Radial Distribution of Electron Quasi-thermal Noise in the Outer Heliosphere

Voyager 1 and 2 are only the two spacecraft that have arrived and passed through the heliospheric boundaries. Based on the plasma data from the Voyager 2 spacecraft, the electron quasi-thermal noise (QTN) is investigated by using of the electron population model consisting of a core with Maxwellian distribution and a halo with kappa distribution. The power spectra of the electron QTN is calculated at different heliocentric distances from 1 to 110 au. The parametric dependence of the QTN power spectra and the effective Debye length on the model parameters, such as the density ratio and temperature ratio of the halo to the core, kappa index and the antenna length, are discussed further. The results show that the electron QTN spectrum consists of a plateau in the low frequency band f < f pt , a prominent peak at the plasma frequency f pt , and a rapid decreasing part in the high frequency band f > f pt . The QTN plateau level basically falls down outwards until the termination shock crossing at about 84 au, after which the plateau rebounds a little near the heliopause. Although the model parameters can be very variable, the QTN plateau level does not present more than the double change in a fairly wide range of the model parameters. The presented results can be useful for future deep-space explorations in the heliosphere and can provide valuable references for the design of onboard detectors.


Introduction
Solar system exploration is nowadays an important topic and a lot of effort has been put into this realm.Voyager 1 and 2 are the only spacecraft that have reached the heliopause, while they are still capable of sending data back to the Earth until now.Voyager 2 crossed the termination shock in 2007 August at 83.7 au (Burlaga et al. 2008) and the heliopause in 2018 November at 119.0 au (Gurnett & Kurth 2019).The plasma parameters are detected by the Plasma Science (PLS) instrument (Bridge et al. 1977) on board Voyager 2, although the PLS instrument of Voyager 1 failed in 1980.The PLS instrument makes use of four Faraday cups to detect the incident plasma flux.By fitting an isotropic Maxwellian proton distribution with a drifting velocity, parameters such as proton density, thermal speed, and solar wind velocity can be deduced.
During Voyager 1ʼs journey shortly before the heliopause crossing, its Plasma Wave System (PWS) (Scarf & Gurnett 1977) data revealed a series of continuous plasma emission lines in the frequency-time spectrogram.The features and possible origins of these lines have been discussed by Ocker et al. (2021) and Gurnett et al. (2021).The quasi-thermal noise (QTN) peak is considered to be the possible explanation to this visible plasma emission line.QTN is a commonly observed noise signal in the solar wind circumstance (Meyer-Vernet 1979;Couturier et al. 1981) , where e, m e , and n e are the electron charge, mass, and number density, respectively, and ε 0 is the vacuum permittivity.First, Meyer-Vernet & Perche (1989) developed the electron QTN spectroscopy to explain this phenomenon, in which the noise voltage power is calculated with Maxwellian distribution or two Maxwellian distributions.Then, Chateau & Meyer-Vernet (1991) and Le Chat et al. (2009) further developed the QTN theory in plasma with kappa electron velocity distributions.
In principle, the electron QTN spectrum can be regarded as the potentials induced on the antenna by the thermal motion of the background electrons.At low frequency (ω < ω p ), the electrons passing by the antenna have enough time to be dressed by the Debye potential and the time they are seen by the antenna is closely equal to the duration of their stay within the Debye sphere, which is about 1/2πf p .Thus, the Fourier transform of the voltage pulse caused by dressed electrons is a constant and results in a plateau at the low-frequency range.At the high frequency (ω > ω p ), on the other hand, there is not enough time for the electrons to be dressed by their mutual coupling effect, so that the number of the electrons being seen by the antenna decreases as the frequency arises.This will cause the QTN spectral power to descend in inverse cube of ω.Finally, a spectral peak is formed just above the plasma frequency ω p due to the interaction of the Langmuir waves and the electrons (see Meyer-Vernet et al. 2017, Section 2).
In a stable and unmagnetized plasma, in general, the electron QTN spectrum is related only to the electron velocity distributions and the antenna geometry.By means of fitting the amplitude of the plateau at the frequency lower than the plasma frequency ω p and the location of the peak above the ω p , one can perform a quick in situ diagnostic on the electron density n e and the electron temperature T e .QTN is a passive measurement, which may help it to avoid disturbing the plasma medium and ensure its relative immunity to perturbations of the spacecraft potential and photoelectrons effect.Moreover, the Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
electron density is deduced by the location of the QTN peak regardless of gains of the signal chain.These advantages make it a fine tool to calibrate the other instruments on the spacecraft, which have been performed with the spacecraft Ulysses, Wind, and Parker Solar Probe (Issautier et al. 2001;Hillan et al. 2010;Moncuquet et al. 2020).
The observations from Helios (Rosenbauer et al. 1977;Pilipp et al. 1987), Ulysses (Hammond et al. 1996;Maksimovic et al. 2000), Wind (Lin 1998;Salem et al. 2003), and Parker Solar Probe (Halekas et al. 2020) show that the solar wind electrons consist of three components, including a relatively cold dense core described by a Maxwellian distribution, a hot thin halo, which is often described by a Maxwellian distribution or a kappa distribution, and a minor beam component called "strahl".In this paper, employing a combined electron distribution function of a cold Maxwellian core and a hot kappa halo, the radial distribution and parametric dependence of the electron QTN are investigated based on the in situ measured plasma data from the Voyager 2 spacecraft at different heliocentric distances in the outer heliosphere from 1 au near the Earth orbit to 110 au in the heliospheric boundary layer.
The rest of this paper is organized as follows.After the basic model for the power spectrum of the electron QTN is introduced in Section 2, Section 3 first presents the numerical results of the electron QTN spectrum from 1 to 110 au and the results at 10 au under different model parameters.The variation of the power density of the QTN spectrum plateau at different heliocentric radial distances and the effects of the model parameters and the antenna length on the effective Debye length and the QTN spectrum are then discussed in this section.Finally, the summary and conclusion are given in Section 4.

Basic Model for Electron QTN Spectrum
The general voltage power density of the QTN spectrum detected by the antenna ports at frequency f ≡ ω/2π can be given as below (Meyer-Vernet et al. 2017) where J is the antenna current distribution in the Fourier space, k is the wavevector, E is the longitudinal electric field fluctuation along k in the Fourier space, and v D is the drifting velocity.Assuming the drifting velocity  v v D T e (the electron thermal speed) and the plasma frequency ω p ? ω ce (the electron gyrofrequency), the general voltage power (1) can reduce to where B(k) is related to the specific electron velocity distribution by The longitudinal dielectric permittivity ε L can be given by where the v ∥ is the velocity parallel to the wavevector k and the i0 + is an infinitesimal positive imaginary part.The antenna response function in the frame of antenna, F(kL), can be determined by the antenna current J.The response function of wire dipole antenna is obtained as follows: and the zero-order Bessel function of first kind J 0 , which can be omitted as 1 if the antenna radius a and length L meet a = L.In the derivation of the last equality above, the linear form of current distribution (Meyer & Vernet 1974 has been used for a set of wire dipole antenna with each length of L and an infinitesimal gap between the two wires.The z direction is parallel to the wire, and r is referred to the radius direction.
For the electrons in the Maxwellian velocity distribution with temperature T e , that is, In space plasmas, the electrons often display the kappa velocity distribution f K (v) in the form of generalized Lorentzian function: and the kappa Debye length where Γ(x) is the gamma function.It is obvious that the kappa distribution reduces to the Maxwellian distribution with the temperature T K and thermal speed v K when the kappa index κ → ∞ .In principle, the kappa index κ can have any value greater than 3/2; however, when κ → 3/2 the kappa distribution will have a non-physical equivalent temperature T K → ∞ for a finite feature "thermal" speed v K .
Chateau & Meyer-Vernet (1991) first adopted the kappa distribution with an integer kappa index κ to calculate the QTN power spectrum.Le Chat et al. (2009) further generalized the calculation of the QTN spectrum to more general cases with any real values of κ > 3/2, that is, where u ≡ L/L DK is the normalized antenna length by the equivalent kappa Debye length, z ≡ ω/kκ 1/2 v K , and the longitudinal permittivity with the function I(z) However, a number of observations show that the electrons in space plasmas often consist of multiple populations (Salem et al. 2023, and references therein), which are difficult to be described simply by a single distribution.In the present work, we employ a combined electron population that consists of a cold core in the Maxwellian distribution f M (v) with the number density n c and temperature T c , and a hot halo in the kappa distribution f K (v) with density n h and temperature T h as follows: where n e = n c + n h is the total electron number density.Based on Equations (2)-( 4), the corresponding power density of the electron QTN spectrum can be derived as follows: where . Similarly, the longitudinal dielectric permittivity can be given by with functions Φ(z c ) and I(z h ) which present in (10) and (16), respectively.

Numerical Results of Electron QTN Spectra and Discussions
Voyager 1 and 2 are the only spacecraft capable of providing in situ measured data of plasma in the outer heliosphere from 1 au near the Earth to the heliospheric boundary layer around 100 au. Figure 1 presents the plasma density n (the black line in the upper panel) and temperature T (the black line in the lower panel) in the outer heliosphere from 1 to 120 au measured by Voyager 2, in which the red lines denote the plasma frequency ( f p in the upper panel) and the Debye length (λ D in the lower panel), respectively.Some disconnections of the lines in Figure 1 are due to the data gaps in the measurements of the proton density and temperature by PLS on Voyager 2. In addition, in the calculation of the plasma frequency and the Debye length the electron density and temperature have been assumed to be equal to the proton density and temperature based on the electric neutrality and isothermal plasma approximations.
From Figure 1, the plasma density is reduced approximately as the inverse square law of heliocentric radial distance until 50 au, from about 10 cm −3 to about 10 −3 cm −3 , and then remains almost invariant but accompanying some ramping up and fluctuating density until the termination shock at about 84 au.These ramps and fluctuations are possibly associated with the stream interaction region (SIR), in which a faster solar wind stream catches up with a slower one.Similarly, the temperature was falling at first but turned to oscillate in the vicinity of 10 4 K away from 20 au, which is probably caused by an energy partition ratio of 0.04 from pickup ions (Wang & Richardson 2001).After the crossing of the termination shock, the density had a trend to rise in the heliosheath, possibly due to the merged interaction region (Richardson et al. 2019), while the temperature jumped to a much greater level, about 10 5 K because of the shock heating, and then came down.The Debye length almost continues to climb up from beginning to end and exceeds 10 m except for a very short duration near Earth, which is well longer than the effective antenna length of Voyager 2 at 7.1 m.
The electron QTN spectrum depends considerably on the velocity distribution functions of the ambient electrons.However, observations show that the electron velocity distribution in the solar wind is difficult to be described by a distribution function for a single component (Salem et al. 2023).In the present calculation, we use a combined distribution function for a compounded plasma of a cold core and a hot halo components, which may be fitted by the Maxwellian distribution f M (v) with the number density n c and temperature T c for a cold core and the kappa distribution f K (v) with the density n h and temperature T h for a hot halo, such as shown in Equation (17).The thermal characteristics of this combined distribution may be determined by the three parameters as follows, the density and temperature ratios α = n h /n c and τ = T h /T c of the halo to the core and the index κ in the kappa distribution function.The total electron density and the equivalent temperature for this compounded plasma can be given by c and 1 , respectively, where á ñ v 2 is the second moment of velocity.Based on Equation (18), Figure 2 presents the electron QTN spectra at different heliocentric distances from 1 au near Earth, 5 au near Jupiter, 10 au near Saturn, 20 au near Uranus, 30 au near Neptune, and 50 au near the aphelion of Pluto to 70, 85 au near the termination shock, 100, and 110 au near the heliopause.To mitigate disturbances from planetary atmospheres or other structures, the plasma density and temperature used in the calculation have been smoothed with a span of 2 au by the "lowess" method, which uses weighted linear least squares and first degree polynomial models for the local regression.For a typical solar wind circumstance with α = 1 and τ ? 1 (Issautier et al. 1999), one may take the parameters α = 0.1 for the halo-core density ratio and τ = 10 for the halocore temperature ratio.As for the parameter κ, the fitting value ranges between 2 and 4 based on the data from Ulysses (Zouganelis 2008), and hence we set κ = 3 in the present calculation.In addition to the parameters of the velocity distribution function, the antenna length L can significantly influence the electron QTN spectra, and the equivalent antenna length L = 7.1 m for the measurement of Voyager 2 has been used in the calculation of Figure 2.
From Figure 2 it can be found that the QTN spectrum consists of three distinct parts: a plateau at the frequency lower than the total plasma frequency , an obvious peak just above f pt , and a rapid decreasing part at higher frequency, where f pc and f ph are the plasma frequencies of the core and halo electrons, respectively.The power spectral density of plateaus shows basically a downward trend as increasing the heliocentric distance from about 5 × 10 −15 V 2 /Hz at 1 au to ∼10 −17 V 2 /Hz near the termination shock at 84 au and then slightly rises to ∼2 × 10 −17 V 2 /Hz near the heliopause at 110 au.Significantly different from the QTN spectra produced by purely Maxwellian velocity distributions (Meyer-Vernet & Perche 1989), the QTN spectrum produced by the compounded distribution in Figure 2 has a clear peak that is much more pronounced than that of the Maxwellian distribution.This pronounced peak can be attributed to the much more considerable high-energy tail of the kappa halo electrons, which may be responsible for the Langmuir turbulence at the plasma frequency f pt .In fact, the QTN peak will possibly not be formed when the velocity distribution function is cut off at its high-energy part (e.g., the "square" distribution) (Chateau & Meyer-Vernet 1989).Moreover, unlike the common belief that the QTN peak is almost invisible when the antenna length L is less than the Debye length L D , in Figure 2 the peaks at the plasma frequency f pt can be clearly recognized though L < L D .Another notable feature of the radial variation in the QTN spectra is that their high-frequency part above f pt becomes flatter at the larger heliocentric distances.This is probably caused by the increase of the Debye length L D for the given antenna length L as shown in Figure 1.
Figure 3 shows further the effects of the model parameters of the compounded distribution (α = n h /n c , τ = T h /T c , κ) and the antenna length (L) on the QTN spectrum at the given heliocentric distance of 10 au.Panel (a) presents the QTN spectra for the different halo-core density ratio α = 1, 0.1, and 0.01 and the given τ = 10, κ = 3 and L = 7.1 m.For a higher α value, i.e., a denser hot kappa halo, the QTN spectrum has a higher plateau in the low frequency lower than the plasma frequency f pt , a higher peak at near f pt , and a steeper descent in the high frequency higher than f pt .The halo-core temperature ratio τ has a similar effect on the QTN spectrum as shown in the panel (b), in which a hotter kappa halo leads to that the QTN spectrum has a higher plateau in the low-frequency region, a higher peak near the plasma frequency, and a steeper descent in the high-frequency region.The effects of the kappa halo component on the QTN spectrum of the compounded solar wind plasma may be attributed to the contribution of the suprathermal electrons of the kappa halo because the highly energetic supra-thermal electrons can produce more powerful Langmuir waves (Le Chat et al. 2009).
Another important parameter that influences highly energetic supra-thermal electrons of the kappa halo is the kappa index κ, which describes the degree that the kappa distribution deviates from the Maxwellian distribution, i.e., the kappa distribution approaches to the Maxwellian distribution with the finite kappa temperature T K when κ → ∞ and when κ approaches to the limit value 1.5 the kappa temperature T K → ∞ , implying that the kappa distribution significantly deviates from the Maxwellian distribution and the major electrons of the kappa halo are distributed in the supra-thermal tail with extremely high energies, which can supply more energy to the Langmuir wave. Figure 3(c) displays the QTN spectra for the kappa index κ = 1.53, 3, 4.5, 10, and 100.For κ = 1.53 close to the limit value 1.5, the QTN spectrum (red line) has a very prominent peak at the plasma frequency and Gurnett et al. (2021) proposed that this prominent peak possibly is responsible for the continuous plasma emission line detected by Voyager 1. Conversely, for κ = 100 ? 1, the QTN spectrum (green line) displays a much weaker peak, which almost consists with the known two Maxwellian broadened peak at the plasma frequency (Meyer-Vernet & Perche 1989).
The detection of the peak and plateau of the QTN spectrum can provide important information of the electron composition in the space plasma as shown above.It is obvious that a longer antenna may provide more effective measurement of the QTN spectrum.Figure 3(d) presents the QTN spectra for the antenna length L = 2 m, 7.1 m, and 15 m but the other parameters are fixed.From panel (d), it is clear that the increasing L brings the antenna a better ability to detect the peak and plateau produced by the present compounded electrons.
To understand further the variation of the QTN plateau in Figure 2, we adopt the low-frequency approximation of the power density V f 2 used by Meyer-Vernet et al. (2017): where V f 2 is the power density of the QTN plateau in the lowfrequency approximation of ω/kv = 1 and F 0 is an integral related to the antenna response function F: = ( ) which monotonically increases as t increases for t < 3.5.In the above expression, T −2 and T −1 are the generalized temperature defined by negative second and first moments of velocity, that is, (Meyer-Vernet 2001): B q q q q 2 with q = −2 and −1, where where q is the order of the velocity moment.For the case of a dilute halo with 1 is approximately equal to the core temperature T c .Since Voyager 2ʼs antenna is immersed mainly in the plasma environment where the Debye length is greater than the antenna length, i.e., L/L D < 1, the QTN plateau in Equation ( 20) is expected to descend because L D ascends before the termination shock near 85 au and then slightly rebounds after the termination shock, as shown in Figure 2.
For more accurate calculation, we can derive the effective Debye length by the distribution function of the compounded plasma in Equation ( 17): DM B e e 0 2 and the negative velocity moments in Equation (24): Based on Equations ( 23) and (24), the power density of the electron QTN plateau of the compounded plasma with the parameters α = 0.1, τ = 10, and κ = 3 can be quantitatively given by Equation (20) and the result is shown in Figure 4, in which the antenna length L = 7.1 m for Voyager 2 has been used.The red and yellow solid line are the trajectory of the 6 plasma parameters measured by Voyager 2 before and after the termination shock (TS) and the black thin lines denote the contour lines of the power density.The locations of representative heliocentric distances are marked with black asterisks.The QTN plateau level mainly falls down until the TS crossing at 83.7 au, after which the plateau rebounds a little near the heliopause.
As shown in Equations ( 23) and (24), the effective Debye length L Deff , the negative first and second velocity moments T −1 and T −2 , and hence the power density of the QTN plateau V f 2 also depend on the model parameters of the compounded distribution the halo-core density ratio α, temperature ratio τ, and the kappa index κ for the hot halo.Figure 5 shows the Debye length L Deff and the power density of the QTN plateau V f 2 at the heliocentric distance of 10 au versus the parameters α (a), τ (b), κ (c), and the antenna length L (d).From Figure 5(a), one can find that for the given parameters τ = 10 and κ = 3, the Debye length L Deff , at first, decreases with α from ∼36 m at α = 0.01 = 1 to ∼21 m when α ∼ 1 and then increases to ∼29 m when α = 100 ? 1.In contrast, for the given parameters α = 0.1 and κ = 3, L Deff increases with τ for τ < 1 but decreases for τ > 1, and hence has its maximum near τ = 1, as shown in Figure 5(b).For the given parameters α = 0.1 and τ = 10, Figure 5(c) indicates that L Deff displays a monotonically increasing trend with κ, especially a rapidly increasing presents when κ is close to the limit value of 1.5 (i.e., k k ¢ º -)  1.5 1 .On the other hand, the plateau power density V f 2 presents an opposite changing trend to the Debye length L Deff , as presented in Figure 5(a) and (c).In Figure 5(b), the increase of V f 2 with τ for τ = 1 can be attributed to the increase of the factor --

T T
2 1 with τ for α = 0.1 = 1.In particular, Figure 5(d) clearly shows that for the given model parameters α = 0.1, τ = 10, and κ = 1.53, the plateau power density V f 2 depends only on the antenna length L through the function F 0 (L/L Deff ) and monotonically increases with the antenna length L for given L Deff , implying that the longer antenna is more favorable for detecting the QTN plateau when L/L Deff < 3.5.
Although it is difficult to determine the model parameters α, τ, and κ due to the high uncertainty of the solar wind plasma flow, the above results presented in Figure 5 show that in a fairly wide range of the model parameters the level of lowfrequency plateau does not present more than double the change.

Summary and Conclusion
In summary, the present work proposed a compounded plasma model of multi-component electrons that consists of a cold Maxwellian core and a hot kappa halo to describe the ambient solar wind plasma and derived the corresponding longitudinal dielectric permittivity ε L and the voltage power density V f 2 for the electron QTN.Based on the in situ measured data by Voyager 2, which is one of the only two spacecraft (Voyager 1 and 2) to have ever fully explored the outer heliosphere, for the given halo-core density ratio α = 0.1 and temperature ratio τ = 10, the kappa index κ = 1.53, and the Voyager antenna length L = 7.1 m, we calculated the electron QTN spectra at 10 different heliocentric distances from the Earth's orbit to the heliospheric boundary layer (i.e., 1, 5, 10, 20, 30, 50, 70, 85, 100, and 110 au).The results show that these QTN spectra have a common feature, i.e., these spectra all consist of three distinct parts: a plateau in the low-frequency band f < f pt , a prominent peak at the plasma frequency f pt , and a rapid decreasing part in the high-frequency band f > f pt .
However, the compounded model of the solar wind plasma and the antenna length both can considerably influence the electron QTN spectrum.As an example, we discussed the effects of the model parameters α, τ, and κ and the antenna length L on the QTN spectrum at the heliocentric distance 10 au.It is found that the denser and hotter kappa halo (i.e., higher α and τ) will produce the stronger QTN spectrum with higher power density.On the other hand, the kappa index κ closer to its limit value 1.5 forms the more prominent peak and when κ → ∞ the QTN spectrum has a much weaker peak.Finally, it is evident that the longer antenna (i.e., larger L) can provide better measurement of the QTN spectrum as expected.
The plateau power density is a good parameter for measuring the QTN spectral strength.Based on the plasma data measured by Voyager 2, we further presented the plateau power density of the QTN spectrum in the low-frequency approximation for the given model parameters α = 0.1, τ = 10, and κ = 1.53 and the antenna length L = 7.1 m.The QTN plateau level basically falls down outwards until the TS crossing at about 84 au, after which the plateau rebounds a little near the heliopause.In general, the model parameters (α, τ, and κ) can be very variable due to the highly dynamic characteristics of the solar wind plasma flow.However, fortunately our results showed the QTN plateau level does not present more than the double change in a fairly wide range of the model parameters.
The deep-space exploration and investigation of the outer heliosphere from the Earth orbit at 1 au to the heliosphere boundary layer at about 100 au is one of main goals of astrophysics and space science.QTN, the electromagnetic fluctuation signal induced by the thermal or quasi-thermal motions of the ambient charged particles, plays a vital role in the measurement of plasma waves and low-frequency radio radiation, as well as in the understanding of the ambient plasma characteristic.On the one hand, in general, only plasma waves above the level of thermal noise have important measurement value.Therefore, the QTN plateau can provide an important reference for the design of the measuring sensitivity and the antenna geometry.On the other hand, the peak of the QTN spectrum at the local plasma frequency f pt is a direct and stable diagnostic signal for the ambient plasma density.This makes it a fine tool to calibrate other plasma measurements onboard spacecraft.The present results are helpful for us to gain a deeper and more comprehensive understanding of the physical characteristics of the QTN spectrum of the solar wind plasma in the outer heliosphere, and may provide valuable references for the design of detection instruments, especially detection antennas, on future deep-space exploration satellites.
In addition, it should be pointed out that the present work considered only the contribution of the electron components to the QTN spectrum.In fact, the ionized component can significantly contribute to the QTN spectrum, especially in the low-frequency range close to the ion plasma frequency m m f e i pt , where m e and m i are the electron and ion mass, respectively.In the following work, we will further consider to include the contribution of ion components to the QTN spectrum.
f/f p is normalized frequency by the plasma frequency f p , u ≡ L/L DM is the normalized antenna length by the Maxwellian Debye length w k B is the Boltzmann constant.The longitudinal permittivity derived with the Maxwellian velocity distribution

Figure 1 .
Figure 1.Plasma density, frequency, temperature, and the Debye length variation from 1 to 118 au.Black lines indicate the plasma density and temperature in the upper and lower panel, while the red lines indicate the plasma frequency and the Debye length, respectively.

Figure 2 .
Figure 2. Electron QTN spectrum with the combined Maxwellian and kappa distribution at different heliocentric distances.The plasma frequency is normalized by f pt , where = + f f f pt pc ph 2

Figure 4 .
Figure 4. Electron QTN plateau level in logarithmic form with wire dipole antenna for Voyager (L = 7.1 m), in a density/temperature plane.The red and yellow solid line indicate the plasma parameters measured by PLS instrument of Voyager 2 before and after termination shock, respectively.Each heliocentric distance presented in the QTN spectra calculation of Figure 2 is marked with an asterisk.

Figure
Figure Dependence of effective Debye length L Deff (blue lines) and the voltage power density at low-frequency plateau (red lines) vs. density ratio α, temperature ratio τ, kappa index κ and antenna length L at 10 au.Note that each panel is plotted with only single parameters varying while the remaining parameters are fixed.For example, panel (a) is plotted as α varying from 0.01 to 100 with τ = 10, κ = 3 and L = 7.1 m.