# Persistent plasma waves in interstellar space detected by Voyager 1

Stella Koch Ocker111corresponding author<EMAIL_ADDRESS>Department of
Astronomy and Cornell Center for Astrophysics and Planetary Science, Cornell
University, Ithaca, NY 14850, USA  James M. Cordes  Department of Astronomy
and Cornell Center for Astrophysics and Planetary Science, Cornell University,
Ithaca, NY 14850, USA  Shami Chatterjee  Department of Astronomy and Cornell
Center for Astrophysics and Planetary Science, Cornell University, Ithaca, NY
14850, USA  Donald A. Gurnett  Department of Physics and Astronomy, University
of Iowa, Iowa City, IA 52242, USA  William S. Kurth  Department of Physics and
Astronomy, University of Iowa, Iowa City, IA 52242, USA  Steven R. Spangler
Department of Physics and Astronomy, University of Iowa, Iowa City, IA 52242,
USA

In 2012, Voyager 1 became the first in situ probe of the very local
interstellar medium 1. The Voyager 1 Plasma Wave System has given point
estimates of the plasma density spanning about 30 au of interstellar space,
revealing a large-scale density gradient 2, 3 and turbulence 4 outside the
heliopause. Previous studies of the plasma density relied on the detection of
discrete plasma oscillation events triggered ahead of shocks propagating
outwards from the Sun and used to infer the plasma frequency and hence density
5, 6. We present the detection of a class of very weak, narrowband plasma wave
emission in the Voyager 1 data that persists from 2017 onwards and enables the
first steadily sampled measurement of the interstellar plasma density over
about 10 au with an average sampling distance of 0.03 au. We find au-scale
density fluctuations that trace interstellar turbulence between episodes of
previously detected plasma oscillations. Possible mechanisms for the
narrowband emission include thermally excited plasma oscillations and quasi-
thermal noise, and could be clarified by new findings from Voyager or a future
interstellar mission. The emission’s persistence suggests that Voyager 1 may
be able to continue tracking the interstellar plasma density in the absence of
shock-generated plasma oscillation events.

##  1 Main

Since crossing the heliopause, Voyager 1 (V1) has measured the plasma density
of the very local interstellar medium (VLISM) by detecting plasma oscillation
events (POEs). The Plasma Wave System (PWS) measures these POEs as narrow
($\approx 0.2$ to $0.4$ kHz wide) features in frequency-time spectra obtained
with the wideband receiver and the plasma wave spectrometer. Fig. 1 is a
composite dynamic spectrum showing all of the high-time resolution data
available from the PWS wideband receiver since V1 crossed the heliopause until
early 2020. The wideband receiver measures the voltage difference at the
antenna terminals at a rate of 28,800 samples s-1, which is converted into a
frequency-time dynamic spectrum by Fourier transforming the voltage time
series. The composite spectrum in Fig. 1 shows salient features between 1.9
and 3.5 kHz and has time and frequency resolutions of 3 days and 0.011 kHz,
respectively.

Eight distinct POEs are visible in the composite dynamic spectrum beginning in
late 2012. While the duration of these events ranges from a couple of days to
a year, they are all believed to be generated via beam-plasma instabilities in
the electron foreshocks of shocks propagating out of the heliosphere 6. The
POEs share key characteristics, including: 1) a spreading in frequency
associated with the excitation of higher wave modes via, e.g., Langmuir
parametric decay 7, 8; 2) an upward frequency drift attributed to the increase
in plasma density as the shock propagates over the spacecraft, which in two
cases has been directly associated with a jump in magnetic field strength
characteristic of a quasi-perpendicular shock 9, 5, 10; and 3) fast ($\sim$
sub-second level) intensity variations accompanied by smoother, broadband
radio emission generated in a nonlinear mode conversion process. Most of these
events are preceded by relativistic cosmic ray bursts detected by the V1
cosmic ray instruments, which indirectly provide energy estimates of $\sim 50$
eV for the electron beams generating the plasma oscillations 6.

In Fig. 2 we show the dynamic spectrum of the PWS wideband data from 2015
through early 2020. Fig. 2a shows the same data as in Fig. 1 with additional
mitigation of telemetry errors. Fig. 2b shows the same dynamic spectrum but
with a different color stretch that shows evidence of a new narrowband signal
spanning the entire period of data available between 2017 and 2020. Fig. 2c
shows the result of convolving a 1D boxcar filter in time with the dynamic
spectrum. This figure also shows complicated POE activity in 2015 and 2016.
The 2015 POE shows possible evidence of multiple shocks, and there is a sharp
drop in the plasma frequency between March and June 2016 when V1 passes into a
low-density region. The 2014 POE persists through December 2014, but it is
unclear whether the spikes of intensity from early-mid 2015 in the boxcar-
filtered spectrum trace the narrowband emission observed later on.

The narrowband signal shown in Fig. 2 constitutes a new class of plasma wave
emission in the Voyager spectrum. Unlike previously detected plasma
oscillations, its narrower width remains roughly constant at about $0.04$ kHz,
and it persists for almost three years (barring data dropouts), corresponding
to a distance traveled by the spacecraft of about 10 au. The signal-to-noise
ratio (SNR) of the narrowband emission is no more than 2 in a single 48 s
epoch and requires some degree of averaging to be measured (see Methods). By
contrast, previously detected POEs are detectable within individual epochs at
the sub-millisecond level with SNRs $>100$, and can vary by more than an order
of magnitude in intensity between epochs spaced a few days apart. There is
also no evidence of the narrowband signal spreading in frequency or
monotonically drifting upwards in frequency. Given its narrow bandwidth, low
amplitude, and years-long persistence, the narrowband plasma wave emission
appears to be distinct from the shock-generated POEs.

The peak frequency of the narrowband emission vs. time is shown in Fig. 3. The
peak frequency was measured by running a modified friends-of-friends (FoF)
algorithm 11 on the spectrum in Fig. 2b. A Gaussian process model was fit to
the frequency time series using Bayesian regression 12 to obtain a continuous
model of the peak frequency vs. time, and is shown in blue in Fig. 3. The
dynamic spectrum was then “de-fluctuated” by aligning the peak frequencies of
the narrowband emission to a reference frequency of 3 kHz (see Fig. 4). The
de-fluctuated spectrum averaged over all epochs in which the narrowband
emission was detected is also shown in Fig. 4. Over a timespan of 70 epochs
(or 210 days), the average amplitude of the narrowband emission is 0.06 (in
units of log10(power) relative to the noise baseline) and the root-mean-square
(rms) of the noise is 0.005. The de-fluctuated spectrum averaged over the
entire duration of the signal shows no evidence of plasma line harmonics above
an rms noise threshold of 0.002. There is no evidence of the narrowband
emission above an rms noise amplitude of 0.005 when the data between the POEs
of 2013, 2014, and 2015 are also averaged over 210-day timespans. Given that
the narrowband emission overlaps with the end of the 2016 POE (see Fig. 2), it
is possible that the narrowband emission is also present in 2015 and 2016.
However, the low SNR of the narrowband emission makes it difficult to
distinguish from the higher-intensity POEs occurring in that period.

Knowledge of the emission mechanism is required to associate the frequency of
the narrowband signal with the plasma frequency ($f_{p}$) and infer the
electron density ($n_{e}$). The behavior of the narrowband signal before and
after the POE of September 2017 is highly suggestive that its frequency traces
the plasma density. The leading edge of the 2017 POE occurs at the same
frequency (3.06 kHz) as the narrowband emission prior to the POE’s onset. In
mid-September 2017 the plasma frequency increases to 3.15 kHz as the
associated shock passes over the spacecraft. During the same period, the
frequency of the narrowband emission also increases, before drifting downwards
over several months after the POE. The frequency of the narrowband emission is
also similar to the plasma frequencies measured from POEs in 2015 and 2016,
which are consistent with $n_{e}\sim 0.11$ cm-3. This electron density is
consistent with the mean density of the Local Interstellar Cloud estimated
from carbon absorption lines in spectra of nearby stars 13.

The weak, narrowband plasma wave emission could be generated through thermal
or suprathermal processes. Thermal density fluctuations with wavelengths
longer than the Debye length ($\sim 20$ m in the VLISM) can induce
electrostatic oscillations at the plasma frequency, and the intensity of the
fluctuations can also be enhanced by suprathermal electrons 14, 15. Thermal
plasma oscillations are routinely observed with incoherent radar scattering in
Earth’s ionosphere and have intensities enhanced by energetic photoelectrons
16, 17, 18. In the VLISM, suprathermal electrons might be contributed from
inside the heliosphere or near the heliopause. It’s possible that an
anisotropy of low-energy cosmic rays from, e.g., a nearby star could induce
plasma oscillations. Any explanation of this nature must account for the
emission’s apparent onset about 15 au away from the heliopause and its years-
long duration.

An alternative explanation for the narrowband signal is quasi-thermal noise
(QTN), which has been routinely observed by plasma wave instruments on
Cassini, Galileo, and other Solar System missions 19. QTN is generated by the
quasi-thermal motions of electrons in a plasma, resulting in a local electric
field. The QTN voltage spectrum peaks at the local $f_{p}$, and its overall
shape depends on the antenna properties, with thin antenna longer than the
Debye length required to mitigate shot noise 19. The Voyager antenna, with an
effective length of 10 meters and width of 12.5 mm, might be able to detect a
marginal peak at $f_{p}$ in the QTN spectrum, but QTN generally predicts a
much broader frequency bandwidth than we observe. A much longer antenna on a
future interstellar probe would be capable of robustly detecting QTN.

The peak frequency of the narrowband emission likely tracks the plasma
frequency, so we use the frequency time series in Fig. 3 to estimate the
spectral coefficient of the electron density power spectrum. The power
spectrum of electron density fluctuations in the ISM follows a roughly power-
law dependence of the form:

$P_{\rm\delta n_{e}}(\mathbf{q})=C_{\rm n}^{2}q^{-\beta},\hskip
7.22743ptq_{\rm o}\leq q\leq q_{\rm i}$ (1)

where $C_{\rm n}^{2}$ is the spectral coefficient, and $q_{\rm o}$ and $q_{\rm
i}$ are the outer and inner scales of turbulence 20. Scattering measurements
from compact radio sources like pulsars have constrained the inner scale of
turbulence in the ISM to be between about 100 and 1000 km 21, 22, 23. A recent
analysis of electron densities measured with V1 from 2012 through late 2016
showed broad consistency with a Kolmogorov power law ($\beta=11/3$), but found
a higher spectral coefficient at the smallest spatial scales $\sim$10s of
meters 4, 24. The magnetic field fluctuations observed by Voyager are
consistent with turbulence in the VLISM being driven at an outer scale $\sim
0.01$ pc 25, while the inner scale may be related to the ion or electron
gyroradii or the Debye length 4.

The densities inferred from the narrowband emission reveal variations on
scales $\sim 1$ au combined with discrete events on $\sim 0.5$ au scales
produced by shocks originating in the heliosphere. The observed variance of
the density fluctuations is directly related to the density power spectrum
(Eq. 1) integrated from 10 au, the largest scale probed in this study (see
Methods). We find $C_{\rm n}^{2}=10^{-1.64\pm 0.02}$ m-20/3, slightly smaller
than the value $C_{\rm n}^{2}=10^{-1.47\pm 0.04}$ m-20/3 found in a previous
analysis of POEs between 2012 and 2019 24. The ratio of the rms density
fluctuations to the mean density is ${\rm
rms}(n_{e})/\bar{n}_{e}=0.0041/0.12=0.034$.

In the local ISM, within about 1 kpc probed by pulsar observations, the
spectral coefficient for $\beta=11/3$ is $C_{\rm n}^{2}\approx 10^{-3.5}$
m-20/3. 26, 27 The electron density of the local ISM has a scale height of
about 1.6 kpc and mean density at mid-plane of 0.015 cm-3 28. Integrating the
Kolmogorov wavenumber spectrum with $C_{\rm n}^{2}=10^{-3.5}$ m-20/3 from a
cutoff of 10 au yields ${\rm rms}(n_{e})=4.8\times 10^{-4}$ cm-3, which
combined with $\bar{n}_{e}=0.015$ cm-3 gives ${\rm
rms}(n_{e})/\bar{n}_{e}=0.032$ for the local ISM, very close to our observed
value for the VLISM. It has been suggested 29, 30 that the VLISM turbulence
spectrum is a superposition of higher and lower amplitude Kolmogorov spectra
of heliospheric and interstellar origin, respectively, which may explain why
the spectral coefficient we infer is much larger than expected for the local
ISM.

The extremely weak, narrowband plasma wave emission reported here persists
over about 10 au of interstellar space traversed by V1. The emission appears
to be distinct from shock-generated plasma oscillations that were previously
used to measure the local density of plasma outside the heliopause, and it may
be generated by thermally or suprathermally excited plasma oscillations. The
persistence of the emission through the most recent data published from V1
suggests that it may continue to be detectable. Continued tracking of the
emission’s frequency, bandwidth, and intensity will likely provide improved
constraints on the emission’s origin, and make it possible for V1 to quasi-
continuously track the electron density distribution of the VLISM along its
trajectory. This future work will improve our understanding of turbulence and
large-scale structure in the VLISM.

Figure 1: Voyager 1 composite plasma wave spectrum. Top: Frequency-time
dynamic spectrum showing all of the Voyager 1 PWS wideband data available
since Voyager 1 crossed the heliopause on August 25, 2012. The time resolution
of the spectrum is 3 days and the frequency resolution is 0.011 kHz. Each
column of pixels corresponds to a 1D spectrum that is the average of all
$\approx 48$ s long observations that fall within that 3-day time bin. The
individual spectra that fall within a given time bin have been equilibrated to
the same noise baseline. The 2.4 kHz supply interference line is masked out,
and the spectrum is smoothed with a 1D Gaussian smoothing kernel with
$\sigma=0.01$ kHz. Bottom: Schematic showing relevant features in the PWS
spectrum, including the locations of previously detected plasma oscillation
events (POEs). Two events have direct associations with shocks detected by the
magnetometer. A magnetic pressure wave was also detected in early 2017. The
lower cutoff frequency of the plasma oscillations corresponds to the local
plasma frequency. The approximate times of relativistic electron bursts
detected by the cosmic ray instruments are also indicated. The model of the
weak, narrowband plasma wave emission presented in this paper is shown in
solid gray, and the plasma frequency inferred from POE activity between 2015
and 2017 is shown in dashed gray. A POE detected in June 2019 is also shown as
a black circle, but was masked in our analysis because it coincided with a
period of severely degraded telemetry performance. Figure 2: Weak narrowband
plasma waves in the Voyager 1 dynamic spectrum. a) Dynamic spectrum showing
the Voyager 1 PWS wideband data from December 2014 through January 2020. The
spectrum was creating using the same methods as Fig. 1, but with a slightly
coarser frequency resolution (about 0.02 kHz) to avoid spectral ringing, and
without Gaussian smoothing. Epochs contaminated by telemetry errors have been
masked out; masked data and data dropouts appear as vertical stripes with zero
logarithmic power. b) Same as top panel, but with a different color stretch
that extends to 0.16 in logarithmic power. c) Same as top panel but smoothed
in time using a 1D boxcar filter and a color stretch extending to a
logarithmic power of 0.055. Figure 3: Peak frequency vs. time of the
narrowband plasma wave emission. The frequencies were extracted by running a
friends-of-friends algorithm on the spectrum in Fig. 2b. Error bars represent
the rms of the noise fluctuations in each column of Fig. 2b. The blue curve
and shaded region show the best-fit Gaussian process model and $95\%$
confidence intervals, respectively. The Gaussian process model is insensitive
to large data gaps and treats the density as roughly constant during these
gaps. Figure 4: De-fluctuated dynamic spectrum of the narrowband plasma wave
emission. Epochs shown are between January 2017 and February 2020 when the
narrowband plasma wave emission was detected. The peak frequency of the
emission was aligned to a reference frequency of 3 kHz. The 4.8 kHz harmonic
of the power supply interference line was masked (white) prior to shifting the
spectrum. The right-hand panel shows the de-fluctuated spectrum averaged in
time with the power-law noise continuum included (dark blue) and with the
continuum removed using a second-order polynomial fit to the logarithmic power
vs. frequency (light blue). There is no evidence of a plasma line harmonic at
6 kHz.

##  2 Methods

###  2.1 Constructing the wideband spectra

The V1 PWS wideband receiver measures the voltage difference across the
spacecraft antenna terminals at a rate of 28,800 samples s-1 for about 48 s to
construct a voltage time series that is stored for later transmission to
ground. Voltage time series are measured about once a week, although over the
course of the mission this time gap varies from about a day to a couple of
weeks. The voltage time series are converted into a frequency-time spectrum by
the Fourier transform method, which allows tuning of the output spectrum’s
frequency resolution. The resulting spectrum shows the relative electric field
intensity for frequencies up to 10 kHz, the frequency roll-off imposed by the
low-pass filter on the receiver. To construct the composite dynamic spectra
shown in Fig. 1 and Fig. 2, the spectra of the voltage time series were sorted
into 3-day long bins and then averaged within each bin. The subsequent time
and frequency resolution of the spectrum in Fig. 1 is 3 days and 0.011 kHz,
respectively. A lower frequency resolution of 0.02 kHz was used to construct
Fig. 2 in order to avoid contamination of the narrowband signal by spectral
ringing.

Due to automatic gain control (AGC), the noise baseline of the voltage time
series measured by the wideband receiver varies between observations. To
correct for this effect, we equilibrated the noise threshold of the spectra
obtained from the voltage time series by subtracting the noise continuum from
each spectrum. The 2.4 kHz supply interference line and its harmonics were
masked prior to noise equilibration. Our spectra therefore do not represent
the absolute electric field intensity and should only be interpreted as the
intensity relative to the noise baseline.

There are two main sources of data quality degradation in the V1 wideband
spectra. The first is spectral interference that persists throughout the
entire timespan since V1 entered the VLISM (and prior): this includes the 2.4
kHz supply interference line and its harmonics, and high-intensity, low-
frequency noise that dominates the spectra up to about 1 kHz. We avoided this
spectral interference by masking each spectrum at the corresponding
frequencies. The second source is degraded telemetry performance that produces
bit errors causing short, temporal ($\lesssim 1$ s), wideband bursts that vary
in intensity and can take up to all of the frequency band. For most of the
interstellar mission, these wideband bursts usually occur just a couple of
times per observation, are about 5 kHz wide, and are only up to an order of
magnitude more intense than the baseline noise. However, at Voyager’s great
distance from the Sun, the SNR of the telemetry signal arriving at the NASA
Deep Space Network (DSN) is increasingly close to the level at which it cannot
be decoded. As a result, since 2018 there has been a dramatic increase in the
number, frequency bandwidth, and intensity of these bursts, which has been
temporarily mitigated by the addition of a fourth antenna to the DSN array.
The subsequent decrease in overall data quality had a minimal effect on the
8-yr long composite spectrum shown in Fig. 1 due to its long timespan, and we
further reduced the minor effects of these bursts by applying a 1D Gaussian
smoothing kernel with $\sigma=0.01$ kHz to the entire spectrum in Fig. 1.

In order to accurately detect the narrowband signal shown in Fig. 2, we masked
the wideband bursts in each 48 s long spectrum before binning and averaging
all of the spectra to create the composite dynamic spectrum in Fig. 2. We also
ignored altogether individual spectra that are completely contaminated by
interference bursts, which corresponds to spectra containing 10 or more
bursts, or equivalently, a noise threshold that is more than 5 times the
equilibrated noise threshold of the composite spectrum. Based on this noise
threshold, zero observations were excised in 2017, 12% of observations were
excised in 2018, and 33% were excised in 2019. The average amount of time
between consecutive, 48 s long observations is 2.9 days. The spectrum must be
averaged over at least one 48 s long observation in order detect the
narrowband emission above a SNR of 2; hence 2.9 days is the shortest timescale
on which the narrowband emission can be deemed continuous.

###  2.2 Plasma density detection and analysis

A modified friends-of-friends (FoF) algorithm 11 extracted the plasma
frequency (and hence density) time series shown in Fig. 3 from the unsmoothed
spectrum of the narrowband signal (see Fig. 2b). The modified FoF algorithm
searched for continuous lines in the spectrum by finding all adjacent local
maxima within a 0.2 kHz-wide band around a given starting frequency. The line
candidates returned by FoF were then sorted by average intensity, and the line
with the highest average intensity corresponded to a line in frequency vs.
time space that passed through all local maxima containing the new narrowband
signal. Outlier maxima picked up by FoF were then excised by calculating a
running mean of the line and removing all points more than 0.04 kHz away from
the mean. The “de-fluctuated” spectrum was calculated by shifting the spectrum
of the narrowband signal so that the peak frequency of the signal extracted by
FoF was aligned to a reference frequency of 3 kHz.

A Gaussian process model was fit to the plasma density time series using the
GaussianProcessRegressor module provided by scikit-learn 12, with the prior
set to a white noise kernel added to a Matern kernel. The Matern kernel takes
the form:

$k(x_{i},x_{j})=\frac{1}{\Gamma(\nu)2^{\nu-1}}\Bigg{(}\frac{\sqrt{2\nu}}{l}d(x_{i},x_{j})\Bigg{)}^{\nu}K_{\nu}\Bigg{(}\frac{\sqrt{2\nu}}{l}d(x_{i},x_{j})\Bigg{)}$
(2)

where $K_{\nu}$ is a modified Bessel function, $\nu$ is set to 1 and $l$ is
set to 10. The time series of mean, posterior values then gave a continuous
temporal model for the frequency of the narrowband emission. In Fig. 3, we
show the Gaussian process model and the 95% confidence intervals of the
posterior. This Gaussian process model is fairly insensitive to the large data
gaps in late 2019 and 2020, resulting in a smooth progression between the
density values bordering these gaps. The model is also shown next to
previously detected POEs in the bottom panel of Fig. 1, but with the largest
data gaps in the density time series indicated.

The variance of electron density fluctuations is directly related to the
density fluctuation power spectrum (Eq. 1) integrated over wavenumber
$q=2\pi/L$, where the length scale $L$ is related to the spacecraft speed (3.6
AU/yr). For an outer scale $q_{\rm o}$ much less than the inner scale $q_{\rm
i}$, the integrated 3D wavenumber spectrum has the form

${\rm Var}(n_{e})=\frac{2(2\pi)^{4-\beta}C_{\rm n}^{2}l_{\rm
o}^{\beta-3}}{\beta-3}\approx 3(2\pi)^{1/3}C_{\rm n}^{2}l_{\rm o}^{2/3},$ (3)

where $\beta>3$ and the second approximation is for $\beta=11/3$. The spectral
coefficient $C_{\rm n}^{2}$ for the VLISM was estimated using the observed
variance of the density time series and $\beta=11/3$. Since the largest length
scale of the density time series is 10 au, we substitute 10 au for $l_{\rm o}$
in Eq. 3 when integrating over the spectrum. The error in $C_{\rm n}^{2}$ was
estimated by simulating the observed density fluctuations as the sum of a
slowly varying signal and normally distributed white noise, with the total
variance of the simulated signal set equal to the observed density variance.
The Var($n_{e}$) was then calculated for 1000 realizations of the simulated
signal and used to calculate the fractional error on Var($n_{e}$) and $C_{\rm
n}^{2}$. The error estimate for $C_{\rm n}^{2}$ thus represents a lower limit
because it only accounts for the error in Var($n_{e}$) due to additive noise.
Eq. 3 was also used to calculate the density fluctuation variance assuming
$C_{\rm n}^{2}=10^{-3.5}$ for the local ISM.

###  2.3 Data Availability

The Voyager 1 data used in this work are archived through the NASA Planetary
Data System (https://doi.org/10.17189/1519903). Data and examples of the PWS
data processing algorithms are also available through the University of Iowa
Subnode of the PDS Planetary Plasma Interactions Node
(https://space.physics.uiowa.edu/voyager/data/).

##  3 Materials and Correspondence

Correspondence and requests for materials should be directed to S.K.O.:
<EMAIL_ADDRESS>
##  4 Acknowledgements

Authors S.K.O., J.M.C., S.C., and S.R.S. acknowledge support from the National
Aeronautics and Space Administration (NASA 80NSSC20K0784). S.K.O., J.M.C., and
S.C. also acknowledge support from the National Science Foundation (NSF
AAG-1815242) and are members of the NANOGrav Physics Frontiers Center, which
is supported by the NSF award PHY-1430284. The research at the University of
Iowa was supported by NASA through Contract 1622510 with the Jet Propulsion
Laboratory.

##  5 Author Contributions

S.K.O. conducted the data analysis and wrote the initial draft of the paper.
J.M.C., S.C., S.R.S., and S.K.O. are NASA Outer Heliosphere Guest
Investigators on the Voyager Interstellar Mission. D.A.G. is the Principal
Investigator of the Voyager PWS investigation and W.S.K. is a Co-Investigator
of Voyager PWS and is responsible for the initial processing of the data at
the University of Iowa. All authors contributed to discussion of the results
and comments on the draft.

##  6 Competing Interests

The authors declare no competing interests.

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