Determination of Solar Wind Angular Momentum and Alfvén Radius from Parker Solar Probe Observations
aa r X i v : . [ a s t r o - ph . S R ] F e b Determination of Solar Wind Angular Momentum and Alfv´enRadius from Parker Solar Probe Observations
Ying D. Liu , , Chong Chen , , Michael L. Stevens , and Mingzhe Liu ABSTRACT
As fundamental parameters of the Sun, the Alfv´en radius and angular mo-mentum loss determine how the solar wind changes from sub-Alfv´enic to super-Alfv´enic and how the Sun spins down. We present an approach to determiningthe solar wind angular momentum flux based on observations from Parker SolarProbe (PSP). A flux of about 0 . × dyn cm sr − near the ecliptic plane and0.7:1 partition of that flux between the particles and magnetic field are obtainedby averaging data from the first four encounters within 0.3 au from the Sun. Theangular momentum flux and its particle component decrease with the solar windspeed, while the flux in the field is remarkably constant. A speed dependencein the Alfv´en radius is also observed, which suggests a “rugged” Alfv´en surfacearound the Sun. Substantial diving below the Alfv´en surface seems plausibleonly for relatively slow solar wind given the orbital design of PSP. Uncertaintiesare evaluated based on the acceleration profiles of the same solar wind streamsobserved at PSP and a radially aligned spacecraft near 1 au. We illustrate thatthe “angular momentum paradox” raised by R´eville et al. can be removed bytaking into account the contribution of the alpha particles. The large protontransverse velocity observed by PSP is perhaps inherent in the solar wind accel-eration process, where an opposite transverse velocity is produced for the alphaswith the angular momentum conserved. Preliminary analysis of some recoveredalpha parameters tends to agree with the results. Subject headings: solar wind — Sun: fundamental parameters — Sun: rotation State Key Laboratory of Space Weather, National Space Science Center, Chinese Academy of Sciences,Beijing, China; [email protected] University of Chinese Academy of Sciences, Beijing, China Smithsonian Astrophysical Observatory, Cambridge, MA 02138, USA LESIA, Paris Observatory, PSL University, CNRS, Sorbonne University, University of Paris, France
1. Introduction
A fundamental issue in solar and stellar physics is how the Sun sheds its angular mo-mentum. In a landmark study, Weber & Davis (1967, hereinafter referred to as the WDmodel) show that the angular momentum loss per unit mass can be simply expressed as L = Ω r A , where Ω is the solar rotation rate. The Alfv´en radius r A is a key parameter of theSun, at which the solar wind radial velocity changes from sub-Alfv´enic to super-Alfv´enic.The WD model predicts that, while there is only a modest tendency for the solar wind toco-rotate with the Sun, the magnetic stresses produce a torque to the Sun corresponding toa rigid co-rotation out to r A . This process provides an efficient way for a star to spin down.Measurements of the solar wind angular momentum flux, however, have given divergentresults. Using data from Mariner 5, Lazarus & Goldstein (1971) find that the magnetic fieldcontribution to the angular momentum flux is dominated by the particle contribution. Thisdiffers from the results of Pizzo et al. (1983) and Marsch & Richter (1984) based on Heliosobservations. Pizzo et al. (1983) obtain an angular momentum flux of about 0 . × dyncm sr − near the ecliptic plane and a distribution of that flux between particles and fieldstresses near 1:3. They also reveal a considerable negative flux in the alpha particles, whichoffsets the protons’ angular momentum. Marsch & Richter (1984) suggest that the ratiobetween the particle and field contributions, which for average wind conditions is about 0.8,depends on the solar wind speed. Compared with the magnetic field’s angular momentumcontent, the solar wind plasma’s angular momentum flux is poorly determined. Previousresults, including those from Wind measurements at 1 au (Finley et al. 2019), show diverseparticle angular momentum fluxes; the field component of the angular momentum flux isrelatively invariant.Launched in 2018 August, the Parker Solar Probe (PSP) mission is intended to dive be-low the Alfv´en surface for the first time (Fox et al. 2016). PSP measurements at the first twoencounters show co-rotational flows up to about 70 km s − around the perihelion (35.7 solarradii), which are much larger than predicted by the WD model (Kasper et al. 2019). Trans-verse velocities exceeding WD predictions have also been reported previously. For example,observations from Mariner 5 at 0.7 au show values of about 10 km s − (Lazarus & Goldstein1971). These large transverse flows challenge our understanding of the angular momentumcarried by the solar wind. A specific problem concerning the PSP measurements is thatthe angular momentum from the high transverse velocities implies an Alfv´en radius largerthan the PSP distance from the Sun. It suggests that PSP may have already crossed theAlfv´en surface during the first two encounters. However, the solar wind radial velocity is stillsuper-Alfv´enic. This discrepancy is called the “angular momentum paradox” (R´eville et al.2020). 3 –In this Letter, we present an approach different from previous studies to determining thesolar wind angular momentum flux based on PSP observations. This new approach enablesus to well constrain the flux and its particle component. In the context of the approach, weillustrate that the “angular momentum paradox” can be removed by taking into account thecontribution of the alpha particles, and also make predictions on PSP crossings of the Alfv´ensurface. Improved results on the solar wind angular momentum flux may be obtained, giventhat (1) effects of stream-stream interactions are minimized at distances of PSP encounters;and (2) transient phenomena such as coronal mass ejections are reduced too during thepresent solar minimum. The outcome of this work helps clarify the angular momentum lossof the Sun as well as the origin of the large proton transverse flow.
2. Methodology2.1. Derivation of Alfv´en Radius and Angular Momentum Flux
In previous studies (e.g., Lazarus & Goldstein 1971; Pizzo et al. 1983; Marsch & Richter1984; Finley et al. 2019; R´eville et al. 2020), the solar wind angular momentum per unit massand the Alfv´en radius are calculated using the expression of L = rv φ − rB r B φ µρv r = Ω r A , (1)where r is the heliocentric distance, ρ the mass density of the solar wind, B r and B φ theradial and azimuthal components of the magnetic field, v r and v φ the corresponding velocitycomponents, and µ the permeability constant. Contribution from the thermal anisotropy hasbeen ignored since it is small (e.g., Weber 1970; R´eville et al. 2020). The angular momentumflux per steradian near the ecliptic plane is L times the mass flux ρv r r , i.e., F = ρv r v φ r − B r B φ r µ . (2)The first term is the contribution from the wind plasma or the particles ( F w ), and thesecond represents the magnetic field contribution ( F m ). Indeed, use of the above formulasis straightforward, as the plasma and field parameters can all be obtained from in situmeasurements. However, the first term often involves complications, such as uncertaintiesin the measurements of the transverse velocity and density, large fluctuations in the protonangular momentum flux arising from stream interactions and transient phenomena, and lackof accurate measurements of the alpha particles that may be a significant factor in modifyingthe angular momentum flux. As a result, the angular momentum flux and its distributionbetween the field and particles are not well constrained. 4 –Here we present a new approach in an attempt to avoid some of those complications.Conservation of mass ( ρv r r ) and radial magnetic flux ( r B r ) leads to a constant quantity M A / ( v r r ), where M A is the radial Alfv´en Mach number defined as M A = v r √ µρ/B r . Basedon this constant quantity and M A = 1 at r = r A , we obtain r A = q v r v rA rM A with v rA beingthe radial velocity at the Alfv´en critical point. The Alfv´en radius can then be reasonablyapproximated as r A ≃ rM A (3)by assuming that v r does not change much from r A to PSP encounter distances. As demon-strated by Weber & Davis (1967), Equation (3) provides a rigorous estimate of the Alfv´enradius. Once the Alfv´en radius is obtained, the angular momentum flux can be derived using F = ρv r r Ω r A ≃ Ω r B r µv r , (4)which is independent of the density. A prerequisite for Equation (4) is L = Ω r A , whichincludes both the particle and magnetic field contributions. It should be emphasized thatthe expression of L = Ω r A is a simple, straightforward derivation from the induction equationand the azimuthal equation of motion, which hold for a variety of circumstances. As pointedout by Pizzo et al. (1983), the expression remains valid even if there are substantial sourcesand sinks of momentum as long as they are confined to the region inside r A .The field component of the angular momentum flux can be calculated using the secondterm of Equation (2), which is independent of the particle parameters. It is relatively invari-ant as seen in the literature and as will be shown below. Taking advantage of the invariantnature of the field’s angular momentum flux, we can then well constrain the flux carried bythe wind plasma using F w = F − F m .An advantage of the above approach is that Equation (4) only requires the radial com-ponents of the velocity and magnetic field, which avoids various complications associatedwith the measurements of the transverse velocity, density, and alpha parameters. A majorapproximation made here is v r ∼ v rA . We will discuss in Section 4 the error that it brings. Inorder to reduce errors, Equation (4) should be used for measurements that are not far awayfrom the Alfv´en critical point. Also note that the expressions used here (e.g., the conservationof mass and radial magnetic flux) represent the average solar wind conditions. We expectthat fluctuations, such as those from inhomogeneities, will average out, as has been done inprevious studies (e.g., Lazarus & Goldstein 1971; Pizzo et al. 1983; Marsch & Richter 1984;Finley et al. 2019).In the following analysis, we use measurements from the FIELDS instrument suite(Bale et al. 2016) and the SWEAP package (Kasper et al. 2016) aboard PSP during the 5 –first four encounters. The respective perihelion is 35.7 solar radii from the Sun’s center forthe first three encounters and 27.9 solar radii for the fourth. All the solar wind parametersare interpolated to minute averages. We cut the data at 0.3 au from the Sun. This isto reduce effects of stream-stream interactions that may develop further out, and also todiminish errors with close-in measurements. There is considerable difficulty for PSP in determining the parameters of the alphaparticles. SWEAP ion instruments include a Faraday Cup (SPC) that looks directly at theSun, and an electrostatic analyzer (SPAN-I) on the ram side of PSP (Kasper et al. 2016;Case et al. 2020; Whittlesey et al. 2020). With SPAN-I, only part of the particle velocitydistribution is observed for most of the time (D. Larson 2019, private communication). Asfor SPC, the alpha peak either strongly overlaps with the proton peak or is too warm tobe clearly resolved against the noise for the majority of the time (M. Stevens 2020, privatecommunication). Although our approach does not involve the alpha parameters, we want toillustrate how the alpha particles carry a significant negative angular momentum flux, whichhelps resolve the “angular momentum paradox” (R´eville et al. 2020).In the presence of the alpha particles, the angular momentum flux of the wind plasmacan be written as F w ≃ n p m p r ( v pr v pφ + 0 . v αr v αφ ) , (5)where the subscripts p and α refer to protons and alphas respectively, n p the proton numberdensity, and m p the proton mass. An average relative abundance of 5% has been assumedfor the alphas. We construct the alpha velocity based on Helios observations between 0 . − v α = v p − sign( B T ) B √ µρ . (6)The term of sign( B T ) can be understood in the following context: since the differentialstreaming is aligned with the Parker spiral field, the alphas would have a negative transversevelocity (i.e., eastward). A natural thought would be whether Equations (5) and (6) could 6 –give a particle angular momentum flux comparable to F − F m (which is the only link in thepresent work between the use of Equation (4) and the analytical development on the alphaparameters). Indeed, this may serve as a test of the alpha parameter construction. However,complications certainly exist in reality. For example, the differential streaming tends tobe around zero in very slow wind (Asbridge et al. 1976; Marsch et al. 1982). Therefore,Equation (6) can only be considered as a very rough estimate. Again, we aim at illustratingif the alphas can offset a considerable amount of the angular momentum flux in the protons,not to the exact value one may expect.This study does not use PSP measurements of the alpha particles, which are not suffi-cient for a statistical sense. We do perform a preliminary analysis of some recovered alphaparameters from SPAN-I, which tends to confirm what we expect from the analytical devel-opment (see discussions in Section 4).
3. Observations and Results
Figure 1 shows the in situ measurements at the first encounter. Transverse velocities, upto about 75 km s − around the perihelion, are observed. With these large co-rotational flowsEquation (1) would give Alfv´en radii above the PSP distance from the Sun, which implies thatPSP may have already crossed the Alfv´en surface. However, the solar wind is always super-Alfv´enic with M A larger than 3 in general. This is called the “angular momentum paradox”(R´eville et al. 2020). The Alfv´en radius derived from Equation (3) is well below the PSPdistance, so the paradox can be removed. Our result from Equation (3) is comparable to theestimate by R´eville et al. (2020) with only the magnetic term of Equation (1). Note thatthey use the SPC densities, which may be underestimated (see below).Negative transverse flows are also seen for enhanced radial speeds, which indicates thatthere might be stream-stream interactions even for distances at the encounter. The inter-action between streams would result in an angular momentum transfer from the faster toslower wind, i.e., the slower wind is deflected in the direction of co-rotation while the fasterwind towards the opposite; the total angular momentum, however, is conserved. The effectsof stream-stream interactions are expected to increase with distance (e.g., Pizzo et al. 1983;Marsch & Richter 1984).The proton angular momentum flux exhibits a large variance, with the standard devia-tion about 4.2 times the average during the time period. Negative values are seen correspond-ing to the eastward flows. The angular momentum flux in the magnetic field is relativelyinvariant with an average of about 0 . × dyn cm sr − , but the standard deviation is 7 –still about 3 times the average. These illustrate a common problem in the determination ofthe solar wind angular momentum flux and its components: the fluctuations can be muchlarger than the average (Pizzo et al. 1983; Marsch & Richter 1984). Again, the fluctuationsare expected to average out. The angular momentum flux, calculated from Equation (4)without invoking the plasma density and the transverse velocities of the protons and alphas,shows an average value of about 0 . × dyn cm sr − . Application of F w = F − F m gives an average particle flux of about 0 . × dyn cm sr − . An anti-correlation is visiblebetween the angular momentum flux and the radial velocity, which can be understood fromEquation (4) if r B r does not vary dramatically. A similar dependence on the radial velocityis also observed in the Alfv´en radius.Figure 1 also indicates that, compared with the electron density from quasi-thermalnoise (QTN) spectroscopy (Moncuquet et al. 2020), the proton density from SPC may besystematically underestimated. Since the electron density is derived from measurements ofthe local plasma frequency, it is thought to be most reliable. For the first four orbits the QTNdensity, on average, is larger than the SPC density by 30 − . We have used the QTN electron density as a proxy of the plasma density throughoutthe study.In situ measurements at the second encounter are shown in Figure 2. While the situationis in general akin to that of the first encounter, we see elevated Alfv´en radii around theperihelion corresponding to an interval of reduced densities, with M A as low as about 1.5.This may suggest conditions in favor of PSP crossings of the Alfv´en surface, i.e., relativelylow densities and radial velocities. The third and fourth encounters are also generally similar(not shown here). For the third encounter, measurements are available only for about a thirdof the time; the radial Alfv´en Mach number approaches 3 towards the perihelion. The fourthencounter yields flagged plasma data around the perihelion (data are problematic), whichare removed from our study; the solar wind is still super-Alfv´enic with M A dipping below 2occasionally.Figure 3 displays the constructed alpha parameters using Equations (5) and (6). Despiteconsiderable fluctuations, the alpha radial velocity is generally larger than that of the protons,but similar values between the two species are also seen for slower wind. Persistent negativetransverse velocities are obtained for the alphas, as expected. However, close to the perihelion Readers are directed to Finley et al. (2020) to see a different proton angular momentum flux from theSPC densities. − . × dyn cm sr − , while the proton’s contribution is about0 . × dyn cm sr − on average. Although their sum is larger than F − F m , the alphasare indeed capable of canceling a significant amount of the angular momentum flux carriedby the protons.Table 1 lists the average Alfv´en radius and angular momentum fluxes for the four en-counters. Again, the angular momentum flux of magnetic stresses is relatively invariantcompared with the flux in the protons. Averaging the whole data yields r A = 9 . F = 0 . × dyn cm sr − , and about 0.7:1 partition of the flux between the particlesand field. These correspond to an average solar wind speed of 333 km s − . It should bestressed that the speed here should not be considered as the final speed, as the solar windmay be still accelerating at distances of the encounters. We see both similarities and dif-ferences, comparing our results with previous ones (Lazarus & Goldstein 1971; Pizzo et al.1983; Marsch & Richter 1984; Finley et al. 2019) as mentioned in Section 1. In particular,our flux is comparable to the lower end of the range obtained by Pizzo et al. (1983), and theratio between the particle and field contributions is similar to that given by Marsch & Richter(1984). The alpha particles, in principle, could cancel a considerable amount of the flux inthe protons. However, we find this more difficult as we move closer to the Sun. It is becauseeither not enough flow states are sampled by the spacecraft, or the alpha velocity deviatesfrom Equation (6). On average, the alphas are anticipated to carry an angular momentumflux of about − . × dyn cm sr − in order to bring down the protons’ flux to the levelof the wind plasma.Table 2 gives the Alfv´en radius and angular momentum fluxes as a function of the solarwind speed. The field component of the flux is remarkably constant over the speed. Thetotal flux decreases with speeds, and so does the flux in the wind plasma given the invarianceof the flux in the field. We see a predominance of the plasma contribution over the field’s forspeeds less than 250 km s − , a near equipartition for speeds between 250 and 350 km s − ,and a reversal for speeds larger than 350 km s − . Interestingly, the wind plasma carries zeroor even negative angular momentum flux when the speed exceeds 450 km s − ; we expect anegative flux of about 0 . × dyn cm sr − in the alphas for such a complete cancellation.Predictions can also be made on PSP Alfv´en surface crossings based on the speed dependenceof the Alfv´en radius. The closest approach to the center of the Sun is 9.86 solar radii andwill occur during the final three orbits (Fox et al. 2016). Substantial diving below the Alfv´ensurface is plausible only for the solar wind with speeds below 350 km s − . For speeds above350 km s − the chance seems low, although PSP may scratch the Alfv´en surface under somecircumstances. 9 –
4. Conclusions and Discussions
The approach that we have presented does not require the transverse velocities of thesolar wind protons and alphas, thus avoiding various complications. This may be helpful forPSP measurements of the Alfv´en radius and angular momentum flux, which exhibit largeproton transverse flows of unknown origin and difficulty in determining the alpha parameters.With the approach the angular momentum flux and its distribution between the magneticfield and particles are well constrained. Averaging data from the first four encounters within0.3 au gives an Alfv´en radius of 9.7 solar radii, a flux of 0 . × dyn cm sr − near theecliptic plane, and a partition 0.7:1 of that flux between the particles and field. The angularmomentum flux in the field is remarkably constant over the solar wind speed, whereas thetotal flux and thus its particle component decrease with speeds. We expect zero or evennegative angular momentum flux in the particles when the speed exceeds 450 km s − . Aspeed dependence in the Alfv´en radius is also observed, so we anticipate a “rugged” Alfv´ensurface around the Sun. PSP’s persistent diving below the Alfv´en surface seems plausibleonly for the solar wind with speeds below 350 km s − .In the context of our approach the “angular momentum paradox” (R´eville et al. 2020)is removed. By constructing the alpha velocity based on observations from 0 . − − . × dyn cm sr − in the alphas is needed to reduce the protons’ flux to the level of the wind plasma. A recentwork by Fisk & Kasper (2020) suggests that the large proton transverse velocity is a resultof interchange reconnection in the corona. If this is true, we would expect both positiveand negative transverse flows everywhere. However, persistent positive flows are observedaround all the perihelions. Our experiment on the alphas indicates that the large transverseflow is likely inherent in the solar wind acceleration process. The acceleration process is suchthat a positive transverse velocity is produced for the protons while a negative one for thealphas, but the angular momentum is conserved.Recovery of the alpha parameters from PSP observations will be key to testing the resultshere and understanding the origin of the large proton transverse flows. Fortunately, aroundthe perihelion of the fourth encounter PSP moves fast enough to shift the majority of thealpha core into the field of view of SPAN-I. Our preliminary analysis of the recovered alphaparameters yields results similar to what we have expected: for the slower wind, the alpharadial velocity resembles that of the protons, and the alpha transverse velocity fluctuatesaround zero; for the faster wind, the alphas show a radial velocity larger than that of theprotons and a significant negative transverse velocity. These results are consistent with aconsiderable negative angular momentum flux in the alphas and our suggestion on the origin 10 –of the large proton transverse flows. We note a recent work by Finley et al. (2021) lookingat the alpha particles from SPAN-I measurements, which is coincident with the time frameof the present study. They perform bi-Maxwellian fits of the truncated velocity distributionfunctions of the particles from the third and fourth encounters. Their results roughly agreewith our preliminary analysis of the recovered alpha parameters, in terms of the velocitydifference between the protons and alphas for both the slower and faster wind. However,the data are limited and not sufficient for a statistical analysis. Studies better tacklingcalibration issues are needed for a definite conclusion.Note that Equation (3) gives a lower limit of the Alfv´en radius, although it should beclose to the real value (Weber & Davis 1967). Detailed error analysis is difficult withoutknowing the velocity profile as a function of distance. For reference, if the radial velocitygrows by 10% from the Alfv´en critical point to PSP encounter distances within 0.3 au, thevalue of the Alfv´en radius will increase by about 5% (i.e., q v r v rA ), and the angular momentumflux will increase by 10%. We have identified the same solar wind streams observed at PSPand a spacecraft near 1 au when they are radially aligned. The results show quite differentacceleration profiles for different solar wind streams. As an extreme case (2020 January 28to 31), the radial velocity increases from about 238 km s − at 29 solar radii to about 512km s − at Wind, implying an average acceleration of about 1.47 km s − per solar radius.Another case on 2019 April 7 to 12 indicates an average acceleration of only about 0.39 kms − per solar radius, i.e., 363 km s − at 39 solar radii versus 432 km s − at Wind. If theacceleration is assumed to be constant from the Alfv´en critical point ( ∼
10 solar radii) to1 au, these two cases suggest an increment in the radial velocity by about 13% and 3%,respectively, from the Alfv´en critical point to the PSP distances.Another issue is that PSP may not have collected enough flow states. We await moredata in the years to come, with improved calibration and in particular considerable recoveryof the alpha parameters.The research was supported by NSFC under grant 41774179, Beijing Municipal Scienceand Technology Commission (Z191100004319003), and the Specialized Research Fund forState Key Laboratories of China. We acknowledge the NASA Parker Solar Probe missionand the SWEAP and FIELDS teams for use of data.
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This preprint was prepared with the AAS L A TEX macros v5.2.
12 –Fig. 1.— PSP measurements at the first encounter. (a) Alfv´en radius in comparison withthe distance of the spacecraft (red). The gray curve represents the Alfv´en radius derivedfrom Equation (1). (b)-(c) Proton transverse and radial velocities. (d) Radial Alfv´en Machnumber. (e) Proton angular momentum flux in comparison with the angular momentumflux of magnetic stresses (red). (f) Angular momentum flux calculated from Equation (4).(g) Magnetic field strength and radial (red) and transverse (blue) components. (h) Protondensity in comparison with the electron density (red) from quasi-thermal noise (QTN). 13 –Fig. 2.— PSP measurements at the second encounter. Similar to Figure 1. The radialAlfv´en Mach number drops to about 1.5 around the perihelion. 14 –Fig. 3.— Constructed helium-ion velocity components and angular momentum flux in com-parison with the proton counterparts (red) for the first encounter. The proton angularmomentum flux is divided by a factor of 5. 15 –Table 1. PSP Measurements of Alfv´en Radius and Angular Momentum FluxEncounter r A F a F m F w b F p F α c ( R ⊙ ) (10 dyn cm sr − )1 8.1 0.13 0.08 0.05 0.37 − .
192 11.8 0.17 0.13 0.04 0.66 − .
083 9.7 0.15 0.07 0.08 0.47 − .
134 9.0 0.13 0.08 0.06 0.82 − . − . a Angular momentum flux calculated from Equa-tion (4). b Angular momentum flux of the solar wind plasmacalculated from F w = F − F m . c Constructed helium-ion angular momentum fluxbased on observations from 0 . − a Percentage r A F F m F w b F p (km s − ) ( R ⊙ ) (10 dyn cm sr − ) v p
250 4.5% 12.3 0.22 0.08 0.14 0.86250 < v p
350 62.3% 10.2 0.16 0.09 0.07 0.65350 < v p
450 27.8% 9.1 0.12 0.09 0.03 0.42 v p >
450 5.4% 7.9 0.09 0.10 -0.01 0.50 a The solar wind at distances of PSP encounters may be stillaccelerating, so the speed here should not be confused with thespeed at 1 au. b Angular momentum flux of the solar wind plasma calculatedfrom F w = F − F mm