Statistical Study of Whistler Waves in the Solar Wind at 1 AU
Yuguang Tong, Ivan Y. Vasko, Anton V. Artemyev, Stuart D. Bale, Forrest S. Mozer
22 Draft version May 23, 2019
Typeset using L A TEX twocolumn style in AASTeX62
Statistical Study of Whistler Waves in the Solar Wind at 1 AU
Yuguang Tong,
1, 2
Ivan Y. Vasko,
1, 3
Anton V. Artemyev,
4, 3
Stuart D. Bale,
1, 2 and Forrest S. Mozer Space Sciences Laboratory, University of California, Berkeley, CA 94720 Physics Department, University of California, Berkeley, CA 94720 Space Research Institute of Russian Academy of Sciences, Moscow, Russia Institute of Geophysics and Planetary Sciences, University of California, Los Angeles, USA
Submitted to ApJABSTRACTWhistler waves are intermittently present in the solar wind, while their origin and effects are notentirely understood. We present a statistical analysis of magnetic field fluctuations in the whistlerfrequency range (above 16 Hz) based on about 801,500 magnetic field spectra measured over threeyears aboard ARTEMIS spacecraft in the pristine solar wind. About 13,700 spectra (30 hours in total)with intense magnetic field fluctuations satisfy the interpretation in terms of quasi-parallel whistlerwaves. We provide estimates of the whistler wave occurrence probability, amplitudes, frequencies andbandwidths. The occurrence probability of whistler waves is shown to strongly depend on the electrontemperature anisotropy. The whistler waves amplitudes are in the range from about 0.01 to 0.1 nT andtypically below 0.02 of the background magnetic field. The frequencies of the whistler waves are shownto be below an upper bound that is dependent on β e . The correlations established between the whistlerwave properties and local macroscopic plasma parameters suggest that the observed whistler wavescan be generated in local plasmas by the whistler heat flux instability. The whistler wave amplitudesare typically small, which questions the hypothesis that quasi-parallel whistler waves are capable toregulate the electron heat flux in the solar wind. We show that the observed whistler waves havesufficiently wide bandwidths and small amplitudes, so that effects of the whistler waves on electronscan be addressed in the frame of the quasi-linear theory. Keywords: solar wind — plasmas — waves INTRODUCTIONWhistler waves, electromagnetic emissions betweenion and electron cyclotron frequencies, are potentiallyregulating several fundamental processes in the colli-sionless or weakly-collisional solar wind. In particular,spacecraft observations of the electron heat flux valuesbelow a threshold dependent on β e were interpreted interms of the heat flux regulation by the whistler heatflux instability (Feldman et al. 1975, 1976; Scime et al.1994; Gary et al. 1999; Tong et al. 2018) and whistler faninstability (Vasko et al. 2019). The observed radial evo-lution of the angular width of suprathermal field-aligned Corresponding author: Yuguang [email protected] electron population (strahl electrons) in the solar wind(e.g., Hammond et al. 1996; Graham et al. 2017) re-quires pitch-angle scattering that can be potentially pro-vided by whistler waves (Vocks et al. 2005; Shevchenko& Galinsky 2010; Vocks 2012; Kajdiˇc et al. 2016; Vaskoet al. 2019). Whistler waves may also suppress the elec-tron heat flux in collisionless or weakly-collisional astro-physical plasma (Pistinner & Eichler 1998; Gary & Li2000; Roberg-Clark et al. 2016; Roberg-Clark et al. 2018;Komarov et al. 2018). The necessity of a heat flux sup-pression mechanism is suggested by observations of thetemperature profile of hot gases in galaxy clusters (e.g.,Cowie & McKee 1977; Bertschinger & Meiksin 1986; Za-kamska & Narayan 2003; Wagh et al. 2014; Fang et al.2018). The understanding of whistler wave origins and a r X i v : . [ phy s i c s . s p ace - ph ] M a y Tong et al. effects requires statistical analysis of whistler wave oc-currence and properties in the solar wind.The magnetic field fluctuations with power-law spec-tra in various frequency ranges are persistently observedin the solar wind and referred to as turbulence (see,e.g., Bruno & Carbone 2013, for review). Early studiesassociate the magnetic field turbulence in the whistlerfrequency range with whistler waves, their power wasshown to decrease with increasing radial distance fromthe Sun and enhance around interplanetary shocks andhigh-speed stream interfaces (e.g., Beinroth & Neubauer1981; Coroniti et al. 1982; Lengyel-Frey et al. 1996;Lin et al. 1998). However, later studies show that thewhistler frequency range of the magnetic field turbulenceis dominated by kinetic-Alfv´en and slow ion-acousticwaves Doppler-shifted into the whistler frequency range(e.g., Bale et al. 2005; Salem et al. 2012; Chen et al. 2013;Lacombe et al. 2017). The whistler wave contributionto the magnetic field turbulence spectrum is still underdebate (e.g., Gary 2015; Narita et al. 2016; Kellogg et al.2018).The modern spacecraft measurements have recentlyshown that whistler waves are intermittently present inthe pristine (not disturbed by shocks or the Earth’s fore-shock) solar wind (Lacombe et al. 2014; Stansby et al.2016; Tong et al. 2019). Whistler waves have beenidentified by a local peak superimposed on a power-lawspectrum of the magnetic field turbulence background.Therefore, these whistler waves should be produced bykinetic instabilities (free energy in the plasma), ratherthan by the turbulence cascade (see Gary 2015, for dis-cussion). In addition to the pristine solar wind, whistlerwaves have been reported around interplanetary shockwaves (e.g., Breneman et al. 2010; Wilson et al. 2013)and in the Earth’s foreshock (e.g., Hoppe & Russell 1980;Zhang et al. 1998).The focus of this paper is the statistical analysis ofwhistler waves produced by kinetic instabilities in thepristine solar wind. The detailed analysis of whistlerwaves in the pristine solar wind has become possibleonly recently due to simultaneous wave and particlemeasurements aboard Cluster and ARTEMIS spacecraft(Lacombe et al. 2014; Stansby et al. 2016; Tong et al.2019). In contrast to WIND and Stereo spacecraft, wavemeasurements aboard Cluster and ARTEMIS are avail-able almost continuously, rather than triggered by high-amplitude events, which typically occur around inter-planetary shocks (e.g., Breneman et al. 2010; Wilsonet al. 2013). Lacombe et al. (2014) have selected abouttwenty 10-minute intervals with whistler wave activityobserved aboard Cluster in the pristine solar wind. Theanalysis of the magnetic field cross-spectra has shown that whistler waves propagate quasi-parallel to the back-ground magnetic field. The simultaneous measurementsof the electron heat flux have been presented to arguethat the whistler waves are produced by the whistlerheat flux instability (WHFI) (see, e.g., Gary et al. 1994,for the WHFI theory). Stansby et al. (2016) have se-lected several 10-minute intervals of ARTEMIS measure-ments to test the whistler wave dispersion relation independence on β e . Tong et al. (2019) have carried out adetailed analysis of wave and particle measurements forStansby et al. (2016) events and demonstrated that thewhistler waves were produced locally on a time scalesof seconds and indeed by the WHFI. The analysis byTong et al. (2019) has proved that the WHFI may in-deed operate in the solar wind and clearly demonstratedthe critical role of the electron temperature anisotropy:the parallel temperature anisotropy may quench theWHFI instability, while the perpendicular temperatureanisotropy favors the instability onset.In spite of some recent progress, the parameters con-trolling the occurrence and properties of whistler wavesin the solar wind have not been considered on a sta-tistical basis. In this paper we present analysis of sev-eral hundred days of ARTEMIS observations in the solarwind (two spacecraft orbiting the Moon, see Angelopou-los 2011, for details). The whistler wave selection pro-duced a dataset of about 13,700 whistler wave spectra( >
30 hours in total) in the pristine solar wind that isthe most representative dataset up to date. The paper isorganized as follows. We describe instrument character-istics, methodology and data selection criteria in Section2. The results of the statistical study are presented inSections 3, 4 and 5. We discuss the statistical resultsin light of whistler wave generation mechanism, electronheat flux regulation and recent particle-in-cell simula-tions in Section 6. The conclusions are summarized inSection 7. DATA AND METHODOLOGYWe use ARTEMIS spacecraft measurements from 2011to 2013 and select observations in the pristine solarwind, that is excluding the Earth’s foreshock and thelunar wake. The Search Coil Magnetometer instrumentprovides Fast Fourier Transform (FFT) magnetic fieldspectra with 8s cadence and covers 64 piecewise linearly-spaced frequency channels between 8 to 4096 Hz (Rouxet al. 2008). We use the spectral power density SPD ⊥ ofthe magnetic field in the spacecraft spin plane (almostecliptic plane), the spectral power density SPD || of themagnetic field component along the spin axis (almostperpendicular to the ecliptic plane) and, the total spec-tral power density SPD = SPD || + 2 SPD ⊥ . The Flux histler waves at 1AU and the electron heat flux parallel to the mag-netic field computed by integrating the electron VDF q e = 12 m e (cid:90) ( v || − (cid:10) v || (cid:11) ) ( v − (cid:104) v (cid:105) ) VDF( v ) d v (1)where m e is the electron mass, v || is the electron ve-locity parallel to the magnetic field and (cid:104) v (cid:105) is the elec-tron bulk velocity. The particle moments available at96s are upsampled to 8s cadence of the magnetic fieldspectra via the linear interpolation. In total we haveanalyzed 801,527 magnetic field spectra, spanning 1,803hours and 359 days in 2011-2013 . In the rest of thispaper, we will refer to each magnetic field spectrum asan independent event. Note that we did not filter outinterplanetary shocks, but looking through the list of in-terplanetary shocks observed on Wind , we found onlyseveral days in our dataset with listed shocks. There-fore, our dataset is dominated by observations in thepristine solar wind. In what follows, we clarify crite-ria for whistler wave selection and demonstrate the dataanalysis techniques.Figure 1 presents the magnetic field spectrum and par-ticle moments for a particular day (July 29, 2011) in ourdataset. Panel (a) shows the total spectral power den-sity from 16 to 300 Hz. The SPD enhancements between20 and 60 Hz appear first around 14:25 UT and con-tinue intermittently thereafter before about 16:30 UT.In terms of a local electron cyclotron frequency f ce , theobserved SPD enhancements are between 0 . . f ce which is in the whistler frequency range. The wave ac-tivity can be characterized by the total magnetic fieldpower in the frequency range between 16 and 300 Hz P B ≡ (cid:90)
300 Hz16 Hz
SPD( f ) df (2) Ground-calibrated particle moments are accessed via twodata products,
THB L2 ESA and
THC L2 ESA which can be foundin https://cdaweb.gsfc.nasa.gov/ . The electron VDF is accessed from http://themis.ssl.berkeley.edu/data/themis/ and thenprocessed by the open-source SPEDAS software (Angelopouloset al. 2019). The data intervals are provided in https://doi.org/10.5281/zenodo.2652949 Panel (b) demonstrates that P B well traces the SPD en-hancements. In the absence of clear wave activities, P B is a mixture of the inherent turbulence background andinstrument noise between 16 and 300 Hz. We divide themagnetic field spectra into two-hour chunks and definethe background power P g as the 20th percentile of P B within every chunk. Panel (c) presents P B /P g , demon-strating thereby that the wave activity corresponds to P B significantly exceeding P g . The amplitude of mag-netic field fluctuations associated with the wave activityis characterized by B w = ( P B − P g ) / . Panel (d) showsthat the amplitude of the magnetic field fluctuationsreaches 0.05 nT, while B w /B , that is the amplitude ofthe magnetic field fluctuations with respect to the back-ground magnetic field, does not exceed 0.01. We empha-size that B w is the amplitude averaged over 8s, while theactual peak amplitude may be larger due to intermit-tent appearance of the magnetic field fluctuations over8s. Panels (e) to (h) present a few plasma parameters: β e = 8 πn e T e || /B , T e ⊥ /T e || is the electron temperatureanisotropy, q e /q is the electron heat flux normalized tothe free-streaming heat flux q = 1 . n e T e (2 T e /m e ) / , v sw is the solar wind proton velocity. In the above pa-rameters, n e is the electron density, T e ⊥ and T e || are theperpendicular and parallel electron temperature, B isthe magnitude of the quasi-static magnetic field. Notewe have used a natural unit system in which temperaturehas the unit eV. The Boltzmann constant is droppedthroughout the paper.Visual inspections of the magnetic field spectra fromour dataset show that SPD enhancements in the whistlerfrequency range are always below 300 Hz. The wavepower P B in the frequency range between 16 and 300Hz is found to be a good indicator of the wave activ-ity. The spectral power density in the first (8 Hz) fre-quency channel is excluded from P B computation, be-cause it provides strong and noisy contribution to P B ,so that the wave activity at f ≥
16 Hz could not beidentified in P B . Another reason for excluding the firstchannel is that it is more likely to be contaminated bylow-frequency magnetic field fluctuations different fromwhistler waves (see below). Visual inspections show that P B > P g is a reasonable empirical criterion for select-ing noticeable wave activities between 16 and 300 Hzand filtering out spectra corresponding to variations ofthe turbulence background. The criterion P B > P g selects 17,050 magnetic field spectra that is about 38hours and about 2% of the original dataset.Although the selected wave activities are in thewhistler frequency range, they do not necessarily rep-resent whistler waves (see Section 1 for discussion).The routinely available ARTEMIS measurements in- Tong et al. f [ H z ] (a) 2011-07-29 P B [ n T ] (b) P B / P g (c) B w / B (d) e (e) T e / T e || (f) q e / q (g) time v s w [ k m / s ] (h) S P D [ n T / H z ] B w [ n T ] Figure 1.
The wave activity in the whistler frequency range observed aboard ARTEMIS on July 29, 2011 (one day from ourdataset): (a) magnetic field spectral power density, 0 . f ce and 0 . f ce are indicated with green and red curves, where f ce is alocal electron cyclotron frequency; (b) the magnetic field power P B in the frequency range between 16 and 300 Hz determinedby Eq. (2); (c) the magnetic field power P B normalized to the background turbulence power P g determined every 2 hours as20th percentile of P B ; the visual inspection of our dataset showed that P B > P g (dashed line) is a reasonable criterion forselecting the wave activity events in the whistler frequency range and filtering out variations of the turbulence background; (d)the amplitude of magnetic field fluctuations evaluated as B w = ( P B − P g ) / (red trace) and B w /B (black trace) that is theamplitude with respect to the local background magnetic field B ; (e)-(h) β e = 8 πn e T e || /B , electron temperature anisotropy T e ⊥ /T e || , the electron heat flux q e normalized to the free-streaming heat flux q = 1 . n e T e ( T e /m e ) / , solar wind velocity v sw . histler waves at 1AU ⊥ and SPD || , along with results of theprevious observations enable us to filter out events con-tradicting the whistler wave interpretation and providea basis to argue that the major part of the selectedevents are whistler waves. The technique relies on theprevious analysis of the magnetic field spectral matrix(spectra and cross-spectra up to 400 Hz) measurementsprovided by Cluster (Lacombe et al. 2014) and theanalysis of magnetic field waveforms (frequencies up to64 Hz resolved) provided by ARTEMIS (Stansby et al.2016; Tong et al. 2019), which both showed that whistlerwaves in the pristine solar wind propagate quasi-parallelto the background magnetic field B . The observationsof quasi-parallel whistler waves are consistent with the-oretical predictions of potential instabilities operatingin the solar wind (Gary et al. 1994, 2012). Obliquewhistler waves may be present in the solar wind, butthey are predicted to be electrostatic and, hence, notidentifiable in the magnetic field spectra (Vasko et al.2019).The whistler wave propagation parallel to the mag-netic field results in a specific relation between SPD || and SPD ⊥ that is dependent on B orientation withrespect to the spin axis (see Figure 2 for schematics).A whistler wave at frequency f propagating parallelto B is a circularly-polarized wave with the magneticfield along b cos(2 πf t ) + b sin(2 πf t ), where b , areunit vectors in the plane perpendicular to B . Thiswave would produce SPD || ( f ) ∝ sin χ and SPD ⊥ ( f ) ∝ (1 + cos χ ) /
2, where χ is the angle between B and thespin axis (Fig. 2), so that the ratio R ≡ SPD || ( f )SPD( f ) (3)would equal to R = 0 . χ . A reasonable agreementbetween the observed R and expected R may allow fil-tering out events corresponding to plasma modes differ-ent from quasi-parallel whistler waves.Figure 3 presents the analysis of the nature of the waveactivity shown in Figure 1. Panel (a) presents angle χ (Figure 2) computed using the quasi-static magneticfield measurements. Panels (b) and (c) present SPD || and SPD ⊥ . For every magnetic field spectrum with P B > P g we identify the frequency channel f w withthe largest total spectral power density, SPD in Figure1a, and compute R using SPD || ( f w ) and SPD( f w ) inEq. (3). Panel (d) shows that R is well consistent with R = 0 . χ , supporting thereby the interpretation Figure 2.
Schematics of the ARTEMIS search coil magne-tometer antennas. The instrument provides spectral powerdensities SPD || and SPD ⊥ of magnetic field fluctuationsalong the spacecraft spin axis and in the plane perpendic-ular to the spin axis. The total spectral power density (Fig-ure 1a) of the magnetic field fluctuations is computed asSPD=SPD || +2 SPD ⊥ . For a whistler wave propagating par-allel to the quasi-static magnetic field B there is a particularrelation between SPD || and SPD ⊥ that depends on angle χ (see Section 2 for details). of the wave activity in terms of quasi-parallel whistlerwaves.Figure 4 presents results of the comparison between R and R evaluated for all 17,050 magnetic field spec-tra with P B > P g . Panel (a) shows that R/ ( R + R )are clustered around 0.5, that is R ≈ R . Most of theevents with R/ ( R + R ) significantly deviating from 0.5are in the three lowest frequency channels at 16, 24 and32 Hz, where low-frequency modes are expected mostlikely to appear due to the Doppler effect. Panel (b)shows that the events with R/ ( R + R ) significantlydeviating from 0.5 have frequencies from 0.02 to 0.5 f ce , demonstrating thereby that the whistler frequencyrange may be populated by plasma modes different fromquasi-parallel whistler waves. We introduce a quan-titative criterion 0 . < R/ ( R + R ) < . Tong et al. [] (a) 2011-07-29 f [ H z ] (b) f [ H z ] (c) t S P D || / S P D (d) R R ( P B / P g > 3) S P D || [ n T / H z ] S P D [ n T / H z ] Figure 3.
The test of the nature of the wave activity observed on July 29, 2011: (a) the angle χ between the magnetic fieldand the spin axis shown in Figure 2 and computed using the quasi-static magnetic field measurements; (b, c) spectra SPD || and SPD ⊥ of magnetic field fluctuations along the spin axis and in the plane perpendicular to the spin axis; (d) the ratio R = SPD || ( f w ) / SPD( f w ) at the frequency channel f w corresponding to the largest SPD, only points at the moments of timewith P B > P g are indicated (red dots); the ratio R expected for a whistler wave propagating parallel to the backgroundmagnetic field is shown by the black curve. P D F (c) (d) C D F R/(R+R ) f w / f c e (b) R/(R+R ) (a) f w [ H z ] R/(R+R ) Figure 4.
Results of testing the nature of the selected ∼ R = SPD || ( f w ) / SPD( f w ), where f w is the frequency channel with the maximum SPD=SPD || +2 SPD ⊥ , and R value (denoted as R ) expected for a whistler wave propagating parallel to the background magnetic field: (a,b) R/ ( R + R ) vs. frequency f w and f w /f ce ; (c,d) the probability and cumulative distribution functions of R/ ( R + R ). The data points within the shaded region,0 . < R/ ( R + R ) < .
6, correspond to wave activity events non-contradicting to the hypothesis of quasi-parallel whistler waves.Panel (d) shows that exclusion of the data points outside of the shaded region filters out less than 20% of the data points. histler waves at 1AU
50 100 150 200 f [Hz] S P D [ n T / H z ] A = (6.4 ± 0.2) × 10 [nT /Hz] f = 40.7 ± 0.4 [Hz]= 9.0 ± 0.4 [Hz] f = 21.2 ± 0.9 [Hz] SPD( f )SPD g ( f )SPD( f ) SPD g ( f )best Gaussian fit Figure 5.
The analysis of the frequency bandwidth of aparticular whistler wave spectrum on July 29, 2011. Themeasured spectral power density at 15:35:33 UT (blue), thebackground spectral power density SPD g ( f ) (green) thatis computed at each frequency f as the 20th percentile ofSPD( f ) at that frequency every two hours. Black dots rep-resent SPD( f )-SPD g ( f ) that is the whistler wave spectrum.The best-fit Gaussian (4) to the whistler wave spectrum isgiven by the red curve. The best fit parameters A and σ areindicated along with the frequency bandwidth estimated asthe width at half maximum, ∆ f = 2 σ (2 ln 2) / . first the background spectral power density SPD g ( f )at frequency f as the 20th percentile of SPD( f ) atthat frequency every two hours. Similarly to P g ,SPD g ( f ) is a combination of the magnetic field turbu-lence background and intrinsic instrument noise level.The whistler wave spectrum SPD( f )-SPD g ( f ) is fittedto the Gaussian model with the peak at f w SPD( f ) − SPD g ( f ) = A exp (cid:20) − ( f − f w ) σ (cid:21) , (4)where A and σ are the best fit parameters. The fre-quency bandwidth ∆ f is estimated as the full width athalf maximum∆ f = 2 σ (2 ln 2) / ∼ . σ Figure 5 presents the analysis of the frequency band-width of a particular whistler wave spectrum with thepeak at f w ∼
40 Hz measured at 15:35:33 UT onJuly 29, 2011 (one spectrum from Figure 1). Thewhistler wave SPD enhancement is about two ordersof magnitude larger than SPD g ( f w ). The Gaussian fitto SPD( f ) − SPD g ( f ) yields the frequency bandwidth∆ f ∼
21 Hz. We restrict the statistical analysis ofthe frequency bandwidth to whistler wave events with f w >
16 Hz, because only in those events we could guar-antee that the peak of the Gaussian is at f w , rather thanat some frequency below 16 Hz. The criterion f w >
16 Hz leaves 5,800 spectra for the frequency bandwidthanalysis that is 42% of the selected 13,700 whistler wavespectra. WHISTLER WAVE OCCURRENCEOut of about 8 × spectra we have associated about13,700 spectra with quasi-parallel whistler waves thatyields a total occurrence probability of whistler wavesof 1.7%. We emphasize that this is the probability ofsufficiently intense whistler waves ( P B > P g ) above16 Hz, i.e. whistler waves that are less intense and atlower frequencies have been excluded. The overall oc-currence of whistler waves in the pristine solar wind iscertainly higher. We demonstrate below that the occur-rence probability of the selected whistler waves dependson macroscopic plasma parameters.Figure 6 presents the analysis of effects of the elec-tron heat flux q e /q and β e on the occurrence probabil-ity of whistler waves. Panel (a) shows the distributionof all ∼ × magnetic field spectra in ( q e /q , β e )parameter plane. The electron heat flux at β e (cid:38) q e /q ∼ /β e that is in agreementwith previous spacecraft observations (Gary et al. 1999;Tong et al. 2018). This heat flux threshold was previ-ously considered as the evidence for the heat flux reg-ulation by the whistler heat flux instability (Feldmanet al. 1976; Gary et al. 1999). Panel (b) shows thedistribution of ∼ ,
700 magnetic field spectra asso-ciated with quasi-parallel whistler waves. Combiningthe distributions shown in panels (a) and (b) we evalu-ate the occurrence probability of whistler waves at var-ious ( q e /q , β e ). Panel (c) shows that the occurrenceprobability does not favor the parameter space nearthe threshold q e /q ∼ /β e and, instead, somewhat en-hances at low heat flux values.Figure 7 presents the analysis of effects of the elec-tron heat flux q e /q and electron temperature anisotropy T e ⊥ /T e || on the whistler wave occurrence probability.Panels (a) and (b) present distributions of all eventsand whistler wave events in ( q e /q , T e ⊥ /T e || ) parame-ter plane. In accordance with previous statistical stud-ies (e.g., ˇStver´ak et al. 2008; Artemyev et al. 2018) so-lar wind electrons at 1 AU most often exhibit paralleltemperature anisotropy, T e ⊥ /T e || <
1. Panels (a) and(b) are combined to compute the occurrence probabilityin ( q e /q , T e ⊥ /T e || ) parameter plane. Panel (c) clearlydemonstrates that the temperature anisotropy quitecritically affects the whistler wave occurrence probabil-ity. At any given q e /q the occurrence probability in-creases with increasing T e ⊥ /T e || . The occurrence proba-bility is less than a few percent at T e ⊥ /T e || (cid:46)
1, but in-creases up to 10-60% at T e ⊥ /T e || >
1. In addition, panel
Tong et al. e q e / q (a) all events e (b) whistler events e (c) occurrence prob Figure 6.
The analysis of whistler wave occurrence in dependence on the electron heat flux q e /q and β e : (a) distribution ofall ∼ × magnetic field spectra in ( q e /q , β e ) parameter plane; (b) distribution of the selected ∼ q e /q = 1 /β e for reference. q e / q T e / T e || (a) all events q e / q (b) whistler events q e / q (c) occurrence prob Figure 7.
The analysis of whistler wave occurrence in dependence on the electron heat flux q e /q and T e ⊥ /T e || : (a) distributionof all ∼ × magnetic field spectra in ( q e /q , T e ⊥ /T e || ) parameter plane; (b) distribution of the selected ∼ histler waves at 1AU q e /q (cid:46) − thetemperature anisotropy should be above 0.75, while at q e /q (cid:38) × − whistler waves may occur at T e ⊥ /T e || as low as 0.5. In addition to the 2D occurrence prob-abilities, we have computed whistler wave occurrenceprobabilities in dependence on individual macroscopicplasma parameters.Figure 8 presents the occurrence probability ofwhistler waves in dependence on q e /q , β e , v sw and T e ⊥ /T e || . The occurrence probability P ( ξ ) of whistlerwaves in dependence on a macroscopic plasma param-eter A is determined as P ( ξ ) = N W ( ξ ) /N ( ξ ), where N W ( ξ ) is the number of whistler wave events with A inthe range ( ξ − ∆ ξ/ , ξ + ∆ ξ/ N ( ξ ) is the totalnumber of events with A in the same range. The binwidth ∆ ξ is chosen so that the number of events withineach bin would be sufficiently large. The uncertaintiesof P ( ξ ) are estimated with the assumption that eachparticle measurement is independent . Panels (a), (c)and (d) demonstrate that the electron heat flux, β e andsolar wind velocity do not significantly affect the occur-rence probability of whistler waves. Panel (b) confirmsthat the whistler wave occurrence probability is criti-cally dependent on the electron temperature anisotropy.The probability is less than 2% at T e ⊥ /T e || < .
9, butincreases from 5 to 15% as T e ⊥ /T e || varies from 0.95 to1.2. WHISTLER WAVE INTENSITYFigure 9 presents the probability distribution func-tions of whistler wave amplitudes B w and B w /B for theslow ( v sw (cid:46)
400 km/s) and fast ( v sw >
500 km/s) solarwind. Our dataset is dominated by the slow solar windevents, fast solar wind events constitute less than 12%of the dataset. Panels (a) and (b) show that whistlerwaves amplitude B w is typically below 0.02 B or inphysical units in the range from 0.01 up to 0.1 nT. Werecall that B w is the amplitude averaged over 8s, so thatthe actual peak amplitudes of magnetic field fluctuationscould be in principle larger due to intermittent presenceof whistler wave over 8s. However, these amplitudesare consistent with previous measurements of whistlerwaveforms aboard ARTEMIS spacecraft (Stansby et al.2016; Tong et al. 2019), indicating thereby that quitelikely whistler waves in the pristine solar wind have am- Assuming that each particle measurement has the same prob-ability to have a whistler companion, and that n measurementsestimate the probability to be p . Then the standard error of p is s p = (cid:112) p (1 − p ) /n . We estimate the uncertainty of p as theuncertainty at the 95% level of confidence δp = 2 s p . plitudes B w much smaller than B . Panels (a) and (b)also demonstrate that there is a bit higher chance to ob-serve intense whistler waves in the slow solar wind thanin the fast solar wind.Figure 10 presents the distribution of the averagedwhistler wave amplitude (cid:104) B w /B (cid:105) in ( q e /q , β e ) and( T e ⊥ /T e || , q e /q ) parameter planes. Panel (a) demon-strates that (cid:104) B w /B (cid:105) is strongest, when both β e and q e /q are high. As a result, the averaged whistler waveamplitude is enhanced in the parameter space around tothe threshold q e /q ∼ /β e . It is interesting to note thatthe whistler wave occurrence probability doesn’t favorthis region in the parameter space (Figure 6). The rea-son is that the occurrence of whistler waves is most crit-ically controlled by the temperature anisotropy, ratherthan q e /q or β e . Panel (b) shows that (cid:104) B w /B (cid:105) en-hances with increasing T e ⊥ /T e || at fixed q e /q , while thepositive correlation between (cid:104) B w /B (cid:105) and q e /q is no-ticeable only at T e ⊥ /T e || (cid:38) B w /B in dependence on individual macro-scopic parameters. The upper panels indicate the meanand median B w /B values in dependence on q e /q , T e ⊥ /T e || , β e and v sw , while the shaded regions coverfrom the 25th percentile to the 75th percentile of B w /B . The bottom panels present the number ofevents within bins used to compute the B w /B distri-butions in the upper panels. Panels (a) and (b) showthat the mean and median values of B w /B are posi-tively correlated with q e /q and T e ⊥ /T e || , though theoverall variation of these values is about 30%. The neg-ative correlation between B w /B and the heat flux at q e /q (cid:38) . B w /B are most strongly correlated with β e , both val-ues increase by about a factor of three as β e increasesfrom 0.1 to 5. Panel (d) shows that the whistler waveamplitude is negatively correlated with the solar windvelocity, varying by a factor of two from the slow to fastsolar wind. WHISTLER WAVE FREQUENCY5.1.
Observations
We consider the frequency channel f w with the largestSPD( f ) or largest enhancement SPD( f ) − SPD g ( f )(both provide the same frequency channel) as the fre-quency of a whistler wave event. We could considerthe frequency channel with the largest relative SPD en-hancement, SPD( f ) / SPD g ( f ), as the whistler wave fre-quency estimate. Because SPD g ( f ) is a monotonicallydecreasing function of the frequency, this approach pro-0 Tong et al. q e / q P r o b w h i s t l e r o cc u rr e n c e (a) T e / T e || (b) e (c)
300 400 500 600 v sw [km/s] (d) Figure 8.
The occurrence probability of whistler waves in dependence on individual macroscopic plasma parameters. B w / B P D F (a) v sw < 400km/s v sw > 500km/s B w [nT] P D F (b) v sw < 400km/s v sw > 500km/s Figure 9.
Probability distribution functions of whistler wave amplitudes B w and B w /B in the slow ( v sw <
400 km/s) andfast ( v sw >
500 km/s) solar wind. e q e / q (a) h B w /B i q e / q T e / T e || (b) h B w /B i Figure 10.
The whistler wave amplitude (cid:104) B w /B (cid:105) averaged over bins in (a) ( q e /q , β e ) and (b) ( T e ⊥ /T e || , q e /q ) parameterplanes. vides frequencies higher than f w , but we have foundthat the difference is less than 50%. We use f w as thewhistler wave frequency estimate, while the use of theother frequency would not affect any of our conclusions.We have found that among various macroscopic plasmaparameters only β e correlates strongly with the normal-ized frequency f w /f ce . Figure 12 demonstrates that there are apparent upperand lower frequency bounds that decrease with increas-ing β e . Below we compare these bounds to theoreticalpredictions of the whistler heat flux instability. To quan-tify the negative correlation between the upper boundon f w /f ce and β e we bin all the whistler wave eventsaccording to β e and select 10% of the highest frequencyevents within each bin. These highest frequency events histler waves at 1AU B w / B (a) meanmedian25 & 75 percentile (b) (c)
300 400 500 6000.0000.0020.0040.0060.008 (d) q e / q c o un t T e / T e || e
300 400 500 600 v sw [km/s] Figure 11.
The whistler wave amplitude B w /B versus (a) the electron heat flux, (b) electron temperature anisotropy, (c) β e ,and (d) the solar wind velocity. The curves represent the median and mean values of B w /B , while the shaded regions coverfrom 25th to 75th percentile of B w /B . e f w / f c e e WHFI maxWHFI min Figure 12.
Whistler wave frequency f w , determined as thefrequency channel with the largest SPD( f ), normalized tothe electron cyclotron frequency f ce versus β e . The blackcurve represent the the best power-law fit to the 10% of thehighest frequency events at various β e . The red and bluecurves represent the maximum and minimum frequencies ofwhistler waves that can be generated by the whistler heatflux instability (see Section 5.2 for details). The presentedfrequencies f w are measured in the spacecraft frame, but theestimates of the Doppler-shift have shown that these frequen-cies differ from the plasma frame frequencies by less than 30%(see Section 5.1 for details). are fitted to a power-law of β e . The best fit (black curve)shown in Figure 12 demonstrates that we generally have f w /f ce (cid:46) . β − . e . The whistler wave frequenciesin Figure 12 are measured in the spacecraft frame anddiffer from those in the plasma frame by the Doppler-shift, ∆ f D = kv sw / π , where k is the whistler wave vec-tor. We have estimated the Doppler-shift for all whistler waves events using the wave vector estimate from thecold dispersion relation, f /f ce = k d e / (1 + k d e ), where d e = c/ω pe is the electron inertial length and ω pe is theelectron plasma frequency (e.g., Stix 1962). We havefound that ∆ f D /f w is less than 0.3, so that the mea-sured frequency can be considered as a good estimate ofthe whistler wave frequency in the plasma frame.Figure 13 presents the frequency bandwidth ∆ f ofabout 5,800 whistler wave events with f w >
16 Hz.Panel (a) shows that ∆ f is typically about 15 Hz,though can be as large as 50 Hz. Panel (b) showsthat the frequency bandwidth normalized to the whistlerwave frequency f w is typically in the range between 0.1and 1. There is a clear positive correlation between∆ f /f w and β e : at β e (cid:28) f /f w ∼ .
2, while ∆ f /f w is typically about 0.5at β e ∼
1. The implications of the frequency width es-timates will be discussed in Section 6.5.2.
WHFI predictions
The linear theory of the WHFI suggests that theelectron velocity distribution function (VDF) consist-ing of bi-Maxwellian core and halo populations, counter-streaming in the plasma rest frame, can be unstable towhistler wave generation at sufficiently large core andhalo bulk velocities (Gary et al. 1975, 1994). Tonget al. (2019) have recently shown for several events thatthe WHFI indeed generates whistler waves in the pris-tine solar wind. In this section we evaluate the max-imum and minimum frequencies of whistler waves ex-pected to be produced by the WHFI in dependence on β e . We consider the simplest electron VDFs consistingof isotropic core and halo populations ( T ⊥ = T || ) and2 Tong et al. e f [ H z ] (a) e f / f w (b) Figure 13.
The frequency bandwidth, in physical units and normalized to f w , of 5,800 whistler wave events, whose frequency f w is above 16 Hz. The frequency bandwidth is presented versus β e .variable values T c /T p n c /n { . , . , . , . } T h /T c { , , , , } ∆ v c /v A {− i/ | i = 0 , , .... } Table 1.
Parameter ranges used for the analysis of the max-imum and minimum frequencies of whistler waves that canbe generated by the whistler heat flux instability (see Section5.2 for details). assume a zero net electron current in the plasma restframe, n c ∆ v c + n h ∆ v h = 0, where n c,h and ∆ v c,h aredensities and bulk velocities of the core and halo popula-tions. Because the bulk velocities are much smaller thanthe corresponding thermal velocities (e.g., Feldman et al.1975; Tong et al. 2019), we have β e ≈ β c + β h , where β c = 8 πn c T c /B , β h = 8 πn h T h /B and T c,h are coreand halo temperatures.The linear growth rate of the WHFI normalized to f ce depends on n c /n , T h /T c , T p /T c , and ∆ v c /v A , where n is the total electron density which is also assumedequal to the proton density, T c,h are the core and thehalo temperatures, T p is the proton temperature, and v A = B / (4 πn m p ) / is the Alfv´en velocity , and m p is the proton mass. The growth rate is almost indepen-dent of the proton to core electron temperature ratio,because in realistic conditions protons do not resonatewith whistler waves produced by the WHFI (Gary et al.1975). In what follows we keep T p /T c = 1 which is areasonable assumption at 1 AU (e.g., Newbury et al.1998; Artemyev et al. 2018). To evaluate the maximumand minimum frequencies of whistler waves that can begenerated by the WHFI instability, we fix β e and vary n c /n , T h /T c and ∆ v c /v A in the ranges typical for thesolar wind at 1 AU (Table 1). For each combination of γ/ω ce a b cf max /f ce > f min /f ce > − f min /f ce > − Table 2.
Values of parameters a, b and c in Eq. (5) thatgives fitting to the maximum and minimum frequencies ofwhistler waves that can be generated by the whistler heatflux instability at various β e . The maximum frequencyquickly converges to some asymptotic value as the growthrate tends to zero, whereas the minimum frequency bounddepends on the growth rate threshold. We present parame-ters for the maximum frequency bound at zero growth rate,and the minimum frequency bounds computed for γ/ω ce > − and 10 − , where ω ce = 2 πf ce . these three parameters we compute the linear growthrate using the numerical code developed by Tong et al.(2015) and identify the frequency of the fastest growingwhistler wave. Then, for each fixed β e we identify themaximum and minimum frequencies of whistler wavesthat can be generated by the WHFI. At a fixed β e theminimum frequency decreases with decreasing thresh-old value on the growth rate. Different threshold val-ues result in different minimum frequency bounds, butthese bounds are of similar shape and almost parallel toeach other in the ( β e , f /f ce ) plane. The maximum andminimum frequency bounds are well fitted to modifiedpower-laws f /f ce = a ( β e + b ) c (5)Table 2 presents the best fit parameters a , b and c forthe maximum frequency bound at zero growth rate andfor the minimum frequency bounds derived for severalgrowth rate thresholds, γ/ω ce > − and 10 − , where ω ce = 2 πf ce .Figure 12 overlays the theoretical maximum and min-imum frequency bounds upon the measured whistler histler waves at 1AU γ/ω ce > − . The frequen-cies of the major part of the observed whistler wavesfall between the minimum and the maximum theoret-ical bounds, demonstrating thereby that the observedwhistler waves could be in principle generated by theWHFI. Moreover, the generation can be local that is thewhistler waves are generated in a local plasma, ratherthan generated in some other region and propagated tothe spacecraft location. DISCUSSIONWe have carried out statistical analysis of whistlerwaves observed in the pristine solar wind using the mostrepresentative dataset collected up to date. We have fo-cused on whistler waves identified by a local peak in thespectral power density of the magnetic field fluctuations,that is why these whistler waves are produced by freeenergy in a plasma, rather than by the turbulence cas-cade. Out of 801,527 magnetic field spectra measured at1 AU aboard ARTEMIS, we have selected about 17,050intense wave activity events in the whistler frequencyrange and associated 13,700 of them with quasi-parallelwhistler waves. Thus, about 80% of the intense events inthe whistler frequency range are consistent with quasi-parallel whistler wave interpretation. This conclusion isin agreement with results of the previous less extensivestudies of waveform and cross-spectra measurements(Lacombe et al. 2014; Stansby et al. 2016; Tong et al.2019). The other ∼
20% of the intense events are highlylikely low-frequency plasma modes Doppler-shifted intothe whistler frequency range, because they are predom-inantly observed in the three lowest frequency channels.The overall occurrence of quasi-parallel whistler wavesin our dataset is about 1.7%, but the actual occurrenceof whistler waves is certainly higher, because we selectedonly sufficiently intense whistler waves above 16 Hz.We have shown that the occurrence probability ofwhistler waves most critically depends on the electrontemperature anisotropy. There is no any drastic depen-dence of the whistler wave occurrence on the electronheat flux, solar wind velocity or β e . The occurrenceprobability is less than 2% when T e ⊥ /T e || (cid:46) .
9, butvaries from 5 to 15% as T e ⊥ /T e || increases from 0.95to 1.2. This correlation is consistent with the recentanalysis by Tong et al. (2019) of several whistler waveevents measured in the burst mode (waveform available)aboard ARTEMIS. Tong et al. (2019) have shown thatwhistler waves in those events were generated locallyby the WHFI, while the temperature anisotropy of thehalo population T h ⊥ /T h || critically affects the instabilityonset: T h ⊥ /T h || sufficiently smaller than unity quenches the instability, while T h ⊥ /T h || > T e ⊥ /T e || corresponds tothe increase of the halo temperature anisotropy, becausetemperature anisotropies of core and halo populationsare positively correlated (Feldman et al. 1976; Pierrardet al. 2016).We have shown that whistler waves in the solar windhave amplitudes B w typically below 0.02 B or in phys-ical units below 0.1 nT. These amplitude estimates areconsistent with the previous less extensive studies, wherewaveform measurements were analyzed (Lacombe et al.2014; Stansby et al. 2016; Tong et al. 2019), but moreextensive waveform analysis should be carried out in thefuture to verify this result. The averaged whistler waveamplitude B w /B is found to be negatively correlatedwith the solar wind velocity. The average B w /B corre-lates positively with the electron heat flux and electrontemperature anisotropy, but the strongest positive cor-relation is found with β e . The variation of q e /q and T e ⊥ /T e || over the observed range results in variation of B w /B by about 30%, while the variation of β e from 0.1to 5 results in variation of B w /B by a factor of three.The presented amplitude estimates and correlations be-tween B w /B and macroscopic parameters should beuseful for future theoretical studies of origin and effectsof whistler waves in the solar wind. At the moment,we note that the whistler wave amplitudes observed at1 AU are much smaller than whistler wave amplitudes B w ∼ B reported in recent Particle-In-Cell simulations(Roberg-Clark et al. 2016; Roberg-Clark et al. 2018), in-dicating thereby that the simulations are initialized withelectron VDFs unrealistic for the solar wind at 1 AU.The fact that the whistler wave amplitudes are rathersmall calls into question their role in the electron heatflux regulation in the solar wind, though this questiondeserves a separate study.We have estimated the frequencies of the observedwhistler waves and bandwidths of the whistler wavespectra. The only electrons that can drive and effi-ciently interact with quasi-parallel whistler waves arethose in the first normal cyclotron resonance (e.g., Shkl-yar & Matsumoto 2009) v || = ω − ω ce k , (6)where v || is electron velocity parallel to the quasi-staticmagnetic field, ω = 2 πf , ω ce = 2 πf ce and k is thewhistler wavenumber. The minimum energy of thecyclotron resonant electrons (e.g., Kennel & Petschek1966)4 Tong et al. e E R [ e V ] (a) e E R / T e || (b) Figure 14.
The minimum energy of electrons to be in the first normal cyclotron resonance with the observed whistler waves.It is given by Eq. (7) with the whistler wave frequencies adopted from Figure 12a. Panel (a) presents the minimum resonantenergy in physical units, while panel (b) presents this energy with respect to the electron temperature T e || . The averagedresonant energies are presented by the red curves. E R = B πn f ce f (cid:18) − ff ce (cid:19) , (7)where we have used cold dispersion relation of whistlerwaves, f /f ce = k d e / (1 + k d e ) (e.g, Stix 1962). Figure14 presents the minimum resonant energy evaluated us-ing Eq. (7) with whistler wave frequencies adopted fromFigure 12a. The minimum resonant energy is negativelycorrelated with β e , because E R ∝ B , while β e ∝ /B .Panel (a) shows that the minimum resonant energy isof a few tens of eV at β e ∼ β e . Panel (b) shows that in terms ofthermal energies the resonant energy is about 3 T e at β e ∼ T e at low β e . We concludethat the observed quasi-parallel whistler waves shouldbe driven by the halo electron population in accordancewith previous theoretical (Gary et al. 1975, 1994) andexperimental (Tong et al. 2019) studies.The estimated bandwidths of the whistler wave spec-tra allow us to evaluate whether the effect of the ob-served whistler waves on electrons could be addressedwithin the quasi-linear theory (QLT) (e.g., Sagdeev &Galeev 1969). The QLT is applicable for a sufficientlywide frequency width of a whistler wave spectrum (e.g.,Karpman 1974): ∆ f /f w (cid:29) ( B w /B ) / ( kv ⊥ /ω ce ) / ,where v ⊥ is the electron velocity perpendicular to themagnetic field. Because the whistler waves interact ef-ficiently with halo electrons, we can assume that v ⊥ isa few times larger than the electron thermal velocity.Using the cold dispersion relation for whistler waves werewrite the QLT applicability criterion∆ ff w (cid:29) (cid:18) B w B (cid:19) / (cid:18) β e f w /f ce − f w /f ce (cid:19) / (8) RHS = ( B w / B ) [ e ( f w / f ce )/(1 f w / f ce )] L H S = f / f w LHS/RHS P D F Figure 15.
Estimated values of the left hand side (LHS) andthe right hand side (RHS) of Eq. (8) using ARTEMIS mea-surements. The red line references equality between LHS andRHS. The probability density function of the ratio LHS/RHSis shown in the inset panel.
Figure 15 presents the test of the QLT applicabilityand shows that ∆ f /f w is always above the right-handside of Eq. (8). The inset panel shows the probabil-ity distribution function of the ratio of ∆ f /f w to theright-hand side and confirms that in the majority of theevents ∆ f /f w is about five times larger than the right-hand side. We conclude that the quasi-linear theoryis likely a good approximation for analysis of effects ofthe observed whistler waves on electrons. At the sametime, we stress that an extensive statistical analysis ofwaveform measurements should be carried out in thefuture to verify that whistler wave amplitudes B w in-ferred from 8s magnetic field spectra do not significantlyunderestimate the actual peak amplitudes of whistler histler waves at 1AU e [ s ] Figure 16.
The quasi-linear relaxation time of unstableelectron VDFs by the observed whistler waves presented ver-sus β e . waves. The statement of the QLT applicability concernsonly whistler waves in the pristine solar wind. Whistlerwaves observed in interplanetary shock waves may berather narrow-band and large-amplitude for the QLT tobe applicable (e.g., Breneman et al. 2010; Wilson et al.2013).Figure 16 presents order of magnitude estimates of thequasi-linear relaxation time of unstable electron VDFsby the observed whistler waves. The relaxation time isgiven by the following expression (e.g., Karpman 1974) τ ≈ πf w β e (cid:18) ∆ ff w (cid:19) B B w , (9)where in deriving this formula we have assumed that f w (cid:28) f ce . The typical relaxation time is a few tensof minutes at low β e to about a minute at β e ∼
1. Inprinciple the relaxation can be as fast as a few seconds.The strong negative correlation between τ and β e is dueto explicit dependence of τ on β e according to Eq. (9)and due to a strong positive correlation between B w /B and β e . During the relaxation time whistler waves maycover spatial distances of a few tens of thousands kilome-ters implying that low-frequency density and magneticfield fluctuations may affect the relaxation process ofthe WHFI (see, e.g., Voshchepynets et al. 2015, for re-laxation of a beam instability in nonuniform solar windplasma).We have shown that the frequency upper bound of theobserved whistler waves is negatively correlated with β e and demonstrated that the frequencies are in effect con-sistent with the theoretical predictions of the WHFI.Thus, in accordance with conclusions of Tong et al.(2019) whistler waves observed in the pristine solar wind can be indeed generated by the WHFI operating in a lo-cal plasma. We have compared the observed frequenciesto predictions of the WHFI theory with electron VDFsconsisting of core and halo electron populations. Thepresence of the anti-sunward strahl population typicalfor the fast solar wind (Pilipp et al. 1987; ˇStver´ak et al.2009) would not affect any characteristics of the WHFI,because whistler waves produced by the WHFI propa-gate anti-sunward and do not resonate with the strahl(e.g., Vasko et al. 2019, for discussion).The original WHFI theory assumed both core andhalo electron populations to be temperature isotropic(Gary et al. 1975). The unstable whistler waves wereshown to propagate parallel to the halo bulk velocityor, equivalently, parallel to the electron heat flux. Inthe realistic solar wind both core and halo populationsexhibit some temperature anisotropies (Feldman et al.1976; ˇStver´ak et al. 2008; Pierrard et al. 2016). Evena slight temperature anisotropy T h ⊥ /T h || > T h ⊥ /T h || > CONCLUSIONIn this section we summarize the results of our statis-tiscal analysis of whistler waves at 1 AU:6
Tong et al.
1. The intense wave activity in the whistler frequencyrange is shown to be dominated (80%) by quasi-parallel whistler waves. The overall occurrence ofquasi-parallel whistler waves in the pristine solarwind is found to be about 1.7%. We emphasizethat only intense whistler waves above 16 Hz havebeen considered in this study, so that the actualoccurrence is certainly higher.2. The occurrence probability of whistler waves in thepristine solar wind is strongly dependent on theelectron temperature anisotropy T e ⊥ /T e || . The oc-currence probability is less than 2% at T e ⊥ /T e || (cid:46) .
9, but varies from 5 to 15% as T e ⊥ /T e || increasesfrom 0.95 to 1.2. There is no apparent dependenceof the whistler wave occurrence on the electronheat flux q e /q , the solar wind velocity v sw or β e .3. Whistler waves in the solar wind have amplitudestypically below 0.02 B , where B is the magni-tude of the quasi-static magnetic field. In physicalunits the amplitudes are in the range from about0.01 to 0.1 nT.4. The average normalized whistler wave ampli-tude B w /B correlates positively with q e /q and T e ⊥ /T e || , but the strongest positive correlationis found with β e . The variation of q e /q and T e ⊥ /T e || over the observed range results in varia-tion of B w /B by about 30%, while variation of β e from 0.1 to 5 results in variation of B w /B bya factor of three. The whistler wave amplitudenegatively correlates with the solar wind velocity,varying by a factor of two from slow to fast solarwind.5. Whistler wave frequencies f w /f ce fall betweensome upper and lower bounds dependent on β e .The upper bound on the whistler wave frequencyis approximately given by 0 . β − . e . The fre- quency bandwidth ∆ f of the whistler waves is de-termined and ∆ f /f w is shown to be positively cor-related with β e .6. We show that the observed whistler wave frequen-cies are consistent with the theoretical predictionsof the whistler heat flux instability, indicatingthereby that whistler waves in the pristine solarwind can be generated by the WHFI. The genera-tion of some of the whistler waves by the temper-ature anisotropy instability can not be ruled out.7. We have shown that the frequency width of thewhistler waves is sufficiently wide so that thequasi-linear theory is likely applicable to describeeffects of the whistler waves on electrons. Thetypical quasi-linear relaxation time in a uniformplasma would be from a minute at β e ∼ β e . In principle therelaxation can be as fast as a few seconds.8. We have estimated the energies of electrons res-onating the whistler waves and shown that thewhistler waves should be driven by suprathemralelectrons, whose minimum energy E R is negativelycorrelated with β e . E R is about a few tens of eV,or equivalently, about three times the thermal en-ergy at β e ∼