Whistler Waves and Electron Properties in the Inner Heliosphere: Helios Observations
Vamsee Krishna Jagarlamudi, Olga Alexandrova, Laura Bercic, Thierry Dudok de Wit, Vladimir Krasnoselskikh, Milan Maksimovic, Stepan Stverak
DD RAFT VERSION A UGUST
7, 2020Typeset using L A TEX preprint2 style in AASTeX61
WHISTLER WAVES AND ELECTRON PROPERTIES IN THE INNER HELIOSPHERE:
HELIOS
OBSERVATIONS V AMSEE K RISHNA J AGARLAMUDI ,
1, 2 O LGA A LEXANDROVA , L AURA B ER ˇ CI ˇ C ,
2, 3 T HIERRY D UDOK DE W IT , V LADIMIR K RASNOSELSKIKH , M ILAN M AKSIMOVIC , AND ˇ S T ˇ EPÁN ˇ S TVERÁK
4, 5 LPC2E/CNRS, 3 Avenue de la Recherche Scientifique, 45071 Orléans Cedex 2, France LESIA, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université, Université de Paris, 5 place Jules Janssen, 92195Meudon, France Physics and Astronomy Department, University of Florence, Via Giovanni Sansone 1, I-50019 Sesto Fiorentino, Italy Astronomical Institute, Czech Academy of Sciences, CZ-14100 Prague, Czech Republic Institute of Atmospheric Physics, Czech Academy of Sciences, CZ-14100 Prague, Czech Republic (Received April 9, 2020; Accepted May 15, 2020)
Submitted to ApJABSTRACTWe present the analysis of narrow-band whistler wave signatures observed in the inner heliosphere (0.3to 1 AU). These signatures are bumps in the spectral density in the 10-200 Hz frequency range of the ACmagnetic field as measured by the search coil magnetometer on-board the
Helios <
500 kms − ) and their occurrenceincreases with the radial distance (R), from ∼
3% at 0.3 AU to ∼
10% at 0.9 AU. In the fast solar wind( >
600 kms − ), whistler activity is significantly lower, whistler signatures start to appear for R > . ∼ .
03% at 0.65 AU to ∼
1% at 0.9 AU. We have studied the variation ofthe electron core and halo anisotropy ( T e ⊥ / T e (cid:107) ), as well as the electron normalized heat flux as a functionof R and of the solar wind speed. We find that in the slow wind electron core and halo anisotropy is higherthan in fast one, and also these anisotropies increase radially in both types of winds, which is in line withthe occurrence of whistler signatures. We hypothesize the existence of a feedback mechanism to explainthe observed radial variations of the occurrence of whistlers in relation with the halo anisotropy. Keywords:
Solar wind - Magnetic fields - Whistler waves - Electrons - Instabilities
Corresponding author: Vamsee Krishna [email protected] a r X i v : . [ phy s i c s . s p ace - ph ] A ug J AGARLAMUDI ET AL . INTRODUCTIONWhistler waves are the most probable electro-magnetic modes observed between the lower hy-brid frequency ( f LH = √ f ce f ci ) and electron cy-clotron frequency ( f ce ) (Gary 1993), where f ci = q p B / π m p , f ce = q e B / π m e , q p is charge of theproton, q e is charge of the electron, m p is massof the electron, m e is mass of the electron and B is magnitude of the magnetic field. These wavesare thought to have a significant contribution in theregulation of the fundamental processes in the so-lar wind, especially for the solar wind electrons.Whistler waves are therefore widely studied phe-nomena and are the key to a better understandingof the global solar wind thermodynamics and en-ergy transport.Whistlers could play a important role in the evo-lution of the solar wind electron velocity distribu-tions (Vocks et al. 2005; Vocks 2012; Kajdiˇc et al.2016; Roberg-Clark et al. 2018; Boldyrev & Ho-raites 2019; Tang et al. 2020) through the pitchangle scattering of the beam-like solar wind elec-tron component, called the strahl (Hammond et al.1996; Graham et al. 2017), which is in turn is ex-pected to affect the second high energy electronpopulation, the halo. Whistler waves are expectedto regulate the heat flux (Gary et al. 1994, 1999a;Roberg-Clark et al. 2018). On the other hand,kinetic simulations (Vocks & Mann 2003; Vocks2012) have shown that whistler waves could alsoprovide a mechanism for the continuous forma-tion of suprathermal electrons in the corona. In theEarth’s magnetosphere, whistler waves have a sig-nificant role in the acceleration and precipitation ofparticles in the radiation belts, e.g. Artemyev et al.(2016).In this study, we focus on whistler waves in theinner heliosphere (for radial distances between 0.3and 1 AU). One of the earliest studies of mag-netic fluctuations at kinetic scales in the inner-heliosphere was done by Beinroth & Neubauer(1981). The authors used the power spectral den-sity of the magnetic field measured by the search coil magnetometer on-board Helios
1. They inter-preted the observed broadband spectrum as due towhistler waves. Later it became clear that thesebroadband spectra represent background turbu-lence (Alexandrova et al. 2012) with wave vectorsmostly perpendicular to the mean field B , k ⊥ (cid:29) k (cid:107) and negligible frequencies in the plasma frame(Lacombe et al. 2017). Instead, the whistler wavesobserved up to date in the free solar wind are nar-row band, quasi-parallel, right handed polarized;in power spectral densities they appear as spec-tral bumps e.g., Lacombe et al. (2014); Kajdiˇcet al. (2016); Roberts et al. (2017). Note that theobserved frequency range in the satellite frameis with in the expected frequency range in theplasma frame, namely, between the f LH and f ce (Gary 1993; Stenzel 1999).Among the early studies which confirmed the ex-istence of whistler waves in the solar wind wasthat by Zhang et al. (1998). They used magneticand electric field high resolution waveform data onboard the Geotail spacecraft upstream of the bowshock. In the free solar wind, the authors observedshort-lived (less than 1 s) monochromatic wave-packets, at frequencies between 0 . f ce and 0 . f ce with RH polarisation and k mostly parallel to B , inanti-sunward direction.An analysis of long-lived (5-10 minutes), narrow-band, quasi-parallel, RH whistlers in the free solarwind was done by Lacombe et al. (2014) using themagnetic spectral matrix routine measurements ofthe Cluster /STAFF instrument, between 2001 and2005. About 20 such events were found in the slowwind, within the [ f LH , . f ce ] frequency range.Stansby et al. (2016) studied the RH wavepack-ets observed in the whistler range using the Artemis electric and magnetic field wave forms measured at128 samples/second in the solar wind. The authorsdetermined empirical dispersion relation and founda good agreement with the linear theory predic-tions for parallel propagating whistlers in a finite β e plasma. They showed that the observed whistlers HISTLER WAVES IN THE INNER HELIOSPHERE
Stereo/WAVES/TDS , whichcapture the most intense signals. The authorsfound quasi-monochromatic, oblique, RH po-larised whistlers around ∼ . f ce . These obliquewhistler waves are mostly associated with thestream interaction regions (SIR’s) and interplan-etary (IP) shocks and have unusual high ampli-tudes in comparison with what was known before(Moullard et al. 2001). The duration of these wavesis found being few seconds to minutes.What are the sources of whistler waves ? Theprimary source of energy in the pristine solar windare instabilities driven by the electron distributionfunctions. In the frame of the solar wind electrons,we expect to see two types of instabilities: thewhistler temperature anisotropy instability, whichdepends on the ratio between perpendicular andparallel electron temperatures T e ⊥ / T e (cid:107) >
1, and thewhistler heat flux instability is expected to developwhen the heat flux is mainly carried by the anti-sunward motion of the halo/strahl electrons rela-tive to the sunward moving core (Gary & Feldman1977; Breneman et al. 2010).Zhang et al. (1998) suggested that the paral-lel short-lived whistlers are generated by the haloelectrons. In the data set of Lacombe et al. (2014),when long-lived whistlers were observed the tem-perature anisotropy of the electrons was T e ⊥ / T e (cid:107) <
1, while the heat flux showed a relative increase atthe time of appearance of the waves, thus indicat-ing that the whistler heat flux instability is prob-ably responsible for the observed whistler activity.Note that the values of the temperatures were thoseof the global electron distribution function, with-out separation between the core, the halo and thestrahl.Wilson et al. (2013) investigated whistler wavesdownstream of super-critical interplanetary shocksand suggested that these waves might be drivenby a heat flux instability and cause perpendicular heating of the halo electrons. Tong et al. (2019b)studied the simultaneous field and particle mea-surements when the field aligned whistler wavesare observed and they showed that the tempera-ture anisotropy of halo electrons significantly af-fects the heat flux instability onset.In this study, we focus on signatures of whistlerwaves within the solar wind (excluding magneticclouds and IP shocks) using search coil magne-tometer (SCM) data from
Helios
1. Spanning thedistances between 0.3 and 1 AU, the
Helios f lh < f < f ce , are signatures of whistlers.Therefore, the presence of a spectral bump in thespectrum is our sole criterion for whistler wave de-tection.The article is structured as follows. In Section 2we describe the Helios DATAWe use magnetic field spectral density data fromthe search coil magnetometer (Dehmel et al. 1975)on-board
Helios B y component of the sen-sor (located in the ecliptic plane) that were col-lected during the period of 12-12-1974 to 20-09-1975 with an 8-second cadence. For a comprehen- J AGARLAMUDI ET AL .sive discussion on
Helios search coil magnetome-ter (SCM) and its data, we refer the reader to theHelios Data Archive.The spectral density of the B y component is mea-sured in 8 logarithmically-spaced frequency bands.Their central frequencies are respectively : 6.8Hz, 14.7 Hz, 31.6 Hz, 68 Hz, 147 Hz, 316 Hz,681 Hz and 1470 Hz. From these we obtain thepower spectral density (PSD) by squaring the spec-tral density. Although the B z component (perpen-dicular to the ecliptic plane) is also measured, itis more affected by the spacecraft electromagneticnoise (Neubauer et al. 1977) and so we discard it.The SCM has two data products: mean and peakmagnetic field spectral density in the consideredtime interval. Depending on the operational modeof spacecraft telemetry system, the mean and peakamplitudes are calculated for intervals of 1.125,2.25, 4.5, 18, 36, 72, 144, 288, 576 or 1152 sec-onds. If the average interval is less than 4.5 s thenthe mean values are compressed to 8 second aver-ages (Helios data archive). Peak amplitudes fre-quently saturate near the Sun; for that reason weconsider mean amplitudes only.We consider the PSD’s that exceed the SCMnoise level by a factor of 2 at least for the first 4bands. The SCM noise is taken from (Neubaueret al. 1977). We obtain 324366 spectra for ouranalysis.For the physical interpretation of our data we usethe proton and electron moments. Proton momentssuch as density, velocity and temperature of 40.5s time resolution are taken from the Helios dataarchive. The electron moments (01-01-1975 to 10-05-1979) are taken from the work of Štverák et al.(2009) and Štverák et al. (2015), here authors haveconsidered only the measurements when the mag-netic field vector is almost in an ecliptic plane. Un-fortunately these electron observations of 40.5 stime resolution are less regularly available.We divided the SCM magnetic spectra observa-tions based on the solar wind bulk velocity ( V sw ),fast solar wind ( V sw >
600 kms − ) and slow solar R ( AU ) V s w ( k m s ) Figure 1.
2d histogram of solar wind bulk speed ( V sw )corresponding to the analyzed spectra for the wholeHelios-1 mission, from December 12, 1974 to Septem-ber 20 1975 as a function of distance ( R ) from the Sun. wind ( V sw <
500 kms − ). In Figure 1 we show the2D histogram of V sw for the analyzed data set as afunction of the distance from the Sun R . We ob-serve that the slow and fast wind SCM data areavailable at all distances (0.3 to 1 AU) for our anal-ysis. There are nearly equal amounts of observa-tions made in the slow and in the fast solar wind:43 % of the spectra are in the slow wind and 37 %in the fast wind. The remaining 20 % correspondto intermediate velocities.The general behavior of the slow and fast solarwind spectra is shown in Figure 2 using the ratioof the amplitude of the PSD between different fre-quency channels as a function of time. In the fastwind, the ratio of amplitudes between two consec-utive channels are almost constant and thereforenearly all the spectral bands show similar behavior.This coherent evolution of the spectra means thatthe spectral shape is conserved and only the am-plitude of the spectrum is varying. A completelydifferent picture emerges from the slow wind, inwhich the channel ratios show frequent intermit-tent bursts, which are signatures of large changesof the spectral shape. HISTLER WAVES IN THE INNER HELIOSPHERE
68 69 70 71 7210 R a t i o o f c h a nn e l s
68 69 70 71 72
Day of the year V s w Figure 2.
The ratio of PSDs between two pairs of fre-quency channels (upper panel) and variation of the so-lar wind velocity (lower panel) around 0.3 AU in 1975from
Helios
WHISTLER WAVE IDENTIFICATIONFigure 3 illustrates typical spectra in the fast(panel (a)) and slow wind (panel (b)) at 0.3 AU.Here one observes:1. Smooth spectra, where amplitude is decreas-ing monotonically with frequency (Figure3 (a)). This is a permanent feature of back-ground magnetic field turbulence as was ob-served at 1 AU (Alexandrova et al. 2012).2. Spectra with local maxima (Figure 3 (b)):background magnetic field turbulence spec-tra influenced by narrow-band waves as in(Lacombe et al. 2014).3. Some spectra which are in between, withouta distinctive local maxima. Those are theones that might have been affected by eithershort lived or very low amplitude waves.We consider those spectra in which one singlelocal maximum (bump) clearly stands out with re-spect to the PSD of the background turbulence.Mathematically speaking, dPSDd f is negative for class1 spectra as we go towards higher frequencies, thiskind of
Helios spectra are analysed in Alexandrovaet al. (2020). However, when the whistler waves influence the spectra, as we go towards higher fre-quencies at a certain frequency dPSDd f will be pos-itive and then automatically again go to negativeas for class 2 spectra. In this way, we automati-cally identify the presence of spectral bump. Us-ing the list of interplanetary shocks (de Lucas et al.2011; Kruparova et al. 2013) and the list of mag-netic clouds (Bothmer & Schwenn 1998) we haveeliminated the interplanetary shocks and magneticcloud intervals from our analysis. In our total an-alyzed samples of N tot =324366 we have observed N w =7287 spectra that have a distinctive spectralbump. This number is small ( ∼ . N tot ). How-ever, this number is comparable to the percentageof whistler waves observed at 1 AU by Tong et al.(2019a), which is ∼ . δ B / B (cid:39) .
05. We observe that adistinctive bump appears for waves that dominatethe background fluctuation during the time of inte-gration of the spectrum. However, even a lifetimeof 1 /
16 in the considered interval can be enoughto observe a bump if the wave dominates the back-ground.The next step is the identification of the cen-tral frequency of the spectral bumps and to ana-lyze whether the observed frequencies are in thewhistler range or not.Figure 4 shows spectra with distinctive localmaxima extracted from the example spectra shownin Figure 3 (b). In Figure 4 (a) we show the spec-tra as a function of the measured f and in Figure J AGARLAMUDI ET AL . (a) Frequency(Hz) P S D ( n T / H z ) Fast wind at 0.3 AU (b) Frequency(Hz) P S D ( n T / H z ) Slow wind at 0.3 AU
Figure 3.
Example of typical Helios1/SCM spectraat 0.3 AU. Panel (a) shows the spectra (217) in thefast wind interval (10:00-10:45, 9-03-1975); panel (b)shows the spectra (136) in the slow wind (19:45-20:30,14-03-1975). f / f ce . From Figure 4 (b) we can infer that for theobserved example spectra, the spectral bumps arein the whistler range, i.e. between ∼ f LH and 0.5 f ce , see two vertical lines. The mean frequency isaround f / f ce = 0 .
1. Now, let us verify the centralfrequency of the bumps for all the spectra.In Figure 5 we show the distribution of normal-ized central frequencies for the N w bumps observedin the Helios1/SCM spectra. Almost all of thebumps are between ∼ f LH and 0 . f ce . There were12 spectra which were observed very close but be-low the whistler range, however, we have includedthem in the analysis, as Doppler shift may be thereason for that.As was discussed in the introduction, Lacombeet al. (2014) have shown that in the free solar (a) Frequency (Hz) P S D ( n T / H z ) (b) f / f ce P S D × f c e ( n T ) Figure 4.
Illustration of 115 spectra with clear bumpsout of 136 spectra of Figure 3 (b) as a function of f inpanel (a) and f / f ce in panel (b). The region betweenthe black vertical lines correspond to whistler range. wind any spectral bump in [ f LH , . f ce ] range cor-responds to right handed (RH), circularly polarizedwhistlers. All the whistlers were observed to bepropagating along the mean magnetic field. Asfar as we have no access to polarization proper-ties, we use the results of Lacombe et al. (2014) asa basis for our assumption that any bump we ob-serve at frequencies f LH < f < . f ce correspondsto whistler waves. WHISTLER WAVE PROPERTIESIn Figure 6 we show an example of the typi-cal plasma parameters around 0.4 AU along withthe amplitudes of the observed whistler wave sig-natures. This figure gives us a glimpse of theplasma conditions when the whistler waves are ob-served. We infer that whistler waves occur inter-mittently and have different amplitudes (the defini-tion of the amplitude of the waves is given in sec-
HISTLER WAVES IN THE INNER HELIOSPHERE f / f ce N u m b e r o f w h i s t l e r s Slow windFast wind
Figure 5.
Histograms showing the distribution of nor-malized frequencies of the spectral bumps. Green his-togram correspond to the slow wind, while the blue his-togram correspond to the fast wind. The red verticallines correspond to f LH / f ce and 0.3 respectively. tion 4.3). Whistler waves appear in the slow solarwind, where the typical conditions when comparedto the fast solar wind are: lower magnetic fieldmagnitude, higher proton density, higher electroncore anisotropy ( A c = T e ⊥ c T e (cid:107) c ), higher electron haloanisotropy ( A h = T e ⊥ h T e (cid:107) h ), lower normalized heat flux( q e (cid:107) / q ), where q = 1 . n e T e ( T e / m e ) / and higher β ec ( n ec k B T ec / B µ ). In the coming subsections, weshall discuss in detail the effects of these proper-ties.4.1. Solar wind bulk speed and whistler waves
Histograms in Figure 7 show the distribution ofsolar wind velocities for all N tot spectra used inour analysis and for the spectra showing the pres-ence of whistler wave signatures. All the analyzedspectra are shown in grey, slow wind whistlers areshown in green, intermediate in red and fast windwhistlers are shown in blue, respectively.Even though we have observed nearly equalnumbers of spectra in the slow and fast wind, themajority of the spectra with signatures of whistlers ∼
93% (6783) were observed in the slow solarwind ( <
500 kms − ). This is similar to the studyby Lacombe et al. (2014) at 1 AU, where the au-thors suggested that a slow wind speed is one of the necessary conditions for the observation of long-lived whistler waves. Interestingly, this conditionat 1 AU also seems to hold in the inner helio-sphere. However, in our study, we also observewhistler waves in the fast solar wind but they arerare, and represent about 3% (237) of the numberof the spectra with wave signatures. The remaining4% are observed in the intermediate wind.Figure 8 shows the percentage of whistler wavesas a function of the wind velocity, i.e. the num-ber of whistlers to the number of spectra availablein the corresponding velocity bin. We observe thatthere is a constant decrease of whistler waves oc-currence as the wind velocity is increasing : thehigher the velocity, the lower the probability of ob-serving whistler waves.4.2. Radial distribution of the observed whistlerwaves
Figure 9 (a) shows the number of spectra withwhistler wave signatures observed at different ra-dial distances from the Sun. We observe that in theslow wind whistler waves are present at all the dis-tances, from 0.3 to 1 AU. In contrast, in the fastwind whistlers only start to appear for R > . Amplitude of the observed whistler waves J AGARLAMUDI ET AL . B ( n T ) a N p ( c m ) b V s w ( k m s ) c A c d A h e q e / q f e c g Day 1975 B y ( n T ) h Figure 6.
Illustration of typical conditions when the whistler waves are observed using a three day interval around0.4 AU on March 21-24, 1975. We show in panel (a) the magnitude of magnetic field, in panel (b) we show theproton density, in panel (c) the proton bulk velocity, in panel (d) the electron core anisotropy ( A c ) and the red linecorresponds to A c = 1, in panel (e) the halo anisotropy ( A h ) and the red line corresponds to A h = 1, in panel (f) thenormalized parallel electron heat flux ( q e (cid:107) / q ) and the red line corresponds to q e (cid:107) / q = 1, in panel (g) the electroncore beta and in panel (h) the amplitudes of the intermittently occurring whistlers waves. HISTLER WAVES IN THE INNER HELIOSPHERE
200 300 400 500 600 700 800 900 V sw (Kms ) N u m b e r o f s p e c t r a a n d w h i s t l e r s Figure 7.
Distribution of the solar wind velocities forthe spectra used in our analysis. The histogram of allanalysed spectra is shown in grey, while green, red,and blue histograms show the fraction with identifiedwhistler waves for slow, intermediate, and fast solarwind, respectively.
300 400 500 600 700 800 V sw (Kms ) % o f w h i s t l e r s Figure 8.
The occurrence of whistler waves as a func-tion of the solar wind bulk speed ( V sw ). For each veloc-ity we show the fraction of PSDs that have the signatureof whistler waves. Error bars correspond to standard er-ror, i.e. 10 (cid:112) N w , V sw / N tot , V sw . Using the contribution from the B y spectral com-ponent we have calculated the approximate RMSamplitude of the fluctuations that are associatedwith whistler waves. The amplitude of whistlerwave is estimated as in the Equation (1), where thepeak value ( PSD y ) of the bump is multiplied withits respective frequency bandwidth ( ∆ f ), whichgives us the mean square amplitude of the fluctu- R (AU) w h i s t l e r s a Slow windFast wind
R (AU) % o f w h i s t l e r s b Slow windFast wind
Figure 9.
Radial variation of number of whistler wavesand their occurrence in the slow (red) and fast (blue)wind as a function of distance from the Sun. Panel(a) shows the number of whistler waves observed inthe slow and fast wind at different distances. Panel (b)shows the percentage of the slow wind whistlers withrespect to the total number of the spectra in the slowwind for the corresponding distance bin and similarlyfor the fast wind. Error bars correspond to standard er-ror, i.e. 10 √ N w / N tot . ation. The square root of mean square amplitudecan be interpreted as the amplitude of the fluctua-tion. Note that by this definition we do not haveexact amplitudes of the wave, but rather the stan-dard deviation of the associated fluctuating wave-field within the 8 seconds of the integration of thespectrum. δ B y = (cid:112) PSD y ∗ ∆ f (1)We have not removed the background turbulencecontribution as the amplitudes of the consideredwhistler waves are large enough that the formerrepresents a minor fraction of the total amplitude( <
10% ).0 J
AGARLAMUDI ET AL ..
AGARLAMUDI ET AL .. R (AU) P S D y ( n T / H z ) Slow windFast wind
Figure 10.
Mean power spectral density of the peakvalue of the spectral bumps in the slow and fast wind atdifferent distances from the Sun. Error bars show thestandard error ( σ √ n ). R (AU) B y ( n T ) a Slow windFast wind0.4 0.5 0.6 0.7 0.8 0.9
R (AU) B y / B b Slow windFast wind
Figure 11.
Panel (a) shows the amplitude of whistlerwaves and panel (b) shows the normalized amplitude ofwhistler waves in the slow and fast solar wind. Errorbars show the standard error ( σ √ n ). Figure 10 shows the radial variation of the meanpower spectral densities corresponding to the peakof the spectral bumps for the case of the slow (red)and fast (blue) solar wind. Figure 11 (a) shows theradial variation of the mean whistler amplitudesseparately for the case of slow (red) and fast (blue)solar wind. We see that the whistler waves havehigher amplitudes in fast than in slow wind and thatthere is no clear radial trend.Figure 11 (b) shows the radial variation of thenormalized whistler amplitudes ( δ B y / B ) for theslow and fast solar wind. We have normalized thewhistler amplitude values with the closest meanmagnetic field values. Interestingly we observethat whistler waves have larger relative amplitudesin the fast wind than in the slow wind. We donot observe a clear trend in the radial evolutionfor both types of the wind. We infer that observedwhistler waves may be considered to be in the lin-ear regime ( δ B / B < . β ec ) and halo electron beta ( β eh )shows the best possible correlation. For β ec and β eh we found to have a weak but statistically positivecorrelation of ∼ .
40 with the relative amplitudes.4.4.
Estimation of phase velocity
Using the cold plasma theory we have the disper-sion relation of the right hand circularly polarizedwhistler waves (Bellan 2006), n = c k ω ≈ + ω pe / ω ( ω ce cos θ kB ω − . (2)For parallel propagating waves with ω (cid:28) ω ce wehave ( ω ce cos θ kB ω − ≈ ω ce ω , thus, the dispersion relation becomes: c k ≈ ω + ω pe ( ω / ω ce ) . (3) HISTLER WAVES IN THE INNER HELIOSPHERE ω (cid:28) ω pe , we obtain c k ≈ ω pe ( ω / ω ce ) , (4)Then, the phase velocity of the quasi parallelwhistler is given by v φ = ω k = c √ ωω ce ω pe . (5)The same equation (5) also holds for anti-parallelwhistler waves.Assuming that the spectral bumps correspond tothe parallel whistlers and the observed frequencyin the satellite frame f is not much influenced bythe Doppler shift and taking in to account low rel-ative frequency of the observed wave signatures( f (cid:39) . f ce ), we can apply expression (5) to esti-mate v φ for our data set.Figure 12 (a) shows the distribution of phase ve-locities ( V φ ) determined from the central frequen-cies of the spectral bumps in the fast wind (blue)and in the slow wind (green). We observe thatphase velocities of the slow and fast solar windwhistlers are in the same range 2 · < V φ < · km/s with the median around (cid:39) km/s.Most of the whistler waves have high phase ve-locities with respect to V sw (not shown) and espe-cially with respect to the Alfvén speed, see Figure12 (b) where one observes 9 < V φ / V A <
21 in thefast wind and 6 < V φ / V A <
21 in the slow wind.We have also looked into how the phase velocityof whistlers is compared to the electron thermalvelocity ( V th , e ), we found that 0 . < V φ / V th , e < . Electron anisotropy and heat fluxcorresponding to the observed whistlerwaves
Studies such as Lacombe et al. (2014); Stansbyet al. (2016); Tong et al. (2019b) have suggestedthat heat flux instability might be acting when thewhistlers are observed. Studies by Gary et al.(1999a); Wilson et al. (2013) have shown that A h > A h
250 500 750 1000 1250 1500 1750 2000
V (kms ) N u m b e r o f w h i s t l e r s a Slow windFast wind V / V A N u m b e r o f w h i s t l e r s b Slow windFast wind
Figure 12.
Histograms of the phase velocity and nor-malized phase velocity of all the observed whistlerwaves. Green corresponds to the slow wind and blue tothe fast wind. In panel (a) we show the phase velocityand in panel (b) we show the phase velocity normalisedto their corresponding Alfvén speed ( V A ). approaches 1 and higher, the plasma becomes un-stable for the heat flux instability. The reason for A h significance will be discussed in detail in thecoming sections.The electron moments on Helios are not alwaysavailable: there are large gaps. To increase thestatistics we have considered the closest electronmoment values to the observed whistlers. We keptthe constraint of maximum 10 minutes from thenearest observed whistler. In Figure 13 we present(a) the normalized electron heat flux q e (cid:107) / q as afunction of the parallel beta of the core of the elec-tron distribution function β e (cid:107) c ; (b) the electron tem-perature anisotropy of the core A c as a function of β e (cid:107) c and (c) the electron anisotorpy of the halo elec-trons A h as a function of β e (cid:107) h . Green dots corre-spond to all the electron moments measured during2 J AGARLAMUDI ET AL ..
Histograms of the phase velocity and nor-malized phase velocity of all the observed whistlerwaves. Green corresponds to the slow wind and blue tothe fast wind. In panel (a) we show the phase velocityand in panel (b) we show the phase velocity normalisedto their corresponding Alfvén speed ( V A ). approaches 1 and higher, the plasma becomes un-stable for the heat flux instability. The reason for A h significance will be discussed in detail in thecoming sections.The electron moments on Helios are not alwaysavailable: there are large gaps. To increase thestatistics we have considered the closest electronmoment values to the observed whistlers. We keptthe constraint of maximum 10 minutes from thenearest observed whistler. In Figure 13 we present(a) the normalized electron heat flux q e (cid:107) / q as afunction of the parallel beta of the core of the elec-tron distribution function β e (cid:107) c ; (b) the electron tem-perature anisotropy of the core A c as a function of β e (cid:107) c and (c) the electron anisotorpy of the halo elec-trons A h as a function of β e (cid:107) h . Green dots corre-spond to all the electron moments measured during2 J AGARLAMUDI ET AL .. e c q e / q a All observationsWhistlers e c A c b WTAFH10 e h A h c WTAFH
Figure 13.
Normalized heat flux and core and halotemperature anisotropy during the period of spectra an-alyzed (12-12-1974 to 20-09-1975). Red dots representthe electron moments corresponding to the whistlersand the green dots correspond to all the electron mo-ments measured during the availability of SCM data. Inpanel (a), we show the normalized heat flux ( q e (cid:107) / q )as a function of electron core parallel beta ( β e (cid:107) c ), theblue line represents the heat flux instability threshold β . c (cid:107) ( γω ci = 10 − ), given by (Gary et al. 1999b). In pan-els (b) and (c), we show A c and A h as a function of β e (cid:107) c and β e (cid:107) h respectively. The maximum growth rate curves( γω ce = 10 − , − , − ) for the core and halo WTA insta-bility are dashed; for the FH instability they are dottedand are taken from the work of Lazar et al. (2018). Theblue curve corresponds to κ = 3 and the black curve to κ = 8. the availability of SCM data; the subset of eventsfor which whistler waves are observed is shown inred.From Figure 13 (a) we infer that whistler heatflux instability (WHF) is probably at work as mostof the data follow the marginal stability thresholdof γω ci = 10 − (Gary et al. 1999b)). The q e (cid:107) / q val-ues corresponding to the whistlers are observedclose to the threshold. However, some valuesare below the threshold which suggests that theWHF instability might not be the sole cause of thewhistlers. Other than q e (cid:107) / q values A h also has animportant role as shown by Tong et al. (2019b) that A h close and higher than one may significantly af-fect the onset of the WHF instability. Indeed, forour data set, we find that in 88% of spectra withsignatures of whistlers, A h >
1. This ascertains theimportance of A h for the whistler wave observa-tions and for their generation through WHF.Figure 13 (b) and (c) show A c and A h as a func-tion of β e (cid:107) c and β e (cid:107) h respectively. We can in-fer that the core and halo whistler anisotropy in-stability is probably at work as most of the dataare well constrained along the instability thresh-old contour γω ce = 10 − . The maximum growth ratecontours for the whistler temperature anisotropy(WTA) and firehose instability (FH) for the differ-ent cases of Kappa ( κ = 3 ,
8) are taken from thework of Lazar et al. (2018), where the authors haveconsidered a bi-Maxwellian core and a bi-Kappahalo of the electron distribution function. All thewhistler points are closer to the whistler anisotropyinstability threshold than to the firehose instabilitythreshold.The large number of events located in the vicinityof the WTA threshold supports the role of the coreand halo WTA instability in generating whistlers.We notice that when whistlers are observed, formost of the cases we find A c > . A h > . A c values corre-sponding to whistlers satisfied A c > A h values satisfied A h > β , the lower HISTLER WAVES IN THE INNER HELIOSPHERE β cases, as in the slow solar wind.To summarize, Figure 13 suggests that the WHFand WTA instabilities are acting and that the threeparameters q e (cid:107) / q , A c and A h could be responsiblefor the observation of whistler wave signatures inthe Helios
SCM spectra. However, due to the lim-itations on the data availability we cannot clearlystate which type of instability is acting at whichtime and for how long, whether is it WHF or coreWTA or halo WTA instability.Previously, to have larger statistics we have con-sidered the closest available electron moment val-ues. Now, let us consider the electron propertiesthat correspond to the observed whistler wave sig-natures. We first look for time intervals in whichwe observe at least 10 consecutive 8-sec spectrawith whistlers, i.e. time intervals that are at least80 s long. Then, we identify the electron momentswhich are measured in those whistler intervals. Bythis method we ensure that moments measured arealways during the presence of whistler waves, wefound only 45 such intervals. Their q e (cid:107) / q valuesare spread below and above the heat flux instabilitythreshold. However, for all the observed whistlerwaves, we observe that A h > .
01 in agreementwith the studies of Gary et al. (1999a); Wilson et al.(2013). Our observations also agree with the workof Tong et al. (2019b), who showed the importanceof the halo anisotropy ( A h ) for the onset of heat fluxinstability.Even though we do not have exact values of q e (cid:107) / q , A h and A c for all the observed spectra withthe signatures of whistler waves, we can have ageneral idea of the conditions around when thewhistler waves appear, i.e. usually in the slowwind. A h and A c values are relatively higher inthe slow solar wind when compared to the adja-cent fast solar wind, while q e (cid:107) / q values are lower.A glimpse of this transformation can be seen froman example interval shown in Figure 6. A c B y / B a A h B y / B b Figure 14.
2d histogram of normalized amplitude asa function of electron core and halo anisotropies. Inpanel (a), we show the normalized amplitude variationas a function of A c and in panel (b), we present the nor-malized amplitude variation as a function of A h . We have also looked into whether relative am-plitudes have any relation to the A h and A c values.From Figure 14 (a) & (b) we can infer that (i) al-most all the whistlers are observed when, 0 . ≤ A c ≤ . . ≤ A h ≤ . δ B y / B are not dependent on the valueof A c or A h . DISCUSSIONS5.1.
Why are whistler waves predominantlyobserved in the slow solar wind ?
Different effects can explain the prevalence ofwhistler waves in the slow solar wind. These canbe observational (whistler are present but cannotbe observed), or physical (conditions are not metfor generating whistlers, or existing whistlers are4 J
AGARLAMUDI ET AL .damped). Let us first consider the observationalones.One of the major differences between fast andslow winds is the higher relative fluctuation levelof the magnetic field in the former, which maytherefore limit the visibility of whistler waves, asnoticed by Lacombe et al. (2014). We performedtests to quantify the impact of the fluctuation level,see Appendix 7.1. While this effect plays a role itis not sufficient to explain the much smaller num-ber occurrence probability of whistlers in the fastsolar wind. Another effect could be the enhancedDoppler shift in the fast wind. We also quantifiedthis effect, which can be ruled out, see Appendix7.1. What is then left over are the conditions forwhistler waves to be generated in the solar wind.5.2.
Presence of whistler signatures and theconditions of whistler generation
The two major sources of whistler wave gen-eration in the solar wind are the whistler heatflux (WHF) and whistler temperature anisotropy(WTA) instabilities (Gary et al. 1994). Observa-tional studies at 1 AU show that whistlers can begenerated by the WHF instability (Lacombe et al.2014; Tong et al. 2019b). We have already shownin Figure 13 (a) that the WHF instability condi-tions are indeed met for some whistlers in the innerheliosphere. Figures 13 (b) and (c) show that theWTA instability conditions are also met for coreand halo electron populations. From this we con-clude that the ratios q e (cid:107) / q , A h and A c are impor-tant parameters for understanding the presence ofwhistler waves in the inner heliosphere. In addi-tion, from the studies of Gary & Feldman (1977);Gary et al. (1999a); Wilson et al. (2013); Tonget al. (2019b) we understand that the ratio T e ⊥ h T e (cid:107) h isanother important parameter for the onset of theWHF instability.Gary & Feldman (1977), assuming a two bi-maxwellian velocity distribution function approx-imation, derived the dispersion relation for the whistler waves, the growth rate is given as γ Ω i ∝ { ( k · v H − ω R ) T e ⊥ h T e (cid:107) h + | Ω e | ( T e ⊥ h T e (cid:107) h − } , (6)where Ω e is the electron cyclotron frequency, Ω i isthe ion cyclotron frequency, v H is the drift velocityfor halo electrons, ω R is the real frequency.In the Equation 6, the first term on the right handside is the contribution of heat flux due to the driftof the electrons and the second term the contribu-tion of the anisotropy to the growth of the instabil-ity. Studies by Gary & Feldman (1977) have sug-gested that when the halo is nearly isotropic, i.e.when A h (cid:39)
1, and the heat flux contribution term ispositive, the WHF instability is driven. An exam-ple of this has been recently reported by Tong et al.(2019b) using the
ARTEMIS data. Gary & Feld-man (1977) have also suggested that when A h > A h > A h > q e (cid:107) / q isabove the whistler heat flux instability threshold(see Figure 13 (a)). A similar observation wasmade by Tong et al. (2019b,a). This leads to theconclusion that the heat flux itself might not al-ways be able to generate whistler waves and a morecomplicated interplay between the heat flux andanisotropy is required.Figure 15 shows the radial evolution of the nor-malized heat flux ( q e (cid:107) / q ), calculated by Štveráket al. (2015) for the period of 01-01-1975 to 10-05-1979, separately for the slow and the fast solar HISTLER WAVES IN THE INNER HELIOSPHERE q e (cid:107) / q val-ues are lower in the slow wind as compared to thefast wind. Furthermore as a function of radial dis-tance q e (cid:107) / q values did not show any clear trendsin the fast wind, whereas in the slow wind we ob-serve a slight decrease in q e (cid:107) / q with R . Theseobservations might be related to the impact of thewhistler waves on the dissipation of the heat fluxthrough the scattering of strahl electrons (Štveráket al. 2015).Figures 16 and 17 presents the radial evolutionof the electron core and halo anisotropies, calcu-lated by Štverák et al. (2009) for the period of 01-01-1975 to 10-05-1979. In Figure 16 we showthe variation of the mean electron core and haloanisotropies, while Figure 17 shows the relativeproportion of electron core and halo anisotropyvalues that are greater than one in the slow andfast winds as a function of radial distance. We seethat the values of (cid:104) A h (cid:105) , (cid:104) A c (cid:105) and also the percent-age of electron core and halo anisotropy values thatare greater than one (% A c > A h >
1) are higherin the slow solar wind throughout the inner helio-sphere as compared to what we see in the fast so-lar wind. We also observe that as we move awayfrom the Sun the value of (cid:104) A h (cid:105) , (cid:104) A c (cid:105) and also the% A c >
1, % A h > A c > A h > A h > A c > R (AU) q e / q Fast windSlow wind
Figure 15.
Mean of q e (cid:107) / q values for the slow (red) andfast (blue) solar wind as a function of distance from theSun for the whole Helios mission. The error bars hereshow the standard error ( σ √ n ) we have a decrease in the percentage of whistlerwaves with velocity. We have also observed a sim-ilar monotonic decrease of (cid:104) A h (cid:105) and (cid:104) A c (cid:105) with theincrease of wind velocity (not shown). We wouldalso like to mention that we have calculated howthe distance from the A h and A c values to theirrespective WTA instability thresholds are varyingfrom 0.3 to 1 AU. We found that as we move far-ther from the Sun, the A h and A c values are closer tothe WTA instability thresholds, as the β e (cid:107) c and β e (cid:107) h are increasing with the distance and thresholds arelower for higher beta. Therefore, from this analy-sis too, we can understand that whistler generationconditions are improving as we move farther fromthe Sun.5.3. Reasons behind the observed A c and A h trends After discussing the observed signatures ofwhistler wave behaviors with the A c and A h val-ues, some other questions can be raised. What canbe the reason for the high A c and A h values in theslow wind as compared to the fast one ? What canbe the reason for the increase in A c and A h as wemove farther from the Sun?6 J AGARLAMUDI ET AL ..
Mean of q e (cid:107) / q values for the slow (red) andfast (blue) solar wind as a function of distance from theSun for the whole Helios mission. The error bars hereshow the standard error ( σ √ n ) we have a decrease in the percentage of whistlerwaves with velocity. We have also observed a sim-ilar monotonic decrease of (cid:104) A h (cid:105) and (cid:104) A c (cid:105) with theincrease of wind velocity (not shown). We wouldalso like to mention that we have calculated howthe distance from the A h and A c values to theirrespective WTA instability thresholds are varyingfrom 0.3 to 1 AU. We found that as we move far-ther from the Sun, the A h and A c values are closer tothe WTA instability thresholds, as the β e (cid:107) c and β e (cid:107) h are increasing with the distance and thresholds arelower for higher beta. Therefore, from this analy-sis too, we can understand that whistler generationconditions are improving as we move farther fromthe Sun.5.3. Reasons behind the observed A c and A h trends After discussing the observed signatures ofwhistler wave behaviors with the A c and A h val-ues, some other questions can be raised. What canbe the reason for the high A c and A h values in theslow wind as compared to the fast one ? What canbe the reason for the increase in A c and A h as wemove farther from the Sun?6 J AGARLAMUDI ET AL .. A c a Slow windFast wind
R (AU) A h b Slow windFast wind
Figure 16.
Illustration of electron anisotropy variationsin the inner heliosphere for the slow (red) and fast (blue)solar wind for the whole
Helios mission. We present inpanel (a) the mean A c ( T e ⊥ c T e (cid:107) c ) and in panel (b) the mean A h ( T e ⊥ h T e (cid:107) h ) values as a function of distance from the Sun.The error bars here show the standard error ( σ √ n ) Štverák et al. (2008) showed that the value of A c increases with the electron collisional age. Colli-sional age was reported to increase with radial dis-tance and to be anti-correlated to the solar windspeed. This is in line with our observations for A c . Collisions are effective at low energies (coreelectrons), but cannot be effective at high ener-gies, such as for halo electron population. We con-jecture that the increase we observe in the A h ( R )with R and also the presence of whistler waves areclosely related to the broadening (scattering) of thestrahl part of the electron distribution. The basis % A c > a Slow windFast wind
R (AU) % A h > b Slow windFast wind
Figure 17.
Illustration of electron anisotropy variationsin the inner heliosphere for the slow (red) and fast (blue)solar wind for the whole
Helios mission. We show inpanel (a) the percentage of A c ( T e ⊥ c T e (cid:107) c ) > A h ( T e ⊥ h T e (cid:107) h ) > for this idea is related to the observed properties ofthe strahl of the electron distribution.Using the Helios electron data, Berˇciˇc et al.(2019) has shown that the strahl is in generalbroader in the slow than in the fast solar wind,and that the width in both cases increases with ra-dial distance. Strahl width was reported to increasewith radial distance by Hammond et al. (1996), us-ing data from
Ulysses mission, and by Grahamet al. (2017), using data from
Cassini .Studies by Maksimovic et al. (2005) and byŠtverák et al. (2009) show that the relative den-sity of the halo and strahl vary oppositely over
HISTLER WAVES IN THE INNER HELIOSPHERE % A c > a R (AU) % A h > b Figure 18.
Illustration of electron anisotropy variationsin the inner heliosphere for different types of wind forthe whole
Helios mission. In panel (a) we show thepercentage of A c ( T e ⊥ c T e (cid:107) c ) > A h ( T e ⊥ h T e (cid:107) h ) > radial distance: while the strahl population is morepronounced closer to the Sun, the halo density in-creases farther from the Sun. These observationsimply that the strahl is scattered into the halo overthe radial distance.The connection between the occurrence ofwhistler waves observed and the width of the strahlelectrons - the strahl is broader and the occurrenceof whistler waves is higher in the slow solar windand farther from the Sun - suggests that the strahlmight be scattered by the whistler waves. Kajdiˇcet al. (2016) in his observational study at 1 AU hadshown that the strahl is highly scattered in the pres-ence of narrow-band whistler waves. In a different context, various numerical studies such as Vockset al. (2005); Boldyrev & Horaites (2019); Tanget al. (2020) had shown that the interaction of elec-tron VDF with the whistler wave turbulence couldlead to the scattering of energetic strahl electronsand to the formation of an electron halo.From the observed relation of whistler waves oc-currence and the A h values, we conjecture that theobserved changes in A h ( R ) might be related to thescattering of the strahl with the whistler waves.5.4. A feedback mechanism
An important and still open question is how themomentum is transferred in this wave-particle in-teraction, to result in an increase of A h .Kajdiˇc et al. (2016) observed a direct link be-tween the scattering of the strahl and the presenceof whistler waves. These whistler waves can in-teract with the strahl electrons through electron cy-clotron resonance. In the following, we make arough estimates of the energy ranges in which thewhistler waves are able to interact resonantly.The equation of electron cyclotron resonance is ω − k · v = ω ce , (7) ω − kv cos θ kv = ω ce , (8)where, ω is the frequency of the wave, k is the wavenumber of the whistler wave, v is the velocity ofthe resonant electrons, θ kv is the angle between thewhistler wave wave vector and the velocity vectorof the resonant electrons and ω ce is the electron cy-clotron frequency.Let us consider the case when resonant electronsare field aligned, as the strahl electrons (Owenset al. 2017). Most of the whistlers observed todate are quasi parallel or anti-parallel (Lacombeet al. 2014; Stansby et al. 2016; Tong et al. 2019b).The condition for resonance is only satisfied whenwhistler waves travel opposite to the electrons as ω ce is always greater than the frequency of the ob-served whistler waves. Therefore, we can under-stand that we have resonance when electrons travel8 J AGARLAMUDI ET AL ..
An important and still open question is how themomentum is transferred in this wave-particle in-teraction, to result in an increase of A h .Kajdiˇc et al. (2016) observed a direct link be-tween the scattering of the strahl and the presenceof whistler waves. These whistler waves can in-teract with the strahl electrons through electron cy-clotron resonance. In the following, we make arough estimates of the energy ranges in which thewhistler waves are able to interact resonantly.The equation of electron cyclotron resonance is ω − k · v = ω ce , (7) ω − kv cos θ kv = ω ce , (8)where, ω is the frequency of the wave, k is the wavenumber of the whistler wave, v is the velocity ofthe resonant electrons, θ kv is the angle between thewhistler wave wave vector and the velocity vectorof the resonant electrons and ω ce is the electron cy-clotron frequency.Let us consider the case when resonant electronsare field aligned, as the strahl electrons (Owenset al. 2017). Most of the whistlers observed todate are quasi parallel or anti-parallel (Lacombeet al. 2014; Stansby et al. 2016; Tong et al. 2019b).The condition for resonance is only satisfied whenwhistler waves travel opposite to the electrons as ω ce is always greater than the frequency of the ob-served whistler waves. Therefore, we can under-stand that we have resonance when electrons travel8 J AGARLAMUDI ET AL .. Resonance Energy(ev) N u m b e r o f w h i s t l e r s Figure 19.
Resonant energy of the electrons interactingwith the whistler waves (assuming that whistler wavestravel opposite to electrons). in the opposite way to whistlers. Assuming thiscondition, i.e. cos θ kv ≈ −
1, we have ω + kv = ω ce , (9) kv ω = ( ω ce − ω ) ω , (10) v = ( ω ce ω − v φ . (11)Equation (11) gives us the velocity of the electronsthat can be resonant with the whistler waves (un-der the assumption of the opposite propagation be-tween the wave and the electrons). The kinetic en-ergy of these resonant electrons, E = 12 m e v , (12)is shown in Figure 19. Most of the resonant elec-tron energies are above 50 eV, which are the ener-gies representative of the strahl and the halo pop-ulations. We conclude that the observed whistlerwaves might be able to scatter strahl and haloelectrons through the electron cyclotron resonance.But how is the energy transferred in this type of in-teraction?Veltri & Zimbardo (1993) have shown that whenresonantly interacting whistler waves scatter the electrons, electron energy is transferred from theparallel to the perpendicular direction. Whichcould be understood as an increase of the tem-perature perpendicular to the magnetic field direc-tion. On the other hand scattering the beam likepopulation towards higher pitch-angles could beinterpreted as scattering the strahl into the halocomponent.We explain this with a feedback mechanism: • Whistler waves scatter the strahl population. • Scattered strahl increases the value of A h . • Increases in the A h favors the creation of thewhistler waves and on.Therefore, we can hypothesize that whistlerwaves could scatter the strahl and create a self-sustaining mechanism that could explain the ob-served whistler properties.However, the A h values cannot keep on increas-ing without a limit as they are bounded by whistlertemperature anisotropy instability thresholds. Thisis the reason why we believe that there is a satura-tion of A h values in the slow wind farther from theSun as shown in Figures 16 and 17.Recently Tang et al. (2020) has shown thatwhistler wave turbulence interaction with suprather-mal electrons can lead to the formation of halofrom strahl and could explain the observed oppo-site variation in the relative density of the halo andstrahl with radial distance. In a future paper, wewould like to verify our suggested feedback mech-anism by modifying the kinetic transport modelused in Tang et al. (2020). We would like totreat halo and strahl as separate populations andstudy the interaction of intermittent narrow-bandwhistler waves with strahl and their influence onhalo population.Even though our explanation appears to be areasonable one, there are some issues related toit. Our calculations are based on the assump-tion that cos θ kv ≈ − HISTLER WAVES IN THE INNER HELIOSPHERE CONCLUSIONOur analysis of
Helios
Cluster , Stereo , Artemis , Geotail ) observations, the spec-tral bumps in
Helios /SCM measurements in thefree solar wind are very likely due to the whistlerwaves. The radial dependence of the properties ofthese waves offer a unique opportunity for investi-gating the connection between their presence andthe properties of the solar wind.These whistler waves are predominantly ob-served in the slow solar wind: their probabilityof occurrence decreases with increase in V sw . Atthe same time, the probability of occurrence of thesignatures of whistlers increases with the radialdistance from the Sun R . In the fast wind close tothe Sun we do not observe any signatures of waves,they start to appear at R > . β for the electron coreand halo populations. For cases where simultane-ous electron measurements are available, we find that the electron core anisotropy A c = T e ⊥ c T e (cid:107) c > . A c and electron halo anisotropy A h arebounded by the WTA instability threshold.We show how the normalized heat flux was vary-ing in the slow and fast solar wind with the radialdistance. The normalized heat flux is found to behigher in the fast wind compared to the slow wind.We observe a decrease in heat flux with the radialdistance. The signatures of whistler waves we havedetected might explain the higher heat flux dissipa-tion in the slow wind as compared to the fast windand also the decrease of heat flux with the radialdistance.The (cid:104) A c (cid:105) , (cid:104) A h (cid:105) and the percentage of events forwhich A c > A h > A h ratio to vary in accordance with theoccurrence of whistlers. Whistler waves scatterthe strahl population, the scattered strahl increasesthe A h ratio which in turn favors the creation ofwhistler waves and so on. This may be consideredas a feedback mechanism.0 J AGARLAMUDI ET AL .New observations by
Parker Solar Probe and
So-lar Orbiter will provide a unique opportunity toverify our observed results with much better re-solved magnetic and electron particle data. Fromour study, we speculate that
Parker Solar Probe and
Solar Orbiter will observe a relatively low oc-currence of whistler waves than what we have ob-served at 0.3 AU as we go closer to the Sun in theslow wind. For the fast wind case, whistler wavesmight not be observed, even if they are observedthey should be sparse. We also speculate that be-cause of the saturation of the strahl, the likelihoodof occurrence of whistler waves is likely to remainstable beyond 5 AU, both in the slow and fast solarwinds. APPENDIX7.1.
Are visibility and high Doppler shift the solereason for negligible fast wind whistlers ? f / f ce P S D × f c e ( n T ) slow wind whistlersFast wind spectra Figure 20.
Spectra observed in the fast wind for 1day (Day 73-74, 1975) and whistler spectra observedin slow wind for 1 day (Day 68.5-69.5, 1975). The ver-tical lines correspond to f lh and 0 . f ce . To look into this issue, we have considered PSDswhich are above 3 times the SCM noise from oneday of slow wind and one day of fast wind at ≈ . (a) f / f ce P S D × f c e ( n T ) ff ce slow wind whistlersFast wind spectra (b) f / f ce P S D × f c e ( n T ) f f ce slow wind whistlersFast wind spectra Figure 21.
Slow wind whistler waves extrapolated con-sidering the Doppler effect in two extreme cases (a)High frequency case (b) Low frequency case. The ver-tical lines correspond to f lh and 0 . f ce . ure 20, in which we plot the PSD as a function of f / f ce , so that magnetic field differences in the slowand fast wind are taken into account.From Figure 20, we get the first glimpse that,if intense whistler waves as in the slow wind arepresent in fast wind, they would be clearly visibleas the turbulence level in the fast wind is not highenough to hide these types of waves.However, there could be another important is-sue of difference in the Doppler shift betweenthe whistlers of slow and fast wind, as velocityof the fast wind is nearly twice that of the slowwind. To verify this, we have considered boththe possible extreme cases, Doppler shift towardshigh frequency (2f) Figure 21 (a) and Dopplershift towards low frequency (f/2) in Figure 21 (b). HISTLER WAVES IN THE INNER HELIOSPHERE ACKNOWLEDGEMENT VKJ is supported by the French Space Agency(CNES) and Région Centre-Val de Loire PhD grant(CNES N Helios data are available on the HeliosData Archive: http://helios-data.ssl.berkeley.edu/.Plots for the article are made using Matplotlib(Hunter 2007).REFERENCES
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