We analytically discuss wave excitation in a homogeneous three component
plasma consisting of solar wind protons, electrons and a beam of cometary
water ions applied to the plasma environment of comet
67P/Churyumov-Gerasimenko. The resulting dispersion relations are studied in
a solar wind rest frame, where a cometary current is solely generated by the
water ion beam, and a cometary rest frame representing the rest frame of the
Rosetta spacecraft. A modified ion-Weibel instability is excited by the
cometary current and predominantly grows perpendicular to this current. The
corresponding water ion mode is connected to a frequency of about

The study of waves in plasma environments is an extensive field in plasma
physics. Special attention is paid to cometary magnetospheres, where plasma
waves and turbulence are one of the most remarkable observations at comets
like 1P/Halley, 21P/Giacobini-Zinner and 26P/Grigg-Skjellerup

Recently, investigations have been focused on comet 67P/Churyumov-Gerasimenko by
the Rosetta mission

In this paper, we derive and discuss the analytic basics of this new type of
low-frequency waves at the comet 67P/Churyumov-Gerasimenko. First, the
applied model is explained and the dispersion relations are deduced in
Sect.

For the dispersion analysis frames with resting comet and with resting solar
wind are used. Finally, three frames are of advantage: the cometary rest
frame CSEQ (comet-centred solar equatorial,

CSEQ is the starting point as observations are presented just in this frame.
As sketched in Fig.

Illustration of the cometary rest frame CSEQ (top), solar wind rest
frame SW (mid) and tilted solar wind rest frame TSW (bottom). Characteristic
velocities of the solar wind

A Galilean transformation along the

For the analytical treatment of the dispersion relation the introduction of a
tilted solar wind rest frame TSW, with the new

The equations of the dispersion relation are related just to the TWS. Later, the results are transformed back to the CSEQ for the discussion and interpretation.

List of plasma characteristics in the environment of
P67/Churyumov-Gerasimenko at heliocentric distances of about 3 AU

Now, the general dispersion tensor

In the previous section and Appendix

Wave propagation parallel to the ambient magnetic field

We start our discussion with wave propagation parallel to the ambient
magnetic field

Wave propagation perpendicular to the cometary current

Wave propagation parallel to the cometary current

Left: estimated growth rate

Next, we discuss the wave propagation perpendicular to both the ambient
magnetic field

Last, wave propagation parallel to the cometary current

Wave phase velocity diagram

So far we derived growth rates parallel and perpendicular to the cometary
current, but not parallel to the ambient magnetic field. Therefore, with the
lower limit for the wave number

Besides the explanation for Rosetta's measurements, our previous discussion
also yields insights to the stabilization of the cometary ion mode. This mode
is most unstable, i.e. each

Following the linear perturbation theory applied in this work
(Appendix

Complementary to our discussion so far real frequencies can also be assumed,
so the solutions of the dispersion relations result in complex wave numbers.
In particular, we find an extremal imaginary wave number with

Illustration of the phase structure in SW. The wave

In the previous subsection we considered the phase velocity

Wave

[t]

Another point of interest is the spatial phase structure of the predominant
waves, which we first discuss in SW and then in CSEQ. As already discussed
perturbations are mainly excited in the plane perpendicular to the current.
As an ansatz for these perturbations we choose a wave

Zoom in Fig.

Variation of the frequency range of

Variation of the frequency range of

Variation of the frequency range of

The isophase structures of Fig.

It should be noted that due to the Galilean transformation to CSEQ the phase velocity also depends on
the inclination

Last, we discuss the effect of changes in the background parameters on the
frequency range of the instability (red) regarding the ambient magnetic field
(

The water ion density is varied from

Last, the variation of the cometary ion velocity from

In this paper we have derived a model for the new type of low-frequency waves
recently detected at 67P/Churyumov-Gerasimenko. Our model results well agree
with measurements of the Rosetta spacecraft in the environment of the comet

The raw data sets underlying the figures are available as supplementary
material to the paper. The data are obtained as numerical solutions of the
dispersion relations resulting from Eq. (

We choose as a general ansatz a cold MHD-plasma with multiple plasma species

The perturbations of the quantities are plane waves, e.g.

We start the further examination with waves that propagate
perpendicular to the magnetic field, so the dispersion relation
Eq. (

We have already seen that a modified O-mode Eq. (

Next, the resonance frequencies for propagation perpendicular to the magnetic
field and the cometary current

Last, the resonance frequencies for propagation parallel to the cometary
current

The last characteristic frequencies, which can be analytically derived, are
the cut-off frequencies

This study was financially supported by the German Bundesministerium für Wirtschaft und Energie and the Deutsches Zentrum für Luft- und Raumfahrt under grant 50QP 1401. The topical editor, C. Owen, thanks M. V. Volwerk and one anonymous referee for help in evaluating this paper.