# On the Plasma Quasi-thermal Noise in the Outer Heliosphere

Nicole Meyer-Vernet, Alain Lecacheux

► To cite this version:

Nicole Meyer-Vernet, Alain Lecacheux. On the Plasma Quasi-thermal Noise in the Outer Heliosphere. Research Notes of the AAS, 2024, 8 (3), pp.92. 10.3847/2515-5172/ad382b . obspm-04919431

HAL Id: obspm-04919431

https://hal-obspm.ccsd.cnrs.fr/obspm-04919431v1

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Distributed under a Creative Commons Attribution 4.0 International License
On the plasma quasi-thermal noise in the outer heliosphere

Nicole Meyer-Vernet¹ and Alain Lecacheux¹

¹LESIA, Observatoire de Paris, PSL Université,
CNRS, Sorbonne Université, Université Paris Cité, 92195 Meudon, France

ABSTRACT

The recent paper by Li et al. on electron quasi-thermal noise in the outer heliosphere is flawed. It assumes the plasma drift speed to be much smaller than the electron thermal speed, even though both quantities are of the same order of magnitude in the outer heliosphere inward of the termination shock, because of the low plasma temperature. In this case, the Langmuir wave dispersion equation and the quasi-thermal noise in the antenna frame are completely changed. Furthermore, these calculations neglect the shot noise, which should produce a large contribution below the plasma frequency with the Voyager antennas in the outer heliosphere.

INTRODUCTION

Plasma quasi-thermal noise (QTN) is routinely observed with wave instruments in space, and spectroscopy of this noise is an efficient tool to measure plasma properties (Meyer-Vernet et al. 1998, 2017). In weakly magnetised plasmas, the electron QTN consists of a plateau below the plasma frequency $f_p$, mainly produced by electrons crossing the plasma sheath surrounding the antenna, a peak close to $f_p$, mainly produced by electrons of speed close to the large Langmuir wave phase speed, and a power decreasing as large frequencies (Meyer-Vernet & Perche 1989). This noise has been suggested to be at the origin of a weak line at the local plasma frequency discovered in spectra from the Voyager 1 Plasma Wave (PWS) instrument in the very local interstellar medium (Ocker et al. 2021; Gurnett et al. 2021). Meyer-Vernet et al. (2022) have shown that this line can be explained by the QTN produced by a minute quantity of suprathermal electrons. Meyer-Vernet et al. (2023) suggested an origin for these electrons and showed that the original interpretation by Gurnett et al. (2021) is questionable. Recently, Li et al. (2024) published calculations of the QTN in the outer heliosphere on an antenna similar to the Voyager one, using classical expressions of the QTN (Meyer-Vernet et al. (2017) and references therein) that assume that the plasma drift speed is much smaller than the electron thermal speed.

THE DRIFT SPEED IS NOT MUCH SMALLER THAN THE ELECTRON THERMAL SPEED

Figure 1 shows the ratio of the drift speed $v_D$ to the electron thermal speed $v_{TH} = (2k_B T/m)^{1/2}$ measured by the Plasma (PLS) instrument (Bridge et al. 1977) onboard Voyager 2 in the solar wind. Here $v_D$ has been estimated from the radial speeds of the solar wind and of the spacecraft, $k_B$ is Boltzmann constant, $m$ is the electron mass, and $T$ is the electron temperature in K, assumed to be close to the proton one. Note that since the Voyager spacecraft speed of about 15 km/s is much smaller than the solar wind speed, the plasma drift speed relative to the antenna is roughly equal to the solar wind speed. Figure 1 shows that the drift speed is of the same order of magnitude as the electron thermal speed and cannot be neglected in most of the outer heliosphere up to the termination shock, contrary to the assumption made in the calculations used by Li et al. (2024).

The voltage power spectrum of the plasma quasi-thermal noise at the terminals of an antenna in a plasma drifting with velocity $v_D$ is

$$V_{\omega}^{2} = \frac{2}{(2\pi)^{3}} \int d^{3}k \left| \frac{\mathbf{k} \cdot \mathbf{J}}{k} \right|^{2} E^{2}(\mathbf{k}, \omega - \mathbf{k} \cdot \mathbf{v}_{D}) \quad (1)$$
<figure><img src="image_2.png" /><figcaption>**Figure 1.** Ratio of the drift speed $v_D$ to the electron thermal speed $v_{TH}$ observed by Voyager 2 as a function of time (in years) and heliocentric distance $R$, approximating the electron temperature by the proton one. The ratio $v_D/v_{TH}$ is close to unity in most of the outer heliosphere, until the crossing of the termination shock where it drops off abruptly.</figcaption></figure>

The first term in the integral in (1) involves the antenna response to electrostatic waves, which depends on the Fourier transform $J(k)$ of the current distribution along the antenna. The second term is the Fourier transform of the autocorrelation function of the plasma electrostatic field fluctuations in the antenna frame. Quasi-thermal noise calculations (Meyer-Vernet et al. (2017) and references therein) used by Li et al. (2024) assume the drift speed $v_D$ to be small enough that $\omega - k \cdot v_D$ in (1) can be replaced by $\omega$. Since the autocorrelation function of the plasma electrostatic field fluctuations is determined by the plasma velocity distributions, the drift speed $v_D$ can be neglected if it is much smaller than the speeds of the particles that determine the QTN, which are mainly the electrons. Since this condition does not hold in most of the outer heliosphere where $v_D$ and $v_{TH}$ are of the same order of magnitude, the QTN will be changed by a large amount. In extreme cases when $v_D > v_{TH}$, the resistance of the antenna could even become negative, producing unreliable electric field measurements (Meyer-Vernet 1989).

## THE SHOT NOISE IS NOT NEGLIGIBLE IN THE OUTER HELIOSPHERE WITH THE VOYAGER ANTENNA

A further problem arises in the calculations published by Li et al. (2024). In the solar wind, the photoelectron flux emitted by electric antennas largely exceeds the collected plasma electron flux, making the floating potential positive (Meyer-Vernet 2007). This holds in most of the outer heliosphere, where the electron flux decreases with heliocentric distance slightly faster than the photoelectric emission. The shot noise on the Voyager antennas, of radius $a$ and length $L$ much smaller than the plasma Debye length, would thus be approximately, if $v_D \ll v_e$ as wrongly assumed by Li et al. (2024)

$$V_{\text{shot}}^2 \approx 2 \times 10^{-16} T^{1/2}(a/L)[\ln(L/a) - 1]^2 (f_p/f)^2 \quad (2)$$

in S.I. units, for $f < f_p$ (Meyer-Vernet & Perche 1989). Equation (2), which holds in a maxwellian plasma, is weakly affected by the presence of suprathermal electrons (Meyer-Vernet et al. 2017). However, the small temperature in the outer heliosphere implies also that the antenna positive potential, equal to a few times the photoelectron temperature in electron volts (eV), can become larger than the electron temperature. Both the antenna potential and the drift speed will increase the shot noise, which is proportional to the electron flux impacting the antennas. With the Voyager antennas of radius $a = 0.635$ cm (Scarf & Gurnett 1977), this yields $V_{\text{shot}}^2 > 10^{-15}(f_p/f)^2 T_{eV}^{1/2} V^2/Hz$, where $T_{eV}$ is
the temperature in eV. This noise should thus exceed the QTN plateau plotted by Li et al. (2024) in most of the outer heliosphere, except with antennas of radius much smaller than on Voyager or being negatively biased (Meyer-Vernet et al. 2017).

CONCLUSION

The QTN spectra published by Li et al. (2024) are invalid in most of the outer heliosphere up to the termination shock, since the calculations assume that the drift speed is negligible compared to the electron thermal speed. Furthermore, the shot noise was neglected in this paper, even though it is significant on Voyager antennas in the heliosphere, and would require much thinner antennas and/or a negative biasing to become negligible.

The PLS data could be obtained from the site: ftp://space.mit.edu/pub/plasma/vgr/v2.

REFERENCES

Bridge, H. S., Belcher, J. W., Butler, R. J. et al. 1977, SSRv, 21, 259

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Kurth, W. S., Burlaga, L. F., Kim, T. et al. 2023, ApJ, 951, 71

Li, Y.-L., Chen, L. & De, J. W. 2024, ApJ, 963, 46

Meyer-Vernet, N. 1989, A&A, 224, L5

Meyer-Vernet, N. 2007, *Basics of the Solar Wind*
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Meyer-Vernet, N., & Perche, C. 1989, J. Geophys. Res., 94, 2405

Meyer-Vernet, N., Hoang, S., Issautier, K., & al. 1998, in
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