Fast Wind Streams Research Articles

On the 1/f Spectrum in Slow Solar Wind Turbulence: The Role of Alfvénicity

The slow solar wind has been recently observed to have a number of intervals that are dominated by large-scale Alfvénic fluctuations, especially within 1 au, with similar turbulence characteristics to those found in fast wind streams, including a 1/f range. These results suggest that the slow solar wind exists in at least two flavors: the typical slow wind that generally does not exhibit a 1/f range and an Alfvénic wind that is more similar to fast wind streams. The Alfvénic slow wind is usually differentiated from the typical slow wind (not dominated by Alfvénic fluctuations) by the normalized cross helicity, σ c . Values of ∣σ c ∣ near unity are associated with Alfvénic fluctuations, whereas values near zero are typically thought of as non-Alfvénic. This classification by cross helicity excludes the case of solar wind fluctuations dominated by balanced Alfvénic turbulence, i.e., the turbulence regime where there is equal energy flux of counterpropagating fluctuations propagating along the mean field. We use a large statistical analysis to isolate intervals of slow wind at 1 au in a 20 yr period of Wind data. These intervals are sorted into subsets corresponding to the type of slow wind via the mean values of their magnetic compressibility and cross helicity. Our analysis finds several intervals of low-cross-helicity slow wind dominated by balanced Alfvénic turbulence, which possess similar characteristics found in high-cross-helicity streams. Our results support the conclusion that a 1/f spectrum may be a property associated with streams dominated by Alfvénic turbulence, whether the turbulence is balanced or imbalanced.

Parker Solar Probe observations of collisional effects on thermalizing the young solar wind

Solar wind ions exhibit distinct kinetic non-thermal features such as preferential heating and acceleration of alpha particles compared to protons. On the other hand, Coulomb collisions in the solar wind act to eliminate these non-thermal features and gradually lead to thermal equilibrium. Previous observations at 1 au have revealed that even though the local Coulomb collisions in the solar wind plasma are rare, the cumulative effect of the collisions during a transit time of a particle can be important in terms of thermalizing the solar wind plasma populations and reducing the ion non-thermal features. Here, we analyze Parker Solar Probe observations to study the effects of Coulomb collisions on the non-thermal features (alpha-to-proton temperature ratio and differential flow) of young solar wind closer to the Sun than previously possible. Our results show that even close to the Sun (∼15Rs), these non-thermal features are organized by collisionality. Moreover, observations at these unprecedented distances allow us to investigate the preferential heating of the alpha particles close to the source for both fast and slow wind streams. We show that the alpha-to-proton temperature ratio is positively correlated with the solar wind speed, which is consistent with Wind observations. Solar wind close to the Sun is less collisionally old than when it reaches 1 au. As such, observed differences in the temperature ratio between slow and fast streams near their solar source suggest causes that go beyond different Coulomb numbers. Our results suggest that slow and fast wind streams, originating from different solar regions, may have different mechanisms for the preferential heating of alpha particles compared to protons.

Quantifying the Energy Budget in the Solar Wind from 13.3 to 100 Solar Radii

A variety of energy sources, ranging from dynamic processes, such as magnetic reconnection and waves, to quasi-steady terms, such as plasma pressure, may contribute to the acceleration of the solar wind. We utilize a combination of charged particle and magnetic field observations from the Parker Solar Probe (PSP) to attempt to quantify the steady-state contribution of the proton pressure, the electric potential, and the wave energy to the solar wind proton acceleration observed by PSP between 13.3 and ∼100 solar radii (R ☉). The proton pressure provides a natural kinematic driver of the outflow. The ambipolar electric potential acts to couple the electron pressure to the protons, providing another definite proton acceleration term. Fluctuations and waves, while inherently dynamic, can act as an additional effective steady-state pressure term. To analyze the contributions of these terms, we utilize radial binning of single-point PSP measurements, as well as repeated crossings of the same stream at different distances on individual PSP orbits (i.e., fast radial scans). In agreement with previous work, we find that the electric potential contains sufficient energy to fully explain the acceleration of the slower wind streams. On the other hand, we find that the wave pressure plays an increasingly important role in the faster wind streams. The combination of these terms can explain the continuing acceleration of both slow and fast wind streams beyond 13.3 R ☉.

Investigating Alfvénic Turbulence in Fast and Slow Solar Wind Streams

Solar wind turbulence dominated by large-amplitude Alfvénic fluctuations, mainly propagating away from the Sun, is ubiquitous in high-speed solar wind streams. Recent observations performed in the inner heliosphere (from 1 AU down to tens of solar radii) have proved that also slow wind streams show sometimes strong Alfvénic signatures. Within this context, the present paper focuses on a comparative study on the characterization of Alfvénic turbulence in fast and slow solar wind intervals observed at 1 AU where degradation of Alfvénic correlations is expected. In particular, we compared the behavior of different parameters to characterize the Alfvénic content of the fluctuations, using also the Elsässer variables to derive the spectral behavior of the normalized cross-helicity and residual energy. This study confirms that the Alfvénic slow wind stream resembles, in many respects, a fast wind stream. The velocity-magnetic field (v-b) correlation coefficient is similar in the two cases as well as the amplitude of the fluctuations although it is not clear to what extent the condition of incompressibility holds. Moreover, the spectral analysis shows that fast wind and Alfvénic slow wind have similar normalized cross-helicity values but in general the fast wind streams are closer to energy equipartition. Despite the overall similarities between the two solar wind regimes, each stream shows also peculiar features, that could be linked to the intrinsic evolution history that each of them has experienced and that should be taken into account to investigate how and why Alfvénicity evolves in the inner heliosphere.

High-frequency heating of the solar wind triggered by low-frequency turbulence

The fast solar wind’s high speeds and non-thermal features require that considerable heating occurs well above the Sun’s surface. Two leading theories seem incompatible: low-frequency ‘Alfvénic’ turbulence, which transports energy outwards and is observed ubiquitously by spacecraft but seems insufficient to explain the observed dominance of ion over electron heating; and high-frequency ion-cyclotron waves, which explain the non-thermal heating of ions but lack an obvious source. Here we argue that the recently proposed ‘helicity barrier’ effect, which limits electron heating by inhibiting the turbulent cascade of energy to the smallest scales, can unify these two paradigms. Our six-dimensional simulations show how the helicity barrier causes the large-scale energy to grow through time, generating small parallel scales and high-frequency ion-cyclotron-wave heating from low-frequency turbulence, while simultaneously explaining various other long-standing observational puzzles. The predicted causal link between plasma expansion and the ion-to-electron heating ratio suggests that the helicity barrier could contribute to key observed differences between fast and slow wind streams.

First Solar Orbiter observation of the Alfvénic slow wind and identification of its solar source

Context.Turbulence dominated by large-amplitude, nonlinear Alfvén-like fluctuations mainly propagating away from the Sun is ubiquitous in high-speed solar wind streams. Recent studies have demontrated that slow wind streams may also show strong Alfvénic signatures, especially in the inner heliosphere.Aims.The present study focuses on the characterisation of an Alfvénic slow solar wind interval observed by Solar Orbiter between 14 and 18 July 2020 at a heliocentric distance of 0.64 AU.Methods.Our analysis is based on plasma moments and magnetic field measurements from the Solar Wind Analyser (SWA) and Magnetometer (MAG) instruments, respectively. We compared the behaviour of different parameters to characterise the stream in terms of the Alfvénic content and magnetic properties. We also performed a spectral analysis to highlight spectral features and waves signature using power spectral density and magnetic helicity spectrograms, respectively. Moreover, we reconstruct the Solar Orbiter magnetic connectivity to the solar sources both via a ballistic and a potential field source surface (PFSS) model.Results.The Alfvénic slow wind stream described in this paper resembles, in many respects, a fast wind stream. Indeed, at large scales, the time series of the speed profile shows a compression region, a main portion of the stream, and a rarefaction region, characterised by different features. Moreover, before the rarefaction region, we pinpoint several structures at different scales recalling the spaghetti-like flux-tube texture of the interplanetary magnetic field. Finally, we identify the connections between Solar Orbiter in situ measurements, tracing them down to coronal streamer and pseudostreamer configurations.Conclusions.The characterisation of the Alfvénic slow wind stream observed by Solar Orbiter and the identification of its solar source are extremely important aspects for improving the understanding of future observations of the same solar wind regime, especially as solar activity is increasing toward a maximum, where a higher incidence of this solar wind regime is expected.

Numerical study of penetration and distribution of charged particles in the magnetosphere during southward IMF case

AbstractThe energetic charged particles penetration in the plasmasphere are carried out using the updated version of 3D Stanford PIC code. We considered slow and fast wind streams to know the penetration of the energetic charged particles (electrons and ions) having different velocities into four regions i.e. cusp, plasmasphere, sunward, and tailward sides. It is observed that the ion penetrations are higher than electrons for solar slow wind streams in the plasmasphere, while it is reverse for the solar fast streams. Also, the results show that the percentage of penetration of the energetic charged particles (both electrons and ions) are the same into the cusp and subsolar point reconnection region. It is different for the higher speed of fast streams; so that the penetrated electrons reached about 10–20 times than penetrated ions. The results show that for the tailward reconnection region, the penetration of ions is 2–3 times higher than the penetration of electrons, but it is the same for the case of higher solar fast speed.

Characteristics of solar wind suprathermal halo electrons

A suprathermal halo population of electrons is ubiquitous in space plasmas, as evidence of their departure from thermal equilibrium even in the absence of anisotropies. The origin, properties, and implications of this population, however, are poorly known. We provide a comprehensive description of solar wind halo electrons in the ecliptic, contrasting their evolutions with heliospheric distance in the slow and fast wind streams. At relatively low distances less than 1 AU, the halo parameters show an anticorrelation with the solar wind speed, but this contrast decreases with increasing distance and may switch to a positive correlation beyond 1 AU. A less monotonic evolution is characteristic of the high-speed winds, in which halo electrons and their properties (e.g., number densities, temperature, plasma beta) exhibit a progressive enhancement already distinguishable at about 0.5 AU. At this point, magnetic focusing of electron strahls becomes weaker and may be counterbalanced by the interactions of electrons with wave fluctuations. This evolution of halo electrons between 0.5 AU and 3.0 AU in the fast winds complements previous results well, indicating a substantial reduction of the strahl and suggesting that significant fractions of strahl electrons and energy may be redistributed to the halo population. On the other hand, properties of halo electrons at low distances in the outer corona suggest a subcoronal origin and a direct implication in the overheating of coronal plasma via velocity filtration.

Large Amplitude Fluctuations in the Alfvénic Solar Wind

Large amplitude fluctuations, often with characteristics reminiscent of large amplitude Alfven waves propagating away from the Sun, are ubiquitous in the solar wind. Such features are most frequently found within fast solar wind streams and most often at solar minimum. The fluctuations found in slow solar wind streams usually have a smaller relative amplitude, are less Alfvenic in character and present more variability. However, intervals of slow wind displaying Alfvenic correlations have been recently identified in different solar cycle phases. In the present paper we report Alfvenic slow solar wind streams seen during the maximum of solar activity that are characterized not only by a very high correlation between velocity and magnetic field fluctuations (as required by outwardly propagating Alfven modes) – comparable to that seen in fast wind streams – but also by higher amplitude relative fluctuations comparable to those seen in fast wind. Our results suggest that the Alfvenic slow wind has a different origin from the slow wind found near the boundary of coronal holes, where the amplitude of the Alfvenic fluctuations decreases together with decreasing the wind speed.

Direct Detection of Solar Angular Momentum Loss with the Wind Spacecraft

Abstract The rate at which the solar wind extracts angular momentum (AM) from the Sun has been predicted by theoretical models for many decades, and yet we lack a conclusive measurement from in situ observations. In this Letter we present a new estimate of the time-varying AM flux in the equatorial solar wind, as observed by the Wind spacecraft from 1994 to 2019. We separate the AM flux into contributions from the protons, alpha particles, and magnetic stresses, showing that the mechanical flux in the protons is ∼3 times larger than the magnetic field stresses. We observe the tendency for the AM flux of fast wind streams to be oppositely signed to the slow wind streams, as noted by previous authors. From the average total flux, we estimate the global AM loss rate of the Sun to be 3.3 × 1030 erg, which lies within the range of various magnetohydrodynamic wind models in the literature. This AM loss rate is a factor of ∼2 weaker than required for a Skumanich-like rotation period evolution ( stellar age−1/2), which should be considered in studies of the rotation period evolution of Sun-like stars.

On slow solar wind with high Alfvénicity: from composition and microphysics to spectral properties

Alfv\'enic fluctuations are very common features in the solar wind and are found especially within the main portion of fast wind streams while the slow wind usually is less Alfv\'enic and more variable. In general, fast and slow wind show many differences which span from the large scale structure to small scale phenomena including also a different turbulent behaviour. Recent studies, however, have shown that even slow wind can be sometimes highly Alfv\'enic with fluctuations as large as those of the fast wind. The present study is devoted to present many facets of this Alfv\'enic slow solar wind including for example the study of the source regions and their connection to coronal structures, large-scale properties and micro-scale phenomena and also impact on the spectral features. This study will be conducted performing a comparative analysis with the typical slow wind and with the fast wind. It has been found that the fast wind and the Alfv\'enic slow wind share common characteristics, probably attributable to their similar solar origin, i.e. coronal-hole solar wind. Given these similarities, it is suggested that in the Alfv\'enic slow wind a major role is played by the super-radial expansion responsible for the lower velocity. Relevant implications of these new findings for the upcoming Solar Orbiter and Solar Probe Plus missions, and more in general for turbulence measurements close to the Sun, will be discussed.

The Role of Proton Cyclotron Resonance as a Dissipation Mechanism in Solar Wind Turbulence: A Statistical Study at Ion-kinetic Scales

Abstract We use magnetic field and ion moment data from the MFI and SWE instruments on board the Wind spacecraft to study the nature of solar wind turbulence at ion-kinetic scales. We analyze the spectral properties of magnetic field fluctuations between 0.1 and 5.4 Hz during 2012 using an automated routine, computing high-resolution 92 s power and magnetic helicity spectra. To ensure the spectral features are physical, we make the first in-flight measurement of the MFI “noise-floor” using tail-lobe crossings of the Earth’s magnetosphere during early 2004. We utilize Taylor’s hypothesis to Doppler-shift into the spacecraft frequency frame, finding that the spectral break observed at these frequencies is best associated with the proton cyclotron resonance scale, 1/k c , rather than the proton inertial length, d i , or proton gyroscale, ρ i . This agreement is strongest when we consider periods where , and is consistent with a spectral break at d i for and at ρ i for . We also find that the coherent magnetic helicity signature observed at these frequencies is bounded at low frequencies by 1/k c , and its absolute value reaches a maximum at ρ i . These results hold in both slow and fast wind streams, but with a better correlation in the more Alfvénic fast wind where the helicity signature is strongest. We conclude that these findings are consistent with proton cyclotron resonance as an important mechanism for dissipation of turbulent energy in the solar wind, occurring at least half the time in our selected interval. However, we do not rule out additional mechanisms.

Production of Sunspots and Their Effects on the Corona and Solar Wind: Insights from a New 3D Flux-Transport Dynamo Model

We present a three-dimensional numerical model for the generation and evolution of the magnetic field in the solar convection zone, in which sunspots are produced and contribute to the cyclic reversal of the large-scale magnetic field. We then assess the impact of this dynamo-generated field on the structure of the solar corona and solar wind. This model solves the induction equation in which the velocity field is prescribed. This velocity field is a combination of a solar-like differential rotation and meridional circulation. We develop an algorithm that enables the magnetic flux produced in the interior to be buoyantly transported towards the surface to produce bipolar spots. We find that those tilted bipolar magnetic regions contain a sufficient amount of flux to periodically reverse the polar magnetic field and sustain dynamo action. We then track the evolution of these magnetic features at the surface during a few consecutive magnetic cycles and analyze their effects on the topology of the corona and on properties of the solar wind (distribution of streamers and coronal holes, and of slow and fast wind streams) in connection with current observations of the Sun.

AN INVESTIGATION OF THE SOURCES OF EARTH-DIRECTED SOLAR WIND DURING CARRINGTON ROTATION 2053

ABSTRACT In this work we analyze multiple sources of solar wind through a full Carrington Rotation (CR 2053) by analyzing the solar data through spectroscopic observations of the plasma upflow regions and the in situ data of the wind itself. Following earlier authors, we link solar and in situ observations by a combination of ballistic backmapping and potential-field source-surface modeling. We find three sources of fast solar wind that are low-latitude coronal holes. The coronal holes do not produce a steady fast wind, but rather a wind with rapid fluctuations. The coronal spectroscopic data from Hinode’s Extreme Ultraviolet Imaging Spectrometer show a mixture of upflow and downflow regions highlighting the complexity of the coronal hole, with the upflows being dominant. There is a mix of open and multi-scale closed magnetic fields in this region whose (interchange) reconnections are consistent with the up- and downflows they generate being viewed through an optically thin corona, and with the strahl directions and freeze-in temperatures found in in situ data. At the boundary of slow and fast wind streams there are three short periods of enhanced-velocity solar wind, which we term intermediate based on their in situ characteristics. These are related to active regions that are located beside coronal holes. The active regions have different magnetic configurations, from bipolar through tripolar to quadrupolar, and we discuss the mechanisms to produce this intermediate wind, and the important role that the open field of coronal holes adjacent to closed-field active regions plays in the process.

Electron energetics in the expanding solar wind via Helios observations

AbstractWe present an observational analysis of electron cooling/heating rates in the fast and slow solar wind between 0.3 and 1 AU. We fit electron velocity distribution functions acquired in situ by Helios 1 and 2 spacecraft by a three‐component (core‐halo‐strahl) analytical model. The resulting radial profiles of macroscopic characteristics (density, temperatures, and heat fluxes) are employed to examine properties of theoretical energy balance equations and to estimate external cooling/heating terms. Our analysis indicates that in contrast to solar wind protons the electrons do not require important heating mechanisms to explain the observed temperature gradients. The electron heating rates are actually found to be negative for both the slow and fast solar wind, namely, due to the significant degradation of the electron heat flux with increasing radial distance from the Sun. Cooling mechanisms acting on electrons are found to be significantly stronger in the slow wind than in the fast wind streams.

DECELERATION OF ALPHA PARTICLES IN THE SOLAR WIND BY INSTABILITIES AND THE ROTATIONAL FORCE: IMPLICATIONS FOR HEATING, AZIMUTHAL FLOW, AND THE PARKER SPIRAL MAGNETIC FIELD

Protons and alpha particles in the fast solar wind are only weakly collisional and exhibit a number of non-equilibrium features, including relative drifts between particle species. Two non-collisional mechanisms have been proposed for limiting differential flow between alpha particles and protons: plasma instabilities and the rotational force. Both mechanisms decelerate the alpha particles. In this paper, we derive an analytic expression for the rate $Q_{\mathrm{flow}}$ at which energy is released by alpha-particle deceleration, accounting for azimuthal flow and conservation of total momentum. We show that instabilities control the deceleration of alpha particles at $r< r_{\mathrm{crit}}$, and the rotational force controls the deceleration of alpha particles at $r> r_{\mathrm{crit}}$, where $r_{\mathrm{crit}} \simeq 2.5 \,\mathrm{AU}$ in the fast solar wind in the ecliptic plane. We find that $Q_{\mathrm{flow}}$ is positive at $r<r_{\mathrm{crit}}$ and $Q_{\mathrm{flow}} = 0$ at $r\geq r_{\mathrm{crit}}$, consistent with the previous finding that the rotational force does not lead to a release of energy. We compare the value of~$Q_{\mathrm{flow}}$ at $r< r_{\mathrm{crit}}$ with empirical heating rates for protons and alpha particles, denoted $Q_{\mathrm{p}}$ and $Q_{\alpha}$, deduced from in-situ measurements of fast-wind streams from the \emph{Helios} and \emph{Ulysses} spacecraft. We find that $Q_{\mathrm{flow}}$ exceeds $Q_{\alpha}$ at $r < 1\,\mathrm{AU}$, and that $Q_{\mathrm{flow}}/Q_{\rm p}$ decreases with increasing distance from the Sun from a value of about one at $r=0.29 - 0.42\,\mathrm{AU}$ to about 1/4 at 1 AU. We conclude that the continuous energy input from alpha-particle deceleration at $r< r_{\mathrm{crit}}$ makes an important contribution to the heating of the fast solar wind.

ON THE ORIGIN OF HIGHLY ALFVÉNIC SLOW SOLAR WIND

Alfvénic fluctuations are a common feature in the solar wind and are found especially in the trailing edges of fast wind streams. The slow wind usually has a lower degree of Alfvénicity, being more strongly intermixed with structures of non-Alfvénic nature. In the present paper we show the first evidence in the interplanetary space of two different kinds of slow solar wind: one coming from coronal streamers or active regions and characterized by non-Alfvénic structures and the other one being highly Alfvénic and originating from the boundary of coronal holes. The Alfvénic character of fluctuations, either outward or inward, can be studied by means of the normalized cross-helicity, , which is an indicator of the alignment. The evolution of toward lower values with increasing radial distance is interpreted both as a decrease of the presence of the outward modes and as a continuous production of inward modes within those regions such as stream shears where some plasma instability is active. On the other hand, the decrease of is often related also to magnetic field and/or density enhancements which specifically act on the destruction of the alignment. In the present analysis we study the role of compressibility presenting both case studies and a statistical analysis over different phases of solar cycle 23. Our findings indicate that the presence of regions of magnetic field compression generally play a major role in the depletion of and thus in the destruction of the alignment.

RADIAL EVOLUTION OF SPECTRAL CHARACTERISTICS OF MAGNETIC FIELD FLUCTUATIONS AT PROTON SCALES

This paper addresses the investigation of the character and the radial evolution of magnetic fluctuations within the dissipation range, right after the high-frequency spectral break, employing observations by Messenger and Wind of the same fast wind stream during a radial alignment. The same event has already been considered in literature to show, for the first time, that the high-frequency break separating the fluid from the kinetic regime moves to lower frequency as the wind expands. The present work aims to analyze the nature of the high-frequency magnetic fluctuations beyond the spectral break and show that their character is compatible with left-hand, outward-propagating, ion cyclotron waves and right-hand kinetic Alfvn waves. It is also shown that the low-frequency limit of these fluctuations follows the radial evolution of the spectral break, which also reflects in the behavior of their intermittency character. Finally, the total power and the compressive character of these two wave populations are analyzed and compared as a function of the heliocentric distance, leading us to conclude that the overall picture is in favor of a radial decrease.

THE SOLAR WIND NEON ABUNDANCE OBSERVED WITHACE/SWICS ANDULYSSES/SWICS

Using in situ ion spectrometry data from ACE/SWICS, we determine the solar wind Ne/O elemental abundance ratio and examine its dependence on wind speed and evolution with the solar cycle. We find that Ne/O is inversely correlated with wind speed, is nearly constant in the fast wind, and correlates strongly with solar activity in the slow wind. In fast wind streams with speeds above 600 km s −1 , we find Ne/O = 0.10 ± 0.02, in good agreement with the extensive polar observations by Ulysses/SWICS. In slow wind streams with speeds below 400 km s −1 , Ne/O ranges from a low of 0.12 ± 0.02 at solar maximum to a high of 0.17 ± 0.03 at solar minimum. These measurements place new and significant empirical constraints on the fractionation mechanisms governing solar wind composition and have implications for the coronal and photospheric abundances of neon and oxygen. The results are made possible by a new data analysis method that robustly identifies rare elements in the measured ion spectra. The method is also applied to Ulysses/SWICS data, which confirms the ACE observations and extends our view of solar wind neon into the three-dimensional heliosphere.

OBSERVATIONAL TEST OF STOCHASTIC HEATING IN LOW-β FAST-SOLAR-WIND STREAMS

Spacecraft measurements show that protons undergo substantial perpendicular heating during their transit from the Sun to the outer heliosphere. In this paper, we use {\em Helios~2} measurements to investigate whether stochastic heating by low-frequency turbulence is capable of explaining this perpendicular heating. We analyze {\em Helios~2} magnetic-field measurements in low-$\beta$ fast-solar-wind streams between heliocentric distances $r=0.29$ AU and $r=0.64$ AU to determine the rms amplitude of the fluctuating magnetic field, $\delta B_{\rm p}$, near the proton gyroradius scale $\rho_{\rm p}$. We then evaluate the stochastic heating rate $Q_{\perp \rm stoch}$ using the measured value of $\delta B_{\rm p}$ and a previously published analytical formula for $Q_{\perp \rm stoch}$. Using {\em Helios} measurements we estimate the `empirical' perpendicular heating rate $Q_{\perp \rm emp} = (k_{\rm B}/m_{\rm p}) B V (d/dr) (T_{\perp \rm p}/B)$ that is needed to explain the $T_{\perp \rm p}$ profile. We find that $Q_{\perp \rm stoch} \sim Q_{\perp \rm emp}$, but only if a key dimensionless constant appearing in the formula for $Q_{\perp \rm stoch}$ lies within a certain range of values. This range is approximately the same throughout the radial interval that we analyze and is consistent with the results of numerical simulations of the stochastic heating of test particles in reduced magnetohydrodynamic turbulence. These results support the hypothesis that stochastic heating accounts for much of the perpendicular proton heating occurring in low-$\beta$ fast-wind streams.