11institutetext: International Centre for Radio Astronomy Research, Curtin
University, Bentley, WA, 6102, Australia
11email<EMAIL_ADDRESS>22institutetext: DUG Technology, 76 Kings Park
Rd, West Perth, 6005, WA, Australia 33institutetext: CSIRO Space & Astronomy ,
PO Box 76, Epping, NSW, 1710, Australia

# Detection of intended and unintended emissions from Starlink satellites in
the SKA-Low frequency range, at the SKA-Low site, with an SKA-Low station
analog

D. Grigg 1122 S. J. Tingay 11 M. Sokolowski 11 R. B. Wayth 11 B. Indermuehle
33 and S. Prabu 11

(Received 2023-08-04; accepted 2023-09-27)

###### Abstract

Context. Intended and unintended radio emissions from satellites can interfere
with sensitive radio telescopes in the frequency ranges of key experiments in
astrophysics and cosmology. We detect strong intended and unintended
electromagnetic radiation from Starlink satellites at the site of the future
SKA-Low facility in Western Australia, using an SKA-low prototype station
known as the Engineering Development Array version 2 (EDA2).

Aims. We aim to show that Starlink satellites are easily detectable utilising
a configuration of low frequency radio antennas representative of an SKA-Low
“station” and that our results complement similar findings with the LOFAR
telescope.

Methods. Utilising the EDA2 at frequencies of 137.5 MHz and 159.4 MHz, we
detect trains of Starlink satellites on 2023-03-17/18 and 2021-11-16/17,
respectively, via the formation of all-sky images with a frequency resolution
of 0.926 MHz and a time resolution of 2 s. Time differencing techniques are
utilised to isolate and characterise the transmissions from Starlink and other
satellites.

Results. We observe Starlink satellites reaching intensities of $10^{6}$
Jy/beam, with the detected transmissions exhibiting a range of behaviours,
from periodic bursts to steady transmission. The results are notable because
they demonstrate that Starlink satellites are detected in the SKA-Low
frequency range, transmitting both intentionally and unintentionally. Follow-
up work and discussion are needed to identify the cause of this unintentional
radiation as it has the potential to interfere with SKA-Low science. It is
likely that the transmission levels will need to be reduced by orders of
magnitude to bring the impact on radio astronomy to potentially manageable
levels.

Conclusions. Our results indicate that both intended and unintended radiation
from Starlink satellites will be detrimental to key SKA science goals without
mitigation. Continued conversation with SpaceX could potentially result in
future mitigations which the EDA2 instrument could efficiently monitor and
characterise at the SKA-Low site.

###### Key Words.:

Radio astronomy, space situational awareness, SKA-low, Starlink, unintended
electromagnetic radiation

##  1 Introduction

As the most sensitive radio telescopes ever conceived are under construction,
satellite constellations are being deployed in low and medium Earth orbit
(LEO/MEO) at an unprecedented scale, in support of global communications,
navigation, and remote sensing (among other
applications)111https://planet4589.org/space/con/conlist.html.

The intended and unintended electromagnetic radiation (IEMR and UEMR) from
satellites in LEO and MEO now form a major source of radio frequency
interference (RFI) for radio telescopes on Earth. This includes those that are
situated in previously pristine locations, many of them protected by Radio
Quiet Zones (RQZs) via their respective national regulators. By design,
satellite constellations are visible from any location on Earth (depending on
constellation parameters such as inclination). Thus, no location can be
protected from electromagnetic radiation emitted from satellites unless
adequate regulatory means are established. Until then, astronomers depend on
the engagement of satellite operators in good faith with our community to
mitigate these impacts.

In this work, UEMR refers to unintentional electromagnetic radiation not
originating from a satellite’s transmitter system, consistent with terminology
used in recent work by Di Vruno, F. et al. (2023). Examples of UEMR could
include radiation from navigation computers, high speed network switches, or
Hall effect thrusters, power supplies, or any other type of electronic
equipment on board a satellite. Some of these sources of UEMR may be weak, but
from a radio astronomy point of view, they can have a serious impact on
science.

IEMR is referred to as intentional electromagnetic radiation originating from
a satellite’s transmitter system which has been approved by relevant governing
authorities. Satellites are legally able to transmit at designated frequencies
that are allocated for this purpose after approval is given via international
governing bodies such as the radiocommunications sector of the International
Telecommunication Union’s (ITU) Radio Regulations
(ITU-R)222https://www.itu.int/pub/R-HDB-22-2013. This can therefore be
referred to as ‘emission’ as per the ITU-R definition.

The specific case considered in this Letter is related to low frequency radio
astronomy and the detection of IEMR and UEMR in this frequency range from
satellites in the Starlink constellation. We have focused on the Starlink
constellation because, at the current time, they are the largest constellation
with more than 4,000 satellites in orbit. In particular, we consider the low
frequency component of the Square Kilometre Array (SKA, Dewdney et al., 2009),
currently under construction at Inyarrimanha Ilgari Bundara, the CSIRO
Murchison Radio-astronomy Observatory in Western Australia.

Within the SKA-Low frequency range (50 - 350 MHz), the only frequency bands
that are protected by the ITU-R for radio astronomy are as follows: 73 - 74.6
MHz, 150.05 - 153 MHz, 225 - 235 MHz, and 322 - 328.6 MHz. It is important to
note that from an international and national regulatory point of view, only
transmissions received by radio telescopes in these frequency ranges in the
designated regions are considered interference.

Despite this, radio astronomy has always depended on being able to observe the
Universe at all wavelengths. In the case of SKA-Low, the frequency range of
100 - 200 MHz is key to the SKA headline science case related to detecting
emissions from redshifted neutral hydrogen from the Epoch of Reionisation
(EoR, the period up to a billion years after the Big Bang, when the first
stars and galaxies formed, Trott et al., 2019).

Hosted at the SKA-Low site, the Engineering Development Array version 2 (EDA2,
Wayth et al., 2021) is an array of 256 low frequency bowtie antennas, arranged
in the configuration of an SKA-Low station (35 m diameter physical footprint).
The EDA2 has been used as a ground-truth system for the evaluation of SKA-Low
stations (Bolli, 2020) and can be operated in an all-sky imaging mode. The
EDA2 has undertaken RFI surveys at the SKA-Low site in the FM band, in the
process detecting both reflected FM emissions and both IEMR and UEMR from
objects in LEO (Tingay et al., 2020; Grigg et al., 2022). Using the EDA2 at
other frequencies (159.4 and 229.7 MHz), in a search for astronomical
transients, Sokolowski et al. (2021) also detected UEMR from satellites.

Of specific relevance is a recent publication describing observations of
Starlink satellites by the Low Frequency Array (LOFAR, van Haarlem, M. P. et
al., 2013), across 110 - 188 MHz. Broadband radiation was detected that
occupies the entire frequency range, as well as narrow-band emissions at 125,
135, 143.05, 150, and 175 MHz (emissions at 143.05 MHz are attributed to
reflections of the GRAVES radar system transmitter) (Di Vruno, F. et al.,
2023). The measured flux densities range between 0.1 and 10 Jy for the
broadband radiation and between 10 and 500 Jy for the narrow-band radiation.
The authors provide a comprehensive description of their observations,
specifically in the context of the regulatory environment for UEMR from
objects in space, as well as a discussion of the likely consequences for radio
astronomy if this radiation cannot be mitigated.

The intent of this Letter is to present relevant data from the EDA2, arising
from our continued work to characterise the RFI environment at the SKA-Low
site (Tingay et al., 2020; Grigg et al., 2022). We briefly present data and
analysis pertaining to the detection of Starlink satellites with the EDA2,
complementing results from the more comprehensive analysis done with the LOFAR
telescope. Our data serve to confirm that radiation from Starlink satellites
is detectable from the SKA-Low site, in the SKA-Low frequency range, and with
an instrument representative of the sensitivity and general characteristics of
an SKA-Low station. Consequently, we are able to qualitatively characterise
potential impacts to observations made with the SKA-Low and show that RQZs
operating at a national level are insufficient for the regulation of radiation
originating from satellites. We also demonstrate a novel “all-sky” approach to
the detection and characterisation of the radiation of interest, which is
different to the approach of the LOFAR work.

Section 2 briefly describes observational parameters and data processing
methods. In Section 3, the results of those observations and data processing
are described, followed by a discussion and conclusions in Section 4.

##  2 Observations and data processing

Two sets of observations were made with the EDA2. The first set took place
between UTC 2023-03-17 12:19:09 and UTC 2023-03-18 07:56:05, a duration of
19.5 hours, at a central frequency of 137.5 MHz. The second set of
observations took place between UTC 2021-11-16 01:24:18 and UTC 2021-11-17
00:37:07, a duration of approximately 23 hours, at a central frequency of
159.4 MHz. In both cases, the bandwidth was 0.926 MHz and both orthogonal
linear polarisations (X - east-west and Y - north-south) of the EDA2 were
utilised.

These frequencies were chosen as part of a larger body of work, where 137.5
MHz is a known ORBCOMM satellite downlink frequency and 159.4 MHz is a well
established test frequency for the EDA2 due to minimal RFI and an optimal
response for the MWA dipole and signal chain.

The voltage data generated by the EDA2 system were used to form cross-
correlation products between all independent pairs of the 256 antennas
(forming XX, YY, XY, and YX correlation products). The correlator output was
averaged over 2 seconds and the full 0.926 MHz band, forming visibility data
at those resolutions, which were calibrated in amplitude and phase as
described by Sokolowski et al. (2021).

Full sky images using the parallel hand polarisation visibilities, XX and YY,
were produced by processing the visibility data with the MIRIAD package (Sault
et al., 1995). For more detail on this procedure, refer to Tingay et al.
(2020). For the 137.5 MHz (159.4 MHz) dataset, the images were 160 $\times$
160 (180 $\times$ 180) pixels in size, each pixel was 1.2∘ (1.0∘) in extent
with an approximate angular resolution of 3.6∘ (3.1∘), and 71,252 (83,426)
images were made, which is an aggregate across both polarisations.

Figure 1:  Three images of Starlink satellites detected at 137.5 MHz. The
images from left to right are at times UTC 2023-03-18 00:59:46, 01:01:04, and
01:02:24. The blue grid is an astronomical projection to show the area of
visible sky. The blue crosses are the TLE predicted positions of all Starlink
satellites detected during this dataset. The red arrows denote detections of
NORAD 55,358, NORAD 55,377, and NORAD 55,340 from left to right. The satellite
in the bottom left corner of all three images is not a Starlink satellite
(NORAD 28,654 - NOAA 18). Arrows indicate the detections referred to. The
median of the Median Absolute Deviation (MAD) value at the zenith in each
image across all difference images at this frequency is 310 Jy/beam and is
high due to there nearly always being a satellite transmitting overhead
throughout the entire dataset. The MAD of images was as low as $\sim$0.6
Jy/beam when there were no transmitters or Milky Way or Sun present.

The images were input to a Python-based detection script. Following our
previous work (Grigg et al., 2022), the images for the two polarisations were
averaged and differenced in time. The differencing of images closely separated
in time removes the majority of astronomical signals, while isolating
detections of faster moving objects in orbit. In this case, we difference
images separated by 16 time steps (32 seconds), such that $\Delta
I_{n}=I_{n}-I_{n-16}$, where $I$ is the image at image index $n$. Satellite
detections appear in the difference images with positive amplitudes if present
in the $I_{n}$ image and with negative amplitudes if present in the $I_{n-16}$
image. Astronomical sources do move across the sky enough to produce weak
residuals in the difference images using a time separation of 32 s, but this
timescale ensures that Starlink satellites close to the horizon (moving with a
lower angular speed) travel far enough to appear in the difference images.
Examples of this technique can be found in Prabu et al. (2020); Tingay et al.
(2020); Grigg et al. (2022).

In order to identify any orbiting objects in our difference images, Two Line
Elements (TLEs) were queried from space-track.org for the entire public
catalogue, over the period 2023-03-16/20 for the 137.5 MHz observations and
over the period 2021-11-15/19 for the 159.4 MHz observations. The TLEs were
then used to predict the positions of all objects in each image of the
datasets.

The results were inspected and analysed using the software tool DUG Insight
(Grigg et al., 2022), which provided an interface to efficiently interrogate
the data. Figures 1 and 3 were produced using this software.

##  3 Results

A comparison of the predicted positions of all satellites from publicly
available TLEs with the emissions detected in the difference images revealed
that Starlink satellites had been detected at both frequencies.

###  3.1 137.5 MHz

An example sequence of difference images from the 137.5 MHz data are shown in
Figure 1. In the difference images, it is apparent that the emission from
these Starlink satellites is highly variable. This is more readily apparent in
a video of the 137.5 MHz difference images, showing the behaviour of the
emission from the Starlink trains333https://youtu.be/TT1hJ2NOZQo.

The detected emissions are a series of flashes from each satellite. The
frequency of the flashing is approximately once per $\sim 100$ seconds (every
50 or 51 images). Three satellites are exceptions. NORADs 55,572, 55,582, and
55,612 flash at a slightly longer period of 120 seconds. In total, over the
observation period, 1,130 flashes were detected from 136 Starlink satellites
in the NORAD # range 53,436 to 55,715.

These detections were made by force-fitting a Gaussian model at the predicted
position of each satellite in each polarisation, and accepting fits where the
percentage errors on the amplitude and width of the Gaussian are $<$10%. No
detections were attempted at elevations below 5∘ or within 2∘ of the Sun.
Where the detection is consistent with two or more Starlink satellites, the
closest on the sky is taken as the identification.

Emission intensities were estimated by correcting the XX and YY measured
intensities using the antenna beam response models (Sokolowski et al., 2017)
and summing the two polarisations together. The flashes reach a maximum
intensity of approximately $10^{6}$ Jy/beam at ranges of $\sim$500 km (EIRP
$\sim$ 30 mW) and a minimum intensity of approximately 2,000 Jy/beam at ranges
of $\sim$2,000 km (EIRP $\sim$ 1mW). EIRPs are calculated assuming the
emission occupies the full 0.926 MHz bandwidth of the system. It is important
to note that these emissions are highly localised and constitute only a minor
portion of the total power entering the EDA2 system meaning that the response
of the EDA2 will still be linear. Multiple passes were observed for the
majority of these objects. The flashes are less than 2 seconds in duration
(since they occupy at most two consecutive images). Thus, pairs of images were
averaged together to ensure 100% of the emission was captured, making the
integration time 4 seconds for the measurement of the flashes.

Figure 2 shows the relationship between peak intensity and range ($r$) for all
flashes detected in the dataset. In general, weaker emissions occur at larger
ranges, but intrinsic variability in the emission strength is also apparent
for individual satellites, as well as between satellites.

Figure 2:  The range of peak intensity measurements for all 1,130 flashes
detected at 137.5 MHz. The red points represent the flash with the highest
intensity for each NORAD identifier, while the blue points represent the
lowest. The yellow points represent the rest of the flashes. An inverse square
law ($\propto\frac{1}{r^{2}}$) is fitted to these data points to account for
the spherical isotropic propagation of the transmission and to guide the
reader’s eye. Figure 3:  Three images of the detected Starlink train at 159.4
MHz. The images from left to right are at times UTC 2021-11-16 10:07:10,
10:07:49, and 10:08:29. The blue grid is an astronomical projection to show
the area of visible sky. The blue crosses are the TLE predicted positions of
all Starlink satellites detected during this dataset. The vertical band on the
right hand side of the image is due to the motion of the Milky Way over the
differencing period. The median MAD value at the zenith across all difference
images is 1.2 Jy/beam.

The 137.5 MHz frequency overlaps with a standard satellite communications
downlink frequency band, meaning that many other satellites were detected in
this dataset. Very high intensity emissions from constellations such as
SPACEBEE, ORBCOMM, and NOAA reached 107 Jy/beam (highest recorded near the
zenith was NOAA 18 at 6.6$\times 10^{7}$Jy/beam), causing extreme sidelobes in
the images which obscured weaker emissions.

SpaceX engineers have confirmed that the Starlink satellites have ‘Swarm’
transmitters, which transmit at 137 MHz (2023, SpaceX, private communication).
These transmitters are used for telemetry, tracking and control, which could
explain why they are observed as sparse, periodic emissions. This frequency
band will continue to be used by Starlink satellites for the foreseeable
future. These emissions are in the class of IEMR.

A report by the European Communications
Committee444https://docdb.cept.org/download/4013 suggests that the Swarm
transmitters have an EIRP downlink limit of -1.55 dBW ($\sim$700 mW) over a 42
- 125 kHz band. Taking our observed 30 mW EIRP near minimum range and
correcting for the bandwidth and our upper limit on the emission duration, we
get a lower limit of $\sim$200 mW (125 kHz transmit band and $<$2 s duration),
consistent with the published value. The document also suggests a 0 dBi
transmit antenna, which appears inconsistent with the scatter in received
power in Figure 2. This can be primarily attributed to variable transmission
power of the transmitters which can be seen in the vertical spread of points
at each range.

###  3.2 159.4 MHz

An example sequence of difference images from the 159.4 MHz data are shown in
Figure 3. The radiation from satellites in the Starlink train is constant,
rather than the flashing observed at 137.5 MHz. A video of the 159.4 MHz data
shows the behaviour of the radiation from this
train555https://youtu.be/FISUgjrCAi4.

Figure 4:  The range of intensity measurements of 3 satellites fitted in the
train of Starlink satellites at 159.4 MHz. The three colours identify the
particular satellite fitted. The dotted lines are for the XX polarisation
images and the solid lines are for the YY polarisation images. The arrows show
the direction of time.

At 159.4 MHz, the steady radiation observed reaches a peak intensity of
$\sim$220 Jy/beam (at a distance of $\sim$380 km, corresponding to an
approximate EIRP of $\sim$2$\mu$W). Again, the measured intensities are
corrected by the beam models. The identification of individual satellites is
challenging, due to the small angular separation of the satellites relative to
the angular resolution of the EDA2. However, predicted positions generated
from TLEs suggest that this train includes 53 Starlink satellites with NORAD
IDs between 49,408 and 49,460. This is a different set of satellites compared
to those detected at 137.5 MHz.

These satellites were launched on 2021-11-13, on a FALCON 9 rocket. The
observations at 159.4 MHz were taken only 3 days after this launch, allowing
an opportunity to study the Starlink satellites at low altitude. The nominal
smallest range to this Starlink train was $\sim$370 km, as predicted by the
TLEs. They later ascended to higher orbital altitudes at 500 - 600 km.

The predicted positions lag behind the observed radiation in the dataset,
which is likely due to the time difference between the epoch of the TLE used
and the time of the detection. The first available TLE for this train was
generated the day after this observation, at UTC 2021-11-17 16:00. This is a
difference of 21 hours. As the Starlink satellites would have been rising to
their operational altitude at this time, it is reasonable that we see a
mismatch between the detected and predicted positions. The same Starlink train
was not detectable in four other passes in the period of the dataset. These
were at significantly lower elevation angles with two being at a range of
$\sim$750 km and the other two at $\sim$1,750 km. Assuming the same power of
the radiation for these other passes, taking into account range and primary
beam attenuation it is not expected that a detection would be made at these
ranges.

Figure 4 shows how the intensity of three satellites in the train varies as a
function of range for each polarisation. As the intensities of satellites in
the train were only high enough to fit a Gaussian model in the YY polarisation
images, the fitting procedure was carried out on just YY polarisation images,
allowing for the determination of the peak location. Subsequently, the
intensity value at this specific pixel position was extracted from both
polarisations and are the values reported in Figure 4. The ranges were
calculated by identifying the Starlink satellite closest to the fit position.
Although there are some uncertainties in the identifications, the ranges are
suitable approximations.

Figure 4 shows a distinctive evolution of the radiation intensities in the YY
polarisation, starting high and going through a transition to much lower
radiation levels at closest approach to the EDA2. The XX polarisation
intensities mirror that behaviour in reverse (start low and transition high),
but at lower intensity levels. This may suggest that geometric effects due to
the transmission surface being obscured at particular orientations as the
satellite passes over could be causing this behaviour.

The authors have been in communication with SpaceX (who owns, builds, and
operates the Starlink constellation), who explain that this radiation is
likely due to the satellites’ propulsion or avionics system and is likely over
50 - 200 MHz (2023, SpaceX, private communication). The propulsion system is
actively engaged during the time this train is detected. This radiation is
therefore in the class of UEMR.

During the period of observation, there were 114 additional Starlink
satellites accessible to our data, of the same generation as the Starlink
satellites we have detected. There are many passes of these objects, with a
closest range of $<$450 km. None of these objects display similar UEMR to the
detected train.

##  4 Discussion and conclusions

The received power of transmissions from the detected Starlink satellites was
often much higher than the brightest astronomical radio sources in the sky.
While the detected IEMR at 137.5 MHz complies with ITU-R regulations, it is
important to note that the UEMR detected at 159.4 MHz poses interference
problems for low frequency radio astronomy. It is important to reiterate that
the frequencies utilised for these observations are not within the ITU-R
protected bands for radio astronomy, but are present in bands that are
relevant to the highest impact astrophysics planned for the SKA-Low facility.

These observations at 137.5 and 159.4 MHz are not simultaneous, so we can draw
no conclusions regarding the frequency extent of the transmissions we detect.
Although the frequency extents of this work are different to the LOFAR work,
both studies detect Starlink satellites across a large fraction of the SKA-Low
frequency range (Di Vruno, F. et al., 2023).

It is of interest to note that the Di Vruno, F. et al. (2023) work did not
detect the Swarm transponder IEMR, which may be due to geographical
restrictions such as not transmitting over Europe. Also, satellites in the Di
Vruno, F. et al. (2023) paper which were still being boosted to their
operational altitude were detected at flux densities up to 10 Jy. Not
mentioned by the authors, multiple generations of Starlink satellites were
detected and could explain the variation in fluxes reported in that work. This
makes directly comparing fluxes between the Di Vruno, F. et al. (2023) work
and this Letter complex, as well as comparing between instruments,
frequencies, timeframes and geographical locations. This being said, the
$\sim$0.1-0.3 Jy signals detected at 550 km range in the Di Vruno, F. et al.
(2023) work would not be expected to be detected in this work as this below
the threshold of even images with the lowest MAD values for each dataset. For
a comprehensive cross-examination between the two studies, further
observations and perhaps collaboration with SpaceX is needed to account for
these differences.

Broad and narrow-band IEMR and UEMR could severely compromise key science for
SKA-Low, such as measurements of the EoR (Furlanetto et al., 2006). Wilensky
et al. (2020) estimated the impact of RFI on the measurement of the EoR power
spectrum (the observational quantity used to detect or constrain the EoR
signal) and determined that the presence of an aggregate apparent 1 mJy of RFI
in the integrated power spectrum is enough to prevent detection for even the
most optimistic theoretical models of the EoR signal. After all forms of
avoidance and RFI mitigation and removal, including attenuation due to
averaging in frequency and time, the apparent RFI remaining in the data needs
to be below $\sim$mJy levels.

An engineering solution to mitigate the IEMR and UEMR at the source on the
satellites may be an option. In this case, reductions in the EIRP of factors
of order 100 to 1,000 may be required to bring the radiation into a regime
that does not challenge key radio astronomy experiments (assuming a notional
temporal occupancy of 10% for 100 Jy radiation over a 1 MHz band within an
overall 100 MHz observation bandwidth). A collaborative effort between
astronomers and satellite operators could explore engineering solutions.

The authors are in close contact with SpaceX, and the company has offered to
continue to discuss possible ways to mitigate any adverse effects to astronomy
in good faith. As part of their design iteration, SpaceX has already
introduced changes to its next generation of satellites which could mitigate
the impact of the Swarm transponder emissions. This highlights the importance
of studies such as this and Di Vruno, F. et al. (2023) at identifying UEMR, so
that policies can be developed based on quantitative information, to ensure
the private LEO space industry can operate whilst minimising its impact on
radio astronomy research.

If engineering solutions cannot be found, or cannot bring the radiation under
an appropriate threshold, RFI identification and flagging algorithms could be
applied to the data collected from radio telescopes. Substantial effort has
been applied in this direction, for example as described in Offringa et al.
(2015) and Wilensky et al. (2019), and in a particularly relevant study, by
Finlay et al. (2023). Flagging of data comes as a last resort, as valuable
information is also lost.

SpaceX’s engagement with the astronomy community has allowed us to confirm
that the measured UEMR is originating from their current generation satellite
platforms. In the absence of suitable regulation, we thus depend on satellite
operators engaging with the radio astronomy community to find solutions. We
encourage all satellite operators to seek our engagement before launching
their satellites, to ensure adequate UEMR mitigation is in place on their
platforms. The appropriate venue to establish this contact is via ITU-R
Working Party 7D (Radio Astronomy), where radio astronomers meet several times
per year to protect the radio spectrum and assess other users’ needs.

In the domain of optical astronomy, significant progress has been made in
mitigating the reflected sunlight from Starlink satellites that interferes
with optical telescope observations. Mallama et al. (2023) reports that
Starlink Generation 2 satellites are a factor of 12 fainter than Generation 1
satellites, despite the larger size of Generation 2 satellites. This
demonstrates that mitigations are possible.

The EDA2 has proven to be a very effective and efficient tool for measuring
and characterising UEMR from objects in LEO, due to its sensitivity, all-sky
imaging capability, and its radio quiet location. The EDA2 will be a very
effective tool for monitoring UEMR from all satellites in the SKA-Low
frequency range, including to assess any improvements realised from
mitigations.

In summary, our conclusions are:

  1. 1.

Using observations made with the EDA2 at 137.5 and 159.4 MHz on two different
occasions, we detect both IEMR (telemetry beacons) and UEMR (propulsion or
avionics; 2023, SpaceX, private communication) respectively from trains of
Starlink satellites.

  2. 2.

Intensities of IEMR at 137.5 MHz are of order 10 $-$ 106 Jy/beam, or in the
$\mu$W - mW EIRP range. Although this IEMR abides by ITU-R guidelines, these
intensities are large compared to the strongest astronomical radio sources in
the sky and will therefore have the potential to disrupt astronomical
observations at SKA-Low frequencies;

  3. 3.

The detected IEMR and UEMR are outside of the frequency bands protected for
radio astronomy, but are at frequencies of great interest for key experiments
for the SKA-Low facility, and at frequencies where RQZ protections at the SKA-
Low site are in place;

  4. 4.

Our results complement recent results published from observations with the
LOFAR telescope. Our results confirm that the SKA-Low site is impacted in the
frequency range of the SKA-Low telescope;

  5. 5.

Mitigations will be required, either to suppress the UEMR at the source or
identify and remove affected sections of data. SpaceX has already introduced
changes to its next generation of satellites which could mitigate the impact
of UEMR; and

  6. 6.

The EDA2 will be a valuable tool for ongoing monitoring and assessment of UEMR
from satellites in LEO into the future.

###### Acknowledgements.

This scientific work uses data obtained from Inyarrimanha Ilgari Bundara / the
Murchison Radio-astronomy Observatory. We acknowledge the Wajarri Yamaji
People as the Traditional Owners and native title holders of the Observatory
site. Support for the operation of the MWA (the host facility for the EDA2) is
provided by the Australian Government (NCRIS), under a contract to Curtin
University administered by Astronomy Australia Limited. We acknowledge the
Pawsey Supercomputing Centre which is supported by the Western Australian and
Australian Governments. Inyarrimanha Ilgari Bundara, the CSIRO Murchison
Radio-astronomy Observatory and the Pawsey Supercomputing Research Centre are
initiatives of the Australian Government, with support from the Government of
Western Australia and the Science and Industry Endowment Fund. We also
acknowledge time used on DUG Technology’s Perth supercomputer and for their
usage of DUG Insight.

## References

  * Bolli (2020) Bolli, P. 2020, IEEE Open Journal of Antennas and Propagation, 1, 253
  * Dewdney et al. (2009) Dewdney, P. E., Hall, P. J., Schilizzi, R. T., & Lazio, T. J. L. W. 2009, IEEE Proceedings, 97, 1482
  * Di Vruno, F. et al. (2023) Di Vruno, F., Winkel, B., Bassa, C. G., et al. 2023, A&A, 676, A75
  * Finlay et al. (2023) Finlay, C., Bassett, B. A., Kunz, M., & Oozeer, N. 2023, MNRAS, 524, 3231
  * Furlanetto et al. (2006) Furlanetto, S. R., Oh, S. P., & Briggs, F. H. 2006, Physics Reports, 433, 181
  * Grigg et al. (2022) Grigg, D., Tingay, S., Sokolowski, M., & Wayth, R. 2022, Astronomy and Computing, 40, 100619
  * Mallama et al. (2023) Mallama, A., Cole, R. E., Harrington, S., et al. 2023, arXiv e-prints, arXiv:2306.06657
  * Offringa et al. (2015) Offringa, A. R., Wayth, R. B., Hurley-Walker, N., et al. 2015, Publications of the Astronomical Society of Australia, 32, e008
  * Prabu et al. (2020) Prabu, S., Hancock, P. J., Zhang, X., & Tingay, S. J. 2020, PASA, 37, e010
  * Sault et al. (1995) Sault, R. J., Teuben, P. J., & Wright, M. C. H. 1995, in Astronomical Society of the Pacific Conference Series, Vol. 77, Astronomical Data Analysis Software and Systems IV, ed. R. A. Shaw, H. E. Payne, & J. J. E. Hayes, 433
  * Sokolowski et al. (2017) Sokolowski, M., Colegate, T., Sutinjo, A. T., et al. 2017, Publications of the Astronomical Society of Australia, 34, e062
  * Sokolowski et al. (2021) Sokolowski, M., Wayth, R. B., Bhat, N. D. R., et al. 2021, Publications of the Astronomical Society of Australia, 38, e023
  * Tingay et al. (2020) Tingay, S. J., Sokolowski, M., Wayth, R., & Ung, D. 2020, PASA, 37, e039
  * Trott et al. (2019) Trott, C. M., Watkinson, C. A., Jordan, C. H., et al. 2019, PASA, 36, e023
  * van Haarlem, M. P. et al. (2013) van Haarlem, M. P., Wise, M. W., Gunst, A. W., et al. 2013, A&A, 556, A2
  * Wayth et al. (2021) Wayth, R., Sokolowski, M., Broderick, J., et al. 2021, The Engineering Development Array 2: design, performance and lessons from an SKA-Low prototype station
  * Wilensky et al. (2020) Wilensky, M. J., Barry, N., Morales, M. F., Hazelton, B. J., & Byrne, R. 2020, MNRAS, 498, 265
  * Wilensky et al. (2019) Wilensky, M. J., Morales, M. F., Hazelton, B. J., et al. 2019, Publications of the Astronomical Society of the Pacific, 131, 114507


$|L^{\prime}|<\min\\{2^{\Omega(d)},\left((w_{k}+\varepsilon)/w_{k}\right)^{\omega(1)}\\}$,
this reduction violates the lower bound of Proposition F.1. Therefore we must
have either $\beta_{k}=\Omega(\sqrt{\log 1/\alpha})$ or
$|L^{\prime}|\geqslant\min\\{2^{\Omega(d)},\left(1/\tilde{w}_{k}\right)^{\omega(1)}\\}$.

Finally, we show that these lower bounds on $\beta_{k}$ and $|L^{\prime}|$
imply the desired lower bound for $\mathcal{A}$. Consider first the case:
$\beta_{k}=\Omega(\sqrt{\log 1/\alpha})$. Note that in the input to algorithm
$\mathcal{A}$ we have $\tilde{w}_{k}=\alpha$. Therefore
$\beta_{k}=\Omega(\sqrt{\log 1/\alpha})$ corresponds to the desired lower
bound in the lemma statement. Consider second the case:
$|L^{\prime}|\geqslant\min\\{2^{\Omega(d)},\left(1/\tilde{w}_{k}\right)^{\omega(1)}\\}$.
We note that, for each $i\in[k-1]$, the original list $L$ must contain some
$\hat{\mu}\in L$ such that $\lVert\hat{\mu}-\mu_{i}\rVert\leqslant C\sqrt{\log
1/w_{\mathrm{low}}}$. Furthermore, because the means $\mu_{i}$ have pairwise
separation larger than $2C\sqrt{\log 1/w_{\mathrm{low}}}$, the original list
$L$ must contain at least $k-1$ means of this kind. However, all of these
means are removed in the pruning procedure, so $|L|\geqslant
k-1+|L^{\prime}|$, so $|L|\geqslant
k-1+\min\\{2^{\Omega(d)},\left(1/\tilde{w}_{k}\right)^{\omega(1)}\\}$. This
matches the desired lower bound in the lemma statement. (The choice to make
the hidden mean the $k$-th mean was without loss of generality, as the
distribution is invariant to permutations of the components.)

###  F.2 Case a): For distributions with $t$-th sub-Gaussian moments

The proof for the case when $D_{i}(\mu_{i})$ has sub-Gaussian $t$-th central
moments employs the same reduction scheme, but reduces from Proposition F.2.

######  Proposition F.2 ([5], Proposition 5.12).

Let $\mathcal{D}$ be the class of distributions on $\mathbb{R}^{d}$ with
bounded $t$-th central moments for some positive even integer $t$, and let
$0<\alpha<2^{-t-1}$. Then any list-decoding algorithm that learns the mean of
an element of $\mathcal{D}$ with failure probability at most $1/2$, given
access to $(1-\alpha)$-additively corrupted samples, must either have error
bound $\beta=\Omega(\alpha^{-1/t})$ or return a list of at least $d$
hypotheses.

Furthermore, in [3], formal evidence of computational hardness was obtained
(see their Theorem 5.7, which gives a lower bound in the statistical query
model introduced by [13]) that suggests obtaining error
$\Omega_{t}((1/\tilde{w}_{s})^{1/t})$ requires running time at least
$d^{\Omega(t)}$. This was proved for Gaussian inliers and the running time
matches ours up to a constant in the exponent.

##  Appendix G Stability of list-decoding algorithms

In this section we discuss two of the existing list-decodable mean estimation
algorithms for identity-covariance Gaussian distributions and show that they
also work when a $w_{\mathrm{low}}^{2}$-fraction of the inliers is
adversarially removed.

First, we consider the algorithm in Theorem 3.1 in [3]. A central object in
their analysis is an “$\alpha$-good multiset", which is a multiset of samples
such that all are within distance $O(\sqrt{d})$ of each other and at least an
$\alpha$-fraction of them come from a $(1-\Omega(\alpha))$-fraction of an
i.i.d. set of samples from a Gaussian distribution $N(\mu,I_{d})$. Then their
algorithm essentially works as long as the input contains an $\alpha$-good
multiset. For our case, after the removal of a $w_{\mathrm{low}}^{2}$-fraction
of inliers, the input essentially continues to contain a
$(1-w_{\mathrm{low}}^{2})\alpha$-good multiset, so the algorithm continues to
work in our corruption model.

Second, we consider the algorithm in Theorem 6.12 in [5]. The main
distributional requirement of their algorithm is that $\mathbb{E}_{x,y\sim
S^{*}}[p^{2}(x-y)]\leqslant 2\mathbb{E}_{g,h\sim N(0,I_{d})}[p^{2}(g-h)]$ for
all degree-$(t/2)$ polynomials $p$, where $S^{*}$ is the set of inliers.
Concentration arguments give with high probability that $\mathbb{E}_{x,y\sim
C^{*}}[p^{2}(x-y)]\leqslant 1.5\mathbb{E}_{g,h\sim N(0,I_{d})}[p^{2}(g-h)]$.
Furthermore, the distribution over $x,y\sim S^{*}$ can be seen as a
$(1-w_{\mathrm{low}}^{2})^{2}$-fraction of the distribution over $x,y\sim
C^{*}$. Then G.1, which follows by standard probability calculations, also
gives that any event under the former distribution can be bounded in terms of
the second distribution:

######  Fact G.1.

For any event $A$,

$\operatorname*{\mathbb{P}}_{x,y\sim
S^{\ast}}(A)\leqslant\operatorname*{\mathbb{P}}_{x,y\sim
C^{\ast}}(A)/(1-w_{\mathrm{low}}^{2})^{2},$ (G.1)

where probabilities are taken over a uniform sample from $S^{\ast}$ and
$C^{\ast}$ respectively.

Overall we obtain

$\mathbb{E}_{x,y\sim S^{*}}[p^{2}(x-y)]\leqslant
1.5/(1-w_{\mathrm{low}}^{2})^{2}\mathbb{E}_{g,h\sim N(0,I_{d})}[p^{2}(g-h)],$

so for $w_{\mathrm{low}}$ small enough we have $\mathbb{E}_{x,y\sim
S^{*}}[p^{2}(x-y)]\leqslant 2\mathbb{E}_{g,h\sim N(0,I_{d})}[p^{2}(g-h)]$ and
their algorithm continues to work in our corruption model.

##  Appendix H Concentration bounds

In this section we prove some concentration bounds essential to our analysis.

######  Lemma H.1.

Let $D$ be a $d$-dimensional distribution with mean $\mu^{*}\in\mathbb{R}^{d}$
and sub-Gaussian $t$-th central moments with parameter $1$. Fix a unit vector
$v\in\mathbb{R}^{d}$. Then

$\operatorname*{\mathbb{P}}_{x\sim D}[|\langle x-\mu^{*},v\rangle|\leqslant
R]\geqslant 1-\left(\frac{\sqrt{t}}{R}\right)^{t}\,.$

###### Proof.

We have that

$\displaystyle\operatorname*{\mathbb{P}}_{x\sim D}[|\langle
x-\mu^{*},v\rangle|>R]$
$\displaystyle\leqslant\frac{\operatorname*{\mathbb{E}}_{x\sim D}\langle
x-\mu^{*},v\rangle^{t}}{R^{t}}\leqslant\frac{(t-1)!!}{R^{t}}\leqslant\left(\frac{\sqrt{t}}{R}\right)^{t}\,,$

where we used that $(t-1)!!\leqslant t^{t/2}=\sqrt{t}^{t}$. ∎

######  Lemma H.2.

Let $D$ be a $d$-dimensional distribution with mean $\mu^{*}\in\mathbb{R}^{d}$
and sub-Gaussian $t$-th central moments with parameter $1$. Let $C^{*}$ be a
set of i.i.d. samples drawn from $D$. Fix a unit vector $v\in\mathbb{R}^{d}$.
Then with probability at least
$1-\exp\left(-2|C^{*}|\left(\frac{\sqrt{t}}{R}\right)^{2t}\right)$,

$\left\lvert\left\\{x\in C^{*}\text{, s.t. }\lvert\langle
x-\mu^{*},v\rangle\rvert\leqslant
R\right\\}\right\rvert\geqslant\left(1-2\left(\frac{\sqrt{t}}{R}\right)^{t}\right)|C^{*}|\,.$

###### Proof.

The result follows by Lemma H.1 and a Binomial tail bound. ∎

######  Lemma H.3.

Let $D$ be a $d$-dimensional distribution with mean $\mu^{*}\in\mathbb{R}^{d}$
and sub-Gaussian $t$-th central moments with parameter $1$. Let $C^{*}$ be a
set of i.i.d. samples drawn from $D$. Fix $m$ unit vectors
$v_{1},\ldots,v_{m}\in\mathbb{R}^{d}$. Then with probability at least
$1-\exp\left(-2|S^{*}|m^{2}\left(\frac{\sqrt{t}}{R}\right)^{2t}\right)$,

$\Big{|}\bigcap_{i\in[m]}\left\\{x\in S^{*}\text{, s.t. }\lvert\langle
x-\mu^{*},v_{i}\rangle\rvert\leqslant
R\right\\}\Big{|}\geqslant\left(1-2m\left(\frac{\sqrt{t}}{R}\right)^{t}\right)|S^{*}|\,.$

###### Proof.

By Lemma H.1 and a union bound over the $m$ directions, we get

$\operatorname*{\mathbb{P}}_{x\sim D}[|\langle
x-\mu^{*},v_{i}\rangle|\leqslant R,\forall i\in[m]]\geqslant
1-m\left(\frac{\sqrt{t}}{R}\right)^{t}\,.$

Then the result follows by a Binomial tail bound. ∎

##  Appendix I Experimental details

#### Adversarial line and adversarial clusters

The following figure illustrates the adversarial distributions used in Figure
2 and further in this section.

True ClusterFake Cluster Figure 4: Two variants of adversarial distribution:
adversarial line (left) and adversarial clusters (right).

#### Data Distribution

We consider a mixture of $k=7$ well-separated
($\lVert\mu_{i}-\mu_{j}\rVert\geqslant 40$) $d=100$ dimensional inlier
clusters whose subgroup sizes range from $0.3$ to $0.02$. The experiments are
conducted once using a Gaussian distribution and once using a heavy-tailed
t-distribution with five degrees of freedom for both inlier and adversarial
clusters. In Figure 6 the latter suggests that our algorithm works
comparatively well even for mixture distributions which do not fulfill our
assumptions. We set $w_{\mathrm{low}}=0.02$ and $\varepsilon=0.12$ so that it
is larger than the smallest clusters but smaller than the largest ones and set
the total number of data points to $10000$. The Gaussian noise model simply
computes the empirical mean and covariance matrix of the clean data and
samples $1200$ noisy samples from a Gaussian distribution with this mean and
covariance. The adversarial cluster model and the adversarial model are as
depicted in Figure 4.

Figure 5: Comparison of five algorithms with three adversarial noise models.
On the left we show worst estimation error of algorithms with constrained list
size and on the right the smallest list size with constrained error guarantee.
We plot the median of the metrics with the error bars showing $25$th and
$75$th percentile. We observe that our method consistently outperforms prior
works in terms of list size and worst estimation error, with the exception of
DBSCAN, which performs at a similiar level.

#### Attack distributions

We consider three distinct adversarial models (see Figure 4 for reference).

  1. 1.

_Adversarial clusters_ : After sampling the inlier cluster means, we choose
the cluster with the smallest weight. Let $\mu_{s}$ denote its mean. Then, we
sample a random direction $v_{c}$ with $\lVert v_{c}\rVert=10$. After that, we
sample three directions $v_{1},v_{2}$ and $v_{3}$ with $\lVert
v_{i}\rVert=10$. Then we put three additional (outlier) clusters with means at
$\mu_{s}+v_{c}+v_{i}$. This roughly corresponds to the right picture in Figure
4. The samples for each adversarial cluster are drawn from a distribution that
matches the covariance of the inlier clusters, with the sample size being
twice as large as of the affected inlier cluster.

  2. 2.

_Adversarial line_ : After sampling the inlier cluster means, we again choose
the cluster with the smallest weight. Let $\mu_{s}$ denote its mean. Then, we
sample a random direction $v_{c}$ with $\lVert v_{c}\rVert=10$. We put three
additional (outlier) clusters with means at $\mu_{s}+v_{c},\mu_{s}+2v_{c}$ and
$\mu_{s}+3v_{c}$, which form a line as shown in Figure 4. The samples are
drawn similarly to the adversarial clusters, with the difference that the
covariance is scaled by a factor of $5$ in the direction of the line.

  3. 3.

_Gaussian adversary_ : Here we simply introduce noise matching the empirical
mean and covariance of all inlier data (i.e., as if all inlier clusters are
generated from the same Gaussian distribution).

Note that in the first and second attack, the adversary creates clusters that
do not respect the separation assumption of the true inlier clusters: either
adversarial clusters are placed around the smallest inlier cluster
(Adversarial Cluster), or the adversarial clusters form a line, pointing out
in some fixed direction (Adversarial Line).

#### Implementation details

We implement the list-decodable mean estimation base learner in our InnerStage
algorithm (Algorithm 3) based on [8]. It leverages an iterative multi-
filtering strategy and one-dimensional projections. In particular, we use the
simplified gaussian version of the algorithm. It is designed for distributions
sampled from a Gaussian but also shows promising results for the experiments
involving a heavy-tailed t-distribution as depicted in Figure 6. The robust
mean estimator used to improve the mean hypotheses for large clusters is
omitted in our implementation.

Figure 6: Worst estimation error and list size comparison for the case where
inlier distributions are heavy-tailed. We can observe numerical stability of
our approach.

#### Hyper-parameter search and experiment execution

The hyper-parameters of our algorithm are tuned beforehand based on the
experimental setup. For the comparative algorithms, hyper-parameter searches
are conducted within each experiment after initial tuning. For our algorithm,
key parameters include the pruning radius $\gamma$ used in the OuterStage
routine (Algorithm 7) and $\beta$ used in the InnerStage (Algorithm 5). In
addition, parameters for the LD-ME base learner, such as the cluster
concentration threshold, also require careful selection, resulting in a total
of 7 parameters. The tuning for these was performed using a grid search
comparing about $250$ different configurations. Similarly, we independently
tune the vanilla LD-ME algorithm, which we run with $w_{\mathrm{low}}$ as
weight parameter. For DBSCAN, we optimize the list size and error metrics by
searching over a range of $100$ values for $\varepsilon$, which controls the
maximum distance between samples considered in the same neighbourhood. The
minimum samples threshold, which validates the density based clusters, is
pretuned beforehand and adjusted based on $w_{\mathrm{low}}$. For $k$-means
and its robust version, utilizing a median-of-means weighting scheme, we
explore $21$ values for $k$, including the true number of clusters. Each
parameter setting is executed $100$ times to account for stochastic variations
in the algorithmic procedures, such as $k$-means initialization. The list size
and worst estimation error for each list of clusters obtained is visualized
exemplarily for one iteration of the experiment in Figure 7. The plot provides
insight into how the different algorithms perform and vary with different list
sizes.

Figure 7: Scatter plot of all results for one iteration of the experiment
using three adversarial noise models.

#### Evaluation details

Note that we have two sources of randomness: the data is random and also the
algorithms themselves are random (except DBSCAN). For a clear comparison, we
sample and fix one dataset for each attack model. we plot the performance of
$100$ runs of each algorithm for each parameter setting, each time recording
the returned list size together with the worst estimation error
$\max_{i\in[k]}\min_{\hat{\mu}\in L}\lVert\mu_{i}-\hat{\mu}\rVert$. Then we
either (i) report the worst estimation error for all runs with constrained
list size (we pick the list size most frequently returned by our algorithm,
specifically $7$ or $10$ in our experiments) (see Figure 5, left), or (ii)
report the smallest list size required to achieve the same or smaller worst
estimation error (we pick the $75$th quantile of errors of our algorithm for a
threshold) (see Figure 5, right). Under size constraint (i), the bar plots
correspond to the median over the runs, with error bars indicating the $25$th
and $75$th quantiles. Under error constraint (ii), the bar plots represent the
minimum list size for which the median over the runs falls below the
threshold, while the error bars show the minimum list size for which the
$25$th and $75$th quantiles meet the constraint. Note that ’n/a’ indicates
that, within the scope of our parameter search, no list size achieves an error
below the specified constraint.

In Figure 6 we study the numerical stability of our approach. In particular,
whether the performance degrades when inlier distribution does not satisfy
required assumptions. We observe that if one uses our meta-algorithm with base
learner designed for Gaussian inliers, we still obtain stable results even in
the case of heavy-tailed inlier distribution.

###  I.1 Variation of $w_{\mathrm{low}}$

To study the effect of varying $w_{\mathrm{low}}$ input on the performance of
our approach and LD-ME, we introduce a new noise model. As illustrated in
Figure 8, we consider a mixture of $k=3$ well-separated clusters: one small
cluster with a weight of $0.045$ and two large clusters, each with a weight of
$0.2$. We place two adversarial clusters (see paragraph on attack
distributions for details): one near the small cluster and another near one of
the large clusters. Furthermore, uniform noise is introduced, spanning the
range of the data generated by the inlier and its nearby outlier cluster and
accounting for $10\%$ of the data in this region. Overall, $\varepsilon=0.56$
and we draw $22650$ samples from this mixture distribution.

For both algorithms we run 100 seeds for each $w_{\mathrm{low}}$ ranging from
0.02 to 0.2, which corresponds to the weight of the largest inlier cluster. In
Figure 9, we plot the median estimation error with error bars showing the
$25$th and $75$th quantiles for the small cluster (top left) and the large
cluster near the outlier cluster (top right). As expected from our theoretical
results, we observe that our algorithm performs roughly constant in estimating
the mean of the large cluster, regardless of the initial $w_{\mathrm{low}}$.
Meanwhile, the estimation error of LD-ME increases as $w_{\mathrm{low}}$
decreases further below the true cluster weight. Furthermore, the plots show
that our approach does consistently outperform LD-ME in terms of both worst
estimation error and list size. Figure 10 also compares the performance of the
clustering algorithms in this experimental setup with results similar to the
ones obtained in the previous experimental settings.

###  I.2 Computational resources

Our implementation of the algorithm and experiments leverages multi-threading.
It utilizes CPU resources of an internal cluster with $128$ cores, which
results in a execution time of about $5$ minutes for a single run of the
experiment for one noise model with $10000$ samples. We remark that classic
approaches like $k$-means and DBSCAN perform fast and the most time-consuming
part is the execution of the LD-ME base learner. Given our experimental setup
with three noise models, it takes about $15$ minutes to reproduce all our
results for one data distribution.

True ClusterFake ClusterUniform noise Figure 8: Setup for $w_{\mathrm{low}}$
variation experiment with clusters contaminated by an adversarial cluster and
uniform noise. Lower color intensities indicate smaller cluster weights.

Figure 9: Comparison of list size and estimation error for small and large
inlier clusters for varying $w_{\mathrm{low}}$ inputs. The experimental setup
is illustrated in Figure 8. The plot on the top left shows the estimation
error for the small cluster and the plot on the top right shows the error for
the large cluster. We plot the median values with error bars indicating 25th
and 75th quantiles. As $w_{\mathrm{low}}$ decreases, our algorithm maintains a
roughly constant estimation error for the large cluster, while the error for
LD-ME increases.

Figure 10: Worst estimation error and list size comparison for the setup used
in the $w_{\mathrm{low}}$ variation experiment.