# Discovery of long-lasting auroral-like radio emission above a sunspot

Sijie Yu  New Jersey Institute of Technology, 323 M.L.K. Blvd, Newark, NJ
07102, USA<EMAIL_ADDRESS>Bin Chen  New Jersey Institute of Technology,
323 M.L.K. Blvd, Newark, NJ 07102, USA  Rohit Sharma  University of Applied
Sciences and Arts Northwestern Switzerland, 5210 Windisch, Switzerland
Timothy S. Bastian  National Radio Astronomy Observatory, 520 Edgemont Road,
Charlottesville, VA 22903, USA  Surajit Mondal  New Jersey Institute of
Technology, 323 M.L.K. Blvd, Newark, NJ 07102, USA  Dale E. Gary  New Jersey
Institute of Technology, 323 M.L.K. Blvd, Newark, NJ 07102, USA  Yingjie Luo
New Jersey Institute of Technology, 323 M.L.K. Blvd, Newark, NJ 07102, USA
School of Physics & Astronomy, University of Glasgow, Glasgow, G12 8QQ, UK
Marina Battaglia  University of Applied Sciences and Arts Northwestern
Switzerland, 5210 Windisch, Switzerland

###### Abstract

Auroral radio emissions in planetary magnetospheres typically feature highly
polarized, intense radio bursts, usually attributed to electron cyclotron
maser (ECM) emission from energetic electrons in the planetary polar region
that features a converging magnetic field. Similar bursts have been observed
in magnetically active low-mass stars and brown dwarfs, often prompting
analogous interpretations. Here we report observations of long-lasting solar
radio bursts with high brightness temperature, wide bandwidth, and high
circular polarization fraction akin to these auroral/exo-auroral radio
emissions, albeit two to three orders of magnitude weaker than those on
certain low-mass stars. Spatially, spectrally, and temporally resolved
analysis suggests that the source is located above a sunspot where a strong,
converging magnetic field is present. The source morphology and frequency
dispersion are consistent with ECM emission due to precipitating energetic
electrons produced by recurring flares nearby. Our findings offer new insights
into the origin of such intense solar radio bursts and may provide an
alternative explanation for auroral-like radio emissions on other flare stars
with large starspots.

## Main text

Stellar magnetism is not only essential for exploring the fundamentals of
stellar magnetic activity and internal dynamo, but also for understanding its
influence on stellar ecosystems, including potentially habitable exoplanets.
Radio observations of coherent emission offer unique diagnostics of physical
parameters in stellar atmospheres. Specifically, radio electron cyclotron
maser (ECM) emission occurs at low harmonics of the gyrofrequency
$\nu_{\mathrm{B}}$, providing direct measurements of magnetic field strengths
above the surface of stars. Radio ECM bursts, commonly observed in planetary
magnetospheres, are attributed to the ECM instability driven by the
precipitation of energetic electrons to the polar regions of the planets[1].
These radio bursts are distinguished by their highly polarized, broadband, and
strongly beamed nature. Similar, albeit more intense, radio emissions have
been identified in low-mass stars[2]. In particular, recent evidence of
periodic broadband radio pulses with high circular polarization from M-dwarfs
[2, 3, 4] favor the planetary hypothesis that these stellar radio bursts are
radio ECM emission generated in a predominately dipolar field’s low-density
cavity above the polar regions, in synchrony with the stellar rotation [5, 6].
If interpreted accurately, these findings offer a new avenue for probing
magnetic fields in the coronae of stellar and sub-stellar objects, a critical
parameter for understanding their internal dynamo and examining exo-space
weather and habitability[7].

A key challenge in comprehending these stellar radio bursts is characterizing
the emission source region due to the absence of spatially resolved
observations, except for rare instances from the Very Long Baseline
Interferometry (VLBI) observations [8, 9]. As the magnetic field topologies of
radio-ECM-emitting stars remain largely undetermined, it is debated whether
these emissions originate from a global dipole magnetic field driven by
auroral-like activities in the magnetosphere[10], or from localized magnetic
field structures, such as starspots, driven by flare-like magnetic activities
in the coronae[11, 12]. The Sun, owing to its proximity, offers valuable
context for studying radio ECM emissions analogous to those in (sub)stellar
systems. Thorough analysis of such radio emissions from the Sun could inform
our understanding of other stars and carry significant implications for
uncovering the origins of intense radio bursts in (sub)stellar systems,
particularly those with sizable starspots. However, while transient radio ECM
bursts have been sporadically recorded on the Sun[13], persistent radio ECM
emissions akin to those documented in stellar literature have yet to be
observed on the Sun.

## Results

We observed the Sun using the Karl G. Jansky Very Large Array (VLA) on 2016
April 9 in both the L band (1–2 GHz) and the S band (2–4 GHz) using two sub-
arrays. In the 1–2 GHz band, we detected a radio emission event from a bipolar
sunspot group, dominated by a large leading spot with negative polarity (Fig.
1). The dynamic spectrum of the radio source (Fig. 2a) obtained by the VLA
reveals long-duration, repeated bursts across a broad frequency range from 1
to 1.7 GHz, persisting for the entire 4.5–h observation, with distinct bursts
occurring at an average rate of $\sim$12 $\mathrm{h^{-1}}$. These bursts are
also found in simultaneous 1 GHz solar radio flux data collected by the
Nobeyama Radio Polarimeters (NoRP) and extend down to 245 MHz over an entire
week when data from the Radio Solar Telescope Network (RSTN) is available
(Extended Data Fig. 1&2), indicating their broadband nature
($\delta\nu/\nu>150\%$). The radio bursts appear “auroral-like”, as their
temporal, spectral, and polarization characteristics, including high circular
polarization fraction and brightness temperature, wide bandwidth, and extended
duration, resemble the documented planetary aurora radio emissions and stellar
radio ECM emissions.

The sunspot group, part of NOAA active region (AR) 12529, is 52∘ east and 10∘
north of the solar disk center. The radio source, $\sim 40^{\prime\prime}$ (30
Mm) from the polarity inversion line (Extended Data Fig. 3), coincides with
sporadic flare activity observed by Geostationary Operational Environmental
Satellites (GOES) in X-ray and the Atmospheric Imaging Assembly (AIA[14])
onboard the Solar Dynamics Observatory (SDO) in (extreme) ultraviolet ((E)UV).
The peak flux density of the radio source in stokes I is approximately 20 sfu
(1 sfu=$10^{4}$ Jy) at 1 GHz and 2000 sfu at 245 MHz (Fig. 3(a)), with an
equivalent radiation brightness temperature of $10^{11}\,\mathrm{K}$ at 1 GHz
and $10^{12}\,\mathrm{K}$ at 245 MHz, based on an inferred source size of
1,000 km (Method). The brightness temperature is several orders of magnitude
greater than the thermal temperature of the flaring plasma in the active
region $\mathrm{T}\approx 6\times 10^{6}\,\mathrm{K}$, constrained by the
emission measure analysis based on the GOES X-ray data (Methods). Notably, the
temporal variation of the radio luminosity from the sunspot source is poorly
correlated with the X-ray luminosity of the flares (see Fig. 2 and Extended
Data Fig. 1&2), suggesting that the bulk of the radio emission is not directly
generated by the thermal and nonthermal electron population trapped in the
flare sites.

The overall radio flux density spectrum of the active region (Fig. 3(a))
consists of two distinct spectral components, exhibiting a minimum near 2 GHz
(Methods). We attribute the radio emission above the frequency minimum to
incoherent gyrosynchrotron radiation emitted by mildly relativistic electrons
energized by the flare process in the AR. The 1-minute VLA snapshot at 3.8 GHz
obtained at 21:52 UT shows a compact source near the bright EUV flare arcade
(Fig. 1) with a relatively low degree of circular polarization $\lesssim 20\%$
(Fig. 3(b)). Its source location is consistent with microwave imaging
spectroscopy observations of other flares at similar frequencies, which have
been interpreted as gyrosynchrotron radiation [15]. Given the proximity of the
source to the flare arcade, low polarization degree, and flux density, we
conclude that the radio emission at frequencies above 2 GHz is mainly
gyrosynchrotron emission.

Figure 1: Auroral-like radio source seen from a sunspot associated with solar
flares. a, Example of a VLA snapshot image at 1.0 GHz (orange) of the radio
emission from the sunspot with the image of the radio-hosting NOAA 12529 AR
observed in EUV wavelengths by the Atomospheric Imaging Assembly (AIA) 171 Å
(blue), 94 Å (cyan) and 1600 Å (red) overlaid on the photospheric image
observed by the Helioseismic and Magnetic Imager (HMI) aboard the Solar
Dynamic Observatory (SDO). Also shown is a VLA 3.8 GHz image (magenta) of the
radio emission from the flare site. b, Closer view of the sunspot region (box
in a). The 1.0 GHz and 3.8 GHz are shown as yellow and magenta contours,
respectively, at 50%, 70%, and 90% of the maximum

.

For frequencies below 2 GHz, the emission is dominated by the highly variable
burst component. From 1.7 GHz to 245 MHz, the emission spectrum shows an
increase of three orders of magnitude in flux density toward lower frequencies
(Fig. 3(a)). The emission is nearly $100\%$ right-hand circularly polarized in
the frequencies between 1.0 to 1.4 GHz (Fig. 3(b)). Moreover, the flux density
of the highly variable burst component does not follow the Güdel–Benz (GB)
radio/X-ray relation, which describes a tight relationship between the
observed X-ray and radio intensity from solar and stellar coronae[16]
(Methods). Specifically, the ratio of the radio to X-ray luminosities for
frequencies below 1 GHz violates the GB relation by up to more than two orders
of magnitude (Fig. 3(c)). Using dynamic imaging spectroscopy with VLA, we have
made images at hundreds of frequencies from 1 to 1.5 GHz at numerous selected
times. During burst periods, all radio images are dominated by a localized
radio source near the sunspot, exhibiting clear dispersion in space at
different frequencies. We derive the source centroid of each radio image made
at a given frequency. In Fig. 4(a), we show the spatial distribution of the
centroid locations by frequency for seven example time intervals, each
spanning a duration of 20 seconds. The frequency-dependent source centroid
locations of each burst form a close-to-linear dispersion in space, with high-
frequency ends oriented towards the sunspot. Despite the overall location of
the radio source relative to the sunspot remaining relatively stationary
throughout the entire duration of the burst group, the distribution of
centroid locations of individual bursts occurring at different times displays
a rapid variation in the frequency-spatial dispersion and an evident spread in
their position angles regard to the sunspot (Fig. 4(h)).

Figure 2: The relationship between X-ray and radio emissions. a, Dynamic
spectra of the right circularly polarized radio emission detected from the
sunspot, as detected by the VLA. b, Time profiles of GOES 1.5–12 keV (black
line) and 3–24 keV X-ray (gray line) X-ray, with corresponding GOES X-ray
flare classes indicated. c, Time profile of the total radiative energy loss
and the in-band (1–2 GHz) radio power for an isotropic emitter. Grey shaded
bars represent gaps in the radio observations. Figure 3: Synoptic radio
spectra of the NOAA 12529 AR. a, Stokes $I$ spectra detected with VLA, RSTN
and NoRP measured between April 9–13, 2016. The peak spectra of the highly
variable burst component measured by RSTN and NoRP at frequencies below 2 GHz
on April 10 and 12 are shown as filled and open blue circles with error bars,
respectively. The spectra at frequencies above 2 GHz during April 9–13,
dominated by incoherent gyrosynchrotron emission, are shown as gray circles.
The error bars represent the $5\sigma_{rms}$ and $1\sigma_{rms}$ uncertainties
at each frequency for frequencies below and above 2 GHz, respectively. b, The
polarization degree (Stokes $V/I$) of the radio emission from the active
region detected with VLA as a function of frequency. c, Güdel–Benz (GB)
relation between radio luminosity $L_{R}$ and X-ray luminosity $L_{R}$ for a
wide range of magnetic active stars (gray symbols). Stellar radio emissions at
frequencies of a few GHz and hundreds of MHz are represented as yellow squares
and circles, respectively. The sunspot radio emission in this study is denoted
by the blue circle. Figure 4: 3D modeling of NOAA 12529 AR and the associated
ECM emission sites. a, Detailed view of the sunspot auroral radio source
(white box in Fig. 1b) at seven example times. The spatial distribution of the
ECM sources appears as the convergent magnetic field lines about the sunspot
is clearly delineated by the ECM sources of different burst events. The
grayscale background is the SDO/HMI continuum image of the photosphere. b, 3D
magnetic field model of the AR, with $X$\- and $Y$-axes pointing to
heliographic longitude and latitude on the solar disk, respectively, and
$Z$-axis pointing radially from the Sun’s center. White curves represent
selected field lines for the closed fields connected to the sunspot umbra.
Similar to Fig. 1, reddish sources on the photospheric surface represent flare
loop footpoints observed by AIA at 1600 Å. c, Side view of the magnetic field
model viewed towards positive $Y$-axis. The green dots above the sunspot
denote the sites of the second harmonic ECM emission sources at
$\nu_{\mathrm{GHz}}=1.2$ at selected field lines, outlining an iso-gauss dome
of $B_{\mathrm{G}}=214$. d, Same as (c), but showing the ECM emission sites
spanning a magnetic field strength from 179 to 268 G, colored according to
field strength $B_{\mathrm{G}}$, which directly maps to ECM frequency
$\nu_{\mathrm{GHz}}$ from 1.0 to 1.5 GHz using the relation
$B_{\mathrm{G}}\approx 357\nu_{\mathrm{GHz}}/s$, where $s=2$ for the second
harmonic ECM emission. e–g, Same as (d), but viewed towards negative $X$-axis,
negative $Z$-axis, and the LOS, respectively. h, Same as (g), but overlaid
with gyro-resonance opacity. Colorbars of magnetic field strength and opacity
shown in (c–h) are displayed alongside. Figure 5: Schematic sketch of the
sunspot auroral radio emission. Physical scenario of electron acceleration by
flaring activity, resulting in sunspot radio aurorae. Individual sunspot-
rooted magnetic field lines filled with trapped electrons are responsible for
individual auroral radio emission events.

The high circular polarization degree and brightness temperature both confirm
that the observed emission is coherent in nature. We employ data-constrained
coronal magnetic field and plasma density modeling to investigate the physical
properties of the emission site (Methods). We find a region above the sunspot
with a ratio of the electron plasma frequency $\nu_{\mathrm{pe}}$ and the
electron gyrofrequency $\nu_{\mathrm{ce}}$ smaller than unity
($R=\nu_{\mathrm{pe}}/\nu_{\mathrm{ce}}<1$; Extended Data Fig. 8(c)).

The distribution of the predicted radio source locations in frequency (Fig.
4(h)) aligns well with the spectro-spatial characteristics of the observed
radio sources (Fig. 5a). The near-linear spatial dispersion as a function of
frequency is well reproduced by individual unresolved emission sites aligned
with the magnetic field lines. The magnetic field strength increase towards
the sunspot accounts for higher-frequency sources (emitted from higher $B_{\rm
G}$ regions) being closer to the sunspot. The overall spectro-spatial
distribution of the source location of all the bursts with different position
angles can be explained by the projection of distinct radio-hosting magnetic
field lines into the plane of the sky (Extended Data Fig. 10 and Methods).

Our analysis suggests most radio-hosting field lines have inclination angles
of $30^{\circ}$ to $60^{\circ}$ relative to the line of sight (Extended Data
Fig. 8(f)&10), consistent with the possible range for second-harmonic ECM
emission to escape through a transparent window at the overlying absorbing
layer at the third harmonic (Extended Data Fig. 8(f)). We note that the
visibility of the observed radio emission is likely a geometric coincidence of
the beaming angle, the source location, and the transparent opacity window,
which are subjected to change with time due to solar rotation and temporal
evolution of the active region. A consequence of the rather stringent
visibility requirement is that the radio emissions are likely only visible for
a small fraction of the time over the entire course of the presence of the
active region on the solar disk. The small filling factor in time may also
explain why the sunspot radio ECM emission is observed only when the sunspot
is between longitudes of $-70^{\circ}$ to $-10^{\circ}$ (east hemisphere) and
sharply diminishes afterward, despite an increased flare activity during the
sunspot’s west hemisphere transit (Extended Data Fig. 2; Methods).

## Discussion

Our interpretation of the observation is illustrated in Fig. 5(c). The
observed coherent radio emission is interpreted as ECM emission induced by
persistent energetic electrons trapped in the large-scale magnetic loops
anchored at the sunspot. Analogous to the case of planetary radio aurora
emissions[17], the converging magnetic field above the sunspot can form a low
$R$ cavity, providing a favorable geometry for radio ECM emission. Energetic
electrons in the loops with initial pitch angles exceeding a threshold reflect
at the magnetic mirror on the sunspot end of the loops, developing an
anisotropic velocity distribution unstable to ECM emission in the cavity. The
source energetic electrons are likely associated with the sporadic flare
activity occurring in the nearby active region, which is responsible for
accelerating electrons to high energies and supplying them into the
overarching magnetic loop system. The persistent radio emission suggests a
continuous, albeit sporadic, operation of maser radiation, necessitating
frequent replenishment of energetic electrons to maintain the electron
anisotropy in the radio source. We speculate that weak yet frequent flare
activity in the AR without apparent X-ray enhancement may drive electron
acceleration[18, 19].

The absence of spatial coincidence and tight temporal correlation between
flare activity and the coherent radio emission implies that the ECM emission
is not directly produced by energetic electrons at the flare site. Instead,
the flare-accelerated electrons may have entered and become trapped in the
overarching magnetic loops before precipitating to the ECM source region.
During this process, transport effects such as collision, scattering, and
trapping in the large-scale loops can act as an electron “reservoir”
decoupling the time correlation between the flare activities and radio ECM
bursts above the sunspot. At the same time, the energetic electrons can be
scattered into the loss cone and enter the region above the sunspot, driving
and modulating the ECM emission. We also note that magnetic field
perturbations in the loop system, potentially caused by propagating
disturbances originating from the sunspot[20, 21] or the flare site[22] with a
period of 3–5 minutes, may affect the strength and direction of the magnetic
field within the source region. Since ECM emission heavily relies on the local
magnetic field, these perturbations could result in additional modulation in
the observed temporal variability of the ECM emission.

The in-band radio power for an isotropic emitter, integrated over the L band
(1–2 GHz), increases of up to 3.5 $\times
10^{18}\mathrm{erg}\,\mathrm{s}^{-1}$ during peaks against the pre-burst
baseline level of $\sim$0.5 $\times 10^{18}\mathrm{erg}\,\mathrm{s}^{-1}$
(Fig. 2(c)). If the radio emission at frequencies below 1 GHz follows the
spectral shape (Fig. 3(a)) recorded on April 12—the most powerful bursts
recorded in the solar rotational cycle—the total radio power would be 100
times higher, reaching $10^{20}\mathrm{erg}\,\mathrm{s}^{-1}$. The flares
release a total radiative energy of an order of $10^{26}\,\mathrm{ergs\
s^{-1}}$ as determined by X-ray diagnostics (Fig. 2(c)). Solar flares
observations suggest that a significant fraction of flare energy is initially
converted into non-thermal electron kinetic energy and subsequently
transformed into various energy forms[23]. Previous studies of solar energetic
electrons observed at 1 AU suggest 0.01–1% of the flare-accelerated energetic
electrons escape from the energy release sites and enter the interplanetary
space[24]. Using this fractional value as the lower limit to estimate the
total power of the energetic electrons trapped in the corona, we obtain an
average power of $>\\!10^{22}$–$10^{24}\,\mathrm{ergs\,s^{-1}}$, more than two
orders of magnitude greater than that of the radio emission. Therefore, we
conclude that the flare-supplied energetic electrons should contain adequate
energy to power the observed coherent radio emission.

Our findings may have important implications for the interpretation of similar
radio emissions observed at other stars and star-planetary systems[25],
because they share similar spectral-temporal properties and the deviation of
the GB radio/X-ray correlation [3, 26, 27]. Stellar ECM emissions are
generally explained within the paradigm of planetary auroral radio emission
from a dipolar magnetosphere, driven by either star-planet interactions[26] or
co-rotation breakdown of a plasma disk[28] for cooler dwarf stars, and
centrifugal breakout reconnection for rapid-rotating early type stars[29].
These stars feature magnetic activity[30, 31] and are notable for the presence
of starspots[32]. If these stars have a continuously replenished population of
energetic electrons due to flaring activity, these electrons would have the
opportunity to be trapped above a starspot and facilitate the growth of ECM
emission, even without a globally dipolar magnetic field geometry. This
scenario may also be viable for interpreting certain radio emissions from the
apparently non-flaring, quiescent stars[2, 26], which show a similar magnitude
of deviation from the tight GB relation in their radio/X-ray correlation: a
large number of small-scale energy release events may be at work to supply
energetic electrons sufficient for powering a persistent radio source under
favorable emission conditions. This scenario is akin to our case in which many
episodes of intense radio bursts are observed during the quiescent period when
there is no flare activity. However, we note that the interpretation may not
be applied toward the ECM radio aurorae on the coolest ultracool dwarfs[33,
34, 35, 36, 37], which lack starspots with a stable loop and flaring energetic
electron acceleration. We also note that our observed bursts above the sunspot
are three orders of magnitude weaker than those reported on certain flare-
active M-dwarfs with extremely high brightness temperatures (up to 1014 K for
those reported by, e.g., [27]; see Methods for more discussions). While such a
difference in radiation intensity is an outstanding topic for further
investigation, we suggest that the extremely high magnetic field strength and
activity level on these M dwarfs could be a contributing factor.

Periodic radio emission occasionally observed from stars, coinciding with
stellar rotation, has been interpreted as emanating from their polar
regions[2, 10, 3, 38, 5, 39, 40], similar to planetary auroral emission. Given
the presence of a persistent starspot lasting for multiple rotations [32] and
recurring flaring activity supplying energetic electrons, we argue that the
sharply-beamed ECM emission above the starspots can also lead to auroral-like
radio bursts with the rotation period of the host star. It is noted that the
emission may also be modulated by the short- and long-term variability of the
flare activity of the host star, which may be responsible for the observed
variations in the burst duty cycle across different timescales detected from
some stars [41, 3, 42, 43, 44] (Methods). Previously observed stellar ECM
emissions may not be interpreted exclusively as magnetospherically-driven
auroral emissions from the polar regions, but could also originate from
regions above strong starspots. In sum, our results not only clarify the
origin of a type of radio burst on the Sun, but also prompt a re-examination
of the physical mechanisms behind certain auroral-like radio emissions from
other stars. This study also highlights the need for more detailed
observations in the future to better understand the intricacies of the ECM
emission on the Sun, including the responsible plasma instability, local
driver of the ECM (e.g., a parallel electric field[45]), and energy efficiency
of the ECM emission, which remains largely unexplored.

## Methods

### GOES X-ray data reduction

The X-ray Sensor (XRS) onboard the Geostationary Operational Environment
Satellite (GOES) continuously monitors the visible solar hemisphere in two
energy bands, one in 1–8 Å (or 1.5–12 keV) and another in 0.5–4 Å (or 3–24
keV), both of which are dominated by thermal continuum emission. The GOES/XRS
data were reduced using the standard analysis tool available in the
SolarSoftWare IDL (SSWIDL) package
(https://www.lmsal.com/solarsoft/ssw_setup.html). Because AR 12529 is the only
flaring AR at the time of the radio observation, we attributed the impulsive
X-ray emission shown in Extended Data Fig. 1(bottom) to thermal Bremsstrahlung
emitted by the coronal plasma heated by the flare activity in this AR. We
subtracted a constant background from the XRS time series, and compute the
total radiative energy loss rate shown in Fig. 2(c) from the expressions [46]
available in the SSWIDL GOES tool. We note The mean square root flux
uncertainty of the 1.5–12 keV and 3–24 keV channels is $5\times 10^{-9}$ and
$2\times 10^{-10}$ $\mathrm{W\,m^{-2}}$, respectively, each of which is two
orders of magnitude lower than the corresponding background level.
Additionally, since the flares exceed the background level by one order of
magnitude at their peak, the detailed choice of the background for subtraction
does not impact the results.

### Radio data reduction: VLA

The observation was made when VLA is in C configuration, for which the longest
antenna baseline is over 3 km, yielding an angular resolution of
$17^{\prime\prime}.3\times 29^{\prime\prime}.3/\nu^{2}_{\mathrm{GHz}}$ at the
time of the observation. We observed the Sun using two subarrays in dual-
polarization mode at 50 ms cadence, with 512 and 1024 frequency channels of 2
MHz spectral resolution, covering 1–2 GHz (L band; 14 antennas) and 2–4 GHz (S
band; 13 antennas), respectively. The flux calibration, bandpass calibration,
and complex gain calibration are performed using a standard celestial
calibrator 3C48. When observing the Sun, 20 dB attenuators were switched into
the signal path of each antenna and each polarization. The delay and amplitude
introduced by the attenuators were removed following the technique described
in Chen et al (2013) [47]. Data were reduced using the Common Astronomy
Software Applications (CASA Release 5.4; http://casa.nrao.edu).

The data contaminated by radio frequency interference, particularly those near
GPS L1 frequency (1.58 GHz), are flagged. The slow-varying background emission
contributed by the quiescent Sun is subtracted from the VLA data. The
spectrotemporal intensity variation (dynamic spectrum) intrinsic to the
sunspot radio source are estimated by averaging the amplitude of the complex
visibility data over short baselines (0.2–0.8 km) that are sensitive to the
spatial scale of the active region. The radio source is located at an angular
distance of $13^{\prime}.4$ from the antenna pointing at solar disk center,
which is comparable to the size of the primary beam of the VLA antennas
$\theta_{\rm PB}=45^{\prime}/\nu_{\rm GHz}$ in the frequency range of 1–4 GHz,
where $\nu_{\rm GHz}$ is observing frequency in GHz. To ensure the absolute
flux density of the source is correct, we scale the observed flux density of
the VLA dynamic spectrum at L and S band up to that at the antenna pointing
center according to the primary beam response (a Gaussian beam pattern is
used). An overview of the radio emission from the sunspot in the L band is
shown in Extended Data Fig. 4 for the 4.5 hr duration of the observation.
Extended Data Fig. 5 shows a detail of the L band dynamic spectrum of 1 minute
observation after 21:48 UT.

Radio imaging of each pixel in the dynamic spectrum where the bursts are found
provides key information on the spatial variation of the radio source as a
function of time and frequency. We produce independent radio images at all the
spectral channels between 1.0 to 1.5 GHz for every four time pixels in the
dynamic spectrum using the CASA task tclean. We note a periodic pattern is
present in the observed visibility phases as a function of time. This is due
to the telescope not updating its geometric delays at a rate that matches the
time-cadence of the data, known as the so-called “delay clunking”, which
introduced an offset with regard to the true phase center a fixed rate of
$-2^{\prime\prime}.5\mathrm{/min}$ in right ascension. This effect is
mitigated by inserting an offset of $2^{\prime\prime}.5\mathrm{/min}$ along
the same direction to the sky coordinates of the radio images. It is known
that the absolute position of the source can be determined to a small fraction
($\sim 10\%$) of the synthesized beam under typical conditions, due to
atmospheric phase stability, the SNR on target, etc. We verified the alignment
accuracy of the radio images (without background subtraction) using features
in the NOAA 12529 active region seen in both radio and EUV wavelengths. Since
the source of the sunspot radio bursts is only marginally resolved by VLA, we
treat the radio source above 50% of the maximum as a point-like source. The
uncertainty of the centroid location of the source $\sigma$ can be determined
by[48]:

$\sigma\approx\theta_{\mathrm{HPBW}}/(\mathrm{SNR}\sqrt{8\ln{2}})\ ,$ (1)

where $\theta_{\mathrm{HPBW}}$ is the half-power beamwidth of the synthesized
beam and SNR is the signal-to-noise ratio of the image. With a typical SNR
value for the bursts exceeding 100, we can infer a position accuracy to
$\lesssim 0^{\prime\prime}.2$. However, the propagation of radio waves through
the inhomogeneous corona toward the observers is known to be susceptible to
scattering effects[49, 50]. Therefore, the estimate of uncertainty given above
should only be considered as a lower limit. In fact, by obtaining the centroid
locations of all frequency–time pixels on the sunspot radio bursts within a
short time period (40 s) and frequency range (40 MHz), we find that they are
distributed rather randomly within an area of FWHM size of $\sim
1^{\prime\prime}.5\times 1^{\prime\prime}.5$ (Extended Data Fig. 10). Hence,
we estimate the actual position uncertainty of the centroids as $\sigma\approx
1^{\prime\prime}.0$.

Extended Data Fig. 1: Radio and X-ray flux of the radio-burst-hosting active
region (NOAA 12529 AR) observed by RSTN, NoRP, and GOES on 2016 April 9–10.
Top panel: The background subtracted solar radio flux at discrete frequencies
monitored by RSTN, NoRP and VLA. The horizontal dotted lines in corresponding
colors indicate equally spaced zero-lines for each frequency channel, set at
20$\times 10^{4}\,\mathrm{Jy}$ (or 20 sfu) intervals. The error bars overlaid
on the zero-flux-lines in corresponding colors show the $3\sigma_{rms}$
uncertainties of each frequency channel. The upward arrows indicate the most
prominent examples of sunspot radio aurorae, while the downward arrows
indicate the occurrence of flare events. The inset shows a blowup of the 1 GHz
flux simultaneously observed by NoRP and VLA, denoted by the dashed box. No
zero-line offset was applied for the inset. Bottom panel: Simultaneous time
profiles of GOES 1.5–12 keV (black line) and 3–24 keV X-ray (grey line), with
corresponding GOES X-ray flare classes indicated. The arrows are the same as
the downward arrows in the top panel. Extended Data Fig. 2: Flux profiles of
radio-hosting active region (NOAA 12529 AR) over one solar rotation cycle. a,
SDO/HMI continuum image sequence showing the transiting sunspot at times
marked by the vertical arrows in (b). b–e, Time history of the heliocentric
angle $\theta$ of the sunspot, the radio, and X-ray emission of the Sun over
one solar rotation. f, Blowup of the synoptic dynamic spectrum of Stokes I
obtained by NoRP and RSTN in (c), showing the arcs (red dashed lines) of radio
emission. g–h, Hierarchical substructures of the emission arcs shown in (f)
obtained by VLA. Extended Data Fig. 3: Photospheric images of NOAA 12529 AR.
a–b, Continuum intensity image at 6173 Å at the solar surface, obtained by the
HMI on board SDO. c–d, Radial component of the photospheric magnetic field
strength. In the right column, the coherent (L-band) and incoherent (S-band)
radio sources are shown as yellow and purple, respectively. Contours for the
two radio sources at 50%, 70%, and 90% are overlaid as black solid and blue
dashed lines. The size of the synthesized beams for the L and S band images
are displayed in the upper-left corner with the corresponding line styles. The
L- and S-band radio images were made using VLA observation on 2016 April 9 at
22:08 and 21:53 UT, respectively. The flare loops and their footprints, as
observed by SDO/AIA 94 and 1600 Å, are shown in blue and red, respectively.
Extended Data Fig. 4: Dynamic spectra of the right circularly polarized radio
emission detected from the sunspot with VLA on April 9, 2016. Extended Data
Fig. 5: Details of the dynamic spectrum between 21:48 and 21:49 UT on April 9,
2016.

### Radio data reduction: RSTN and NoRP

USAF Radio Solar Telescope Network (RSTN) monitors the sun-as-a-star radio
flux at 8 fixed frequencies — 0.245, 0.41, 0.61, 1.415, 2.695, 4.995, 8.8 and
15.4 GHz — at a cadence of one second. The data in use were recorded by the
RSTN Learmonth site. The daily radio fluxes at frequency $\nu_{\rm GHz}>1$
show an overall inverted-U shape, which is mainly due to the varying zenith-
angle-dependent opacity owing to atmospheric absorption of the radio waves. To
correct for this effect, we first fit the day-to-day variation of the local
noon radio flux at each frequency by using a third-order polynomial function
over one solar rotation cycle from 2016 April 6 to 2016 May 2. Next, the daily
inverted-U shape of the radio light curves in each frequency due to
atmospheric absorption were fit by using an eighth-order polynomial function
and subsequently removed. Finally, the resulting light curves are scaled to
the fitted day-to-day noon flux curve. In addition, the Nobeyama Radio
Polarimeters [51] (NoRP) perform full-stokes polarimetric observations of the
Sun at six frequencies — 1, 2, 3.75, 9.4, 17, 35 GHz — with a temporal
resolution of 0.1 second. Data was reduced using the NoRP package in the
SolarSoftWare IDL package (https://lmsal.com/solarsoft/ssw/radio/norh/). We
used a similar data processing method to correct the daily shape presented in
the NoRP data.

In the absence of flares, the gradually varying background presented in the
reduced RSTN and NoRP radio data at frequencies above 2 GHz is contributed
mainly by the quiescent active region and the solar disk due to thermal free-
free and gyroresonance emission. Since the background variations have a time
scale that is much longer than that of flares, a high-pass filter was then
used to remove the slow-varying components from the RSTN and NoRP data. To
determine the flux uncertainty of each frequency channel, we calculated the
root mean square ($\sigma_{rms}$) of the temporal fluctuations during two
quiescent time intervals of 3–4 UT and 6–7 UT on April 8.

The Extended Data Fig. 1(top) presents the resulting daily solar radio fluxes
on April 9–10, 2016. The frequency range between 1 to 2 GHz is dominated by
the repetitive radio bursts with a typical cadence of 5–10 minutes, which are
not well correlated with the X-ray flux, as depicted in Fig. 2a. The Extended
Data Fig. 1(top) and its inset panel show the background-subtracted data at 1
GHz obtained simultaneously by both the VLA and NoRP. The 1 GHz light curves
from NoRP show a deviation no greater than 50% from that obtained by VLA,
indicating a high degree of agreement between the two datasets and further
verifying the robustness of the background subtraction approach. Despite the
VLA observation ending at 22:50 UT on 2016 April 9, the same emission feature
observed by the VLA is still evident in the NoRP at 1 GHz after 22:50 UT.
Furthermore, we show the fluxes below 1 GHz obtained by RSTN in Extended Data
Fig. 1(top). The burst component in the 1 GHz flux, denoted by the upward
arrows, is also notably presented in fluxes at lower frequencies down to 0.245
GHz. The temporal characteristics of the light curves when the bursts occur
are similar, indicating the broadband nature of the bursts that should extend
to well below 1 GHz. This broadband nature aligns with the rising spectrum
toward lower frequencies shown in the VLA data of 1–2 GHz (L band). However,
it is worth noting that the RSTN data lacks polarization measurements. As a
result, in our interpretation, we have heuristically assumed that the radio
bursts observed in the non-VLA data at frequencies below the L band are
circularly polarized coherent radio emissions. We made this assumption based
on two facts: 1) the bursts observed by the VLA, NoRP, and RSTN have a similar
temporal profile at distinct frequencies below 2 GHz, and 2) the flux density
of the bursts is too high for incoherent radio emission without major solar
eruptive events.

Overall, the observations presented in Extended Data Fig. 1 demonstrate a
consistent temporal behavior across a broad range of frequencies. The
agreement between the VLA and NoRP data at 1 GHz, along with the detection of
bursts at lower frequencies by the RSTN, provides compelling evidence for the
broadband nature of these bursts. However, further studies incorporating
polarization measurements and high-spectral resolution in a wider frequency
range are necessary to gain a comprehensive understanding of these bursts and
their polarization properties at different frequencies.

Extended Data Fig. 2 shows the time history of the solar radio and X-ray
emission for an entire solar rotation period. The temporal and spectral
variation of the radio emission indicates that the same coherent emission
feature persists for over an entire week (or $\sim$1/4 of the rotation period;
demarcated by the yellow shaded area in Extended Data Fig. 2(b, d–e)). The HMI
continuum image series in Extended Data Fig. 2a show that AR 12529, which the
sunspot belongs to, is the only AR on the disk when the sunspot radio emission
is present, therefore we attributed the impulsive X-ray emission shown in
Extended Data Fig. 2(e) to the flare activity in this AR. The GOES X-ray flux
level displays sporadic flare activity during the time when the sunspot moves
across the solar disk, and drops to the background level after the sunspot
moves to the back side of the Sun. The radio emission shows little correlation
with that of the X-ray emission from flare activity in the AR. In fact, the
radio emission is only visible for a limited duration during the rotation
cycle, specifically when the sunspot is located between longitudes of -70∘ to
-10∘. The temporal, spectral, and polarization characteristics of the radio
emission, including broadband arc-like structures in the time-frequency domain
(Extended Data Fig. 2f), closely resemble ECM emissions reported in the
literature from (sub)stellar systems[5, 4]. Most notably, the sun-as-a-star
light curves throughout a solar rotation, primarily modulated by the coherent
sunspot radio bursts (Extended Data Fig. 2(c–d)), display a striking
similarity to rotationally-modulated periodic radio aurora from M-dwarfs[6].
The similarity is mainly evident in the temporal patterns in radio emissions.
In both cases, the radio emissions exhibit rotationally modulated behavior,
which corresponds to the rotation of the respective celestial bodies.
Specifically, in the solar case, the burst duty cycle—the fraction of time
spent in bursting—is approximately 20% in the L and P bands. This value is
comparable to those of coherent radio bursts observed in M-dwarfs within the
same frequency range, as demonstrated in previously reported stellar
instances.[6, 25].

The radio bursts from the sunspot contribute significantly to the flux
variations at frequencies below 2 GHz, while at frequencies above 2 GHz, the
temporal fluctuations are mainly caused by incoherent gyrosynchrotron emission
generated by energetic electrons resulting from solar flares. To obtain the
RSTN and NoRP data points in the radio burst spectra shown in Fig. 3, two
different approaches were adopted for frequencies below and above 2 GHz,
respectively. The upward arrows in Extended Data Fig. 1(top) denote examples
of sunspot radio bursts at 1 GHz that occurred on April 9–10. We easily
identified the burst component against the noise levels of the time series,
and used the peak-to-trough amplitude of the most prominent burst that
occurred at 03:30 UT on April 10 as the radio flux density of the
corresponding frequencies. Nonetheless, it is important to highlight that this
method inherently eliminates any constant flux component of the sunspot radio
emission. As such, the flux density at frequencies below 2 GHz should be
interpreted as a lower limit only. The resulting spectrum in the 1–2 GHz range
deviates no more than $\pm$50% from the maximum of the VLA background
subtracted spectra obtained between 21:45 and 22:50 UT on April 9. We used the
same method to obtain the spectrum of the strongest sunspot radio burst in
this solar rotation cycle, which was recorded at 22:58 UT on April 12. The
downward arrows in Extended Data Fig. 1 denote the X-ray flare events that
occurred on April 9–10. For frequencies above 2 GHz, the radio light curves of
the flares are just above or even comparable to the noise levels. Therefore,
we took the peak fluxes of individual frequencies in the entire time interval
when the sunspot radio emission is present (April 9–13). However, we note that
the low signal-to-noise ratio may introduce bias in the flux estimation for
frequencies above 2 GHz. Thus, the flux estimated at these frequencies should
be considered as upper limits.

### The radio–X-ray Güdel–Benz relation

The Güdel–Benz (GB) empirical relation connects the radio and X-ray emission
for a broad range of magnetically active stars, with the relationship
$L_{R}\approx 10^{-15.5\pm 0.5}\leavevmode\nobreak\ L_{X}\leavevmode\nobreak\
\mathrm{Hz}^{-1}$, where $L_{R}\leavevmode\nobreak\
\mathrm{[erg\leavevmode\nobreak\ s^{-1}\leavevmode\nobreak\ Hz^{-1}]}$ denotes
radio luminosity and $L_{X}\leavevmode\nobreak\
\mathrm{[erg\leavevmode\nobreak\ s^{-1}]}$ represents X-ray luminosity within
soft X-ray energy range [16]. In Fig. 3c, we present the best linear fit to
the data from Benz & Güdel (1994)[16], as given by Berger et al. (2010)[52],

$L_{R}=1.36(L_{X}-18.97).$ (2)

However, as discovered in the past two decades, this correlation may not hold
true for certain late-type dwarf stars. The radio/X-ray correlation of stellar
ECM radio emission, spanning a broad frequency range between hundreds of MHz
and a few GHz as documented in the literature[52, 53, 27], are evidently
divergent from the GB relation. This deviation is exemplified by the yellow
scatters depicted in Fig. 3c.

The simultaneous radio and X-ray observations allow us to assess the observed
sunspot radio ECM bursts in the context of the GB relation. In order to draw a
comparison between the sunspot radio luminosity $L_{R}$, which reached its
peak at 245 MHz, and the solar X-ray $L_{X}$ luminosity, we calculate the
$L_{X}$ using the following approach. The $L_{X}$ of the Sun at the time of
the observation is dominated by the flare activity in NOAA AR 12529. We used
the SSWIDL GOES tool to model the background-subtracted X-ray emission and
estimate the total emission measure $EM$ and a temperature $T$ with an
isothermal model and photospheric elemental abundance. Subsequently, the
calculated $EM$ and $T$ were integrated into the Raymond-Smith plasma
model[54] to determine the total $L_{X}$ in the energy band of 0.2–2.0 keV—a
common used energy band in the literature[52, 53, 27] (Extended Data Fig. 6).
The radio/X-ray correlation of the sunspot radio emission is shown as the blue
circle in Fig. 3c, displaying a clear deviation from the GB relation by more
than two orders of magnitude, comparable to the observations of stellar ECM
radio emission documented in previous research[52, 53, 27].

Extended Data Fig. 6: Temporal evolution of the flare activity in NOAA 12529
AR on April 9th (left) and 12th (right). Displayed sequentially from top to
bottom are GOES 1.5–12 keV (black line) and 3–24 keV X-ray (grey line), the
derived plasma temperature $T$ and total emission measure $EM$ determined
based on the background-subtracted X-ray emission, as well as the X-ray
luminosity in 0.2–2.0 keV ($L_{X}$) employing the Raymond-Smith plasma model.

### Coronal magnetic field model

The coronal magnetic field above the sunspot is not directly measured using,
e.g., the Zeeman splitting technique, because the corona is extremely tenuous
and optically thin. To reconstruct the coronal magnetic field in three
dimensions, we perform a nonlinear force-free field (NLFFF) extrapolation
based on measurements of vector magnetic field at the photosphere. More
details of the assumptions and procedures for the NLFFF technique can be found
in other works[55]. In short, in the solar corona, the magnetic pressure
dominates over non-magnetic forces. Therefore, under a quasi-static
configuration (which is applicable to our case with a quasi-steady active
region), the corona can be approximated as in a force-free state. In this
case, reconstructing the coronal magnetic field can be well described as a
boundary value problem under the force-free field approximation[56]. The
vector photospheric magentic field measurements were recorded by the
Helioseismic and Magnetic Imager (HMI) instrument on board SDO [57] at 22:00
UT, processed with the standard pipeline[58]. In addition, a pre-processing
procedure was applied to reduce the net force and torque of the photospheric
magnetic field[59].

The original photospheric vector magnetogram data has an angular resolution of
$0.03\,^{\circ}\,\mathrm{pixel}^{-1}$ in heliographic coordinate system[60],
which was binned to $0.06\,^{\circ}\,\mathrm{pixel}^{-1}$ (equivalent to $\sim
720\,\mathrm{km}\,\mathrm{pixel}^{-1}$) for the subsequent extrapolation
performed in a Cartesian grid of $400\times 224\times 224$.

### Coronal density model

The corona density model above the sunspot is approximated using the plane-
parallel atmosphere assumption. The coronal electron number density as a
function of height $h$ above the solar surface $n_{\mathrm{e}}(h)$ is
prescribed using the Baumbach-Allen formula originally derived from white-
light observations during solar eclipses [61]:

$n_{\mathrm{e}}(h)=m\left[0.036(\frac{h+R_{\odot}}{R_{\odot}})^{-1.5}+1.55(\frac{h+R_{\odot}}{R_{\odot}})^{-6}+2.99(\frac{h+R_{\odot}}{R_{\odot}})^{-16}\right]\times
10^{8}\,\mathrm{cm}^{-3},$ (3)

where $m$ is a factor to account for the enhanced overall density above the
sunspot-hosting active region.

To determine an appropriate $m$ factor for our case, we note that the sunspot
first appears on the east limb two days prior to the time of the events under
study. The sunspot could be clearly distinguished as it rotated to its
position on April 9 2016 from the east limb and does not show any significant
morphological changes in the two-day period (Extended Data Fig. 2(a)),
implying that the coronal magnetic and density structures above the sunspot
should not vary significantly within the period. Thus, we further constrain
the density model using multi-band EUV images obtained by SDO/AIA when the
sunspot is located at the east limb. A differential emission measure (DEM)
analysis is performed using the regularized inversion method[62] based on
imaging data at six SDO/AIA EUV bands (94 Å, 131 Å, 171 Å, 193 Å, 211 Å, 335
Å). The results of the analysis are a distribution of the line-of-sight-
integrated total emission measure $\xi(x,y)=\int n_{e}^{2}(x,y)dz\approx
n_{e}^{2}(x,y)\,D$ in the plane of the sky, where $D$ is the column depth
weighted by the square of the density along the line of sight. In order to
derive the density distribution $n_{e}(x,y)$ from the DEM results, we take a
constant column depth of $D=50\times 10^{3}\,\mathrm{km}$. With the derived
limb-view $n_{e}(x,y)$ map shown in Extended Data Fig. 7(b), we can derive the
$n_{\mathrm{e}}(h)$ distribution above the active region shown in Fig. 7(c),
in which each data point at $h$ above the limb is obtained by averaging the
$n_{\mathrm{e}}$ values from the pixels enclosed in the black contours in
Extended Data Fig. 7(b). We find the sixfold Baumbach–Allen model is an
excellent fit to the $n_{\mathrm{e}}(h)$ inferred from the DEM analysis. We
caution, however, that the actual density distribution depends on the
selection of the column depth along the line of sight, which is largely
unknown. The column depth we adopted should only be considered as a rough
order of magnitude estimation. Therefore, the derived $n_{\mathrm{e}}(h)$
distribution may vary by a scale factor of a few if we select a different
column depth.

Extended Data Fig. 7: Electron number density distribution in the corona above
the sunspot on 2016 April 7 when the sunspot was located at the east solar
limb. a, SDO/AIA 171 Å passband image at 08:10 UT. b, electron density
$n_{\mathrm{e}}$ derived by EUV diagnostics. c, distribution of
$n_{\mathrm{e}}$ and the corresponding electron plasma frequency
$\nu_{\mathrm{pe}}$ in height $h$ obtained from regions marked in (b) for a
series of equally spaced radial heights above the sunspot. The red line
represents the sixfold Baumbach–Allen model. The error bars are obtained by
with the column depth $D$ ranging from 15–150 Mm.

### Emission mechanism of the sunspot radio bursts

There are two major competing emission mechanisms that can generate coherent
radio emissions in the solar corona. Coherent plasma radiation often dominates
the solar radio emission of solar flares in a wide frequency range spanning
from $\sim 1$ GHz all the way to tens of MHz. They are driven by the non-
linear growth of plasma instabilities and followed by a wave-wave conversion,
radiating at a frequency close to the fundamental or second harmonic electron
plasma frequency $\nu_{\mathrm{pe}}\approx 8980\sqrt{n_{e}}\ \mathrm{Hz}$. In
contrast, the electron cyclotron maser (ECM) emission likely dominates the
emission in regions of high magnetic field strength and/or low plasma density,
where the electron plasma frequency ($\nu_{\mathrm{pe}}$) is less than the
electron gyrofrequency ($\nu_{\mathrm{ce}}$) at the source site [63]. The ECM
mechanism has been suggested as the primary emission mechanism of auroral
radio emission in planetary magnetospheres, such as the terrestrial auroral
kilometric radiation [64], Jovian decametric radiation[65], and Saturn’s
kilometric radiation[66], as well as strong radio radiation with high circular
polarization in a variety of magnetically active stars such as M dwarfs [2, 3,
5]. ECM emission is generated near low harmonics of the electron cyclotron
frequency $s\nu_{\mathrm{ce}}=2.8\times 10^{6}sB\,\mathrm{Hz}$ (where $s$ is
the harmonic number and $B$ is the magnetic field strength in Gauss) due to
non-linear wave growth under a variety of plasma instabilities including loss-
cone, ring shell, or horseshoe instabilities[17]. We recognize that details of
the ECM model, including the responsible plasma instability, local driver of
ECM (e.g., a parallel electric field, as suggested by Ergun et al. 2000[45]),
and energy efficiency of the ECM emission remain largely unexplored.
Addressing these details requires further, and preferably, in situ
measurements to characterize the local magnetic and electric field, electron
momentum and angular distribution, and other plasma parameters at the radio
emitter[45].

Extended Data Fig. 8: Physical characteristics in a 2D slice on the $X$–$Z$
plane of the 3D model of NOAA 12529 AR. The location of the $X$–$Z$ plane is
indicated by the semitransparent gray shade in Extended Data Fig. 4(a). a,
electron plasma frequency $\nu_{\mathrm{pe}}$. b, electron cyclotron frequency
$\nu_{\mathrm{ce}}$. The red contours from top to bottom show the frequencies
of 0.125, 0.25, 0.5, 1.0, 2.0 GHz. c, ratio of the electron plasma frequency
to electron cyclotron frequency $R=\nu_{\mathrm{pe}}/\nu_{\mathrm{ce}}$. The
green contour lines show the ratio $R$ of 0.25, 0.5, 1.0, 1.5, and 2.0, from
bottom to top. The same contours are shown in panel (b). d, magnetic field
scale height LB. e, magnetic field angle $\theta$ relative to the line-of-
sight. f, gyro-resonance opacity $\tau$ of $O$ mode emission at
$\nu_{\mathrm{GHz}}=1.0$ along the line-of-sight. The black arrow indicates
the direction towards an Earth-based observer. The dotted, solid, and dashed
lines indicate the direction with $\theta$ equals to $30^{\circ}$,
$45^{\circ}$, and $60^{\circ}$. The corona that is transparent to the $s=3$
gyro-resonance absorption layer is outlined as the tilted white shade. The
green contour denotes the $s=2$ ECM source that is inferred from the observed
spatial distribution of the source locations of the coherent radio bursts. The
yellow cross marks the approximate location of the solar flares.

Although establishing a comprehensive emission model is challenging and beyond
the scope of the current study due to a lack of in situ data, we can
investigate the emission mechanism using remote-sensing data provided by
multi-wavelength observations. By employing a data-constrained 3D coronal
magnetic field model and electron number density model, we calculate the ratio
$R=\nu_{\mathrm{pe}}/\nu_{\mathrm{ce}}$ in the 3D volume above the active
region to identify regions more likely for the plasma radiation or ECM
emission. The method has been previously employed to investigate the emission
mechanism of coherent radio bursts in the solar corona[67, 68]. The resulting
distributions of $\nu_{\mathrm{pe}}$, $\nu_{\mathrm{ce}}$, and their ratio $R$
at the $Y=0$ plane in the model are shown, respectively, in the left column of
Extended Data Fig. 8. Although both the electron number density $n_{e}$ and
magnetic field strengths $B$ decrease with increasing height in the corona,
and so do the plasma frequency $\nu_{\mathrm{pe}}$ and cyclotron frequency
$\nu_{\mathrm{ce}}$, the $B$ decreases more rapidly in the region above the
sunspot. This leads to a low $R$ cavity just above the sunspot (Extended Data
Fig. 8(c)), in which the electron maser instability is likely to be excited.
The ECM emission frequency $\nu$ in the $R\lesssim 1$ cavity ranges from above
1.5 GHz to 250 MHz (Extended Data Fig. 8(b)). This agrees with the
observations that the sunspot auroral bursts are presented in a wide frequency
range from $\sim$ 2 GHz all the way to the lowest frequency of RSTN at 245 MHz
(Extended Data Fig. 1(a)). We acknowledge that $n_{e}$ and $\nu_{\mathrm{pe}}$
are reliant on the column depth $D$, which should be regarded as a rough
estimate. Nevertheless, it is important to note that the electron plasma
frequency gradually varies with the fourth root of column depth $D$, following
the relationship $\nu_{\mathrm{pe}}\propto{n^{0.5}_{e}}\propto 1/D^{0.25}$.
Consequently, changes in $D$ have only a minimal impact on the electron plasma
frequency $\nu_{\mathrm{pe}}$. Varying $D$ from 15 to 150 Mm leads to a 30%
difference in $\nu_{\mathrm{pe}}$ with respect to the value derived using
$D=50\,\mathrm{Mm}$ (Extended Data Fig. 7(c)). This variation does not
significantly affect the inferred ECM harmonic and the ratio $R$.

Owing to the relatively high plasma density in the solar corona compared to,
e.g., planetary magnetospheres, the detectability of the ECM emission is also
affected by gyro-resonant absorption due to the ambient thermal plasma at
higher harmonics of the electron gyro-frequency[63]. The opacity of the gyro-
resonance layers depends on the magnetic field geometry, the thermal electron
number density $n_{e}$ and the viewing direction[69]:

$\tau(s,\nu,\theta)=\frac{\pi
e^{2}}{2m_{e}c}\frac{n_{e}L_{B}(\theta)}{\nu}\frac{s^{2}}{s!}\left(\frac{s^{2}}{2\mu}\right)^{s-1}F(\theta)$
(4)

where $\theta$ is the angle between the LOS and the local magnetic field
direction, $L_{B}(\theta)=B/|\nabla B|$ is the magnetic scale height along the
LOS direction, $\mu=m_{e}c^{2}/k_{B}T$ is the square of the ratio between the
speed of light and electron thermal speed, and $F(\theta)$ is a function of
angle $\theta$:

$F(\theta)=\frac{\sin^{2s-2}\theta\left(\sin^{2}\theta+2s\cos^{2}\theta+\sigma\sqrt{\left.\sin^{4}\theta+4s^{2}\cos^{2}\theta\right.}\right)^{2}}{\sin^{4}\theta+4s^{2}\cos^{2}\theta+\sigma\sin^{2}\theta\sqrt{\sin^{4}\theta+4s^{2}\cos^{2}\theta}}$
(5)

with $\sigma=1$ for extraordinary mode ($X$ mode) or $\sigma=-1$ for ordinary
mode ($O$ mode).

The nearly 100% right circularly polarized radio emission is closely
associated with the sunspot in negative magnetic polarity, suggests that the
emission is likely in the ordinary mode, or $O$ mode. With Equation 4, we
evaluate the absorption of ECM radiation produced at a harmonic of $s_{e}$ by
the overlying gyro-resonance layer at $s_{a}=s_{e}+1$ (the absorption at
larger harmonics is much smaller and can be neglected) for a given observing
frequency $\nu$. In Extended Data Fig. 8(d)–(f), we show, respectively, maps
of $L_{B}$, $\theta$ and the opacity $\tau$ of the $s_{a}=1,2,3$ layer for
o-mode radiation at a representative frequency of $\nu_{\mathrm{GHz}}=1.0$
along the line of sight at the $Y=0$ plane. Our model shows that the $s_{a}=2$
layer is mostly optically thick except for a narrow window at small $\theta$
values with respect to the line of sight where the ECM emission produced at
the fundamental harmonic $s_{e}=1$ can escape. On the other hand, the model
suggests that ECM emission produced at the second harmonic $s_{e}=2$ is only
partially affected by the gyro-resonance absorption layer at $s_{a}=3$. The
latter shows a morphology similar to the $s_{a}=2$ layer, but has a much wider
transparent window covering the sunspot umbra and penumbra regions. Therefore,
the second-harmonic ECM emission can escape from the transparent window in the
$s_{a}=3$ layer and be observed, as demonstrated in the schematic picture in
Fig. 4(h). We note that while the opacity structure model is dependent
linearly on electron number density $n_{e}$, the latter varies with the square
root of column depth $D$. Changes in $D$ only have a minimal impact on opacity
structure, following the relationship $\tau\propto D^{-0.5}$. Varying $D$ from
15 to 150 Mm results in a scale factor of 0.7 to 2.2 for the opacity value,
which is not significant enough to affect the overall opacity structure for
the ECM emission and its observability (Extended Data Fig. 9).

Extended Data Fig. 9: Physical characteristics in a 2D slice on the $X$–$Z$
plane of the 3D model of NOAA 12529 AR for three different column depths. The
left and right columns show the same plots of the ratio
$R=\nu_{\mathrm{pe}}/\nu_{\mathrm{ce}}$ and the gyro-resonance opacity $\tau$
as Extended Data Fig. 8(c&f), but for column depths $D$ of 150, 50, and 15 Mm
from top to bottom, respectively.

We end this section with a brief discussion of the growth of ECM emission in
$X$ and $O$ mode, which depends heavily on the value of
$R=\nu_{\mathrm{pe}}/\nu_{\mathrm{ce}}$. Parametric investigations on ECM
instability in the literature suggests that at the fundamental $O$ mode or
second harmonic $X$ mode radiation would be dominant when $0.3\lesssim
R\lesssim 1$, and the second harmonic O-mode would dominate the radiation when
$1.4\lesssim R\lesssim 2$[70, 71]. In the regions where the radio source at
1–2 GHz, the ratio $R$ derived from the magnetic field model and coronal
density model generally falls in the range of $0.3\lesssim R\lesssim 1$
(Extended Data Fig. 8(b)), which would favor fundamental $O$ mode or second
harmonic $X$ mode. However, as we described in the coronal density model
section, the corona density depends strongly on the selection of the column
depth. The absolute coronal density can be scaled up by a factor of a few if
we assumed a smaller column depth, so that the ratio $R$ can satisfy required
values for second harmonic $O$ mode to dominate. We caution that such
uncertainty in the density model impedes an accurate determination of the
ratio $R$, so as for the preferable emission wave mode. For emission below 1
GHz, we choose not to discuss the wave mode because of lacking imaging
observation and polarization measurement at frequencies below 1 GHz.
Nevertheless, Chen et al[72] found the corona may consist of many unresolved
overdense loops based on the discrepancy between the density inferred from the
frequency of type III radio bursts and that inferred from EUV observations. It
is suggested that there are overdense magnetic loops above the sunspot with a
density enhancement by up to an order of magnitude against the tenuous
background, the ratio $R$ in which can be greater by the square root of the
same factor. If the observed coherent radio bursts can be interpreted as ECM
emission originating in these overdense loops, the second harmonic $O$ mode
radiation would be more probable to dominate in these loops with a higher
plasma density. The overdense loops scenario offers an alternative avenue for
second harmonic $O$ mode ECM radiation to dominate in a general low $R$
environment ($R$<1).

Extended Data Fig. 10: Distribution of the frequency-dependent source centroid
locations for seven individual 20-second time integrations. Top panel: The
frequency distribution of radio source locations for the example times (same
as these used for Fig. 4a). The close-to-linear distribution of radio sources
in space as a function of frequency is attributed to radio sources along the
respective loop, which is outlined by a solid line. Bottom panel: the plane-
of-the-sky projected distance $d(\nu)$ of radio source locations from the
source centroid at 1 GHz along the direction of the flux tubes. The predicted
$d(\nu)$ dependence along flux tubes of three orientations with viewing angle
$\theta$ equals to $30^{\circ}$, $45^{\circ}$, and $60^{\circ}$ as the dotted,
solid, and dashed lines, respectively.

### Spatial distribution of the ECM source

With spatially and temporally resolved imaging spectroscopy, we can derive the
centroid location of each observed source at any given time $t$ and frequency
$\nu$, provided that it has a sufficient signal-to-noise-ratio against the
quiescent background. The top panels in Extended Data Fig. 10 shows examples
of the distribution of the frequency-dependent source centroid locations for
seven individual 20-second time integrations. Intriguingly, the spatial
distribution of each burst forms a close-to-linear distribution in a way
similar to those derived from electron-beam-driven type III radio bursts [72,
18].

As the emission frequency $\nu$ due to the ECM mechanism depends only on the
magnetic field strength $B$, the observed spatial distribution of the radio
source centroid location at a given frequency $\nu$ in projection
$\vec{r}(\nu)$ is governed by the magnetic field distribution above the
sunspot $\vec{r}(B)$. Our observations have suggested that, for each radio
burst, the distribution of the frequency-dependent radio source centroids is
likely aligned with a single magnetic loop. Therefore, we can reduce the
distribution of the radio sources to a one-dimensional dependence of $d(\nu)$,
where $d$ denotes the plane-of-the-sky projected distance from a reference
location along the respective loop, selected as that of the source centroid at
1 GHz. We construct the $d(\nu)$ dependence of the seven example times, shown
in the bottom panel of Extended Data Fig. 10 as colored circles (whose sizes
are scaled by their peak flux). With the 3D coronal magnetic field model to
provide the mapping of $d(B)$ for each magnetic loop in projection, we can
predict the expected $d(\nu)$ dependence in different viewing angles of
$\theta$. In the bottom panel in Extended Data Fig. 10, we show the predicted
$d(\nu)$ dependence along three directions with $\theta$ equals to
$30^{\circ}$, $45^{\circ}$, and $60^{\circ}$ as the dotted, solid, and dashed
lines, respectively. The results suggest that the viewing angle $\theta$ of
the radio sources with the observed $d(\nu)$ dependence should lie in the
range of $30^{\circ}$–$60^{\circ}$. This range of viewing angles demarcates a
region in the vicinity of the apex of the $s=2$ gyroresonance dome (denoted as
the green contour in Extended Data Fig. 8(e&f)), corresponding to the
predicted ECM emission sites that are co-spatial with the transparent window
of the $s_{a}=3$ gyroresonance layer in projection. The range of viewing
angles is also consistent with the values of $\theta$ in the inferred source
region (green contour in Extended Data Fig. 8(e).) We note that the beaming
angle of 30∘–60∘ is generally smaller than observed values (75∘–80∘) in the
case of Jovian radio auroral radiation[73], but similar to the beaming angles
($\gtrsim 40^{\circ}$) of the auroral kilometric radiation in the terrestrial
case[45]. The latter is often attribute to the refraction effect that the
radio waves is initially emitted to the perpendicular direction but is ducted
away through refraction at the borders of the density depletion cavity in
which the emission is produced[45]. This does not seem to be the case for the
sunspot radio emission. For the ducting to occur, the plasma frequency of the
surrounding boundary of the low-density cavity should be significantly higher
than the emitting frequency of the radiation. This requires a plasma density
of at least a few times of $10^{10}\,\mathrm{cm}^{-3}$, which is much higher
than the inferred plasma density near the source region. One possibility is
that the small beaming angle is a result of ECM processes coupled with Alfvén
waves. Wu et al (2012)[74] found that preexisting Alfvén waves can modify the
classical ECM processes by affecting the velocity distribution of energetic
electrons via pitch-angle scattering. Indeed, Alfvén waves is expected in the
quiet solar corona[75, 76] and flare regions[77]. The beaming angle of second
harmonic $O$ mode ECM emission, if driven by nonthermal electrons in a
horseshoe distribution, can be as low as 30∘. The interpretation of the
observed coherent radio bursts as second harmonic $O$ mode ECM emission is
consistent with the small beam opening angles predicted in the analyticla
study[71]. Nevertheless, we acknowledge that the observed beaming angle is not
fully consistent with the values of previously observed planetary ECM
emissions. To produce a fully self-consistent model of ECM emission from the
sunspot requires not only a better understanding the corona context using a
data constrained MHD model, but also a fully kinetic simulations to achieve a
realistic nonthermal electron distribution, both of which are beyond the scope
of this study.

On the other hand, plasma emission radiates at a frequency close to the
fundamental or second harmonic electron plasma frequency $\nu_{\mathrm{pe}}$.
Plasma radiation at frequencies in the range 1-1.5 GHz suggests the ambient
plasma density above the sunspot in the range $1$–$3\times
10^{10}\,\mathrm{cm^{-3}}$, which is more than an order of magnitude higher
than coronal density model constrained by the DEM analysis (Extended Data Fig.
7(c)). It is worth asking, however, whether the observed radio bursts are
plasma radiation originating from the aforementioned overdense magnetic loops
in the corona. As the emission frequency $\nu$ due to the plasma radiation
mechanism is proportional to the square root of plasma density
($\nu_{\mathrm{pe}}\propto\sqrt{n_{\mathrm{e}}}$), the $d(\nu)$ of the plasma
radiation source is determined by the plasma density distribution above the
sunspot. The plasma frequency $\nu$ varies from 1.0 to 1.5 GHz over $d(\nu)$
of 3–6 Mm corresponds to the plasma density varying from $1\times 10^{10}$ to
$3\times 10^{10}\,\mathrm{cm^{-3}}$ over a height of 4–8 Mm, assuming the
overdense loops have inclination angles of $\sim 45^{\circ}$. This is
equivalent to a density scale height
$\mathrm{L}_{n}=n_{\mathrm{e}}/(dn_{\mathrm{e}}/d\vec{r})$ of 4–8 Mm under
hydrostatic equilibrium, which are at least an order of magnitude smaller than
the hydrostatic value at typical coronal temperature ($\mathrm{L}_{n}\approx
46T_{\mathrm{MK}}\,\mathrm{Mm}$, or 94 Mm for $T=2\,\mathrm{MK}$[78]). Such
steep density profiles, in other words, steep pressure gradients, are
inconsistent with the expectation for the quasi-static magnetic structures
above the sunspot. Therefore, we reaffirm the conclusion that plasma radiation
is unlikely the responsible mechanism for the observed radio bursts.

### Observability of the sunspot ECM emission

ECM generates radiation along a thin, conical sheet. Consequently, sunspot ECM
emissions are only observable when the narrow radiation beam crosses the line
of sight due to the Sun’s rotation. Although ECM emissions should also appear
when the sunspot is in the symmetric phase on the western side of the solar
disk, they are only visible for a specific heliocentric angle range on the
eastern side, as shown in Extended Data Fig. 2. In this study, we attribute
this phenomenon to the influence of solar flares on the observability of
sunspot ECM emissions. During the observed sunspot radio bursts, the emission
source region was far from the flare sites. However, as the sunspot rotates to
the other side of the solar disk, the potential emission source (the green
contour line in Extended Data Fig. 11) would become much closer to the flare
sites. The close spatial proximity of source sites to flares can significantly
alter the local plasma and magnetic environment, affecting ECM generation and
escape. A more detailed discussion of various impacts is provided below:

Extended Data Fig. 11: Similar as Extended Fig. 8(f), this figure displays the
gyro-resonance opacity $\tau$ of $O$ mode emission at $\nu_{\mathrm{GHz}}=1.0$
along the line-of-sight, considering the AR is located to the opposite
longitude with respect to the solar disk center and assuming a 10-fold
increase in electron number density $n_{\mathrm{e}}$ to account for the impact
of the nearby flare region. The black arrow indicates the direction towards an
Earth-based observer. The corona that is transparent to the $s=3$ gyro-
resonance absorption layer is outlined as the tilted white shade. The green
contour represents the potential source location of the $s=2$ ECM radio
emissions. The yellow cross marks the approximate location of the solar
flares.

  * •

ECM condition: The ECM mechanism requires a specific condition of $R\lesssim
1$ at the source site [63]. Given the source site’s proximity to flares, the
impact of solar flares on coronal density structures must be considered. The
thermal plasma density distribution during flares is highly dynamic and
complicated, which required specialized modeling efforts to simulate its
temporal evolution[79]. While this is beyond the scope of this study, a rough
estimate can still be performed. Solar flares can increase coronal plasma
density by a factor of over 10 within tens of seconds[80]. This over 10-fold
increase in density near flare regions could raise the $R$ value in the nearby
ECM source region by a factor of a few, as $R$ is proportional to
$\sqrt{n_{\mathrm{e}}}/B$. In this case, the source region near the flare
sites may not meet the ECM condition and cannot produce ECM emission.

  * •

Opacity: The solar corona’s highly structured nature means that the opacity of
the gyro-resonance absorption layer, which depends on the viewing direction
and 3D distributions of the coronal magnetic field and thermal plasma[63],
will change over time as the Sun rotates. This change subsequently affects the
emissivity and polarization of sunspot ECM emissions. As mentioned earlier,
the opacity varies linearly with the electron number density $n_{\mathrm{e}}$.
A more than 10-fold increase in $n_{\mathrm{e}}$ near the flare regions could
significantly alter the opacity of the nearby gyro-resonance layer in the line
of sight (Extended Data Fig. 11), restricting the escape of ECM radiation.

  * •

Magnetic configuration: In our interpretation, the generation of the sunspot
radio bursts requires closed magnetic loops trapping energetic electrons.
Recurring non-eruptive flares are favored because they can continuously
replenish the population of energetic electrons in the overarching radio-
hosting magnetic field lines above the energy release site without disrupting
the overall magnetic field geometry. In contrast, eruptive flares are expected
to open up the magnetic field lines above [81]. The newly accelerated
electrons by solar flares can easily escape to interplanetary space along the
opened field lines without getting trapped in the low corona. Moreover, the
opening of the overarching magnetic field lines results in the destruction of
the electron reservoir where energetic electrons accelerated by previous
flares are trapped, further interrupting the generation of the sunspot radio
bursts. As demonstrated in Extended Data Fig. 2(e), NOAA 12529 AR became more
violent on 2016 April 14, producing an M class flare and a few sub-M class
flares. In contrast, the AR only produced a few C class flares earlier.

One or more of the factors mentioned above may influence the observability of
the sunspot ECM emission as the viewing angle varies, in addition to the
effect of stellar rotation. Understanding the viewing geometry effects
requires long-term broadband dynamic imaging spectropolarimetry to study in
detail the time-dependent variations of the morphology, location, and
polarization of the radio source. The emissivity of sunspot coherent
radiation, including but not limited to the emission intensity, polarization
and wave mode, depends heavily on the magnetic field distribution and coronal
plasma distribution, which are subjected to change with time due to the
gradual evolution of the radio-hosting active region and abrupt energy release
events. To establish the relationships of coherent radio bursts with the
structure and dynamics of the corona, including plasma density inhomogeneity,
magnetic field topologies, and their variations, broadband dynamic imaging
spectroscopy observations in the frequency range of 200 MHz to 2 GHz are
required. These observations should be complemented by other multi-wavelength
observations, including optical, EUV, and X-ray. Moreover, with the aid of
data-constrained coronal 3D magnetic and thermodynamic structure, imaging
spectropolarimetry would allow us to characterize and constrain coronal
thermal parameters, magnetic field properties, and the energy distribution of
nonthermal electrons that drive ECM emission. Detailed context data at other
wavelengths, coupled with radio imaging observations, would enable the
examination of the relationship between the rotational modulations of sunspot
ECM emission and other significant observables, such as photometric and
Doppler spectroscopic measurements[32, 82]. Such observations will be crucial
for improving our understanding of the generation and rotational modulation
effects of ECM bursts from the Sun and other stars.

An important corollary to the discussion of observability of the sunspot ECM
emission is the potential influence of viewing effects on the duty cycle of
stellar ECM emissions. If these stellar emissions were associated with the
presence of starspots on the surfaces of these (sub)stellar systems, viewing
effects may play a role in determining the temporal pattern within a single
duty cycle of these emissions. Like the solar case, many previously reported
stellar ECM emissions only appear in a certain range of stellar rotational
phase[10, 82]. Depending on the relative positions of the starspot and its
neighboring flare sites with respect to the light of sight, as well as the
inclination of the rotation axis relative to the line of sight as viewed from
the Earth, stellar ECM emissions may manifest a second time in symmetric phase
with respect to the stellar-centric $0^{\circ}$ longitude line. Similar cases
have also been observed in other stars[44, 6]. Furthermore, as a type of
transient radio source, the duty cycle of stellar ECM emissions could also
vary as a consequence of the short-term evolution of the starspot and its
magnetic connectivity to other parts of the stellar surface. Long-term
variations, such as the emergence of a starspot, may lead to the appearance of
ECM radio emission on a star that previously lacked such radio bursts in
earlier observations, and vice versa. Understanding these viewing effects and
their impact on the observability of sunspot and starspot ECM emissions can
inform future survey-type observations of stellar ECM emissions.

In concluding this section, we explore the potential applicability of our
model to early-type stars (spectral type B or A). Rotationally modulated ECME
emissions have been discovered in several early-type stars, exhibiting
consistent phases over extended periods [44, 82]. The prevailing hypothesis
suggests that global dipole-like magnetic fields can create magnetic-mirror-
like conditions necessary for maser emission[83]. These stars often possess
magnetic fields with kG strengths and simple topologies, often well
approximated by a dipole[84]. Stellar wind[44] or magnetic reconnection in the
magnetosphere[85] may provide energetic electrons required to drive ECM
emissions. These electrons then become trapped in the stellar magnetospheric
pole regions, producing magnetic-mirror-like conditions that facilitate ECME
emission. However, recent studies reveal that radio ECM emission can emerge
from stars with surface magnetic fields significantly deviating from an
axisymmetric dipole[86], leaving the role of the dipole magnetic field
topology in producing radio ECM emission remains an open question. Analogous
to the magnetospheric pole regions and the electromagnetic engine in the
magnetosphere, a localized magnetic structure such as a starspot and nearby
flare activity can also produce radio ECM emissions, similar to the solar
case. Recent time-series photometry from space missions like Kepler, K2, and
TESS challenges this belief, suggesting that rotational modulation is probable
in stars with radiative envelopes [87]. The observed light variations can be
most simply interpreted as rotational modulation induced by surface brightness
inhomogeneities, such as magnetic spots. These localized magnetic fields might
be associated with magnetic fields generated in subsurface convection zones
[88]. Additionally, it is plausible that differential rotation in A and B
stars may be sufficient to create local magnetic fields via dynamo action
[89]. Due to the extreme stability of the magnetic fields in these types of
stars[90], the periodic ECME emission can be remarkably stable, unlike the
case for the highly dynamic magnetic field of the Sun. If the rotational
modulation observed in early-type stars indeed results from persistent
magnetic spots, our model could offer an alternative approach to generating a
stable burst duty cycle over extended periods in these stars. However, this
proposition depends on the presence of persistent magnetic spots on early-type
stars. Further investigation is necessary to assess the efficacy of the model
in accounting for the long-term radio emission behavior exhibited by these
stars.

### Source size and brightness temperature

The broadband radio bursts emitting from the sunspot are detectable across
frequencies ranging from 245 MHz to 1.7 GHz. The maximum flux density of these
recorded bursts in stokes I, observed at 245 MHz at approximately 23 UT on
April 12, is around 2000 sfu, or 2$\times 10^{10}$ mJy. Were the observed
bursts to be viewed from a stellar distance, the flux density at 245 MHz can
be expressed as $0.5(D_{\rm pc})^{-2}$ mJy, where $D_{pc}$ is the stellar
distance in parsec. If we take the median distance of 20 parsec in the recent
LOFAR observations[27], the flux density would approximate 1 $\mu$Jy. Although
the total flux density is two or three orders of magnitude lower than the
median values reported in the recent LOFAR observations at hundreds of MHz
[27], we argue that the brightness temperature of the sunspot radio source can
still be very high. This is owing to the likely compact and unresolved nature
of many individual bursts that comprise the observed radio source. We consider
the magnetic loops anchored at the sunspot where impulsively accelerated
electrons are injected and an unstable loss-cone distribution is set up by
magnetic mirroring on the sunspot end of the loops. the brightness temperature
for a continuously operating maser is given by[63],

$T_{\mathrm{b}}=\frac{m_{e}v_{0}^{2}}{4\pi
k_{\mathrm{B}}}\frac{c^{2}}{\nu^{2}Lr_{0}}\approx 2\times
10^{12}\left(\frac{E}{1\mathrm{\leavevmode\nobreak\
keV}}\right)\left(\frac{\nu}{200\mathrm{\leavevmode\nobreak\
MHz}}\right)^{-2}\left(\frac{L}{R_{\odot}}\right)^{-1}\mathrm{\leavevmode\nobreak\
K}$ (6)

where $v_{0}$ is the velocity of the emitting electrons, $\nu$ is the emission
frequency, $L$ is the length scale of the magnetic trap with respect to the
solar radius $R_{\odot}$, $r_{0}$ is the classical electron radius, $m_{e}$ is
the electron mass, $k_{\mathrm{B}}$ is the Boltzmann constant, and $c$ is the
speed of light. Using a magnetic loop length scale of approximately 1/10 of
the solar radius $R_{\odot}$ and electron energy $E$ of 20 keV, the brightness
temperature is estimated to be approximately $10^{14}$ K at 245 MHz and
$10^{13}$ K at 1 GHz for the fundamental emission. For the second harmonic ECM
emission, the values are $10^{12}$ K at 245 MHz and $10^{11}$ K at 1 GHz,
considering the latter’s growth rate potentially being two orders of magnitude
lower than the fundamental[71].

High brightness temperatures are not uncommon for solar radio bursts, as
evidenced by solar decimetric spikes[91] whose source size scale is suggested
to be below 1000 km. For the 20 sfu radio observed by VLA at 1 GHz, the
brightness temperature corresponds to a source size of 1000$\times$1000 km.
Since the observed radio source of the 1 GHz image is marginally resolved by
VLA, a single 1000$\times$1000 km source can not account for the asymmetric
structure seen in the VLA 1 GHz image. We speculate that the observed radio
source may comprise many small, unresolved sources instead of a single
1000$\times$1000 km source. To illustrate such a possibility, we employed a
hundred ($N=100$) randomly distributed $10^{11}$ K point sources in the second
harmonic gyroresonance of $\nu_{\mathrm{GHz}}=1$ at selected field lines. We
note that the choice of the source number $N$ is arbitrary, but it does not
affect our subsequent analysis. Each point source has a uniform diameter of
$1000/\sqrt{N}\mathrm{\leavevmode\nobreak\ km}$ and a flux density of $20/N$
sfu. The model is shown in Extended Data Fig. 12a. The input model is first
convolved with a $40^{\prime\prime}$ Gaussian beam to account for angular
scattering at 1 GHz by coronal turbulence[49]. Subsequently, the model is
convolved with the synthesis beam of the VLA at 22 UT on April 9 to create a
simulated VLA image. The result, shown in Extended Data Fig. 12b,
qualitatively agrees with the actual VLA image at this frequency in terms of
source location and size. It is important to note that the argument for
multiple sources is consistent with the inhomogeneous and fragmentary nature
of magnetic reconnection, as evidenced by recent VLA solar observations[18].
Under this scenario, semi-relativistic electron beams can be emanated into
distinct field lines, even though they are accelerated within an extremely
compact region ($\sim 600\mathrm{\leavevmode\nobreak\ km}$) in the low solar
corona over a brief time scale (1–2 seconds). Assuming that each ECM source is
associated with a different electron beam energized within the same
acceleration region, they can, in principle, arrive at various field lines
above the sunspot, resulting in the scattered distribution of ECM sources as
portrayed in the model. While our observations show that the radio bursts
could originate from numerous distinct magnetic field lines, we note that
these field lines are all rooted at the umbra region of the sunspot with a
negative polarity. Consequently, at any particular instance, these field lines
share similar line-of-sight angles with regard to the observer, which gives
rise to a high degree of circular polarization of the bursts over our
observational period of several hours.

Nevertheless, explaining the observed radio bursts on certain M dwarfs with
orders of magnitude higher flux density than our case remains challenging.
However, we note that the plasma environment on these stars are much more
extreme than that on the Sun. First, the average surface magnetic field on
these stars can exceed kilo Gauss [92, 93, 94], compared to $\sim$1–10 Gauss
on the Sun. The magnetic fields in these starspots may exceed those in
sunspots by orders of magnitude, potentially boosting the intrinsic brightness
of the ECM emission[37]. Second, the level of flare activity on these fully
convective stars is much higher than that on the Sun, allowing a possibility
to pertain a much larger spatial and temporal filling factor for these bursts
over the stellar disk. This speculation is reflected in the radio/X-ray
correlation in the context of the GB relation, where the sunspot radio source
exhibits similar deviation from the radio/X-ray relation as other stellar
radio ECM emissions, despite its relatively low magnetic activity level (X-ray
luminosity) among the others. Both may potentially contribute to the much
brighter bursts observed on these active stars, although further
observational, theoretical, and modeling investigations are required.

Extended Data Fig. 12: Simulation of the 20 sfu sunspot radio image at 1 GHz.
a: The input model of the radio ECM sources at 1 GHz displayed in a context of
the sunspot and the corresponding gyroresonance opacity layer, similar to Fig.
4h. The model consists of a hundred randomly distributed point sources with a
brightness temperature of $10^{11}$ K at selected field lines. b: similar
Extended Data Fig. 3, but showing the simulated VLA image at
$\nu_{\mathrm{GHz}}=1.0$ overlaid on the HMI continuum intensity image.

## Data availability

The data that support the plots and other findings within this paper are
available at https://data.nrao.edu/portal (VLA; Project ID: 16A-377),
https://www.sws.bom.gov.au/World_Data_Centre (RSTN),
https://solar.nro.nao.ac.jp/norp (NoRP),
https://www.swpc.noaa.gov/products/goes-x-ray-flux (GOES), and
http://jsoc.stanford.edu (SDO), or from the corresponding author upon
reasonable request.

## Code availability

The magnetic field extrapolation[55] software packages are available through
IDL SolarSoft at https://sohowww.nascom.nasa.gov/solarsoft. The regularized
inversion code for differential emission measure (DEM) calculation[62] is
available at https://github.com/ianan/demreg. Public software packages
utilized in this study include SunCASA https://github.com/suncasa/suncasa-src,
CASA[95] https://casa.nrao.edu, SunPy[96] https://sunpy.org, and Astropy[97]
https://www.astropy.org.

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## Acknowledgements

The VLA is operated by the National Radio Astronomy Observatory (NRAO), a
facility of the National Science Foundation (NSF) operated under a cooperative
agreement by Associated Universities, Inc. S.Y. and B.C. are supported by NASA
HSOC and ECIP grants (80NSSC20K1283/SV0-09025 and 80NSSC21K0623), and NSF
grant AGS-1654382 to NJIT. R.S. acknowledges the support of the Swiss National
Foundation, under grant 200021_175832. R.S. also acknowledges Dr. Säm Krucker
and Dr. André Csillaghy, FHNW, for their support. D.E.G. acknowledges support
from NASA HSR grant 80NSSC18K1128. The authors acknowledge Dr. Arnold Benz for
the useful discussions.

## Author contributions statement

S.Y. conceived the study, carried out the data reduction, analysis,
visualization, interpretation and manuscript preparation. B.C. led the VLA
observing proposal (VLA/16A-377) and planned and conducted the VLA
observations. T.S.B. and D.E.G. participated in the VLA proposal and the
observing programs. R.S. conducted an independent VLA data reduction and cross
checked the results. B.C., R.S., T.S.B., S.M., and D.E.G. contributed to the
theoretical interpretation. S.Y. and B.C. led the manuscript writing, and all
authors discussed the results and commented on the manuscript.

## Competing Interests Statement

The authors declare no competing interests.