Observations of waves can give us information of the plasma in which they are generated and through which they have propagated. Waves are also of general interest in plasma physics as they provide a means for energy transfer and because they affect the charged particle distributions through wave–particle interaction processes. When comets 21P/Giacobini–Zinner and 1P/Halley were visited by spacecraft in the 1980s, a variety of plasma waves were reported . Among these observations were ion acoustic waves, detected both in the bow shock region and upstream .

The Rosetta spacecraft accompanied comet 67P/Churyumov–Gerasimenko for 2 years from August 2014 to September 2016. Shortly after the spacecraft reached the comet, low-frequency (f≲100mHz), long-wavelength (100km≲λ≲700km) waves were detected in the magnetic field data . These were named “singing comet” waves; they have been interpreted in terms of a modified ion-Weibel instability , found to be compressional and detected as far as 800 km from the nucleus . Waves in the lower hybrid frequency range (f≲15Hz) were found by and . Lower hybrid waves were frequently seen in bursts in connection with density gradients, oscillating on minute timescales . These minute timescale oscillations are known as steepened waves and were observed outside the diamagnetic cavity . Electric field measurements showed waves in the lower hybrid frequency range on both sides of the diamagnetic cavity boundary, indicating a mode conversion between lower hybrid waves and ion acoustic waves .

Ion acoustic waves are compressional plasma waves that are weakly damped only when Te≫Ti, where Te and Ti are the electron and ion temperatures respectively, and the frequency is below the ion plasma frequency. In this limit, the angular frequency ω is proportional to the wave number k and the phase speed is cs=kBTe/mi, where kB is Boltzmann's constant and mi is the ion mass see, for example,. As the frequency approaches the ion plasma frequency, they become increasingly heavily damped, and also the phase speed decreases. If the ion and electron temperatures are similar, this also leads to heavy damping, and ion acoustic waves are usually not detectable in that regime. Charged particle distributions become unstable when one population drifts at a large enough speed relative to another population, and this results in the growth of waves. In particular, this applies to the current-driven ion acoustic instability, where the electron and ion populations are in relative motion. The current-driven ion acoustic instability has been studied in laboratory experiments , and mapped out the unstable parameter regimes theoretically. At comet 67P/Churyumov–Gerasimenko, ion acoustic waves were observed by the Rosetta spacecraft on 20 January 2015 at approximately 2.5 au from the Sun, before the diamagnetic cavity had formed seefor a discussion of diamagnetic cavity formation, and also in the diamagnetic cavity near perihelion . The ion acoustic waves seen in the cavity were interpreted as a result of part of the current at the diamagnetic cavity boundary closing through bulges on that boundary and generating waves through a current-driven instability . Ion acoustic waves can be identified by measuring either the variations in plasma density or in the electric field. At comet 67P/Churyumov–Gerasimenko, detected the waves in electric field oscillations and in density variations. In this work, the detection relies on density variations.

In this article, we examine ion acoustic waves detected by Rosetta during its close flyby of comet 67P on 28 March 2015. The comet was at a heliocentric distance of 2.0 au at the time, and the gas production rate varied between 3×1026 and 9×1026s-1 during the day. Magnetic pileup and draping have been studied before for this flyby, both in observational studies and using hybrid simulations . The magnetic field piled up near the nucleus, causing the solar wind protons to be deflected out of the ecliptic plane. This in turn caused the draped magnetic field in the region near the nucleus to align itself with the deflected solar wind flow. No sign of a diamagnetic cavity was seen during the flyby, and that was likely due to it not having formed yet . The hybrid simulations of the flyby presented by show the presence of an infant bow shock approximately 100 km from the nucleus, but that is farther out than the spacecraft reached on that day. Thus, the spacecraft was situated in the inner coma, where the plasma was of cometary origin. The plasma parameters, although not exactly the same, were in a range similar to that of the diamagnetic cavity observations near perihelion. The magnetic field environment was dominated by magnetic pileup and draping during the flyby, while the diamagnetic cavity has its own peculiar magnetic field environment, with a sharp discontinuity at the boundary.

We use the comet-centred solar equatorial coordinate system (CSEQ) throughout this article. In this system, the x axis points from the comet to the Sun, the z axis is the component of the rotation axis of the Sun that is perpendicular to the x axis, and the y axis is directed to complete the right-handed coordinate system original definition in the SPICE kernel;. The spacecraft moved from negative to positive y and z values at a nearly constant x=11km. The spacecraft trajectory is illustrated in Fig. .

The closest approach occurred at 13:05 UTC, and then Rosetta was at a cometocentric distance of 15 km. The spacecraft 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 Figs.  and h.

Because of the uncertainty at which both electron and ion distribution functions are known, we have calculated dispersion relations under several different assumptions about these distributions. We then compare the results of the calculations with the wave observations in order both to arrive at an explanation for how the waves are generated and to put constraints on what we can say about the electron and ion 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  for nine test cases used to model the distribution near closest approach and two cases for the outbound trajectory.

We use the simple pole expansion method to compute the dispersion relations . In a comet context it was reviewed by also providing the computer code for the computations. Each component of the distribution function is modelled by an approximate Maxwellian, 5Mm(v)=1+v-vD22vt2+…+1m!v-vD22vt2m-1, where v is velocity, vt is the thermal speed, vD is the drift speed, and m is the number of terms included in the expansion. The procedure used to calculate dispersion relations is briefly described in Appendix . The expression inside the brackets of Eq. () is the reciprocal of a Taylor expansion of exp⁡v-vD2/2vt2. As m tends to infinity, Mm(v) approaches a Maxwellian, and for small values of m the distributions have suprathermal tails. In the distributions in Table , m=3 for the ions and m=5 for the electrons. The influence of suprathermal tails is evaluated in Appendix .

Figure  shows dispersion relations for the nine test cases that correspond to the observations near closest approach. We are assuming a real wave number k and a complex angular frequency ω. Panel (a) shows the real part of ω, and panel (b) shows the damping rate γ. Negative values of γ correspond to wave growth. Panel (c) shows the shaded rectangle in panel (b) in more detail. Several of the curves are so similar they fall on top of each other and are difficult to distinguish in the figure. In all cases the least damped or fastest growing mode is the ion acoustic mode and that is the one shown.

The density of the warm ion population is varied in distributions 1–3. In distribution 1 the warm ion density is 4 cm-3 as estimated in Sect. . In distribution 2 the warm ion density is assumed to be zero, and in distribution 3 the warm ion density is 10 times higher than the estimate in Sect. . The real part of the dispersion relation is indistinguishable among the three cases, as seen in Fig. a. The damping rates in Fig. c are very close in the three cases, although it can be seen 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 density is set to zero in the rest of the distribution functions.

We have used the current density estimate, J=4.9µAm-2, obtained in Sect.  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, |vD|=|J/(en)|=39.3kms-1, to one of the electron populations. Here, n denotes the density of the electron population in question. 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 distribution 2. We conclude that to drive the ion acoustic waves unstable, the current cannot be carried by the warm electrons alone. In distribution 9 the current is carried by both electron populations, which each are given a drift velocity of |vD|=19.6kms-1. This yields a lower growth rate than for distribution 2, but the mode is still unstable for a range of k values.

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 2 eV warm electrons but otherwise is equal to distribution 5. In distribution 8 the electron temperature has been increased to 4 eV with all other parameters the same as in distributions 2 and 5. The growth rate for distribution 8 is higher than that for the other two distributions. However, in all three cases the waves are unstable over a similar wavelength range.

Distributions 6 and 7 have 0.01 and 0.04 eV cold ions, respectively; that is to say, in distribution 6 the ions are colder and 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. c). Also the real part of ω is affected, as shown in Fig. a, but this is significant only for k values larger than the k which corresponds to maximum growth. The influence of suprathermal ions on the dispersion relations and growth rates is evaluated in Appendix , and it is found to be similar to the difference between distribution with 0.01 and 0.02 eV ions. The distribution function of the cold ions 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 nine 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 , are shown in Fig. . 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 line in Fig.  was recorded at 17:56:54. In distribution A the cold electrons have been given a drift velocity corresponding to the current density J=1.9µAm-2 measured when the spacecraft was moving away, and the dispersion relation corresponding to distribution A is very similar to that 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.  we see no large-scale change near 18:00. That the field remains constant, except for small fluctuations, while the spacecraft moves, means that there is no large-scale magnetic field gradient in this region and hence no large-scale current either. Thus, the 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 waves.

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 fm in the moving medium is Doppler-shifted to frequency fsc in the spacecraft frame according to 6fsc=vphfm+ucos⁡αvphfmfm, where vph is the phase speed given by the dispersion relation, u is the speed of the moving medium, and α is the angle between the wave direction of propagation and the velocity u.

Figure a shows the H2O+ ion plasma frequency and the frequency of maximum growth, Doppler-shifted to the spacecraft frame according to Eq. () for the dispersion relations that correspond to distributions 2, 7, and 9 and with u determined by Eq. (). 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 an interpretation in which observations and theory are consistent. This is explained in what follows.

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 . For our near closest approach sample spectrum shown by the black curve in Fig.  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 the dispersion relation corresponding to distributions 2 or 9 (solid red curve in Fig. a), this means that α≲56∘ and in the case of distribution 7 (solid black curve in Fig. a) the angle is restricted to α≲49∘. 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. In 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. a is above 2 kHz for 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. . The dashed blue curve corresponding to the frequency of maximum growth for distribution 9, in which cold and warm electrons both carry current in equal measure, falls between the other two. 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.

For the outbound part of the spacecraft trajectory, Fig. b 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 overlaps in Fig. , 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.  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 at which the wave power peaks tells us more about the source region than about the conditions at the spacecraft position. The blue curve in Fig.  has fallen to the noise floor at approximately 3 kHz, and from the solid curve in Fig. b the dispersion relation is seen to be in agreement with data for angles in the range α≲70∘.

We have analysed data obtained during Rosetta's close flyby of comet 67P on 28 March 2015. The multiple instruments used were RPC-LAP, RPC-ICA, RPC-MAG, and RPC-MIP, all part of the Rosetta Plasma Consortium. Waves which we interpret as current-driven ion acoustic waves were recorded by the Langmuir probe instrument RPC-LAP as probe current variations. 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. could observe field line draping until the rapid magnetic field change that occurred at 20:42 when Rosetta was at 34 km cometocentric distance.

Data from RPC-LAP indicate the presence of two electron populations, one cold at temperature around 0.2 eV and one warm at ∼2 eV. Furthermore, the RPC-LAP characteristics show the presence of a cold ion population drifting with a speed between 3 and 3.7kms-1. Theoretical estimates of acceleration by a radial ambipolar electric field lead to ion bulk speeds in this range in agreement with observations . The cold component of the ion distribution went undetected by the ion spectrometer RPC-ICA. Instead, a warm, several electronvolts in temperature ion distribution was detected by RPC-ICA, but its density is not sufficient for it to have any significant influence on the waves.

We are not able to measure the fine details of the electron and ion 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 kBT=0.04eV. This is the warmest cold distribution that we tried, but it is still a very low temperature compared to all the other charged particle populations. The same result may be obtained with a lower temperature, if there also are suprathermal ions 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 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 to the spacecraft position, the ions may undergo some heating, either through an increased bulk temperature or by forming suprathermal tails. The rapidly decreased growth with increasing cold ion temperature from 0.01 via 0.02 to 0.04 eV also shows that the cold ion temperature is of this order (a few hundredths of electronvolt). For a warmer cold ion population, the distribution would be stable and no waves generated, and as seen in Sect. , if the cold ion population is 0.01 eV or colder, the higher growth rate leads to a worse agreement with observations. The growth rate is also influenced by what fraction of the current is carried by the cold electrons. This, in turn, depends on the relative speed of the two electron populations and how the electron density is distributed between them. To summarise the result of computing dispersion relations for the different distributions we have tried, it is distribution 7 in Table  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 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 , 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 , but those ions were faster than the (3–3.7) kms-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 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. The use of wave observations in combination with wave theory complements the other measurements, particularly the cold ion population that is inaccessible to the particle instruments. It also lets us confirm the interpretation of probe data concerning the electron populations and the interpretation of the observed variation of the magnetic field as a spatial gradient. Yet 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. Obtaining 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 .