Ion acoustic waves near a comet nucleus: Rosetta observations at comet 67P/Churyumov-Gerasimenko

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The closest approach occured at 13:05 UTC, and then Rosetta was at a cometocentric distance of 15 km. The spacecraft 50 moved slowly (with a relative speed to the comet below 1 ms −1 ) and was in the the vicinity of the nucleus for several hours as shown in Fig. 1 and in panel h of Fig. 2.

Instrumentation
The data used in this article was obtained by instruments belonging to the Rosetta Plasma Consortium (RPC) (Carr et al., 2007).
For the wave observations (Sect. 2.2) we used the Rosetta Langmuir probe instrument (RPC-LAP)  to and each time series contains 1600 samples, corresponding to a time series length of 85.3 ms. This process was repeated every 160 s. Each of the two probes obtained 295 such time series during the day. The power spectral density for each time series is computed, using Welch's method (Welch, 1967), averaging segments that are 256 samples long with an overlap of 65 % 60 ( Fig. 2a and b). The probes were held at fixed potentials with respect to the spacecraft: probe 1 at +30 V and probe 2 at −30 V.
The Langmuir probe instrument was also used to measure the bulk speed of the ions and the electron temperature by sweeping the probe potential and measuring the probe current as described in Sect. 2.3.
We use the Mutual Impedance Probe (RPC-MIP)  to obtain the plasma density during the flyby. The RPC-MIP instrument observes the plasma frequency, from which the plasma density is derived (Fig. 2f). The ion populations Fig. 2b. The colour coded quantity is the logarithm of the power spectral density (PSD) of the probe currents. The lowest frequency bins are at risk of picking up low-frequency noise, and we therefore show the spectrum for frequencies above 200 Hz. There may be other waves present at low frequencies, but in this article we only consider ion acoustic waves above 200 Hz.
A high amplitude wave signal is seen during the close flyby and it falls off as the spacecraft moves away from the nucleus. 75 The power spectral density of the positively biased probe 1 is several orders of magnitude higher than that of the negative probe 2. This means that the probe 1 signal is dominated by the electron current and that the signal is proportional to the density variation of the wave. The probe was thus operating in the same regime as when waves were observed in the diamagnetic cavity when the comet was at perihelion (Gunell et al., 2017a). Also the maximum PSD value is similar to those observations and the plasma density, shown here in Fig. 2f, was in both cases somewhat above 1000 cm −3 . The situation differs from the first ion 80 acoustic wave observations at comet 67P (Gunell et al., 2017b) when the plasma density was an order of magnitude smaller and the waves coupled capacitively to the probe through the displacement current instead of a particle current. The difference between the probes is also seen in Fig. 2c, which shows the integral of the power spectral density over frequencies from 200 Hz up to the Nyquist frequency. Fig. 3 shows four sample spectra of the probe 1 current for frequencies from 200 Hz up to the Nyquist frequency. There 85 is wave power starting at the low end of this frequency range with a broad maximum in the vicinity of 1 kHz, and at higher frequencies the PSD declines toward the noise floor. The black curve shows the PSD at 13:24:54, which is near closest approach to the comet nucleus at a cometocentric distance of 15 km. As seen in Fig. 2c the total wave power fluctuated but remained at a generally high level while the spacecraft was in the near-nucleus environment. The PSD obtained at 15:16:54 (red curve in Fig. 3) is another example from this period. The spacecraft was at 17.5 km cometocentric distance and the wave power was even 90 higher than that shown by the black curve. The wave power declined as the spacecraft moved to larger cometocentric distances. This process started approximately at 17:45 when Rosetta was at 24 km from the centre of the nucleus. Two examples from the declining phase are shown in Fig. 3: the spectrum obtained at 17:56:54 at 25 km (blue curve) and one spectrum from 19:08:54 at 29 km (green curve) when the wave power had fallen even more. The two curves that will be used for comparison with wave theory in Sect. 3 are the black curve (13:24:54) for closest approach and the blue curve (17:56:54) for the outbound case. The 95 peaks at multiples of 1 kHz seen in the frequency range where the wave power is low, both in Fig. 3) and Fig. 2 are artefacts generated by the spacecraft.

Plasma properties
To analyse the waves we need to know the basic properties of the plasma. The plasma density obtained by the mutual impedance probe, RPC-MIP, is shown in Fig. 2f. The density peaks around closest approach and then falls off as the spacecraft moves 100 away from the nucleus. The scattered instantaneous plasma density values are a signature of strong plasma inhomogeneities of approximately 10 % around closest approach. For the calculations in Sect. 3 we estimate a plasma density of n e = 1600 cm −3 at closest approach and n e = 1000 cm −3 for the outbound case. Fig. 2g shows an energy spectrum of the ions observed by RPC-ICA. Starting at E/q ≈ 20 V is a warm (k B T i ≈ 6 eV around the time of closest approach) water ion population, which has been accelerated toward the spacecraft due to the negative 105 spacecraft potential. Some accelerated water ions are seen at higher energies, but the vast majority of the ions seen in Fig. 2g belong to the warm, low energy, population. Fitting the observed flux to a Maxwellian distribution we arrive at a density estimate of about 4 cm −3 for this ion population. However, this is far below the 1600 cm −3 plasma density measured by RPC-MIP. It was shown by Bergman et al. (2020a, b) that low energy (down to 5 eV) ions describe complicated orbits in the potential well around the spacecraft. Therefore the field of view of the instrument may be far from what is nominally expected, and the 110 low energy part of the observed distribution functions can be very inaccurate. The field of view for ions with lower energies than the 5 eV lower limit considered by Bergman et al. (2020a, b) is even more limited, and the fraction of that population that is detected may not be distinguishable from ions belonging to the warm population. Thus, the discrepancy between the RPC-ICA measured ion density and the plasma density measured by RPC-MIP may be explained by a cold water ion distribution that is invisible to  This is confirmed by Langmuir probe characteristics shown in Fig. 4. The left panel shows the part of the characteristics dominated by the ion current. For a cold ion population drifting at a bulk speed u the probe current I depends on the probe to where r p = 2.5 cm is the radius of the probe, m i is the ion mass and n i the ion density. We fit a line to the linear part of the 120 curve, and taking the derivative of Eq. (1) and rearranging we can determine the drift velocity from the slope dI/dV of that line: Taking the ion density to be equal to the plasma density measured by RPC-MIP we arrive at an ion drift speed of 3 km s −1 near closest approach and 3.7 km s −1 at 17:52:06 when the spacecraft was moving outward as shown in Fig. 4. These numbers 125 are within the range of those observed by Odelstad et al. (2018). The velocity obtained from Eq. (2) is a upper limit, as it is based on the assumption of cold, zero temperature, ions. The ion temperature can be estimated from the neutral temperature, as ions are created by ionisation of the neutrals. Biver et al. (2019) found neutral temperatures in the 50-200 K range, which corresponds to approximately 0.02 eV, and that is well below the 1 eV kinetic energy, corresponding to the 3-3.7 km s −1 drift speeds obtained above.

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The right-hand panel of Fig. 4 shows the part of the probe characteristic where the current is dominated by the electrons.
The dashed lines have been fitted to the high probe potential part of the sweep. Here, the current varies linearly with voltage Figure 5. Magnetic field magnitude during a period around closest approach. The red lines are fitted to the data in order to derive the current densities J = 4.9 µA m −2 (inbound) and J = 1.9 µA m −2 (outbound). (Swift and Schwar, 1970) and the cold electron temperature is (Engelhardt et al., 2018) The slopes of these lines correspond to temperatures of k B T e = 0.2 eV for both the closest approach and outbound curves.

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The curves also show that the plasma potential is between 12 and 14 volts above the spacecraft potential, approximately.
For the spacecraft to become that negatively charged there must be an additional electron population which is warmer. The estimate we use in Sect. 3 is that the electron distribution is constituted by two contributions with equal densities: one cold with temperatures as estimated in Fig. 4 and one warm with a temperature of 4 eV. This follows previous Langmuir probe sweep interpretations from when the comet was near perihelion Gunell et al., 2017a;Odelstad et al., 2018) 140 with the difference that the cold electrons are not quite as cold here as the 0.1 eV that was estimated near perihelion. These two electron temperature values are within the range of those observed by RPC-MIP at similar heliocentric distances in 2016 (Wattieaux et al., 2020). from magnetic pileup and field line draping, but there are also other changes in the magnetic field that can be seen in Fig. 2d and e.
To estimate the current associated with the non-uniformity of the magnetic field we fit lines to the magnitude of the magnetic field as shown in Fig. 5. Then we estimate the magnitude of the current density by where |∆B| is the change in the fitted magnetic field magnitude and |∆r| is the distance the spacecraft moved during the same period of time. This yields a current density of J = 4.9 µA m −2 when the spacecraft was approaching the nucleus and lower than what was observed by Rosetta. The difference could be attributed to the limited resolution or the use of an averaged outgassing profile in the simulations . For the magnitude of B the difference only amounts to 10-20 %, but the simulation does not follow how the plasma quantities develop in time. Fig. 5 shows that the magnetic field changed on much shorter timescales than those of our linear approximations during the flyby. From a single spacecraft measurement we cannot determine whether these magnetic field fluctuations are due to local variations of the current in the plasma or whether 160 the whole inner region of the ionised coma is undergoing oscillations. Thus, the current density may have been both higher and lower than these average values during the flyby. 40 nT approximately. This corresponds to electron cyclotron frequencies between 0.6 and 1.1 kHz, which is in the middle of the observed frequency range. However, the wave frequency does not follow the changes in the magnetic field, which rules out electron cyclotron waves. The ion cyclotron frequency is (0.02 − 0.03) Hz, which is below the frequencies we can resolve.

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The spacecraft was at 15 km cometocentric distance at closest approach and at 25 km at 18:00 when the wave amplitude started to decrease. Thus, the typical length for the variation in wave amplitude is about 10 km. Assuming a typical B of 30 nT warm ions at 6 eV would have a gyroradius of 50 km. Cold ions are picked up by the electric field, moving along trajectories with a radius of curvature that is even larger. The ions can thus be seen as unmagnetised. Warm electrons at 4 eV have gyroradius of 225 m and for cold 0.2 eV electrons the gyroradius is 50 m approximately.

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In Sect. 3 we use kinetic theory to compute dispersion relations for electrostatic waves in an unmagnetised plasma. This is applicable if the wavelength is much shorter than the gyroradii of the particles so that the influence of magnetic forces on particle motion is negligible on wavelength scales. In Sect. 3 it is seen that the phase speed for ion acoustic waves is approximately 1.7 km s −1 . Thus, a wave at 200 Hz (the lower limit of the spectrum shown in Fig. 2a) has a wavelength of 8.5 m, which is far below all the gyroradii reported above. The assumption that the plasma is unmagnetised for wave purposes 180 holds above that limit, and these are the waves considered here. For waves at the very lowest frequencies, below the range considered here, the wavelength is longer, and electromagnetic effects would have to be taken into account. can say about the charged particle distributions. The total distribution function is composed of a cold and a warm electron and a cold and a warm ion distribution. The parameters are shown in Table 1 for 7 test cases used to model the distribution near closest approach and 2 cases for the outbound trajectory. We use the simple pole expansion method to compute the dispersion relations (Löfgren and Gunell, 1997;Skiff, 2001, 2002;Tjulin et al., 2000;Tjulin and André, 2002). In a comet where v t is the thermal speed, v d is the drift speed, and m is the number of terms included in the expansion. In the distributions in Table 1, m = 3 for the ions and m = 5 for the electrons. The influence of suprathermal tails is evaluated in Appendix A. The density of the warm ion population is varied in distributions 1-3. In distribution 1 the warm ion density is 4 cm −3 205 as estimated in Sect. 2.3. In distribution 2 the warm ion density is assumed to be zero, and in distribution 3 the warm ion density is ten times higher than the estimate in Sect. 2.3. The real part of the dispersion relation is indistinguishable among the three cases, as seen in Fig. 6a. The damping rates in Fig. 6c are very close in the three cases, although it can be descried that distribution 3, with the highest warm ion density, has a slightly smaller growth rate than the other two. However, the difference is small, and we conclude that the warm ion population only has a negligible influence on the waves. Therefore, the warm ion 210 density is set to zero in the rest of the distribution functions.
We have used the current density estimate, J = 4.9 µA m −2 , obtained in Sect. 2.3 for the inbound part of the flyby. The dispersion relations are computed in the rest frame of the ions and the current is modelled by assigning a drift velocity, |v D | = 39.3 km s −1 , to one of the electron populations. In distribution 2 and distribution 4 that drift speed is given to the cold and warm electron distribution, respectively. For distribution 4 the ion acoustic mode is damped, while it is growing for 215 distribution 2. We conclude that to drive the ion acoustic waves unstable the current must be carried by the cold electrons.
In distribution 5, the temperature of the warm electrons has been decreased to 1 eV. This leads to a decreased growth rate compared to distribution 2, which has 4 eV warm electrons but otherwise is equal to distribution 5. However, in both cases the waves are unstable over approximately the same wavelength range.
Distributions 6 and 7 have 0.01 eV and 0.04 eV cold ions, respectively, that is to say, in distribution 6 the ions are colder and 220 in distribution 7 warmer than they are in the otherwise equal distribution 2. This affects the growth rate so that distributions with colder ions grow faster and over a wider k range than distributions where the ions are warmer (Fig. 6c). Also the real part of ω is affected, as shown in Fig. 6a, but this is significant only for k values larger than the k which corresponds to maximum growth. The influence of suprathermal particles on the dispersion relations and growth rates is evaluated in Appendix A, and it is found to similar to the difference between distribution with 0.01 and 0.02 eV ions. The distribution function of the cold ions 225 cannot be measured directly, and hence effects caused by the shape of the distribution cannot be distinguished from effects caused by the temperature alone. However, we may conclude from all 7 cases that any process that gives the ions higher or the electrons lower energy will lead to decreased growth or increased damping.
Dispersion relations for distributions A and B, detailed in Table 1, are shown in Fig. 7. These distributions correspond to the plasma parameters obtained during the outbound passage of the spacecraft, and close to when the PSD represented by the blue Figure 6. Dispersion relations at closest approach for the seven assumed distributions specified in Table 1  current density J = 1.9 µA m −2 measured when the spacecraft was moving away, and the dispersion relation corresponding to distribution A is very similar to those at closest approach. In distribution B none of the populations have been assigned a drift velocity. This leads to a stable distribution, and the waves are weakly damped instead of growing. Examining the magnetic field in Fig. 2 we see no large scale change near 18:00, which means that there was no large scale current present. Thus, the 235 change in the plasma that affects the waves is the absence of a current, and this indicates that the reason why the wave spectrum fades out as the spacecraft moves away from the nucleus is the decline of the current density. Around 18:00 the waves likely were propagating to the spacecraft from a source region closer to the nucleus, where the current density was still high enough to generate the waves.

Doppler shift 240
The dispersion relations are computed in the ion frame of reference and the observations are, by necessity, performed in a spacecraft-fixed frame. A frequency f m in the moving medium is Doppler shifted to frequency f sc in the spacecraft frame according to where v ph is the phase velocity given by the dispersion relation, u is the speed of the moving medium, and α is the angle 245 between the wave direction of propagation and the velocity u. Fig. 8a shows the H 2 O + ion plasma frequency and the frequency of maximum growth, Doppler shifted to the spacecraft frame according to Eq. (5) for the dispersion relations that correspond to distributions 2 and 7, and with u determined by Eq.
(2). The frequencies are shown as functions of the angle α. The angle is not known from observations, but by comparing the Doppler shifted frequencies to the observed spectrum we can assess what values of α would lead to a reasonable spectrum.

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The dispersion relations show that the damping is considerable at the ion plasma frequency. This has also been seen in experiments with current-driven ion acoustic waves, where the power declines with frequency and reaches the noise floor at frequencies well below the ion plasma frequency (Kawai et al., 1978). For our near closest approach sample spectrum shown by the black curve in Fig. 3 this happens at approximately 5 kHz. The range of angles α consistent with the observed spectra can then be constrained to those for which the ion plasma frequency is mapped to frequencies above 5 kHz. If the waves follow 255 the dispersion relation corresponding to distribution 2, (solid red curve in Fig. 8a) this means that α 56 • and in the case of distribution 7 (solid black curve in Fig. 8a) the angle is restricted to α 48 • . We will round this off to α 50 • .
For a particular dispersion relation to be in agreement with observations, there should be significant wave power at the Doppler shifted frequency of maximum growth. With this regard distribution 7 is in better agreement with observations than distribution 2, because the spectrum has fallen significantly at 2 kHz and the dashed red curve in Fig. 8a is above 2 kHz for 260 most of the relevant angle range of α 50 • determined above. The dashed black curve is close to 1 kHz in this range, and it is in good agreement with the peak of the spectrum in Fig. 3. Of the different dispersion relations we have examined it is the one corresponding to distribution 7 that best fits the Rosetta data. However, several distributions can lead to similar growth rates at similar frequencies, and we cannot constrain the distribution function closely. What we can say is that distributions that lead to moderate growth rates are in better agreement with the data than those that show very rapid growth.
265 Fig. 8b shows the Doppler shifted ion plasma frequency and frequency of maximum growth for distribution A. The real part of the dispersion relation for distributions A and B overlap in Fig. 7, and therefore the Doppler shifted ion plasma frequency will be the same for distribution B as for distribution A. For distribution B the waves are damped everywhere, and there is no frequency of maximum growth. We have already concluded in Sect. 3 that distribution B is more likely than distribution A, and that the waves that were observed as the spacecraft moved away were not generated at the spacecraft location. The frequency 270 where the wave power peaks tells us more about the source region than about the conditions at the spacecraft position. The blue curve in Fig. 3 has fallen to the noise floor at approximately 3 kHz, and from the solid curve in Fig. 8b the dispersion relation is seen to be in agreement with data for angles in the range α 70 • .

Discussion and conclusions
We have analysed data obtained during Rosetta's close flyby of comet 67P on 28 March 2015. Waves which we interpret as 275 current-driven ion acoustic waves were recorded by the Langmuir probe instrument RPC-LAP. These waves were seen all the time the spacecraft was close to the nucleus and the wave power started to decrease at approximately 24 km cometocentric distance. We estimated the current density from magnetic field measurements and found that the same currents that are involved in draping and pileup of the magnetic field  are sufficient to drive the ion acoustic mode unstable, according to the kinetic model we have used to compute dispersion relations. Koenders et al. (2016) could observe field line have any significant influence on the waves.
We are not able to measure the fine details of the particle distributions. However, by testing different assumptions about the distributions it is possible to say something about it. We have seen that the best agreement between the theoretical dispersion relations and the observed wave spectra is obtained when the growth rate is moderate. This can be achieved with a cold ion distribution with k B T = 0.04 eV. This is the warmest cold distribution that we tried, but it is still a very low temperature 290 compared to all the other charged particle populations. The same result may be obtained with a lower temperature, if there also are suprathermal particles present. To accurately measure distribution functions at such low energies would represent a challenge in space-based instrumentation. These cold ion temperatures are reasonable, considering that Biver et al. (2019) found neutral temperatures up to approximately 0.02 eV between the nucleus and 15 km cometocentric distance. The ion distribution is formed by ionisation of the neutrals, and initially the neutral and ion temperatures are the same. On their way out 295 to the spacecraft position, the ions may undergo some heating either through an increased the bulk temperature or by forming suprathermal tails. To summarise the result of computing dispersion relations for different distributions, it is distribution 7 in Table 1 that shows the best agreement with observations. It has the warmest cold ion distribution (0.04 eV), the current carried by the cold electrons, and no warm ion component as that was found to be negligible.
By computing the Doppler shift and comparing observed spectra with wave theory and known properties of current-driven 300 ion acoustic waves we can estimate the angle between the bulk velocity of the cold ions and the propagation direction of the waves to be α 50 • for closest approach and α 70 • farther out when Rosetta was moving away and the wave power decreasing. Previous estimates have shown that ions move away from the centre of the comet, predominantly in a radial direction (Odelstad et al., 2018) as would be expected if they are accelerated by the ambipolar field present in the inner coma . There are also observations of ions with an anti-sunward velocity component (Berčič et al., 2018), but 305 those ions were faster than the (3-3.7) km s −1 we have observed here. If we assume that the ions move radially outward, the estimate of α 50 • for the waves near closest approach will also apply to the angle between the direction of propagation and the radial direction. Waves should propagate in the direction of the relative velocity between the electrons and the ions, and our angle estimates must not be seen as general results. They apply only at the position of the spacecraft during the flyby and for the orientation of the current at the time. During the outbound pass of the spacecraft, the angle of propagation cannot be 310 restricted more than to say that it is below 70 • . Here, Rosetta was likely outside the source region, and the waves propagated to the spacecraft from a source located closer to the nucleus.
We only have information about the waves along a single spacecraft trajectory, and what we know about the current comes from crude estimates based on single spacecraft magnetic field observations. To obtain a more complete picture of currents and waves in the inner coma would require the comet to be accompanied by multiple spacecraft collecting data at the same time 315 (Götz et al., 2019).
Code and data availability. The Rosetta data sets are available at the ESA Planetary Science Archive <https://archives.esac.esa.int/psa>.
The specific data set used in this article is available at <https://doi.org/10.5281/zenodo.3973232> together with computer codes to produce the figures (Gunell et al., 2020). Note: the doi has been reserved and the data set will be published there upon acceptance of this article.
Appendix A: The influence of suprathermal particles In both cases the cold ion temperature was 0.02 eV, and for comparison the dispersion relation for distribution 6, which has 0.01 eV cold ions, is also shown in Fig. A1. The results are very similar, and we conclude that the influence on the dispersion 325 relation from the suprathermal tails is similar to the difference between distributions with 0.01 and 0.02 eV ions. Since we cannot directly measure the distribution function at these low energies we cannot tell the two effects apart. The distributions with higher m indeces shown in Fig. A1 and those in Fig. 6 both agree with observations within the limits of experimental uncertainty.
Author contributions. HG performed the analysis in collaboration with CG, who also was the one to identify the flyby as an item of interest for wave studies, and EO, who in particular contributed to the plasma characterisation based on Langmuir probe data. All authors contributed to the writing of the final manuscript.