Non-equilibrium ionization (NEI) is a key process often times neglected when modeling astrophysical plasmas with thermodynamical timescales much shorter than the timescales for ionization and recombination. In this paper, we perform NEI modeling on a magnetohydrodynamic (MHD) simulation of the solar wind during the Whole Sun Month (Carrington Rotation 1913 from 1996 August 22 to September 18), and compare the resulting charge state distributions with in situ measurements made with the Solar Wind Ion Composition Spectrometer (SWICS) on Ulysses. We trace the wind trajectory back to 20$R_\odot$ using the velocity measured by Ulysses at about 4 AU, and obtain the plasma flow trajectory within 20$R_\odot$ using the MHD simulation data performed by Predictive Science Inc. We assume that the wind started from the solar surface and was initially in ionization equilibrium, and analyze the time-dependent ionization state of solar wind along each of the trajectories. In our analysis we: (1) obtain charge state densities and ratios for slow and fast winds based on the NEI model, and compare them with in situ observations; and (2) measure the "freeze-in" distance for several ions observed by SWICS to determine a possible correlation between when the ionization states become fixed and the electron density and outward velocity of the plasma at the freeze-in height. This study provides a stringent test on comparing charge state distributions from the outer corona predicted from simulations with in situ measurements made in the far heliosphere. This showcases the challenges in matching plasma conditions near the corona to those observed in interplanetary space.
The shape of the solar wind bubble within the interstellar medium, the so-called heliosphere, has been explored over six decades. As the Sun moves through the surrounding partially-ionized medium, neutral hydrogen atoms penetrate the heliosphere, and through charge-exchange with the supersonic solar wind, create a population of hot pick-up ions (PUIs). The Termination Shock (TS) crossing by Voyager 2 (V2) data demonstrated that the heliosheath (HS) (the region of shocked solar wind) pressure is dominated by suprathermal particles. Here we use a novel magnetohydrodynamic model that treats the f…
Neutral hydrogen has been shown to greatly impact the plasma flow in the heliopshere and the location of the heliospheric boundaries. We present the results of the Solar-wind with Hydrogen Ion Exchange and Large-scale Dynamics (SHIELD) model, a new, self-consistent, kinetic-MHD model of the outer heliosphere within the Space Weather Modeling Framework. The charge-exchange mean free path is on order of the size of the heliosphere; therefore, the neutral atoms cannot be described as a fluid. The SHIELD model couples the MHD solution for a single plasma fluid to the kinetic solution from for neut…
The scaling of the turbulent spectra provides a key measurement that allows to discriminate between different theoretical predictions of turbulence. In the solar wind, this has driven a large number of studies dedicated to this issue using in-situ data from various orbiting spacecraft. While a semblance of consensus exists regarding the scaling in the MHD and dispersive ranges, the precise scaling in the transition range and the actual physical mechanisms that control it remain open questions. Using the high-resolution data in the inner heliosphere from Parker Solar Probe (PSP) mission, we fin…
A recent study by Opher et al. (2015) suggested the heliosphere has a "croissant" shape, where the heliosheath plasma is confined by the toroidal solar magnetic field. The "croissant" heliosphere is in contrast to the classically accepted view of a comet-like tail. We investigate the effect of the "croissant" heliosphere model on energetic neutral atom (ENA) maps. Regardless of the existence of a split tail, the confinement of the heliosheath plasma should appear in ENA maps. ENA maps from the Interstellar Boundary Explorer (IBEX) have shown two high latitude lobes with excess ENA flux at high…
In the near future, Parker Solar Probe and Solar Orbiter will provide the first comprehensive in-situ measurements of the solar wind in the inner heliosphere since the Helios mission in the 1970s. We describe a reprocessing of the original Helios ion distribution functions to provide reliable and reproducible data to characterise the proton core population of the solar wind in the inner heliosphere. A systematic fitting of bi-Maxwellian distribution functions was performed to the raw Helios ion distribution function data to extract the proton core number density, velocity, and temperatures par…
The anisotropy of solar wind turbulence is a critical issue in understanding the physics of energy transfer between scales and energy conversion between fields and particles in the heliosphere. Using the measurement of \emph{Parker Solar Probe} (\emph{PSP}), we present an observation of the anisotropy at kinetic scales in the slow, Alfvénic, solar wind in the inner heliosphere. \textbf{The magnetic compressibility behaves as expected for kinetic Alfvénic turbulence below the ion scale.} A steepened transition range is found between the inertial and kinetic ranges in all directions with respect…
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…
The classic accepted view of the heliosphere is a quiescent, comet-like shape aligned in the direction of the Sun's travel through the interstellar medium (ISM) extending for 1000's of AUs (AU: astronomical unit). Here we show, based on magnetohydrodynamic (MHD) simulations, that the tension (hoop) force of the twisted magnetic field of the sun confines the solar wind plasma beyond the termination shock and drives jets to the North and South very much like astrophysical jets. These jets are deflected into the tail region by the motion of the Sun through the ISM similar to bent galactic jets mo…
Robustly identifying the solar sources of individual packets of solar wind measured in interplanetary space remains an open problem. We set out to see if this problem is easier to tackle using solar wind measurements closer to the Sun than 1 AU, where the mixing and dynamical interaction of different solar wind streams is reduced. Using measurements from the Helios mission, we examined how the proton core temperature anisotropy and cross helicity varied with distance. At 0.3 AU there are two clearly separated anisotropic and isotropic populations of solar wind, that are not distinguishable at …
We combine different measurements of the polytropic index of the proton plasma in the heliosphere: i) near-adiabatic index in the inner heliosphere ~1AU, ii) subadiabatic indices in the outer heliosphere ~20-40AU, and iii) near-zero indices in the inner heliosheath. These observations are unified by a single theoretical model of the polytropic index throughout its radial extent in the heliosphere; the corresponding fitting reveals the decreasing trend of the polytropic index with increasing heliocentric distance R. We anticipate that with increasing R, (i) the Debye length and mean-free-path d…
Coronal holes (CHs) are regions of open magnetic flux which are the source of high speed solar wind (HSSW) streams. To date, it is not clear which aspects of CHs are of most influence on the properties of the solar wind as it expands through the Heliosphere. Here, we study the relationship between CH properties extracted from AIA (Atmospheric Imaging Assembly) images using CHIMERA (Coronal Hole Identification via Multi-thermal Emission Recognition Algorithm) and HSSW measurements from ACE (Advanced Composition Explorer) at L1. For CH longitudinal widths $Δθ_{CH}<$67$^{\circ}$, the peak SW velo…
Data from the Voyager probes and the Interstellar Boundary Explorer have revealed the importance of pick-up ions (PUIs) in understanding the character and behavior of the outer heliosphere, the region of interaction between the solar wind and the interstellar medium. In the outer heliosphere PUIs carry a large fraction of the thermal pressure, which effects the nature of the termination shock, and they are a dominate component of pressure in the heliosheath. This paper describes the development of a new multi-ion, multi-fluid 3-D magnetohydrodynamic model of the outer heliosphere. This model h…
Using the Parker Solar Probe data taken in the inner heliosphere, we investigate the power and spatial anisotropy of magnetic-field spectra at kinetic scales (i.e., around sub-ion scales) in solar wind turbulence in the inner heliosphere. We find that strong anisotropy of magnetic spectra occurs at kinetic scales with the strongest power in the perpendicular direction with respect to the local magnetic field (forming an angle theta_B with the mean flow velocity). The spectral index of magnetic spectra varies from -3.2 to -5.8 when the angle theta_B changes from 90 to 180 (or 0) deg, indicating…
The first computation of the compressible energy transfer rate from $\sim$ 0.2 AU up to $\sim$ 1.7 AU is obtained using PSP, THEMIS and MAVEN observations. The compressible energy cascade rate $\varepsilon_C$ is computed for hundred of events at different heliocentric distances, for time intervals when the spacecraft were in the pristine solar wind. The observational results show moderate increases of $\varepsilon_C$ with respect to the incompressible cascade rate $\varepsilon_I$. Depending on the level of compressibility in the plasma, which reach up to 25 $\%$ in the PSP perihelion, the diff…
We investigate the solar wind energy flux in the inner heliosphere using 12-day observations around each perihelion of Encounter One (E01), Two (E02), Four (E04), and Five (E05) of Parker Solar Probe (PSP), respectively, with a minimum heliocentric distance of 27.8 solar radii ($R_\odot{}$). Energy flux was calculated based on electron parameters (density $n_e$, core electron temperature $T_{c}$, and suprathermal electron temperature $T_{h}$) obtained from the simplified analysis of the plasma quasi-thermal noise (QTN) spectrum measured by RFS/FIELDS and the bulk proton parameters (bulk speed …
Ensemble Modeling of the Solar Wind Flow with Boundary Conditions Governed by Synchronic Photospheric Magnetograms. I. Multi-point Validation in the Inner Heliosphere
The slow solar wind is generally believed to result from the interaction of open and closed coronal magnetic flux at streamers and pseudostreamers. We use 3-dimensional magnetohydrodynamic simulations to determine the detailed structure and dynamics of open-closed interactions that are driven by photospheric convective flows. The photospheric magnetic field model includes a global dipole giving rise to a streamer together with a large parasitic polarity region giving rise to a pseudostreamer that separates a satellite coronal hole from the main polar hole. Our numerical domain extends out to 3…
Plasma processes occurring in the corona and solar wind can be probed by studying the thermodynamic properties of different ion species. However, most in-situ observations of positive ions in the solar wind are taken at 1 AU, where information on their solar source properties may have been irreversibly erased. In this study we aimed to use the properties of alpha particles at heliocentric distances between 0.3 and 1 AU to study plasma processes occurring at the points of observation, and to infer processes occurring inside 0.3 AU by comparing our results to previous remote sensing observations…
Dynamics of the Upwind Heliosphere Due to Data-Driven, Solar Wind and Magnetic Field Variations and Implications for Wave Propagation into the Very Local Interstellar Medium
We use a three-dimensional magnetohydrodynamic simulation of the solar wind to calculate cosmic ray diffusion coefficients throughout the inner heliosphere ($2~R_\odot - 3$ AU). The simulation resolves large-scale solar wind flow, which is coupled to small-scale fluctuations through a turbulence model. Simulation results specify background solar wind fields and turbulence parameters, which are used to compute diffusion coefficients and study their behavior in the inner heliosphere. The parallel mean free path is evaluated using quasi-linear theory, while the perpendicular mean free path is det…
Wave-particle instabilities driven by departures from local thermodynamic equilibrium have been conjectured to play a role in governing solar wind dynamics. We calculate the statistical variation of linear stability over a large subset of Helios I and II fast solar wind observations using a numerical evaluation of the Nyquist stability criterion, accounting for multiple sources of free energy associated with protons and helium including temperature anisotropies and relative drifts. We find that 88\% of the surveyed intervals are linearly unstable. The median growth rate of the unstable modes i…
The mixture/interaction of anti-sunward-propagating Alfvénic fluctuations (AFs) and sunward-propagating Alfvénic fluctuations (SAFs) is believed to result in the decrease of the Alfvénicity of solar wind fluctuations with increasing heliocentric distance. However, SAFs are rarely observed at 1 au and solar wind AFs are found to be generally outward. Using the measurements from Voyager 2 and Wind, we perform a statistical survey of SAFs in the heliosphere inside 6 au. We first report two SAF events observed by Voyager 2. One is in the anti-sunward magnetic sector with a strong positive correlat…
Voyager 1 and 2 crossed the heliopause at $\sim$122 AU in 2012 and $\sim$119 AU in 2018, respectively. It was quite a surprise because the thickness of the inner heliosheath obtained by the existing at that time models of the global heliosphere was significantly larger (by 20-40 AU). Until now, the problem of the heliosheath thickness has not been fully resolved. Earlier in the frame of an oversimplified toy model of nearly isothermal solar wind plasma it has been shown that the effect of electron thermal conduction may significantly reduce the thickness of the inner heliosheath. In this paper…
The outer heliosphere is a dynamic region shaped largely by the interaction between the solar wind and the interstellar medium. While interplanetary magnetic field and plasma observations by the Voyager spacecraft have significantly improved our understanding of this vast region, modeling the outer heliosphere still remains a challenge. We simulate the three-dimensional, time-dependent solar wind flow from 1 to 80 astronomical units (AU), where the solar wind is assumed to be supersonic, using a two-fluid model in which protons and interstellar neutral hydrogen atoms are treated as separate fl…
The solar wind, a continuous flow of plasma from the sun, not only shapes the near Earth space environment but also serves as a natural laboratory to study plasma turbulence in conditions that are not achievable in the lab. Starting with the Mariners, for more than five decades, multiple space missions have enabled in-depth studies of solar wind turbulence. Parker Solar Probe (PSP) was launched to explore the origins and evolution of the solar wind. With its state-of-the-art instrumentation and unprecedented close approaches to the sun, PSP is starting a new era of inner heliospheric explorati…
The origins and generation mechanisms of the slow solar wind are still unclear. Part of the slow solar wind is populated by "number density structures", discrete patches of increased number density that are frozen in to and move with the bulk solar wind. In this paper we aimed to provide the first in-situ statistical study of number density structures in the inner heliosphere. We reprocessed in-situ ion distribution functions measured by Helios in the inner heliosphere to provide a new reliable set of proton plasma moments for the entire mission. From this new data set we looked for number den…
Absorption of the Lyman-$α$ radiation on Interstellar Neutral Hydrogen (ISN H) atoms in the heliosphere is a potentially important effect to account for in precise gas distribution simulations. In this paper we develop a method to estimate the magnitude of absorption of solar Lyman-$α$ radiation inside the solar wind termination shock and to include absorption effects in the Warsaw Test Particle Model (WTPM) by an appropriate modification of radiation pressure. We perform calculations of absorption effects on a 3D grid in the heliosphere and present a set of parameters to model absorption effe…
We analyze the evolution of the interplanetary magnetic field spatial structure by examining the inner heliospheric autocorrelation function, using Helios 1 and Helios 2 "in situ" observations. We focus on the evolution of the integral length scale (λ) anisotropy associated with the turbulent magnetic fluctuations, with respect to the aging of fluid parcels traveling away from the Sun, and according to whether the measured λis principally parallel (λ_parallel) or perpendicular (λ_perp) to the direction of a suitably defined local ensemble average magnetic field B0. We analyze a set of 1065 24-…
We investigate the radial dependence of the spectral break separating the inertial from the dissipation range in power density spectra of interplanetary magnetic field fluctuations, between $0.42$ and $5.3$ AU, during radial alignments between MESSENGER and WIND for the inner heliosphere and between WIND and ULYSSES for the outer heliosphere. We found that the spectral break moves to higher and higher frequencies as the heliocentric distance decreases. The radial dependence of the corresponding wavenumber is of the kind $κ_b\sim R^{-1.08}$ in good agreement with that of the wavenumber derived …
NASA's Parker Solar Probe (Parker) spacecraft reached its first perihelion of 35.7 solar radii on November 5th, 2018. To aid in mission planning, and in anticipation of the unprecedented measurements to be returned, in late October, we developed a three-dimensional magnetohydrodynamic (MHD) solution for the solar corona and inner heliosphere, driven by the then available observations of the Sun's photospheric magnetic field. Our model incorporates a wave-turbulence-driven (WTD) model to heat the corona. Here, we present our predictions for the structure of the solar corona and the likely {\it …
The role of solar wind expansion in generating whistler waves is investigated using the EB-iPic3D code, which models solar wind expansion self-consistently within a fully kinetic semi-implicit approach. The simulation is initialized with an electron velocity distribution function modeled after Parker Solar Probe observations during its first perihelion at 0.166 au, consisting of a dense core and an anti-sunward strahl. This distribution function is initially stable with respect to kinetic instabilities. Expansion drives the solar wind into successive regimes where whistler heat flux instabilit…
We make use of the Parker Solar Probe (PSP) data to explore the nature of solar wind turbulence focusing on the Alfvénic character and power spectra of the fluctuations and their dependence on distance and context (i.e. large scale solar wind properties), aiming to understand the role that different effects such as source properties, solar wind expansion, stream interaction might play in determining the turbulent state. We carry out a statistical survey of the data from the first five orbits of PSP with a focus on how the fluctuation properties at the large, MHD scales, vary with different sol…
Alfvénic fluctuations in the solar wind display many properties reflecting an ongoing nonlinear cascade, e.g. a well-defined spectrum in frequency, together with some characteristics more commonly associated with the linear propagation of waves from the Sun, such as the variation of fluctuation amplitude with distance, dominated by solar wind expansion effects. Therefore both nonlinearities and expansion must be included simultaneously in any successful model of solar wind turbulence evolution. Because of the disparate spatial scales involved, direct numerical simulations of turbulence in the …
The nature of the plasma wave modes around the ion kinetic scales in highly Alfvénic slow solar wind turbulence is investigated using data from the NASA's Parker Solar Probe taken in the inner heliosphere, at 0.18 Astronomical Unit (AU) from the sun. The joint distribution of the normalized reduced magnetic helicity $σ_m (θ_{RB}, τ)$ is obtained, where $θ_{RB}$ is the angle between the local mean magnetic field and the radial direction and $τ$ is the temporal scale. Two populations around ion scales are identified: the first population has $σ_m (θ_{RB}, τ) < 0$ for frequencies (in the spacecra…
Instabilities described by linear theory characterize an important form of wave-particle interaction in the solar wind. We diagnose unstable behavior of solar wind plasma between 0.3 and 1 au via the Nyquist criterion, applying it to fits of $\sim1.5$M proton and $α$ particle Velocity Distribution Functions (VDFs) observed by \emph{Helios I} and \emph{II}. The variation of the fraction of unstable intervals with radial distance from the Sun is linear, signaling a gradual decline in the activity of unstable modes. When calculated as functions of the solar wind velocity and Coulomb number, we ob…
We present the analysis of narrowband whistler wave signatures observed in the inner heliosphere (0.3-1 au). These signatures are bumps in the spectral density in the 10-200 Hz frequency range of the AC magnetic field as measured by the search coil magnetometer on board the Helios1 spacecraft. We show that the majority of whistler signatures are observed in the slow solar wind (<500 km/s) and their occurrence increases with the radial distance (R), from ~3% at 0.3 au to ~10% at 0.9 au. In the fast solar wind (>600 km/s), whistler activity is significantly lower; whistler signatures start to ap…
The linear scaling of the mixed third-order moment of the magnetohydrodynamic fluctuations is used to estimate the energy transfer rate of the turbulent cascade in the expanding solar wind. In 1976 the Helios 2 spacecraft measured three samples of fast solar wind originating from the same coronal hole, at different distance from the sun. Along with the adjacent slow solar wind streams, these represent a unique database for studying the radial evolution of turbulence in samples of undisturbed solar wind. A set of direct numerical simulations of the MHD equations performed with the Lattice-Boltz…
The solar wind in the inner heliosphere has been observed by Parker Solar Probe (PSP) to exhibit abundant wave activities. The cyclotron wave modes in the sense of ions or electrons are among the most crucial wave components. However, their origin and evolution in the inner heliosphere close to the Sun remain mysteries. Specifically, it remains unknown whether it is an emitted signal from the solar atmosphere or an eigenmode growing locally in the heliosphere due to plasma instability. To address and resolve this controversy, we must investigate the key quantity of the energy change rate of th…
We analyzed intermittency in the solar wind, as observed on the ecliptic plane, looking at magnetic field and velocity fluctuations between 0.3 and 1 AU, for both fast and slow wind and for compressive and directional fluctuations. Our analysis focused on the property that probability distribution functions of a fluctuating field affected by intermittency become more and more peaked at smaller and smaller scales. Since the peakedness of a distribution is measured by its flatness factor we studied the behavior of this parameter for different scales to estimate the degree of intermittency of our…
We report proton temperature anisotropy variations in the inner heliosphere with Parker Solar Probe (PSP) observations. Using a linear fitting method, we derive proton temperature anisotropy with temperatures measured by the Solar Probe Cup (SPC) from the SWEAP instrument suite and magnetic field observations from the FIELDS instrument suite. The observed radial dependence of temperature variations in the fast solar wind implies stronger perpendicular heating and parallel cooling than previous results from Helios measurements made at larger radial distances. The anti-correlation between proton…
Deviations of the interplanetary magnetic field (IMF) from Parker's model are frequently observed in the heliosphere at different distances r from the Sun. Usually, it is supposed that the IMF behavior corresponds to Parker's model overall, but there is some turbulent component that impacts and disrupts the full picture of the IMF spatial and temporal distribution. However, the analysis of multi-spacecraft in-ecliptic IMF measurements from 0.29 AU to 5 AU shows that the IMF radial evolution is rather far from expected. The radial IMF component decreases with the adiabatic power index (|Br| \si…
The Interstellar Boundary Explorer (IBEX) has now operated in space for 7 years and returned nearly continuous observations that have led to scientific discoveries and reshaped our entire understanding of the outer heliosphere and its interaction with the local interstellar medium. Here we extend prior work, adding the 2014-2015 data for the first time, and examine, validate, initially analyze, and provide a complete 7-year set of Energetic Neutral Atom (ENA) observations from ~0.1 to 6 keV. The data, maps, and documentation provided here represent the 10th major release of IBEX data and inclu…
The first two orbits of the Parker Solar Probe (PSP) spacecraft have enabled the first in situ measurements of the solar wind down to a heliocentric distance of 0.17 au (or 36 Rs). Here, we present an analysis of this data to study solar wind turbulence at 0.17 au and its evolution out to 1 au. While many features remain similar, key differences at 0.17 au include: increased turbulence energy levels by more than an order of magnitude, a magnetic field spectral index of -3/2 matching that of the velocity and both Elsasser fields, a lower magnetic compressibility consistent with a smaller slow-m…
The shape of the electron velocity distribution function plays an important role in the dynamics of the solar wind acceleration. Electrons are normally modelled with three components, the core, the halo, and the strahl. We investigate how well the fast strahl electrons in the inner heliosphere preserve the information about the coronal electron temperature at their origin. We analysed the data obtained by two missions, Helios spanning the distances between 65 and 215 R$_S$, and Parker Solar Probe (PSP) reaching down to 35 R$_S$ during its first two orbits around the Sun. The electron strahl wa…
We report results of a theoretical investigation of polarization of the X-ray emissions induced in charge-exchange collisions of fully stripped solar wind ions C$^{6+}$ and O$^{8+}$ with the heliospheric hydrogen atoms. The polarization of X-ray emissions has been computed for line-of-sight observations within the ecliptic plane as a function of solar wind ion velocities, including a range of velocities corresponding to the slow and fast solar wind, and Coronal Mass Ejections. To determine the variability of polarization of heliospheric X-ray emissions, the polarization has been computed for s…
The solar wind carves a bubble in the surrounding interstellar medium (ISM), known as the heliosphere. Charged interstellar dust grains (ISDG) encountering the heliosphere may be diverted around the heliopause or penetrate it depending on their charge-to-mass ratio. We present new calculations of trajectories of ISDG in the heliosphere, and the dust density distributions that result. We include up-to-date grain charging calculations using a realistic UV radiation field and full 3-D magnetohydrodynamic fluid + kinetic models for the heliosphere. Models with two different (constant) polarities f…
Models play an important role in our understanding of the global structure of the solar wind and its interaction with the interstellar medium. A critical ingredient in many types of models are the charge-exchange collisions between ions and neutrals. Some ambiguity exists in the charge-exchange cross-section for protons and hydrogen atoms, depending on which experimental data is used. The differences are greatest at low energies, and for the plasma-neutral interaction in the outer heliosheath may exceed 50\%. In this paper we assess a number of existing data sets and formulae for proton-hydrog…
We propose a transport theory for the kinetic evolution of solar-wind electrons in the heliosphere. We derive a gyro-averaged kinetic transport equation that accounts for the spherical expansion of the solar wind and the geometry of the Parker-spiral magnetic field. To solve our three-dimensional kinetic equation, we develop a mathematical approach that combines the Crank--Nicolson scheme in velocity space and a finite-difference Euler scheme in configuration space. We initialize our model with isotropic electron distribution functions and calculate the kinetic expansion at heliocentric distan…
Switchbacks are discrete angular deflections in the solar wind magnetic field that have been observed throughout the heliosphere. Recent observations by Parker Solar Probe (PSP) have revealed the presence of patches of switchbacks on the scale of hours to days, separated by 'quieter' radial fields. We aim to further diagnose the origin of these patches using measurements of proton temperature anisotropy that can illuminate possible links to formation processes in the solar corona. We fitted 3D bi-Maxwellian functions to the core of proton velocity distributions measured by the SPAN-Ai instrume…
