Alfven waves in the solar atmosphere. Alfvenic fluctuations are introduced on the bottom boundary of the extremely large simulation box that ranges from the photosphere to far above the solar wind acceleration region. Our model is ab initio in the sense that no corona and no wind are assumed initially.The numerical experiment reveals the quasi-steady solution that has the transition from the cool to the hot atmosphere and the emergence of the high speed wind. The global structure…
Stochastic heating is a non-linear heating mechanism driven by the violation of magnetic moment invariance due to large-amplitude turbulent fluctuations producing diffusion of ions towards higher kinetic energies in the direction perpendicular to the magnetic field. It is frequently invoked as a mechanism responsible for the heating of ions in the solar wind. Here, we quantify for the first time the proton stochastic heating rate $Q_\perp$ at radial distances from the Sun as close as $0.16$ au, using measurements from the first two Parker Solar Probe encounters. Our results for both the amplit…
Ultraviolet spectroscopy of the extended solar corona is a powerful tool for measuring the properties of protons, electrons, and heavy ions in the accelerating solar wind. The large coronal holes that expand up from the north and south poles at solar minimum are low-density collisionless regions in which it is possible to detect departures from one-fluid thermal equilibrium. An accurate characterization of these departures is helpful in identifying the kinetic processes ultimately responsible for coronal heating. In this paper, Ultraviolet Coronagraph Spectrometer (UVCS) measurements of the H …
Context. Spacecraft observations early revealed frequent multiple proton populations in the solar wind. Decades of research on their origin have focused on processes such as magnetic reconnection in the low corona and wave-particle interactions in the corona and locally in the solar wind.Aims.This study aims to highlight that multiple proton populations and beams are also produced by magnetic reconnection occurring locally in the solar wind. Methods. We use high resolution Solar Orbiter proton velocity distribution function measurements, complemented by electron and magnetic field data, to ana…
We study the propagation, reflection, and turbulent dissipation of Alfven waves in coronal holes and the solar wind. We start with the Heinemann-Olbert equations, which describe non-compressive magnetohydrodynamic fluctuations in an inhomogeneous medium with a background flow parallel to the background magnetic field. Following the approach of Dmitruk et al, we model the nonlinear terms in these equations using a simple phenomenology for the cascade and dissipation of wave energy, and assume that there is much more energy in waves propagating away from the Sun than waves propagating towards th…
In solar wind, magnetohydrodynamic (MHD) discontinuities are ubiquitous and often found to be at the origin of turbulence intermittency. They may also play a key role in the turbulence dissipation and heating of the solar wind. The tangential (TD) and rotational (RD) discontinuities are the two most important types of discontinuities. Recently, the connection between turbulence intermittency and proton thermodynamics has been being investigated observationally. Here we present numerical results from three-dimensional MHD simulation with pressure anisotropy and define new methods to identify an…
A chromosphere is a universal attribute of stars of spectral type later than ~F5. Evolved (K and M) giants and supergiants (including the zeta Aurigae binaries) show extended and highly turbulent chromospheres, which develop into slow massive winds. The associated continuous mass loss has a significant impact on stellar evolution, and thence on the chemical evolution of galaxies. Yet despite the fundamental importance of those winds in astrophysics, the question of their origin(s) remains unsolved. What sources heat a chromosphere? What is the role of the chromosphere in the formation of stell…
One proposed mechanism for heating the solar wind, from close to the sun to beyond 10 AU, invokes low-frequency, oblique, Alfven-wave turbulence. Because small-scale oblique Alfven waves (kinetic Alfven waves) are compressive, the measured density fluctuations in the solar wind place an upper limit on the amplitude of kinetic Alfven waves and hence an upper limit on the rate at which the solar wind can be heated by low-frequency, Alfvenic turbulence. We evaluate this upper limit for both coronal holes at 5 solar radii and in the near-Earth solar wind. At both radii, the upper limit we find is …
We examine the long-term time evolution (1965-2015) of the relationships between solar wind proton temperature (Tp) and speed (Vp) and between the proton density (np) and speed using OMNI solar wind observations taken near Earth. We find a long-term decrease in the proton temperature-speed (Tp-Vp) slope that lasted from 1972 to 2010, but has been trending upward since 2010. Since the solar wind proton density-speed (np-Vp) relationship is not linear like the Tp-Vp relationship, we perform power law fits for np-Vp. The exponent (steepness in the np-Vp relationship) is correlated with the solar …
The modeling of the heliosphere requires continuous three-dimensional solar wind data. The in-situ out-of-ecliptic measurements are very rare, so that other methods of solar wind detection are needed. We use the remote-sensing data of the solar wind speed from observations of interplanetary scintillation (IPS) to reconstruct spatial and temporal structures of the solar wind proton speed from 1985 to 2013. We developed a method of filling the data gaps in the IPS observations to obtain continuous and homogeneous solar wind speed records. We also present a method to retrieve the solar wind densi…
The steady, supersonic outflow from the Sun we call the solar wind was first posited in the 1950s and initial theories rightly linked the acceleration of the wind to the existence of the million-degree solar corona. Still today, the wind acceleration mechanisms and the coronal heating processes remain unsolved challenges in solar physics. In this work, I seek to answer a portion of the mystery by focusing on a particular acceleration process: Alfven waves launched by the motion of magnetic field footpoints in the photosphere. The entire corona is threaded with magnetic loops and flux tubes tha…
This Letter investigates the spectral characteristics of the interplanetary magnetic field fluctuations at proton scales during several time intervals chosen along the speed profile of a fast stream. The character of the fluctuations within the first frequency decade, beyond the high frequency break located between the fluid and the kinetic regime, strongly depends on the type of wind. While the fast wind shows a clear signature of both right handed and left handed polarized fluctuations, possibly associated with KAW and Ion-Cyclotron waves, respectively, the rarefaction region, where the wind…
Orbits six through nine of the Parker Solar Probe have been studied to show that solar wind core electrons emerged from 15 solar radii with temperatures that were constant to within ~10% although the solar wind speed varied from 300 to 800 km/sec. After leaving 15 solar radii, the core electrons were isotropically heated as much as a factor of two below 30 solar radii by triggered and natural ion acoustic waves. To distinguish this wave heating from processes occurring inside 15 solar radii, the electron temperature is modeled as having two components, the base temperature observed at 15 solar…
