Zone of Preferential Heating for Minor Ions in the Solar Wind

The properties of minor ions in the solar wind are important indicators of the state of the solar corona and the heating processes it undergoes as it expands. Evidence of preferential heating of minor ions has been observed in the solar wind, and is believed to be active up to a certain radial boundary, beyond which thermalization due to Coulomb collisions predominates. Building on previous works that calculated the location of this boundary for alpha particles, this work calculates the outer radial boundary of the zone of preferential heating for selected heavy ions in the solar wind. This analysis uses ion data from the Solar Wind Ion Composition Spectrometer and proton data from the Solar Wind Electron, Proton, and Alpha Monitor aboard the Advance Composition Explorer spanning the years from 1998 through to 2011. Observations of proton and ion temperatures, velocities, and densities, and fixed parameters derived from temperature, density, and velocity scaling laws are used in a model function to predict the radial boundary and excess temperature at the boundary via reduction of the χ 2/degrees of freedom statistic. In this study, the values of the radial boundary of the preferential heating zone were quite high when compared to what was previously found for alpha particles, but a clear scaling relationship between excess temperature and ion properties was observed.


Introduction
A plasma in thermodynamic equilibrium is expected to have equal temperatures and velocities across all its constituent species.Remote imaging of the solar atmosphere and inner corona as well as measurements at 1 au reveal the highly nonthermal nature of the solar wind, with thermal in this context referring to the state of thermodynamic equilibrium and thermalization, the processes driving the plasma toward thermodynamic equilibrium.Alpha particles and heavy ions (mass > 4 amu) that are hotter than protons and electrons and differential streaming proportional to the Alfvén speed are indicative of dissipation mechanisms that preferentially heat some species more than others.Minor-ion properties may be key in tracing preferential heating in the solar wind, and could potentially help to identify the combination of mechanisms heating the corona.The launch of Parker Solar Probe (PSP) in 2018 and Solar Orbiter (SO) in 2020 have provided the opportunity to compare theory/models to in situ measurements of solar-wind densities, velocities, and temperatures at unprecedented solar distances.The in situ plasma instruments aboard both spacecraft provide a combination of proton and alpha measurements, with SO providing additional minor-ion measurements (Müller et al. 2013;Fox et al. 2016;Kasper et al. 2016;Owen et al. 2020).These missions have and will continue to provide direct insight into key signatures of coronal heating and expansion.Measurements made by spacecraft at 1 au, however, are still needed in combination with theory to predict and add context to data from PSP and SO.
As temperatures within the transition region of the solar atmosphere rise to 10 3 times higher than photospheric levels, the plasma remains highly collisional.Equal kinetic temperatures are maintained for ions and electrons in the inner corona up to about 0.3 solar radii (R S ) above the surface due to the frequent collisions that serve to thermalize the plasma (Neugebauer 1976;Geiss et al. 1995;Ko et al. 1997;von Steiger & Zurbuchen 2011).Above the low corona, the frequency of ion-ion Coulomb collisions drops due to expansion driven by rising temperatures; consequently, electron collisions begin to dominate at larger radii.Differences in kinetic temperature between electrons, protons, and heavy ions are seen to develop and grow with distance from the Sun in the presence of some dissipative process that preferentially heats some species over others (Landi & Cranmer 2009;Verscharen et al. 2019).These differences may be due to the dependence of the heating processes on mass, charge, or a combination of both, thus making the study of heavy ions, with their multiple charge states, critical to unraveling the heating mechanisms in the solar corona and solar wind.
Heavy ions make up only a small fraction of the plasma in the corona and solar wind (<0.1%).Their varying mass and charge make them useful test particles for tracing physical processes in the plasma.Many of these heavy ions start off neutral in the photosphere.As they expand up into the chromosphere, increasing electron temperatures and densities create an environment where ionization (and recombination) can proceed to strip electrons from these atoms until their charge states reflect the conditions in the corona (Hundhausen et al. 1968;Hundhausen 1969;Bochsler 2007).As the plasma expands out of the corona, eventually the electron density is low enough that collisional ionization and recombination cease.The charge states of the heavy ions become frozen-in and remain unchanged on their outward journey into the heliosphere (Esser et al. 1998;Landi et al. 2012).While ionization and recombination cease, the weakly collisional solar wind still undergoes infrequent Coulomb collisions, which act to thermalize the plasma, driving the temperatures of ions and protons to approach each other (Marsch & Goldstein 1983;Marsch 2006).It is important to differentiate between the electron temperatures that can be derived from heavy-ion charge states observed in the solar wind, which reflect temperatures in the corona where freeze-in occurs, and the kinetic temperatures of ions and protons, which reflect the thermal spread of their velocity distributions.In this work, we explore the kinetic temperatures of preferentially heated heavy ions.
Observations of temperature ratio variation in relation to the prevalence of Coulomb collisions can be traced back to determine temperatures of minor ions close to the Sun.By tracing this evolution from 1 au back to within a few R s of the solar surface, one can determine the radial distance at which the effect of Coulomb thermalization supersedes that of any preferential heating mechanism.For a schematic of this process, wherein the expanding solar wind is divided into three radial zones, a collisional zone, a preferential heating zone (PHZ), and a relaxation zone, see Figure 3 in Kasper et al. (2017).
In the absence of direct observations of the dissipation mechanism responsible for coronal heating, the nonthermal signatures in the solar-wind plasma can provide information about the dominant processes heating the corona.The solar wind emerging from the inner corona is highly nonthermal, as evidenced in observations of unequal temperatures across species, proton beams in proton velocity distribution functions (VDFs), and large temperature anisotropies in protons and alpha particles by the Helios 1 and 2 spacecraft (Marsch et al. 1982;Hefti et al. 1998;Tu et al. 1998;Marsch 2012).These nonthermal features are either generated locally or remnants of processes that occurred closer to the Sun.Statistical studies have shown that the intensity and occurrence rate of these nonthermal properties are a function of the Coulomb collisions that occur as the solar wind propagates to the observing spacecraft, suggesting that Coulomb collisions thermalize the solar wind when their effects exceed any remaining preferential heating processes (Maruca et al. 2013;Kasper et al. 2013).
Measurements of ion temperatures from the inner corona out to 1 au reveal mass or supra-mass proportional temperatures across species.(e.g., Kasper et al. 2008;Tracy et al. 2016;Wilson et al. 2018).Differential streaming between protons and minor-ion species proportional to the Alfvén speed has been observed in the inner heliosphere (Alterman et al. 2018;Šafránková et al. 2021).Such departures from local thermodynamic equilibrium are indicative of a preferential heating process potentially driven by differences in particle mass and charge.Vetting out the actual relationship between temperature profiles and ion properties could aid in the identification of the predominant heating mechanism(s).
Among the proposed heating theories is the generation of ion cyclotron waves by anisotropic dissipation of large-scale MHD waves perpendicular to the background magnetic field (Hollweg & Isenberg 2002;Isenberg & Vasquez 2007;Cranmer & van Ballegooijen 2003).Perpendicular heating and proton-alpha drifts could also be driven by dissipation of high-frequency Alfvén/ion cyclotron waves generated by plasma instabilities (Klein et al. 2018).Stochastic heating, wherein fluctuations on the scale of ion gyroradii disturb ion orbits perpendicular to the magnetic field, leads to perpendicular kinetic energy diffusion (Chandran & Hollweg 2009;Chandran et al. 2013).While this list is not comprehensive (see Velli et al. 2015, Cranmer & Winebarger 2019, and Verscharen et al. 2019 and the references listed therein for an overview of coronal heating models), the above theories identify signatures produced in plasma properties that are measurable and traceable.This work is focused on evaluating mechanisms driven by wave-particle interaction, which should impart dependencies on mass m or charge-to-mass ratio (q/m) in trends of unequal temperatures against kinetic properties and derived quantities.
Given the collisional frequency between two species, the Coulomb number, N c , is the estimated accumulation of Coulomb thermalization times as the plasma travels at the bulk speed from the Sun to the point of observation.A distinction must be made between calculations of Coulomb collisions using plasma conditions at 1 au, referred to herein as N c , and calculations taking into account the radial variation of the plasma kinetic properties, referred to as A c .Using measurements from the Solar Wind Ion Composition Spectrometer (SWICS) aboard the Advanced Composition Explorer (ACE), and following the aforementioned formalism, Tracy et al. (2015) showed that the temperature ratios T i /T p of various species of minor ions to protons in the solar wind are well ordered by N c , which can be expressed as follows: where ν ip is the collisional frequency between some ion species i and protons p, and U is the solar-wind bulk speed.Note here the use of the subscripts "i" and "p," which will be used throughout this paper, and may differ from those used in referenced literature.Calculations of collisional frequencies of all possible species pairings detected by SWICS revealed that the most frequent, and thus most significant, Coulomb interactions occur between ions and protons.Tracy et al. (2016) extended their initial study to determine the mass dependency of the T i /T p for collisionally young solar wind, finding that T i /T p ∝ 4/3m i /m p .Kasper et al. (2017) developed and fit an analytical model of solar-wind heating to Wind Solar Wind Experiment (SWE) measurements to determine the proposed radial boundary of the PHZ, determining that coronal heating is likely an ongoing process that extends to 20-40 R s for He 2+ (α) ions.Kasper & Klein (2019) later compared the time evolution of their calculated radial boundary, finding a correlation with average sunspot number and that the PHZ boundary varies with the solar cycle along with the Alfvén surface, with a correlation coefficient greater than 0.9.
Here, the same procedure described in Kasper et al. (2017) is employed to find the outer boundary of the zone of preferential heating for minor ions.To gain insight into the heating sources that may be present, mass and charge-per-mass scaling relationships with various properties are explored.The remainder of the paper proceeds as follows.Section 2 gives a brief description of the data set and adjustments made to the model; Section 3 gives the radial boundary and excess temperature decay profiles versus collisional age for selected ions; and the results, sources of error, and next steps are discussed in Sections 4 and 5, respectively.

SWICS and SWEPAM Data Sets
The SWICS aboard ACE uses a combination of time-offlight, total energy, and energy-per-charge measurements to identify and characterize protons, alphas, and 102 heavy ions in the solar wind (Gloeckler et al. 1998;Stone et al. 1998;von Steiger et al. 2000).This study uses the 2 hr averaged density, velocity, and thermal velocity derived from the VDFs of all heavy ions detected by the SWICS instrument.Proton measurements come from the 12 minutes SWICS/Solar Wind Electron, Proton, and Alpha Monitor (SWEPAM) merged proton data product made available through the ACE Science Center, and are averaged down to match the 2 hr resolution of the minor-ion data.The data sets cover the period from 1998 August through to 2011 January, with interplanetary coronal mass ejections removed.

Implementation of the Radial Boundary Model
The following is a brief summary of the theory behind and implementation of the radial boundary model for use with heavy ions.For an in-depth description, see Kasper et al. (2017).The purpose of the derivation is to determine the evolution of excess temperature with collisional age.As such, the radial evolution of the equilibrium temperature excess, ε = T i /T p − 1, must be related to A c , which, as previously stated, can be thought of as the radial accumulation of Coulomb collisional times.The model is based on several major assumptions about the preferential heating of ion species in the solar corona; namely, that there is a zone of preferential heating in the corona, below which some unspecified heating processes dominate over thermalizing Coulomb collisions, resulting in nonthermal characteristics and preferential heating of heavy-ion species relative to protons.This preferential ion heating brings each species to a particular value of ε.At the outer boundary, R b , of this zone, where each species has a welldefined temperature excess, ε b , the cumulative effects of Coulomb collisions begin to surpass the contribution of the heating source, which may be localized or ongoing, resulting in a decay in excess temperature to some residual value, ε r , observed at the spacecraft.
Beginning with the energy equation for each individual species, Kasper et al. (2017) derive an expression for the radial evolution in excess temperature, assuming that while other heating processes may be ongoing, the preferential heating source cuts off at R b , that the radial density of both species follows the same function, and that for all ion species, the most significant Coulomb collisions occur between ions and protons, as determined by Tracy et al. (2015).These assumptions allow for cancellations and substitutions that allow dε/dr to be written in terms of Coulomb collisions, bulk speed, and ion-toproton mass density ratio: Here, ν ip is the ion-to-proton collisional frequency, and ν pi is the reversed interaction.Ions and protons are denoted by i and p, respectively.U is the solar-wind bulk speed, and F is the ionto-proton mass density ratio.Coulomb collisions between two species are accounted for according to the expression derived in Hernández et al. (1987): Using this frequency, a more rigorous expression for the accumulated effects of Coulomb thermalization compared to Equation (1) accounting for the radial variation (A c ) in the plasma parameters can be derived.Note that Kasper et al. (2017) used m i = 4 amu for alpha-particle mass.This value is kept as a variable here to generalize to all minor ions.Aside from this, the equation shown below is identical to Kasper et al. (2017): where ˜is a reduced collisional frequency, and all subscripts specific to alpha particles have been generalized to i. See Kasper et al. (2017) for the full expansion of the left side of the differential equation.Here, δ and σ are fixed parameters accounting for the radial dependence of the ion temperature and velocity scaling, respectively.The scaling factors fall into a range reported by Marsch et al. (1982) and Hellinger et al. (2011).Kasper & Klein (2019) report a linear relationship δ = 0.813-1.037σbetween these values that minimizes rms error between calculated values of the proposed boundary of preferential heating and the Alfvén point as well as a δ = 2/3-4/3σ relation based on the collisional age equation.Variation of δ and σ within observed ranges and along these lines did not result in significantly different R b or trends in mass ratio and q/m.Values of δ = 0.814 and σ = −0.05are used here for consistency with previous work.Equation (4) is fitted to the observed decay in ε with A c to determine the best-fit values of R b , ε b , and ε r .

Data Selection
Of the 103 ions characterized by SWICS, analysis for the following are included in this paper: He 2+ , C 4-6+ , O 5-8+ , Ne 8-9+ , Mg 6-12+ , Si 6-12+ , and Fe 9-13+ .Density, speed, and thermal speed, calculated from moments of each ion's VDF, are used at each valid 2 hr timestep to determine excess temperature and collisional frequency.The number of valid data points at all solar-wind bulk speeds varies between about 40,000 and 50,000 for the most abundant ions.This work was completed in two speed ranges: over all solar-wind speeds and on v < 500 km s −1 .The exclusion of solar-wind speeds above 500 km s −1 to focus on collisional slow wind yields a broader range of A C values (see Figure 2 in Kasper et al. 2017), but also reduces the data set number by about 10,000 timesteps.
The binning routine employed here excludes bins in A c that do not contain enough data points, or those with an irregular distribution of data.This requirement only excludes the highest and lowest A c bins for all ions, so though the curves might be truncated, there is no need for any interpolation within endpoints.Nearly all of the ions are skewed toward low A c , leading to sparsely populated bins at high A c and increasing uncertainty at the tail end of the decay curve.In some cases, this limits the model's ability to converge on a fit; these ions are considered nonviable and are not used in this study.

Minor-ion Temperature Profiles
The outer boundary of the PHZ for a given ion is determined by fitting the modeled temperature decay profile to the observed excess temperature when binned by collisional age via reduction of the χ 2 /degrees of freedom (dof) statistic.Excess temperature decay profiles and model fits calculated for selected ions from the SWICS data set at all solar-wind speeds are shown in Figure 1.The contents of Figure 2 are the same, aside from the solar-wind bulk speed, which is limited to all speeds below 500 km s −1 .In these figures, the black diamonds represent excess temperature calculated and binned from SWICS data, where temperatures were derived in the manner described by Shearer et al. (2014).The gray, vertical crosshairs represent the 1σ uncertainty in ε for individual bins, while the horizontal are representative of the overall average of the uncertainties in A c .The red curves depict the modeled decay, and the total number of binned data points is denoted by "Npts."Free parameters R b , boundary ε b , and residual ε r are printed on each plot, though in keeping with the convention introduced in Kasper et al. (2017), ε 1 denotes residual excess temperature (ε r ) and ε 0 denotes the difference between the boundary and residual excess The left sides of Figures 1(a)-(f) and 2(a)-(f) represent collisionally young solar wind, and the right sides collisionally old wind.Limiting solar-wind speeds to below 500 km s −1 causes the modeled and observed curves to shift to slightly higher A c , as the observed data in each bin has a higher concentration of collisionally old wind.This is consistent with the known properties of the slow and fast wind.Though profiles for only six ions are shown in this paper, the shift is present in all the curves analyzed.Nearly every ion also exhibits slightly lower ε values at low A c under the slow-wind limit, likely due to either preferential heating being more prevalent in the fast wind than the slow, or truncation in the lowest A c wind.Although these trends are consistent, for most ions, they are small enough to fall within the error margins for both ε and A c .
A "bump" that occurs at low A c in some ions, as reported by Tracy et al. (2015Tracy et al. ( , 2016)), does not appear to be present when excluding solar-wind speeds above 500 km s −1 .This is evident when comparing Figures 1(d) with 2(d), Figures 1(e) with 2(e), and Figures 1(f) with 2(f).The bump is a clear deviation from the expected exponential decay profile, and its occurrence in collisionally young wind may impact both predictions of the radial boundary and excess temperature at the boundary.For every analyzed ion, the bump begins around A c = 0.01 and peaks at about A c = 0.1.All or the majority of this range, 0.01 A c 0.1, is present in the observed range for all ions in this study, regardless of the slow-wind limit.This indicates that the flattening of the bump is not due to the truncation of collisionally young wind, but rather the removal of a subset of the wind with properties that may be independent of collisional age, the collisionless fast wind.Currently, the model does not account for this feature, but is still capable of reproducing observed boundary and residual excess temperature values (bins farthest to the left and right along the curve), and to an extent the shape of the relaxation curve at high Ac.Trends in the R b prediction with ion features are shown in Figure 3.
Ideally, fully thermalized, collisionally old solar wind would reach an asymptotic ε ∼ 0. Previous work indicates the existence of a nonzero, residual excess temperature at high A c corresponding to larger radial distances from the Sun (Kasper et al. 2013(Kasper et al. , 2017;;Tracy et al. 2016).For most cases in this work, the modeled relaxation curve appears to level out by A c ∼ 10, while the excess-observed excess temperature bins still exhibit a marked decrease, albeit a less shallow reduction than at intermediate A c .Most minor ions in this study never achieve excess temperature values <1, and that residual increases with increasing mass and charge-per-mass, as can be seen in the right panels of Figures 4 and 5.
Regardless of solar-wind speed, the R b predictions here extend to a much broader range than the results of Kasper et al. (2017).χ 2 /dof and rms values tend to be higher and more variable than what was found for alpha particles using Wind/ SWE data.R b for alpha particles lies at about 96 R s , and heavier ions extend to an even greater radial distance according to a weak charge-to-mass ratio and mass scaling.That the alphaparticle R b reported here is above the upper limit may be explained by differences in implementation: velocity range, σ and δ variables, and also by inherent differences between the SWICS and SWE data sets.This model is sensitive to the time resolution of the data, and SWICS and SWE are known to report different proton densities.The years included in these studies are different, comprised of varying proportions of solar cycles 22 (Wind only), 23, and 24.Accordingly, the R b values reported here for minor ions using the SWICS data set can be taken as an upper limit on the PHZ boundary, rather than an exact prediction.
R b versus charge-to-mass ratio, R b versus mass, and R b versus charge-squared-to-mass ratio appear in Figures 3(a), (b), and (c), respectively.In Figures 3(a (2019) for alpha particles.These plots show results for speeds below 500 km s −1 .Table 1 shows the r 2 correlations for all ions as well as correlation coefficients of the trends in observed boundary and residual excess temperature.The top row includes all speeds, while the second row only shows wind speeds below 500 km s −1 .Though there is a visible relationship between ion mass and R b , the r 2 correlation is only 0.27 for slow speeds, while the correlation coefficient of ò b and ion mass is high.Aside from R b versus q/m, the correlation coefficients and r 2 are quite similar between the two wind types.The correlation coefficients of ε b with ion mass and q/m are both higher in the slow wind.High charge-squared-to-mass ratio and high ion masses produce the highest radial boundary values.The strongest relationship is with mass, and Figure 3(c) demonstrates that, for a given element, R b tends to increase with increasing charge state in the heaviest ions and decrease with charge state in lower-mass ions, such as N, C, and O.The charge-to-mass and charge-squared-to-mass ratio plots show that, with some deviations, some species fall along lines with unique slopes.These relationships may be explored further in future work.
The scaling in excess temperature with mass is compared to a model of stochastic heating in the corona and inner heliosphere.When counterpropagating, low-frequency, high-  temperatures as a is a constant related to the velocity power spectrum of the turbulence and is taken to be 1/4 in this case.Proportionality constants of 0.71 for protons and 1.0 for ions were used by Chandran (2010).Ion mass and charge are A and Z, respectively.It must be noted that the Chandran (2010) model focuses on r < 15 R s , and the scaling factor found therein was chosen to match observed perpendicular temperature ratios in the corona.Here, 0.71 is not used and the scaling factor is left as a free parameter.
Figure 4 displays the excess temperature versus charge-tomass ratio at ACE, along with predictions of the trend in temperature anisotropy for various masses made by the stochastic heating model presented in Chandran (2010).predictions of the ε curve with q/m at each of the masses used in this study, using the equations described above from the Chandran (2010) model.Ions of the same species would be expected to fall along the gray line that corresponds to their mass.At the PHZ boundary (Figure 4(a)), only He, C, and a few heavier ions with charge-to-mass ratio approaching 0.5 fall near their mass lines (first from the bottom for alphas, second from bottom for C, third from bottom for N, etc.).Ions of some other elements compare much less favorably with the predictions.In the extreme case, for Fe ions (red stars), the lowest charge states fall between the mass 20 and mass 24 line (rather than 56) but follow a similar trend with q/m.The higher charge states follow an opposite trend with q/m, crossing several mass lines.Because no directional distinction is made for the observed SWICS ion temperatures, their distribution and evolution are not due purely to a perpendicular heating source.It is known that alpha perpendicular temperatures decay more rapidly than parallel as the radial distance from the Sun increases (Marsch et al. 1982).The flatter observed ε curves for each species could be reflective of a reduced decay rate due to the presence of parallel temperatures.At 1 au, none of the ions land very near to the curve corresponding to their mass, though a trend with a q/m ratio does remain.Here, in both the PHZ boundary (Figure 4(a)) and the 1 au (Figure 4(b)) cases, ε has a charge state dependence, though that dependence is less distinct for thermalized wind at 1 au. Figure 5 shows the relationship between ε values and ion-toproton mass ratio m i /m p at the preferential heating boundary and at au.Here, the symbols represent each ion, but the color gradient corresponds to different charge states, with the lowest charge states marked in the lightest blue and the highest charge states in the darkest blue.Ions of the same element appear in columns.The excess temperatures shown here are in good agreement with the T i /T p ∼ 4/3 * m i /m p found in Tracy et al. (2016), depicted in Figures 5(a

Discussion
Using the SWICS data set, it has been shown here that the heating model from Kasper et al. (2017) predicts a massdependent outer radial boundary of the PHZ, though with significantly higher uncertainty than it does for alphas measured by Wind/SWE.There are a number of potential causes for this.Note.Charge states of C, N, O, Ne, Mg, Si, S, and Fe are calculated separately for q/m and q 2 /m dependencies.For all solar wind speeds, r 2 values and correlation coefficients for all species are depicted in the top row.Observed excess temperature (ε) could decay with collisional age (A c ) in a manner that differs too greatly from the modeled exponential decay for some ions.Tracy et al. 2016) found that many of the most abundant minor ions exhibit a bump in their temperature decay versus collisional age profiles at low A c .The low number density of the minor ions coupled with high ion mass leads to a lower collisional frequency and thus a higher concentration of data at low A c and sparse data at high A c when compared to what is observed for alpha particles in Kasper et al. (2017).As such, the data fed into the model from 1 au might not yield high enough A c to capture the full decay curve or some ions.The values used for the residual ε free parameter are between 0.5 for He 2+ and 30 for Fe ions, much higher than the 0.2-0.3detected by Wind/SWE.These high residuals could have a major impact on the predicted R b values, which are calculated assuming full thermalization at 1 au.
It could be that some minor ions are not fully thermalized by the time they have reached 1 au, reflected in the remaining mass dependence observed in residual excess temperatures.The ion heating model, which assumes full thermalization, would then predict R b using a truncated data set.Additionally, this model assumes an unspecified preferential heating source that cuts off at the outer boundary.Such an assumption does not preclude the presence of some other, ongoing heating source.One must also consider how the results vary due to differing measurement techniques between the Heavy Ion Sensor ACE/SWICS and Wind/SWE.Due to differences in the size of the SWICS and SWE data sets, this analysis could not be completed in 25 km s −1 windows, as was done in Kasper (2017).Instead, the model was run on all solar-wind speeds and again with 500 km s −1 as the upper limit.The exclusion of solar-wind speeds higher than 500 km s −1 did reduce the overall available events to be binned by about 10,000.This reduced amount only impacted the quality of observed data for the least abundant ions.The exclusion of the fast wind, as expected, shifted the observed and model ε decay curves to higher collisional ages and higher temperature ratios.In most cases, the calculated R b was lowered by 1−30 R s .The magnitude of this reduction had no apparent dependence on ion properties.The bump feature in the relaxation curve either became present or more prominent when wind speeds greater than 500 km s −1 were included, which significantly impacted reported χ 2 /dof values.
Though the χ 2 /dof < 2 found by Kasper et al. (2017) was not achieved by all the ions included in this study, ε decay curves for alphas were fitted with χ 2 /dof ∼ 3.9 in the slow wind, and χ 2 /dof ∼ 5.1 in all speeds wind.χ 2 /dof values of below ~5 were found for more abundant ions with a few exceptions, indicating that the model may not be capturing aspects of the temperature ratio decay for some heavy ions.The model either did not converge or resulted in poor fits with very high χ 2 /dof values for less abundant ions, as is the case with C 4+ .Among the abundant ions, there are still deviations from monotonic exponential decay, such as the bump feature seen at lower A c in O 6+ and Mg 9+ in Figures 1(c) and (d).The bump is either significantly reduced or not present when considering wind below 500 km s −1 , and though the inclusion of higher speeds does increase both predicted R b and χ 2 /dof, it does not significantly affect overall trends in R b versus ion mass and charge-to-mass ratio.Note also the slight steepening of the curve at the highest and lowest A c values for some of the ions, such as C 6+ .Less abundant ions such as C 4+ , O 5+ , Ne 9+ , and Mg 6+ all have ε curves that deviate significantly from expected exponential decay, likely driving the uncertainty.The standard deviation in each bin of observed ε is also much higher than what is found in the more abundant ions.
The right panels of Figures 4 and 5 reveal that at 1 au none of the ions included in this study completely relax to the residual values found in Kasper & Klein (2019).In fact, there is a clear relationship between ε r and charge-to-mass ratio as well as ε r and m i /m p , which may partially be an instrumental effect.As SWICS steps through higher energy-per-charge, ) deflection voltage windows on the entrance system (here ρ is a step-dependent growth factor and i signifies an E/q step), the gaps between the window centers grow (Gloeckler et al. 1998;Shearer et al. 2014).Ions with higher m/q (or lower q/m for the sake of consistency with the rest of the paper) are measured in higher-energy windows.Figure 6(a) depicts the range of energy windows that would be used to measure typical ion speeds, 200-800 km s −1 .The two ions shown, 2+ and Fe 8+ , represent the high and low end of the range of q/m detected by SWICS, respectively.Note that this is not the full range, as protons are detected in a different channel from the heavy ions, and lower charge states of Fe are far less common than Fe 8+ .Due to the logarithmic spacing of the E/q steps, the lower E/q values are closer together than the higher E/q values, which, in turn, implies that the resolution of the VDF width varies across the energy-per-charge scan.This variation creates uncertainty in the widths of the distributions in both E/q and m/q, with the lower m/q (or higher q/m) ions having lower uncertainty.The thermal spread of the distributions in E/q, with wider thermal widths having less uncertainty than narrow, propagates through as temperature is determined from the second moment of each ion's VDF.It may be that this uncertainty is on order of ε r , and calculations of excess temperature for fully or nearly thermalized minor ions are not possible with the SWICS instrument.
In order to quantify the effects of the gaps between the logarithmically spaced energy windows sampled by the SWICS instrument, an idealized, continuous VDF was compared to one built from SWICS energy window centers.The continuous VDFs were assigned varying ion speed peaks and thermal widths.The centers chosen for this were 400, 600, 800, and 1000 km s −1 , and the thermal speeds were 60, 120, and 180 km s −1 .These parameters were then used to model a VDF from SWICS noncontinuous sampling, and the recovered moments were calculated.The recovered thermal speeds were then compared to the input, shown in Figure 6(b).This was completed for alphas and Fe 8+ to look at the effects on a broad range of charge-to-mass ratios.The same was done for Fe 8+ to show the full range in recovered v th .Keep in mind that high-q/m ions like alphas are measured in lower E/q windows, while low-q/m ions like Fe 8+ are measured in higher E/q windows.Alphas and Fe 8+ traveling at the same speed would be measured in two very different E/q windows, and low-q/m ions are measured over a much broader range of E/q.At low speeds, or in lower-energy windows, the ratio between recovered v th and input v th remains close to one, but for higher-energy windows, which would be used to sample lowerq/m ions, this ratio increases, becoming higher at lower thermal speeds than it was for alphas.Of the thermal widths and ion speeds used here, the lowest v th,rec /v th is about 1.005, and the highest is 1.11, with a clear energy dependence.The high estimates, however, are only achieved by using nonphysical values in the solar wind (very fast and very cold).As such, the highest error likely to be introduced in real solar wind is about 5%.This is not enough to fully account for the highest residuals seen in the SWICS data.

Conclusions
The results shown here demonstrate that the PHZ boundary model from Kasper et al. (2017) can be applied to SWICS 2 hr data to provide an upper limit on the boundary of preferential heating for minor ions.Although the R b values are somewhat higher than expected, the more abundant ions have produced results qualitatively consistent with heating theories.Comparison with a model of stochastic heating highlights the chargedependent stratification of temperatures within each species.The high R b and high ε r are likely tied to each other, and, depending on their cause, could also be driving up χ 2 /dof values.Recent work by Johnson et al. (2023) has shown, using PSP and Wind data, that the preferential heating of alpha particles persists up to about 0.27 au, beyond which Coulomb collisions provide the dominant thermalizing effect.While this is lower than the alpha-particle result shown here with SWICS data, it is within the range of SWICS R b predictions for ions with a q/m of about 0.5.Further analysis is needed to determine the impact of the 2 hr versus 12 minute cadence on the model predictions.Direct measurements of temperature ratios from the Heavy Ion Sensor (HIS) aboard SO will be able to verify predictions for heavy ions with predicted R b > 60 R s (Owen et al. 2020).
This subset of the SWICS data set does exhibit boundary dependencies on the associated ion properties, which could be an indication that the heating mechanism is indeed tied to ion gyrofrequencies and dependent on q/m.Again, verification of the higher R b values predicted here will aid in discerning this.
We have quantified the effects of ACE/SWICS E/q stepping when compared to an idealized, continuously sampling instrument.The uncertainty introduced by this is only about 5%, not enough to fully account for the observed trends in ε r versus mass ratio and charge-to-mass ratio.Other explanations have been considered for this trend but will be reserved for future work.The fact that the q/m dependence is still present, though diminished at 1 au, could indicate that by 1 au minor ions have yet to be fully thermalized and, as such, traces of the preferential heating source are still present.
D 'Amicis & Bruno (2015) have demonstrated that solar-wind behavior can be further categorized by distinguishing between Alfvénic, non-Alfvénic slow, and fast wind by identifying periods where magnetic field and bulk speed are well correlated.Stansby et al. (2018) show that the high variability in plasma parameters in the slow solar wind could be due to the presence of wind originating from the same sources as fast wind.This highly Alfvénic slow wind, which aside from traveling at slower speeds, displays qualities more consistent with fast wind.Implementing such a distinction on the SWICS data set in conjunction with the Kasper et al. (2017) preferential heating model may lend greater insight into the heating of minor ions.At closest approach, PSP will fly within 10 R s , while its farthest perihelion was at 35 R s .This range puts PSP at the perfect distance to investigate the solar wind's transition from the preferential ion heating zone to the relaxation zone.SO's orbit will reach perihelion around 60 R s , placing it at a distance to see the edge of the PHZ for the more nonthermal ions.This investigation, along with the implementation of near-Sun PSP data and HIS data from SO, will be reserved for future work.
) and (b), the colors are associated with different elemental species, while in Figure3(c), shades of blue correspond to charge state.The pink dashed line indicates the range of R b found in Kasper et al. (2017) and Kasper & Klein

Figure 1 .
Figure 1.Excess temperature vs. collisional age for six selected ions at all solar-wind speeds.Black diamonds are observed excess temperature, while red lines are model fits.Faint gray lines signify errors based on variance within each bin.

Figure 2 .
Figure 2. Excess temperature vs. collisional age for six selected ions for solar-wind speeds below 500 km s −1 .
Figures 4(a) and 5(a) show the predictions at the boundary, and Figures 4(b) and 5(b) show conditions at 1 au.The symbols in Figures 4(a) and (b) represent each ion in this study, with colors corresponding to ion mass and all charge states for a given mass shown in the same color.The faint gray lines are

Figure 3 .
Figure 3.The top-left panel shows R b dependence on the charge-to-mass ratio.The top-right panel shows the R b charge-squared-to-mass dependence, and the bottom panel shows the mass ratio m i /m p .All panels in this figure refer only to v < 500 km s −1 .The pink dashed lines represent the R b range predicted by Kasper et al. (2017) and Kasper & Klein (2019) at the charge-to-mass ratio of alphas (0.5).
Figure5shows the relationship between ε values and ion-toproton mass ratio m i /m p at the preferential heating boundary and at au.Here, the symbols represent each ion, but the color gradient corresponds to different charge states, with the lowest charge states marked in the lightest blue and the highest charge states in the darkest blue.Ions of the same element appear in columns.The excess temperatures shown here are in good agreement with the T i /T p ∼ 4/3 * m i /m p found inTracy et al. (2016), depicted in Figures5(a) and (b) with the green dashed line.The pink dashed line corresponds to the linear fit of the modeled ε values.The slight divergence in slopes at the highest ion masses are well within the uncertainty in ε r and ε b found by the model.The difference in slopes in Figure 5(b) is expected, as the excess temperature values depicted by the symbols here are the model's prediction of residual excess temperature.When considering the evolution of ε for all ions with A c ,

Figure 5 .
Figure 5. Excess temperature vs. mass ratio calculated at preferential heating boundary and 1 au.
Table 1Radial Boundary and Excess Temperature Correlations with Ion Properties Wind Type r 2 (Rb vs. mass) r 2 (R b vs. q/m) r 2 (R b vs. q 2 /m) cc (E b vs. mass) cc (ò r vs. mass) cc (ò b vs. q/m) cc (ò r vs. q/m)

Figure 4 .
Figure 4. Excess temperature vs. charge-to-mass ratio at the radial PHZ boundary and at ACE. Gray lines denote the Chandran et al. (2013) prediction for each mass.The bottom curve is for 4 amu, then 12 amu, 14 amu, etc.

Figure 6 .
Figure 6.(a) Visualization of the full range in energy-per-charge associated with typical solar wind velocities for high q/m alphas and low q/m Fe 8+ ions.(b) The ratio of recovered vs. input thermal speeds from idealized, continuous sampling vs. logarithmically spaced energy windows.