Radial Evolution of the Near-Sun Solar Wind: Parker Solar Probe Observations

A statistical study of the radial evolution of the solar wind within 0.3 au is shown in this Letter based on Parker Solar Probe observations. We show the radial distribution of the main solar wind parameters, including the solar wind speed V sw, magnetic field ∣B∣, the number density of electrons N e , protons N p , and α particles N α , and the temperature of protons T p and α particles T α . The power-law fitting results of these parameters in the near-Sun solar wind are compared with previous radial models. We also show the radial distribution of the angle between the magnetic field B , solar wind V sw, and radial vector R . In the solar wind within 0.3 au, θBVsw , and θ BR mainly concentrate around 135°, and θRVsw almost concentrates in the region less than 20°. Furthermore, we also present the radial distribution of the relative values between the solar wind parameters, including the electric neutrality estimation ((2 × N α + N p )/N e ≃ 1 within 0.2 au), relative number density ratio (N α /N p ≃ 0.02 in slow solar wind and N α /N p ≃ 0.04 in faster solar wind), relative temperature ratio (T α /T p decreases with the increase of heliocentric distance and its decay rate is larger in faster solar wind), and differential speed (both V α − V p and (V α − V p )/V A are larger in the faster solar wind and decrease as heliocentric distance increases).


Introduction
The solar wind is the continuous outflow plasma from the solar corona (Parker 1958(Parker , 1963)).Based on observations, it can be divided into three types: first, the steady fast wind that originates from the open magnetic field lines in the coronal holes; second, the unsteady slow wind coming from the tips and edges of temporarily open streamers or from the opening loops and active regions; third, the transient wind, which is also known as coronal mass ejections (CMEs; Marsch 2006).Usually, we classify the solar wind by its velocity: fast (>600 km s −1 ), intermediate (400 ∼ 600 km s −1 ), and slow (<400 km s −1 ) solar wind (Marsch et al. 1982;Ďurovcová et al. 2017, 2021;Mostafavi et al. 2022).
The solar wind consists of protons, α particles, electrons, and very few heavy ions (von Steiger et al. 2000;Marsch 2006).In situ measurements of modern ion spectrometers on various spacecraft can measure the three-dimensional velocity distribution functions (VDFs) of major particles (protons, α particles, and electrons; Kasper et al. 2016;Durovcová et al. 2019;Ďurovcová et al. 2021;Abraham et al. 2022) and the chemical composition and ionization state of various heavy ions (von Steiger et al. 1997(von Steiger et al. , 2000)).Based on these in situ observations, some studies have learned the radial evolution of some solar wind parameters mainly beyond 0.3 au.Marsch et al. (1982Marsch et al. ( , 1982) ) and Durovcová et al. (2019) comprehensively analyzed the evolution of solar wind plasma parameters from 0.3 to 1 au using Helios observations.The radial evolution results beyond 1 au are mainly revealed by Ulysses observations (Neugebauer et al. 1996;Goldstein et al. 2000;Reisenfeld et al. 2001).
Unlike previous space missions, the in situ measurements of the Parker Solar Probe (PSP; its nearest heliocentric distance is about 10 R e ) are able to extend into the near-Sun solar wind (<0.3 au; Fox et al. 2016).Based on PSP observations, some radial properties of the near-Sun solar wind are revealed.Abraham et al. (2022) exhibited the radial evolution of electrons in the slow solar wind from 0.13 to ∼ 0.5 au through fitting electron VDFs observed by PSP.Šafránková et al. (2023) presented the first comprehensive statistical study of the evolution of compressive and noncompressive magnetic field fluctuations in the inner heliosphere.Based on observations from PSP perihelia of Encounters 3-7, Mostafavi et al. (2022) statistically revealed the alpha-proton differential flow in the near-Sun solar wind.
A statistical study of the evolution of the solar wind will help us understand the origin of the solar wind and the physical mechanisms involved.Especially, considering the near-Sun solar wind may carry more coronal source characteristics, the accurate radial evolution of the near-Sun solar wind may be not consistent with the prediction of previous models at larger heliocentric distance (Halekas et al. 2020;Abraham et al. 2022;Mostafavi et al. 2022;Šafránková et al. 2023;Liu et al. 2023).Therefore, in this Letter, we aim to present the radial evolution of solar wind parameters below 0.3 au by analyzing PSP observations from Encounters 1-15 statistically.This Letter is organized as follows.Section 2 introduces the data and methodology.Section 3 presents the radial distributions of the solar wind parameters.The discussion and summary are shown in Sections 4 and 5, respectively.

Data and Methodology
In this Letter, we statistically analyze the radial distributions of the solar wind parameters, including the magnetic field strength, number density, temperature and bulk flow speed of protons and α particles, and number density of electrons.Therefore, both the magnetic field measured by the FIELDS instrument (Bale et al. 2016)  SWEAP instrument has two ion sensors, the Solar Probe Cup (SPC; Case et al. 2020) and the Solar Probe Analyzer for Ions (SPAN-I; Livi et al. 2022).The SPC is a Faraday cup that points directly to the Sun, which is optimized for the measurement of positive ions in the outer phases of the encounter where solar-wind flows are primarily radial in the spacecraft frame (Livi et al. 2022).The SPAN-I consists of a ram-facing electrostatic analyzer and time-of-flight section, which is optimized for ion observations near the closest approach, where the inflow may be strongly nonradial in the comoving frame due to the extremely high orbital speed of the spacecraft, which will be as high as 190 km s −1 (Livi et al. 2022).During Encounters 1-3, the orbits of PSP are relatively large and the solar wind ion flow is mainly in the radial direction.As PSP gets closer to the Sun in the next encounters, its increasing lateral velocity will cause the solar wind ion flow in the spacecraft frame to move into its ram-facing side and, consequently, is within the field of view of SPAN-I.We consider data during PSP perihelia of Encounters 1-15.
Therefore, we collect ion data (the bulk speed, number density, and temperature of protons and α particles) measured by the SPC in Encounters 1-3 and SPAN-I in Encounters 4-15.The number density of electrons is from quasi-thermal noise.
In this Letter, we collect parameters of the solar wind every 20 s in each encounter of PSP.In every 20 s, if there are multiple data for a parameter, we take their average.Moreover, if there are no data in a time interval, we record the parameter here as NaN.Therefore, based on this method, we obtain 1,256,940 data for each parameter in the solar wind.All data cover a total of about 300 days.
In our statistical analyses, we also consider the deferential speed V α − V p between protons and α particles by using the bulk speed of both.This method is the same as that in Mostafavi et al. (2022).The normalization of this deferential speed is based on the local Alfvén speed V A , which is defined by the strength of the background magnetic field and the number density of protons and α particles

Radial Distributions of Solar Wind Parameters
Figure 1 shows the radial distribution of parameters in the solar wind.In this figure, except for the results of all statistical data, we also show the results in slow (V p < 400 km s −1 ), intermediate (400 V p 600 km s −1 ), and fast (V p > 600 km s −1 ) solar wind.
Figures 1( a1)-(a4) show the radial distribution of the proton bulk speed V p (solar wind speed).The magenta lines in all panels are our power-law fitting results.The black and cyan lines are the Sheeley-like empirical model ) km s −1 , where r is the heliocentric distance in unites of au) and power-law fitting results (V sw = 226.1 × (215r) 0.13 km s −1 ) of Helios observations shown in Figure 1(b) of Bale et al. (2016), respectively.By comparing our fitting results with these two models from Helios observations, we can see that these two models are only close to the results of intermediate solar wind beyond 0.3 au observed by PSP.The fitting results of total data and slow solar wind data are smaller than the previous two models.However, for intermediate and fast solar wind, our fitting results in the near-Sun solar wind are larger than Helios predictions.Bale et al. (2016).For cases of the intermediate and fast solar wind, the electron number density predicted by Helios observations is larger than that from PSP observations in the near-Sun solar wind.For cases of total data and slow solar wind, the results of the Helios model are larger than PSP observations when the heliocentric distance is less than ∼0.15 au, and smaller than those when the heliocentric distance is beyond ∼0.15 au.
Figures 1(d1)-(d4) show the radial distribution of the number density of protons N p .The solid and dotted cyan lines are fitting results of slow (N p = 10 0.89 × r −2.05 + 10 −0.22 × r −2.15 cm −3 ; here N p is the sum of the number density of proton core and beam) and fast (N p = 10 0.41 × r −1.96 + 10 −0.51 × r −1.53 cm −3 ) solar wind shown in Durovcová et al. (2019), respectively.Due to the Sun-facing thermal protection system (TPS) heat shield on board PSP, the VDFs of protons and α particles observed by PSP are incomplete, especially when PSP is at a large heliocentric distance (Kasper et al. 2016;Livi et al. 2022).Therefore, we only fit the parameters of protons and α particles below 0.3 au.Our fitting results reveal that the proton number density in the near-Sun solar wind is not consistent with the predictions of Helios observations.In the near-Sun solar wind, Helios predictions in the slow solar wind are relatively close to our fitting results of the total data and slow solar wind data, and Helios predictions in the fast solar wind are a bit close to our fitting results of intermediate and fast solar wind data.
Figures 1( e1)-(e4) show the radial distribution of the number density of α particles N α .The solid and dotted cyan lines are fitting results of slow (N α = 10 −0.70 × r −2.42 cm −3 ; here N α is the number density of the alpha core) and fast (N α = 10 −1.00 × r −1.91 cm −3 ) solar wind shown in Durovcová et al. (2019), respectively.In the near-Sun solar wind, only the prediction of fast solar wind from Helios observations seems to be close to PSP observations.
Figures 1( f1)-(f4) show the radial distribution of proton temperature T p .The solid cyan lines are a power-law fitting model (T p = 332.6 × (215r) −0.67 eV) of Helios observations in Figure 1(d) of Bale et al. (2016).For all data and slow solar wind, the prediction of the Helios observation is almost consistent with that of the PSP observation.However, for intermediate solar wind and fast solar wind, there is a large deviation between Helios predictions and PSP observations.
In this part, based on PSP observations, we present a more accurate radial distribution model of the near-Sun solar wind.The power-law fitting results (with a function in the form Y = aX b ± c, where a and b are coefficients obtained from power-law fitting, c is the overall uncertainty) are shown in Table 1.Note that our simple power-law fitting results are mainly based on PSP observations within 0.3 au.Hence, their correctness within 0.3 au is unquestionable.However, when these fitting results are extended to further radial distances, the farthest reasonable extension distance is about 1 au.This is because the radial evolution trend of the solar wind changes with increasing radial distance (more detailed analysis can be seen in Maruca et al. 2023).
We also show the radial distribution of the angle between the magnetic field B, solar wind V sw , and radial vector R, which is shown in Figure 2. In the near-Sun solar wind, q BV sw mainly concentrates around 135°, which means the magnetic field is almost quasi-antiparallel to the direction of the solar wind.When the heliocentric distance is beyond 0.3 au, the distribution of q BV sw becomes relatively chaos.The radial distribution of θ BR is almost the same as that of q BV sw .The q RV sw almost concentrates in the region less than 20°.
The radial distribution of the relative values between the solar wind parameters is shown in Figure 3. Figures 3(a1)-(a4) are electric neutrality estimations (2 × N α + N p )/N e of solar wind.We can see the value of (2 × N α + N p )/N e decreases with the increase of heliocentric distance and is ∼1 when the heliocentric distance is less than 0.2 au.The reason for this phenomenon is that the Sun-facing TPS of PSP causes the VDFs of protons and α particles measured at large heliocentric distances to be incomplete (Kasper et al. 2016;Livi et al. 2022).Therefore, we mainly focus on the results <0.3 au.
Figures 3( b1)-(b4) are the relative number density ratio between α particles and protons N α /N p .For all data and data in the slow solar wind, the median value of N α /N p is around 0.02.In intermediate and fast solar wind, the median value of N α /N p is larger and around 0.04.
Figures 3( c1)-(c4) are the relative temperature ratio between α particles and protons T α /T p .The median value of T α /T p decreases with the increase of heliocentric distance and the decay rate in the slow solar wind (from ∼7.5 at 0.05 au to ∼3.5 at 0.3 au) is significantly less than that in the faster solar wind (from ∼11 at 0.05 au to ∼5.5 at 0.3 au).
Figures 3(d1)-( d4) and (e1)-(e4) are differential speeds between α particles and protons V α − V p and (V α − V p )/V A , respectively.The results in Figures 3(d1)-(d4) indicate that the differential speed between α particles and protons decreases with the increase of the heliocentric distance and its decay rate in the intermediate solar wind (from ∼160 km s −1 at 0.05 au to ∼38 km s −1 at 0.3 au) is larger than that in the slow solar wind (from ∼60 km s −1 at 0.05 au to ∼20 km s −1 at 0.3 au).For the normalized differential speed (V α − V p )/V A , its median value also tends to decay with the heliocentric distance.Moreover, the (V α − V p )/V A in faster solar wind tends to be larger, which is <0.5 in the slow solar wind and ∼0.5 in the intermediate solar wind.Furthermore, due to the fact the distribution of V α − V p in the near-Sun solar wind is mainly concentrated in the region >0, the bulk speed of α particles is almost larger than that of protons at the heliocentric distance less than 0.3 au, which is also revealed by Mostafavi et al. (2022).

Uncertainty of Data
As shown in Figures 3(a1)-(a4), the electric neutrality condition of the solar wind can not be satisfied in the relatively large heliocentric distance.This is not a physical phenomenon and is a flaw in PSP plasma measurements.In the relatively large heliocentric distance, the Sun-facing TPS of PSP blocks the particle flow in some directions, resulting in incomplete measurements of PSP ion VDFs.However, the data of the magnetic field, quasi-thermal noise electron number density, and bulk flow speed of ions are almost unaffected.Therefore, in this Letter, the number density and temperature of ions within 0.3 au are more reliable, and the corresponding fitting results in Figure 1 use data within 0.3 au.For the radial distribution in other figures, the results within 0.3 au can be trusted.

Insufficient Data Size
In this Letter, not only do we present the statistical results of all data, but we also discuss the statistical results in different solar winds according to the solar wind speed.The data of slow and intermediate solar wind within 0.3 au are relatively sufficient.However, the fast solar wind data are relatively not enough.Therefore, the statistical results shown in the fast solar wind may have errors caused by insufficient statistical quantity.

Summary
In this Letter, based on PSP observations, we present a radial distribution of the solar wind parameters in the near-Sun solar wind.Through comparing with previous models, we find some insufficiency of previous models.For proton bulk speed V p (or solar wind speed V sw ), the fitting results of total and slow solar wind data are smaller than the previous two models from Helios observations, and our fitting results in the intermediate and fast solar wind are larger than Helios predictions to the near-Sun solar wind.For the magnetic field strength |B|, the power-law fitting results of Helios observations are relatively close to our fitting results within 0.3 au.Our fitting results of electron number density N e are smaller than predictions of Helios observations in the near-Sun solar wind.For the proton number density N p within 0.3 au, although Helios predictions are not consistent with our fitting results, the results in the corresponding solar wind are relatively close.For α number density N α in the near-Sun solar wind, only the prediction of fast solar wind from Helios observations seems to be close to PSP observations.For proton temperature T p , our fitting results of total and slow solar wind data are almost consistent with Helios predictions.However, for the intermediate solar wind and fast solar wind, there is a large deviation between Helios predictions and PSP observations.For α temperature T α within 0.3 au, only Helios predictions for the fast solar wind are close to our fitting results of the intermediate solar wind, and there is a large deviation in other cases.Our power-law fitting results for these solar wind parameters are listed in Table 1.
In Figure 2, we also show the radial distribution of the angle between the magnetic field B, solar wind V sw , and radial vector R.In the near-Sun solar wind, q BV sw and θ BR mainly concentrate around 135°, and q RV sw almost concentrates in the region less than 20°.
Furthermore, we also present the radial distribution of the relative values between solar wind parameters, which is shown in Figure 3.The electric neutrality analysis of the solar wind indicates that the ion VDFs observed by PSP in relatively large heliocentric distances are incomplete.The relative number density ratio N α /N p is almost unchanged with the increase of heliocentric distance (N α /N p ∼ 0.02 in the slow solar wind and N α /N p ∼ 0.04 in the faster solar wind).The median value of T α /T p decreases with heliocentric distance and the decay rate in the slow solar wind is less than that in the faster solar wind.Moreover, the average value of T α /T p in the slow solar wind is smaller than that in the faster solar wind.The radial characteristics of differential speed V α − V p are quite like that of T α /T p .Both V α − V p and (V α − V p )/V A tend to decay with the increase of the heliocentric distance, and the decay rate in the intermediate solar wind is larger than that in the slow solar wind.Furthermore, both V α − V p and (V α − V p )/V A in the faster solar wind are relatively bigger.

Figure 1 .
Figure 1.Radial distribution of the solar wind parameters: (a) the proton bulk speed V p ; (b) the strength of background magnetic field |B|; (c)-(e) the number density of electrons N e , protons N p , and α particles, respectively; (f)-(g) the temperature of protons T p and α particles T α , respectively.From left to right in every row are the results based on the total data, data in slow solar wind (V p < 400 km s −1 ), data in intermediate solar wind (400 V p 600 km s −1 ), and data in fast solar wind (V p > 600 km s −1 ).The filled points represent the median of each variable (the second quartile) in each radial bin, and the lower/upper error bar denotes the difference between the lower/upper quartile and the second quartile.The magenta lines are our power-law fitting results.The lines in other colors are radial distribution models from other papers.
Figures 1(b1)-(b4) show the radial distribution of the strength of background magnetic field |B|.The black and cyan lines are the Parker Spiral models ( = B and power-law fitting results (21.5 × 10 3 r −1.51 nT) of Helios observations shown in Figure 1(a) of Bale et al. (2016), respectively.The green lines are the estimated models (B = 250 × (0.1/r) ) in Equation (1) of Šafránková et al. (2023).Generally, in the near-Sun solar wind (<0.3 au), the power-law fitting results of Helios observations are relatively close to our fitting results.It is worth noting that the fitting results of PSP and SoLO observations given by Šafránková et al. (2023) are smaller than our fitting results.This may be related to the average selection of observational data (20 minutes for Šafránková et al. 2023 and 20 s for this paper).Figures 1(c1)-(c4) show the radial distribution of the number density of electrons N e .The cyan lines are power-law fitting results (1.25 × 10 6 × (215r) −2.26 cm −3 ) of Helios observations shown in Figure 1(c) of
and plasma parameters measured by the Solar Wind Electrons Alphas and Protons (SWEAP) instrument (Kasper et al. 2016) are taken into account.The Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

Table 1
Power-law Fitting Results