Anticorrelation between the Bulk Speed and the Electron Temperature in the Pristine Solar Wind: First Results from the Parker Solar Probe and Comparison with Helios

When immersed in a space plasma, an antenna will measure the electrostatic fluctuations induced by the quasi-thermal motion of the ambient electric charges which surround it. The theory of antenna-plasma coupling in space environments out of thermal equilibrium and its use for actual measurements is now well established (Meyer-Vernet 1979; Meyer-Vernet & Perche 1989; Meyer-Vernet et al. 2017). The so-called QTN spectroscopy has been applied successfully to several past missions operating in the solar wind such as ISSE 3 (Meyer-Vernet 1979; Couturier et al. 1981), Ulysses (Maksimovic et al. 1995; Issautier et al. 1999), Wind (Maksimovic et al. 1998), or STEREO (Martinović et al. 2016). It has been implemented on the FIELDS instrument to provide accurate measurements of the electron density and temperature of the outer corona down to 10 solar radii (Bale et al. 2016).

The first results of the QTN measurements on PSP are described in detail in this special issue by Moncuquet et al. (2020). These authors present a method that yields the total electron density N_e, the core temperature T_c, and a suprathermal temperature. Note that in this study, this latter suprathermal temperature is actually the total contribution of the halo + strahl thermal pressure.

As a complementary method, we use, in the present paper, the high-frequency part of the QTN spectra recorded by RFS, above the plasma peak. For this frequency range, the contributions of the proton Doppler-shifted thermal noise and the electron shot noise, including the current-biasing of the antennas, which is performed for DC electric fields measurements, are negligible (Meyer-Vernet et al. 2017). Moreover our QTN analysis is mostly dependent on the electron total thermal pressure. For frequencies f satisfying $f\gg {f}_{p}{L}_{{\rm{D}}}/L$ where f_p and L_D are the local plasma frequency and Debye length, respectively, and L is the physical length of one arm of the dipole, the QTN is proportional to ${N}_{e}{T}_{e}/{f}^{3}$ (Meyer-Vernet & Perche 1989).

Another important contribution that needs to be considered in our analysis and which can be clearly observed by the RFS radio receiver is the galactic radio background (Novaco & Brown 1978; Cane 1979). This background radiation is relatively constant and covers a 4π solid angle roughly isotropically. Note that modulations of the galactic background as a function of the observed solid angle are less than 20% in the frequency range of consideration (Manning & Dulk 2001). To obtain the pure QTN spectrum at high frequency, we have to properly subtract the radio galactic background from the total signal.

Figure 1 displays a typical power spectral density (PSD) RFS spectrum between 100 kHz and 10 MHz of the FIELDS $| V1-V2|$ antenna dipole. These observations, represented by black diamonds, have been obtained by merging the RFS/HFR and RFS/LFR spectra in one. We did the same for all the data of the first perihelion used here. The dotted horizontal line represents the RFS pre-deployment internal noise, which is ${V}_{\mathrm{noise}}^{2}\sim 2.2\times {10}^{-17}\,{{\rm{V}}}^{2}\,{\mathrm{Hz}}^{-1}$ in the considered frequency range (Pulupa et al. 2020). The black dashed line on Figure 1 represents the RFS PSD V²_galaxy corresponding to the Novaco & Brown (1978) radio galaxy model that we have computed using the detailed method described by Zaslavsky et al. (2011). More precisely we have used an RFS spectrum measured when PSP was close to 1 au. At this distance the plasma peak (around 20 kHz) is much lower than the typical f_p for the current study. The corresponding QTN spectrum is therefore negligible in the 1–10 MHz range at 1 au. Then taking the observed value of the RFS PSD in the range 2–3 MHz and subtracting the above defined receiver noise one obtains a value of $\sim 2.3\times {10}^{-17}\,{{\rm{V}}}^{2}\,{\mathrm{Hz}}^{-1}$ for the galaxy PSD in this frequency range. Finally, using the Novaco & Brown (1978) model, which yields a radio brightness of $1.1\times {10}^{-20}\,{\rm{W}}\,{{\rm{m}}}^{-2}\,{\mathrm{Hz}}^{-1}\,{\mathrm{sr}}^{-1}$ for the corresponding range and applying formula (13) from Zaslavsky et al. (2011) we obtain a reduced effective length of ${\rm{\Gamma }}{L}_{\mathrm{eff}}=1.17$, which we can then use for computing ${V}_{\mathrm{galaxy}}^{2}(f)$ on Figure 1.

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We can now fit the RFS QTN spectra using the following procedure for each individual RFS $| V1-V2|$ spectrum. We first select all the data frequency points verifying ${V}_{\mathrm{obs}}^{2}(f)\gt 3\times {V}_{\mathrm{noise}}^{2}$ in the frequency range 300 kHz–20 MHz. The 300 kHz lower-frequency limit has been chosen to be well above f_p so that, as described previously, the proton QTN and the shot noise can be neglected. Then we remove the receiver noise and the galaxy spectrum in order to define the observed QTN spectra ${V}_{\mathrm{obs} \mbox{-} \mathrm{QTN}}^{2}(f)={V}_{\mathrm{obs}}^{2}(f)-{V}_{\mathrm{noise}}^{2}-{V}_{\mathrm{galaxy}}^{2}(f)$ to which we can apply the QTN spectroscopy. The blue curve on Figure 1 represents ${V}_{\mathrm{obs} \mbox{-} \mathrm{QTN}}^{2}(f)$ for the considered spectrum. In this particular case ${V}_{\mathrm{obs} \mbox{-} \mathrm{QTN}}^{2}(f)$ is only visible up to 2.3 MHz. Above this frequency, it is lower than the sum of the receiver noise and the galaxy spectrum and thus cannot be measured. The displayed ${V}_{\mathrm{obs} \mbox{-} \mathrm{QTN}}^{2}(f)$ spectrum is defined on 38 frequency points. The spectra we fit in this study are defined on a number of frequency points, which is usually comprised between ∼10 and ∼50 and are never defined on frequencies over 3.5 MHz, because of the galactic background. The final step now consists of fitting ${V}_{\mathrm{obs} \mbox{-} \mathrm{QTN}}^{2}(f)$ using a full and comprehensive QTN model with Lorentzian VDFs for the electrons (Chateau & Meyer-Vernet 1989; Zouganelis 2008; Le Chat et al. 2011). This kind of model is well adapted for the high-frequency part of the spectra, which is mostly sensitive to the total electron thermal pressure and less to the precise shape of the model VDFs. The red curve on Figure 1 represents the theoretical Lorentzian QTN spectrum, which fits the best ${V}_{\mathrm{obs} \mbox{-} \mathrm{QTN}}^{2}$. Note that for our fitting procedure we only have two free parameters, which are the total temperature T_e and the index κ of the electron VDFs (see Chateau & Meyer-Vernet 1991; Zouganelis 2008; Le Chat et al. 2011). The plasma frequency for each fitted spectrum is set to be the one obtained by the peak tracking method used by Moncuquet et al. (2020) with an uncertainty of 4%, which is the standard frequency resolution of the RFS. As for T_e and the index κ, the uncertainties indicated on the figure correspond to the resolution of the model grid that was used for our fitting. Finally, the green straight line on Figure 1 represents an ${f}^{-3}$ variation that compares well to ${V}_{\mathrm{obs} \mbox{-} \mathrm{QTN}}^{2}(f\geqslant 1.5\,\mathrm{MHz})$.

The QTN temperatures ${T}_{e,\mathrm{QTN}}$ are represented as the black line on Figure 2 for 12 days around the date of the first perihelion. ${T}_{e,\mathrm{QTN}}$ is ranging between 3 and 6 × 10⁵ K in good agreement with an extrapolation of typical 1 au electron temperatures in the range 1–2 × 10⁵ K and with a typical radial gradient of ${T}_{e}\propto {R}^{-(0.6-0.8)}$ observed in the inner heliosphere (Maksimovic et al. 2000). Note that we have displayed in Figure 2 only 50% of the data for which the fittings yield a ${\chi }^{2}$, which is lower than the median of all ${\chi }^{2}$.

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The SPAN instrument is composed of two electron sensors on the ram and anti-ram faces of the spacecraft. They together measure the majority of the three-dimensional electron VDF as explained in detail by P. Whittlesey et al. (2020, in preparation). The first SPAN observations obtained during the first two PSP perihelia show that there is a large strahl/halo density ratio at  35 R_s (Halekas et al. 2020). This is in good agreement with the expectations from Maksimovic et al. (2005) and Štverák et al. (2009).

In this paper we are using data provided by Halekas et al. (2020), who have developed a robust fitting procedure to retrieve the electron core temperature ${T}_{c,\mathrm{SPAN}}$. This temperature is displayed in Figure 2 as the blue line. In order to provide a better visual comparison between ${T}_{e,\mathrm{QTN}}$ and ${T}_{c,\mathrm{SPAN}}$ we have multiplied arbitrarily the latter by 1.47, which is the median value of ${T}_{e,\mathrm{QTN}}/{T}_{c,\mathrm{SPAN}}$ in the considered time interval. This factor measures the actual contribution of the suprathermal population (halo and strahl) to the electron thermal pressure. Overall, the agreement between ${T}_{e,\mathrm{QTN}}$ and ${T}_{c,\mathrm{SPAN}}$ is quite good. The linear Pearson correlation between the two data sets is equal to 0.57 for the whole time interval of Figure 2. One should note, however, that there is a discrepancy between the two temperatures during a time interval ranging between 2 and 4 days after the perihelion, which occurred on 2018 November 6 at 03:27. The reason for this discrepancy is still under investigation and is outside the scope of the present paper.

On Figure 2 we have also displayed the solar wind bulk speed V, in red, as measured by the SWEAP Faraday cup instrument (Kasper et al. 2016). A clear anticorrelation between V and both ${T}_{c,\mathrm{SPAN}}$ and ${T}_{e,\mathrm{QTN}}$ is visible across the time interval. This anticorrelation, which is more pronounced between V and ${T}_{c,\mathrm{SPAN}}$ (linear Pearson correlation of −0.59) than between V and ${T}_{e,\mathrm{QTN}}$ (correlation of −0.20), despite the good correlation between the two temperatures, has also been reported by Halekas et al. (2020). We now explore this property by revisiting the Helios data.

In this paper we use three different data sets from the ions and electron electrostatic analyzers on board the Helios 1 and 2 spacecraft (Schwenn et al. 1975), which performed in-ecliptic measurements in the heliocentric distance range between 0.3 and 1 au. The first data set contains ∼1,877,000 measurements of proton density N_p, temperature T_p and bulk speed V measurements. This constitutes the original Helios plasma data set available at ftp://cdaweb.gsfc.nasa.gov/pub/data/helios/ helios1/merged/.

The second and third data sets contain electron density N_e and temperature T_e measurements obtained by fitting the velocity distributions recorded by the I2 electron analyzer (Rosenbauer et al. 1977; Pilipp et al. 1987). This analyzer was designed to measure only 2D distribution functions with an aperture pointing in the ecliptic plane.

The second data set we use has been obtained by Štverák et al. (2009). These authors used only those measurements for which the magnetic field vector is close enough to the ecliptic plane, that is, when ${B}_{z}/| B| \lt 0.1$. This way they restricted their analysis to the good measurements of the VDFs in the (${v}_{\perp },{v}_{\parallel }$) plane, where the directions ⊥ and ∥ are with respect to the local magnetic field vector. This data set contains ∼66000 measurements of N_e and T.

Finally, the last data set we used in this work is obtained in a similar way to that of Štverák et al. (2009), but using a slightly different model to describe the velocity distribution of electrons. Details on the data processing can be found in Section 3.2 of Berčič et al. (2019). This last data set is not limited to the times when the magnetic field lies within the 2D field of view of the electron instrument and therefore only provides good estimations of the perpendicular part of the electron temperature. On the positive side, it gives better statistics, with approximately three times as many point as the Štverák et al. (2009) data set.

If one displays, for the full Helios data set, the solar wind bulk speed as a function of the heliocentric distance one can note that, while the fastest solar wind flows are bounded by a constant value of about 800 km s⁻¹, the slowest solar wind streams increase with radial distance between 0.3 and 1 au.

In order to properly quantify this actual acceleration of the slow solar wind we use a procedure illustrated by Figure 3. We first split the data into 20 heliocentric distance bins. For statistically representative radial coverage, we define these radial bins so that they each contain an equal number of data points, which is 1,877,000/20 ∼ 93,800. In each of the radial bins we compute the median value of the heliocentric distance and assign it to the radial bin. We then make the assumption that the lowest quintile of the speed data distribution in the first radial bin at ∼0.3 au (black histogram bins on Figure 3) contains the same solar wind streams as the lowest quintile of the data in the following radial bin and so on until the last one at ∼0.98 au. We proceed the same way for the second quintile of the bulk speeds and so on. In this manner we separate the solar wind into five wind families that we display using five color codes (black, red, green, blue, and cyan in order of increasing speed values) throughout the paper. For each of these families we compute the median bulk speeds of the five quintiles in each of the 20 radial bins. This leads to 20 $({R}_{i},{V}_{i})$ data points, which we display in the upper panel of Figure 4. We proceed the same way with the proton density $({R}_{i},{N}_{{pi}})$ and total temperature $({R}_{i},{T}_{{pi}})$ data.

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In the upper panel of Figure 4 it can be clearly seen that while the bulk speed of the fastest solar wind (cyan) is approximately constant, the bulk speed for the slowest wind (black) is clearly increasing. In order to quantify this apparent increase, we have performed a linear fit in the form $V(R)=A\times {R}_{\mathrm{AU}}+{V}_{0}$ for each of the five wind families. The lower panel of Figure 4 displays variation of the parameters A as a function of V₀. This V₀ fitting parameter is used in the following in order to identify the wind families. Except for the fastest wind (${V}_{0}\approx 600$ km s⁻¹), all the wind families with ${V}_{0}\lt 500$ km s⁻¹ exhibit an increase with radial distance, which is more important as V₀ decreases. The slowest wind family (black dots on the upper panel of Figure 4) with a ${V}_{0}\approx 250$ km s⁻¹ exhibits an increase of speed up to $\approx 90$ km s⁻¹ per au.

In Figure 5 we display from top to bottom the proton densities $({R}_{i},{N}_{{pi}})$, proton temperatures $({R}_{i},{T}_{{pi}})$, and the electron temperatures $({R}_{i},{T}_{{ei}})$ from the (Štverák et al. 2009) data set for the five wind families. Note that since there are less data points in the electron data set than in the proton one, we have defined 10 radial bins for the latter case instead of 20. The well-known correlations between the proton bulk speed, density, and temperature can clearly be seen in Figure 5. The fastest is the solar wind (cyan dots), the largest is its proton temperature and the less dense wind. In order to better visualize these correlations, we fit the wind families' proton temperatures with power laws of the form ${T}_{p}={T}_{p0}\times {R}_{\mathrm{AU}}^{{\alpha }_{p}}$. The resulting fits are displayed in Figure 5 by full lines with the same colors used for the wind families. We do the same for the electron temperatures.

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We display the outputs of the above power-law fittings in Figure 6. One au temperatures T_p0 and T_e0 (upper panel) and power-law indexes ${\alpha }_{p}$ and ${\alpha }_{e}$ (lower panel) are displayed as a function of V₀ for the protons (in red) and the electrons (in black), respectively. We can see in this figure one of the major results of this article. While the proton temperature gradients are very similar for the five families, with a average power index of $\approx -0.9$, this is not the case for the electron gradients. The temperature gradients for the electrons are quite different, with a clear tendency for the slow wind electron to cool down more steeply (${\alpha }_{e}\approx -0.8$) than the fast wind ones (${\alpha }_{e}\approx -0.3$). This result is consistent with those obtained by S̊tverák et al. (2015), who show that for the slow wind the electron temperature power-law radial dependence is ${T}_{e}\propto {r}^{-0.59}$, while it is ${T}_{e}\propto {r}^{-0.31}$ for the fast wind. It is also consistent with the outcome of the study by Marsch et al. (1989), who applied a slightly different technique for binning the data with respect to the bulk speed. Marsch et al. (1989) actually used the same interval limits for the speed bins at all radial distances. This technique, although similar to ours, does not account for the acceleration of the solar wind, especially for the slowest one. The red triangles connected by a dashed line in the lower panel of Figure 6 represent the ${\alpha }_{e}$ power-law index as a function of V from Marsch et al. (1989). Note also that the proton temperature power-law indexes we obtain are very similar to those retrieved by Totten et al. (1995), who used the same technique as Marsch et al. (1989) for binning the data by bulk speed. The black triangles connected by a dashed line in the lower panel of Figure 6 represent the ${\alpha }_{p}$ power-law index as a function of V from Totten et al. (1995). Considering the values of T_p0 and T_e0, which are the proton and electron modeled temperatures at 1 au, we retrieve the well-known and established correlation between the proton temperature and the wind bulk speed and the lack of such a correlation for the electrons.

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We now explore how these correlations evolve with radial distance. First, we downsample the proton data set to times when the electron measurements in the Štverák et al. 2009) data set were performed. We thus present correlations on the same statistical basis for protons and electrons. As done for the latter previously, we now compute the median values $({V}_{i},{T}_{{pi}})$ on 10 radial bins R_i instead of 20. The upper panel of Figure 7 displays the proton temperature as a function of the bulk speed for all the points (light dots) and the medians (thick dots) for the first (black), the sixth (red), and the tenth (blue) radial bins. These three radials bins correspond to the following heliocentric distance ranges, respectively: $0.29\lt {R}_{\mathrm{AU}}\lt 0.31$, $0.55\lt {R}_{\mathrm{AU}}\lt 0.69$, $0.95\lt {R}_{\mathrm{AU}}\lt 0.98$. Because the T_p gradients are similar for all the wind families the 1 au $(V,{T}_{p})$ correlation is maintained at all radial distances. This is true both for the medians and for all the data points within a radial bin. This can be seen quantitatively in Table 1, where we have reported the linear Pearson correlation coefficients between V and T_p for the radial bin data defined above. While the correlation coefficient between the proton temperature and speed is equal to 0.75 close to 1 au, it falls down to 0.55 close to 0.3 au, but still remains.

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Table 1.  Linear Pearson Correlation Coefficients for (${T}_{p},V$) and (${T}_{e},V$) for Three Different Radial Distances

	Protons	Electrons	Electrons
		(Štverák et al. 2009)	(Berčič et al. 2019)
$0.29\lt {R}_{\mathrm{AU}}\lt 0.31$	0.55	−0.70	−0.94
$0.55\lt {R}_{\mathrm{AU}}\lt 0.69$	0.72	−0.50	−0.81
$0.95\lt {R}_{\mathrm{AU}}\lt 0.98$	0.75	0.31	−0.32

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For the electrons the tendency is the opposite, as one can see in the lower panel of Figure 7. There is a strong $(V,{T}_{e})$ anticorrelation at 0.3, with a correlation coefficient of −0.70. This property is most likely the remnant of the coronal electron temperature mapping. However, as the T_e gradients are quiet different for the different wind families, the pristine $(V,{T}_{e})$ anticorrelation is cleared as the wind expands. At 1 au the anticorrelation has completely disappeared on the Helios data, with a very low $(V,{T}_{e})$ correlation coefficient of 0.31. The same trend is also observed using the Berčič et al. (2019) data set, for which we have computed the correlations between V and ${T}_{e\perp }$, the total electron temperature in the perpendicular direction to the magnetic field.

Finally, in Figure 8 we have combined the Helios electron median temperatures displayed in Figure 7 with those measured by PSP and discussed in Section 2. For PSP we have only selected data in the radial range $35.7{R}_{S}\lt R\lt 45{R}_{S}$ and binned them on quartiles according to their bulk speeds. ${T}_{e,\mathrm{QTN}}$ is displayed in magenta, while $1.47\times {T}_{c,\mathrm{SPAN}}$ is displayed in orange. Note that the limit of the slow wind observed by Helios at 66 R_s is shifted to even slower wind speeds at 35 R_s, illustrating once more the strong wind acceleration that occurs for the slow wind streams. Another very interesting curve is displayed in green on Figure 8. It corresponds to the measurements of the total electron temperatures obtained with the 3DP electrostatic analyzer on board Wind (Lin et al. 1995). For this curve we have accumulated four years of Wind data (1995–1998) and displayed the medians of T_e for 8 bins of bulk speed V. The Wind T_e variations as a function of V superimpose almost perfectly with those of Helios at 1 au in the range $350\lt V\lt 600$ km s⁻¹. Above 600 km s⁻¹ the Wind T_e is anti-correlated with V, demonstrating that for these highest speeds the pristine electron temperature versus bulk speed anticorrelation is conserved up to one au.

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As the final conclusion of our analysis, we show in Figure 8 that this tendency for different electron thermal gradients for different speed regimes is also valid at the radial distances covered by the Ulysses probe. For this reason we have used the electron moments of the Ulysses electron analyzer (Bame et al. 1992). We have restricted our analysis to the period comprised between the launch in 1990 October and the end of the first orbit on 1998 February. We have first divided these data into in-ecliptic regions, with an absolute value of the spacecraft heliolatitude λ smaller than 20°, and high-latitude regions, with $| \lambda | \gt 50^\circ$. The high-latitude data correspond to the well-known period, during solar minimum, when Ulysses was immersed in a very fast wind emanating from polar coronal holes (McComas et al. 2003). The in-ecliptic electron data are represented by the black triangles, squares, and crosses in Figure 8, while the high-latitude data are represented by the diamonds. For the in-ecliptic electron data we have applied the same binning technique for speeds and radial distances as previously used. The black triangles in Figure 8 represent the $({V}_{i},{T}_{{ei}})$ medians for the radial range 1.42 < au < 1.95, the black squares are for 3.01 < au < 3.54, and the crosses are for 4.60 < au < 5.13. It is striking to see in this figure how the primordial anticorrelation in the near solar corona turns into a rough global correlation between V and T_e very far from the Sun. As for the high-latitude data, the radial distance covered is comprised between 1.5 and 3.7 au. When applying our binning technique for these data, and even with different numbers of radial bins, we observe that the $({V}_{i},{T}_{{ei}})$ medians are all superimposed on each other. We have thus defined four velocity bins for all the high-latitude electron data, independently of the radial distance, and displayed them in Figure 8 with diamonds. As one can see, and contrary to the Ulysses in-ecliptic data, V and T_e are anti-correlated when the bulk speed is large, even far from the Sun.