11institutetext:  Institut für Astronomie & Astrophysik, Eberhard Karls
Universität Tübingen, Sand 1, 72076 Tübingen, Germany 22institutetext: INAF -
Osservatorio Astronomico di Palermo, Piazza del Parlamento 1, 90134 Palermo,
Italy 33institutetext: Space Telescope Science Institute, 3700 San Martin
Drive, Baltimore, MD, USA 44institutetext: Center for Astrophysical Sciences,
Johns Hopkins University, 3701 San Martin Drive, Baltimore, MD USA
55institutetext: Laboratory for Atmospheric and Space Physics, University of
Colorado Boulder, Boulder, CO, USA 66institutetext: Department of Astronomy,
University of Washington, Seattle, WA USA

# Transitions in magnetic behavior at the substellar boundary

E. Magaudda 11 B. Stelzer 1122 R. A. Osten 3344 J. S. Pineda 55 St. Raetz 11
M. McKay 3366

We aim at advancing our understanding of magnetic activity and the underlying
dynamo mechanism at the end of the main-sequence. To this end, we have
embarked in collecting simultaneous X-ray and radio observations for a sample
of M7..L0 dwarfs in the solar neighborhood using XMM-Newton jointly with the
Jansky Very Large Array (JVLA) and the Australia Telescope Compact Array
(ATCA). We supplemented the data from these dedicated campaigns with X-ray
data from the all-sky surveys of the ROentgen Survey with an Imaging Telescope
Array (eROSITA) on board the Russian Spektrum-Roentgen-Gamma mission (SRG).
Moreover, we complement this multi-wavelength data set with rotation periods
that we measure from light curves acquired with the Transiting Exoplanet
Survey Satellite (TESS). We limit the sample to objects with rotation period
$<1$ d, focusing on the study of a transition in magnetic behavior suggested
by a drastic change in the radio detection rate at $v\sin{i}\approx 38\,\rm
km/s$, corresponding to $P_{\rm rot}\approx 0.2$ d for a typical UCD radius of
$R_{\star}=0.15\,R_{\odot}$. Finally, to enlarge the target list we have
compiled archival X-ray and radio data for UCDs from the literature, and we
have analysed the above-mentioned ancillary eROSITA and TESS observations for
these objects analogous to the targets from our dedicated X-ray/radio
campaigns. We compiled the most up-to-date radio/X-ray luminosity ($L_{\rm
R,\nu}-L_{\rm x}$) relation for $26$ UCDs with rotation periods ($P_{\rm
rot}$) lower than $1$ d, finding that rapid rotators lie the furthest away
from the “Güdel-Benz” relation previously studied for earlier-type stars.
Radio bursts are mainly (although not exclusively) experienced by very fast
UCDs ($P_{\rm rot}\leq 0.2$ d), while X-ray flares are seen by objects
spanning the whole range of rotation. Finally, we examined the $L_{\rm
x}/L_{\rm bol}$ vs $P_{\rm rot}$ relation, where our sample of UCDs spans a
large range of activity level, i.e. $\log(L_{\rm x}/L_{\rm bol})=-5.5$ to
$\log(L_{\rm x}/L_{\rm bol})=-3$. Although they are all fast rotating,
evidently X-ray activity decouples from that of normal dynamos. In fact, we
found no evident relation between the X-ray emission and rotation, reinforcing
previous speculations on a bimodal dynamo across late-type dwarfs. One radio-
detected object, 2MJ0838, has a rotation period consistent with the range of
auroral bursting sources; while it displays moderately circularly polarized
emission, there is no temporal variation in the polarized flux. A radio flare
from this object is interpreted as gyrosynchrotron emission, and it displays
X-ray and optical flares. Among the $10$ UCDs observed with the dedicated
X-ray/radio campaigns, we found a slowly rotating apparent auroral emitter
(2MJ0752), that is also one of the X-ray brightest radio-detected UCDs. We
speculate that this UCD is experiencing a transition of its magnetic behavior
since it produces signatures expected from higher mass M dwarfs along with
emerging evidence of auroral emission.

###### Key Words.:

stars: low-mass; stars:coronae; stars: activity; stars: rotation; stars:
magnetic field; X-rays: stars

##  1 Introduction

Stellar magnetic activity is a fundamental ingredient in defining the observed
spectral flux of low-mass stars. Magnetism powers chromospheres and coronae,
which exhibit emissions from X-ray to radio wavebands (see review by Linsky,
2017), but also generates variability through, for example, photospheric
spots, active regions, and flaring reconnection events (see within Solanki et
al., 2006). Observations of these emissions serve as critical diagnostics of
the dynamic plasma of the stellar atmosphere, and must be physically linked
with the internally generated magnetic dynamo (e.g., Browning, 2008). This
connection is well evinced by multi-wavelength results demonstrating rotation-
activity-age correlations (e.g., Wright et al., 2018; Newton et al., 2017;
Magaudda et al., 2020; Pineda et al., 2021), tying the activity to long-term
angular momentum evolution. Despite concerted effort studying stellar
activity, many aspects of the underlying physics remain uncertain (e.g.,
dynamo mechanism, winds, coronal heating, etc.), especially in low-mass stars
which often show stronger magnetic fields (e.g., Shulyak et al., 2017) and
more energetic eruptions (e.g., Günther et al., 2020) relative to what has
been observed from the Sun.

Understanding these effects of the stellar magnetic field are more important
now than ever before because of the proliferation of detected exoplanetary
systems around low-mass stars. These stars host many of the extrasolar planets
most favorable for atmospheric characterization (e.g., Morley et al., 2017),
and interpreting those results requires understanding the high-energy stellar
emissions, and their evolutionary history (e.g., Johnstone et al., 2021). Of
particular interest are very-low mass stars at the end of the main sequence
which are capable of hosting multiple temperate Earth-sized planets (e.g.,
TRAPPIST-1; Gillon et al., 2017), and can display enhanced magnetic activity
for several billions of years (e.g., West et al., 2008).

However, relative to warmer stars, the nature of the stellar activity must
change in this mass range, and toward lower mass brown dwarfs, as effective
temperatures cool and the atmospheres become more neutral, decoupling the
atmospheric dynamics from the magnetic field (e.g., Mohanty & Basri, 2003;
Rodríguez-Barrera et al., 2015). This regime of ultracool dwarfs (UCDs;
spectral types $\geq$M7) displays clear deviations from the standard coronal
paradigm defined by Solar activity (see within Pineda et al., 2017), best
demonstrated by their remarkable radio emissions.

For a wide range of stars (main-sequence flare stars, RS CVn binaries, weak-
line T Tauri stars, and Algol binaries) there is a tight relation between the
luminosity in the radio and in X-rays, underlining the close connection
between non-thermal gyrosynchrotron radio emission and thermal X-rays (Gudel
et al., 1993; Güdel & Benz, 1993). This ‘Güdel-Benz’ relation (hereafter GB-
relation), however, breaks down in the UCD regime. Several UCDs with sensitive
X-ray and radio measurements are overluminous in radio with respect to the GB-
relation by factors of $>$ 1000 (e.g., Williams et al., 2014a; Pineda et al.,
2017). Super-imposed on this quiescent, quasi-steady (synchrotron) emission,
tracing radiation belt structures (Kao et al., 2023; Climent et al., 2023),
are highly circularly polarized periodically pulsed bursts (e.g., Burgasser &
Putman, 2005; Hallinan et al., 2006, 2008; Route & Wolszczan, 2012; Williams
et al., 2015; Kao et al., 2016; Vedantham et al., 2020; Rose et al., 2023).
This ‘bursting’ radio component has been attributed to the electron cyclotron
maser instability (ECMI; Hallinan et al., 2008), a phenomenon seen
predominately in solar system giant planets that is tied to auroral electron
precipitation (e.g., Zarka, 2007).

Conversely, the short-term variability of typical UCDs in the X-ray band
exhibits the typical signatures of flares that are likely caused by magnetic
reconnection events (e.g., Stelzer et al., 2006; Berger et al., 2008; Williams
et al., 2014b). Long-term optical photometric monitoring of this population
shows the capacity for these stars to also show white-light flares (e.g.,
Paudel et al., 2018; Murray et al., 2022). These behaviors suggest that much
UCD magnetic activity remains analogous to that of higher-mass GKM stars.

The existing radio and X-ray data thus present a heterogeneous picture for
UCDs. Indeed, the accumulation of data over the past decade have only
reinforced the suggestion by Stelzer et al. (2012) of a dichotomy between
radio-bursting / X-ray faint and radio-quiet / X-ray flaring UCDs, although
the boundary may not be rigid. Still, the nature of the onset of this behavior
near the end of the stellar main sequence is unclear. The emergence of UCD
ECMI radio emission is most likely shaped by magnetic field structure/strength
and the dwarf rotation rate (see within Stelzer et al., 2012; Pineda et al.,
2017; Pineda & Hallinan, 2018).

The confirmation of resolved quiescent radio emission in the UCD LSR
J1835+3259 from energetic electrons trapped in large-scale magnetic loops
reinforces the idea of large-scale dipolar topologies being a critical
ingredient for the radio-bright UCDs (Kao et al., 2023; Climent et al., 2023).
Theoretical treatments of the engines powering the radio-bright UCDs strongly
depend on the object’s rotation rate (e.g., Nichols et al., 2012; Turnpenney
et al., 2017; Saur et al., 2021). Moreover, Pineda et al. (2017) used
spectroscopic rotation measurements to reveal a sharp rise in the detection
fraction of UCD radio emission: above $v\sin{i}\approx 35$ km s-1 (rotation
period, $P_{\rm rot}\sim 3.5$ hr) the UCD detection fraction at radio
wavelengths rises steeply, 4-5 times the overall rate of unbiased surveys
($\sim$$10\%$; e.g. Lynch et al., 2016). However, the unknown inclination
angle makes it difficult to ascertain the precise role of rotation in observed
UCD radio emissions.

The Transiting Exoplanet Survey Satellite (TESS; Ricker et al., 2015), through
its broadband optical monitoring all-sky survey enables examination of
photometric rotation periods across a broad sample of stars, including nearby
UCDs. With TESS, we are now in the position to put reliable constraints on all
relevant parameters: rotation, X-ray and radio emission, and clearly test
whether the dichotomy of UCD X-ray/radio behavior emerges strictly across the
rotation period boundary suggested by Pineda et al. (2017).

In this article, we build a mini-survey of GHz radio observations and X-ray
data for a carefully-selected sample of UCDs accessible to TESS. Our multi-
wavelength analysis is designed to examine UCDs on both sides of this rotation
boundary ($\sim$3.5 hr), and assess the nature of their magnetically driven
emissions. This sample is presented in Sect. 2, with its radio, X-ray and
optical data analysis in Sect. 3. In Sects. 4 we present interesting radio
emissions for specific objects and the coronal/magnetospheric models,
respectively. We finally discuss our sample across all three wavebands adopted
throughout this paper (Sect. 5) and propose our conclusions in Sect. 6.

[b]

Table 1: Overview of the UCDs from our X-ray/radio campaigns: cols.1&2 show
the 2MASS ID and the alternative name from Simbad, and the last two columns
inform about the instruments used for the X-ray and radio observations.

2MASS ID | Name | X-ray1 | Radio2
---|---|---|---
J07522390$+$1612 | LP 423-31 | X/eR | VLA
J03510004$-$0052 | LP 593-68 | X/eR | VLA
J17571539$+$7042 | LP 44-162 | X | VLA
J04402325$-$0530 | LP 655-48 | X/eR | Ant13
J03061159$-$3647 | SSSPM J0306-3648 | X/eR | VLA
J08380224$-$5855 | SCR J0838-5855 | X/eR | ATCA
J04351612$-$1606 | LP 775-31 | X/eR | Ant13
J06521977$-$2534 | DENIS J065219.7-253450 | X | JVLA
J02150802$-$3040 | LP 885-35 | X/eR | Ant13
J10551532$-$7356 | WISEA J10551571-735611.3 | eR | ATCA

  * 1

X: XMM-Newton; eR: eROSITA

  * 2

VLA: Very Large Array; JVLA: Jansky Very Large Array; ATCA: Australia
Telescope Compact Array; Ant13:Antonova et al. (2013).

##  2 Sample

The focus of this work is to investigate the activity of UCDs across the
boundary in terms of rotation rate, $v\sin{i}\sim 35$ km/s, at which previous
data has suggested a drastic change of the radio properties Pineda et al.
(2017). For a typical late-M dwarf this corresponds to a rotation period
$P_{\rm rot}\sim 0.15$ d ($\approx 3.5$ hr), justified by most radio bursters
being seen equator-on. Our focus is on quantifying the multi-wavelength
behavior of UCDs in the areas just above and below this threshold.

To this end, we have obtained deep X-ray and radio data for $10$ nearby UCDs.
The targets were selected with the criterion that $P_{\rm rot}$ be $<1$ d,
drawing from a larger list of UCDs for which we had preliminarily determined
the rotation period from TESS light curves. Considering the decay of activity
in the L dwarf regime which pushes current instruments beyond their
sensitivity limits, for our campaigns we focused on UCDs with spectral types
from M7 to L0. Objects with small distances were preferred to guarantee the
highest achievable sensitivity for the activity measurements. The target list
and the X-ray and radio instruments used for each of them is provided in Table
1. The data were obtained in three dedicated X-ray/radio campaigns carried out
in 2012-2013, 2014-2015 and 2020-2021.

The X-ray observations were carried out with the X-ray Multi-Mirror mission
(XMM-Newton) and the radio data were obtained from the Jansky Very Large Array
(JVLA) and the Australia Telescope Compact Array (ATCA). For four UCDs X-ray
and radio data were acquired simultaneous or nearly simultaneous, see Tables 3
& 8. To enlarge the data base we have searched the literature for UCDs
observed in both the X-ray and the radio band, and we found $16$ objects.

[b]

Table 2: Stellar parameters and Gaia-DR3 data of the UCDs studied in this
work, see text in Sect. 2.

2MASS ID | SpT | $J$ | $T_{\rm eff}$ | $\log L_{\rm bol}$ | $R_{\star}$ |  Gaia-DR3 IDs | $d$ | $PM_{\rm\alpha}$ | $PM_{\rm\delta}$ | $G$ | $G_{\rm bp}$ | $G_{\rm rp}$
---|---|---|---|---|---|---|---|---|---|---|---|---
|  | [mag] | [K] | [$L_{\odot}$] | [$R_{\odot}$] |  | [pc] | [mas/yr] | [mas/yr] | [mag] | [mag] | [mag]
New UCD sample |
2MJ0752 | M7.0 | 10.9 | 2683 | -2.67 | 0.20$\pm$0.02 | 666988221840703232 | 18.88$\pm$0.01 | 183.1 | -350.3 | 14.75 | 17.22 | 13.33
2MJ0351 | M7.5 | 11.3 | 2611 | -3.02 | 0.14$\pm$0.01 | 3257243312560240000 | 14.67$\pm$0.01 | 10.9 | -469.9 | 15.31 | 18.28 | 13.84
2MJ1757 | M7.5 | 11.5 | 2611 | -2.93 | 0.17$\pm$0.01 | 1638180413086979840 | 19.13$\pm$0.08 | 6.8 | 326.6 | 15.65 | 18.92 | 14.14
2MJ0440 | M7.0 | 10.7 | 2683 | -3.20 | 0.11$\pm$0.01 | 3200303384927512960 |  09.74$\pm$0.01 | 334.5 | 127.9 | 14.90 | 18.05 | 13.40
2MJ0306 | M8.5 | 11.7 | 2467 | -3.32 | 0.12$\pm$0.01 | 5047423236725995136 | 13.26$\pm$0.01 | -172.1 | -669.2 | 15.99 | 19.53 | 14.46
2MJ0838 | M6.0 | 10.3 | 2830 | -2.98 | 0.13$\pm$0.01 | 5302788969815543936 | 11.11$\pm$0.03 | -56.9 | -313.6 | 14.52 | 17.33 | 12.87
2MJ0435 | M7.0 | 10.4 | 2683 | -3.04 | 0.14$\pm$0.01 | 3171631420210205056 | 10.60$\pm$0.02 | 160.1 | 315.1 | 14.6 | 17.92 | 13.09
2MJ0652 | L0.0 | 12.7 | 2248 | -3.57 | 0.10$\pm$0.01 | 2920995300823950720 | 16.06$\pm$0.02 | -235.5 | 87.9 | 17.43 | 20.64 | 15.87
2MJ0215 | M7.5 | 11.6 | 2611 | -3.24 | 0.12$\pm$0.01 | 4971892010576979840 | 14.05$\pm$0.01 | 768.6 | -360.0 | 15.8 | 19.13 | 14.32
2MJ1055 | M7.0 | 10.6 | 2683 | -2.93 | 0.16$\pm$0.01 | 5225863906515477376 | 12.90$\pm$0.08 | 163.1 | -216.3 | 14.72 | 17.84 | 13.22
Literature sample |
2MJ0036 | L3.5 | 12.4 | 3286 | -3.97 | 0.03$\pm$0.01 | 2794735086363871360 | 8.74$\pm$0.01 | 901.5 | 124.3 | 17.5 | 20.44 | 15.91
2MJ0523 | L2.5 | 13.0 | 3554 | -3.78 | 0.03$\pm$0.01 | 2985035874544160384 | 12.73$\pm$0.02 | 107.5 | 161.47 | 18.03 | 21.27 | 16.48
2MJ0602 | L1.0 | 12.3 | 4150 | -3.73 | 0.03$\pm$0.01 | 3457493517036545280 | 11.66$\pm$0.02 | 157.6 | -506.4 | 17.23 | 20.8 | 15.66
2MWJ1507 | L5.0 | 12.8 | 2990 | -4.26 | 0.03$\pm$0.01 | 6306068659857135232 | 7.41$\pm$0.01 | -151.8 | -896.0 | 17.95 | 21.17 | 16.34
BRI0021-0214 | M9.5 | 11.5 | 2314 | -3.44 | 0.12$\pm$0.01 | 2541756977144595712 | 12.45$\pm$0.03 | -74.6 | 140.2 | 16.58 | 20.02 | 15.03
DENIS J1048 | M9.0 | 9.5 | 2349 | -3.53 | 0.10$\pm$0.01 | 5393446658454453632 | 4.05$\pm$0.01 | -1179.3 | -988.1 | 14.01 | 17.62 | 12.46
Gl 569 B | M9.1 | 11.1 | 2380 | -3.17 | 0.15$\pm$0.01 | 1187851653287128576 | 9.94$\pm$0.01 | 279.1 | -117.9 | 9.12 | 10.31 | 8.03
Kelu 1 | L2.0 | 13.4 | 3722 | -3.6 | 0.04$\pm$0.01 | 6187779556809793024 | 20.39$\pm$0.3 | -305.4 | -27.3 | 18.74 | 21.16 | 16.79
LHS2065 | M9.0 | 11.2 | 2394 | -3.51 | 0.10$\pm$0.01 | 5761985432616501376 | 8.66$\pm$0.01 | -516.6 | -199.6 | 15.89 | 19.11 | 14.35
LP 349-25 | M7.9 | 10.6 | 2553 | -2.78 | 0.21$\pm$0.02 | 2799992744809482112 | 14.13$\pm$0.09 | -490.3 | -843.5 | 14.96 | 18.31 | 13.38
LP 412-31 | M7.9 | 11.8 | 2546 | -3.26 | 0.12$\pm$0.01 | 0056252256123908096 | 14.65$\pm$0.02 | 392.7 | -186.6 | 16.1 | 19.22 | 14.58
LP 944-20 | M9.5 | 10.7 | 2321 | -3.84 | 0.07$\pm$0.01 | 4860376345833699840 | 6.43$\pm$0.01 | 309.0 | 269.0 | 15.44 | 18.94 | 13.89
LSR J1835 | M8.4 | 9.5 | 2474 | -3.4 | 0.11$\pm$0.01 | 2091177593123254016 | 5.69$\pm$0.01 | -72.6 | -755.1 | 14.85 | 18.46 | 13.31
TVLM513 | M8.5 | 11.8 | 2467 | -3.65 | 0.08$\pm$0.01 | 1262763648230973440 | 10.73$\pm$0.02 | -43.1 | -65.1 | 16.51 | 20.13 | 14.95
vB 10 | M8.0 | 9.9 | 2539 | -3.39 | 0.10$\pm$0.01 | 4293315765165489536 | 5.92$\pm$0.01 | -598.8 | -1366.1 | 14.3 | 17.75 | 12.77
vB 8 | M6.9 | 9.8 | 2690 | -3.19 | 0.12$\pm$0.01 | 4339417394313320192 | 6.49$\pm$0.01 | -813.0 | -870.6 | 13.81 | 17.05 | 12.32

Our nearby targets have high proper motion, and for the match of the objects’
optical position with X-ray and radio images their apparent space motion needs
to be taken into account. We, thus, have to propagate the optical positions to
the X-ray/radio observing date. To this end, we first search for the Gaia
counterparts (CTPs) of our targets. We started with identifying all Gaia
objects in their vicinity and then narrowed down the search as described in
the following. First, we matched the J2000 coordinates of our UCD sample with
the coordinates given in the Gaia-DR3 catalog (epoch J2016, Gaia Collaboration
et al., 2022) within a radius of $30^{\prime\prime}$. Within this radius we
found $54$ and $137$ Gaia-DR3 CTPs for the $10$ UCDs from our dedicated
X-ray/radio observations and the literature sample, respectively, shown in
gray in Fig. 1. We then performed the proper motion111The uncertainties on the
proper motions range from $0.03$ mas/yr to $0.3$ mas/yr and are neglected
here. (P.M.) correction propagating the J2016 coordinates of all these
potential Gaia-DR3 CTPs backwards to the J2000 epoch. Then we calculated again
the separation between the position of the UCDs in our input catalog and the
position of the Gaia sources. In Fig. 1 we show how the separation between the
Gaia-DR3 CTPs and the input coordinates of the UCD sample has changed after
applying the P.M. correction (before and after the P.M. correction in gray and
in turquoise, respectively). Each UCD has now its closest Gaia-DR3 CTP within
a radius of $1^{\prime\prime}$, except for one target from the literature with
separation $\leq 6^{\prime\prime}$. Given this excellent match, we adopted for
each UCD this closest Gaia-DR3 CTP after the P.M. correction. We confirmed our
associations checking the Gaia-DR3 source_id on
Simbad222https://simbad.u-strasbg.fr/simbad/sim-fid, from where we retrieved
also the spectral types (SpTs).

We also made use of the All-Sky survey data of the ROentgen Survey with an
Imaging Telescope Array (eROSITA, Predehl et al., 2021) on board the Russian
Spektrum-Roentgen-Gamma mission (SRG, Sunyaev et al., 2021). Due to the data
sharing policy our consortium eROSITA_DE has access to the western galactic
half of the sky that comprises all but one of the UCDs in our sample and $7$
out of $16$ of the literature sample. Thus, we performed a match within
$30^{\prime\prime}$ between the P.M. corrected coordinates of our samples with
the boresight position of the merged eRASS:4 catalog
(all_s4_SourceCat3B_221031_poscorr_mpe_clean.fits) provided by the German
consortium, that combines the data from all $4$ eROSITA All-sky surveys. The
details about how the official catalogs are compiled will be published with
the first data release of eROSITA within the work by Merloni et al. (2024) on
the first all-sky survey (eRASS1). Eight UCDs of our sample are detected in
the eRASS:4 catalog, specifically the eRASS proved to be useful for one
object, for which no observational time was allocated with XMM-Newton during
our campaigns. Among the objects of the literature sample only $2$ are
associated with an eROSITA source listed in eRASS:4.

For both samples we derived the bolometric luminosity ($L_{\rm bol}$) and the
effective temperature ($T_{\rm eff}$) that we used to calculate the radius
($R_{\star}$) through the Stefan-Boltzmann law. First, we computed $T_{\rm
eff}$ and the bolometric correction in the $J$ band ($BC_{\rm J}$, $J$ band
magnitude from the Two Micron All-Sky Survey – 2 MASS, Cutri et al., 2003)
with the polynomial fits as a function of spectral type from Filippazzo et al.
(2015). Then, we calculated the absolute $J$ band magnitude ($M_{\rm J}$) with
the Gaia-DR3 (Gaia Collaboration et al., 2022) distances listed in Table 2.
Finally, we used the distances combined with $BC_{\rm J}$ to estimate $L_{\rm
bol}$. The uncertainties for $L_{\rm bol}$ and $T_{\rm eff}$ are taken from
Filippazzo et al. (2015) and they are $0.133$ dex and $113$ K, respectively.
We computed those of the radius by performing the error propagation.

The two samples count $26$ UCDs in total. Their Gaia-DR3 parameters and
stellar parameters are reported in Table 2. In particular, in col.1 we show a
reduced 2MASS name when only the epoch and the first three digits of the right
ascension are displayed333The source designation for objects in 2MASS Catalogs
represents the epoch, right ascension and declination of the source during the
observation: Jhhmmss[.]ss$\pm$ddmmss[.]s (Cutri et al., 2003).. Spectral types
are in col.2, $J$ band magnitude and effective temperature in cols.3&4,
bolometric luminosity and radii in cols.5&6\. From cols.7 to 13 we report the
Gaia-DR3 source_id with distance, proper motion and photometry.

Figure 1: Separation between the Gaia-DR3 (J2016) and the J2000 positions for
the $10$ UCDs from our dedicated X-ray/radio campaigns and the $16$ UCDs from
the literature sample. All $191$ Gaia-DR3 counterparts within
$30^{\prime\prime}$ are shown in gray. In turquoise there are the same objects
but after the application of their P.M. correction. The red histogram
represents the subsample of closest Gaia-DR3 CTPs to all UCDs after the P.M.
correction.

Figure 2: Background-subtracted XMM-Newton EPIC/pn light curves in the energy
band of $0.2-2.0$ keV of the detected UCDs (blue) and the light curves of the
background (green) both with $1$ ks binsize. The mean count rate and its
standard deviation are shown as horizontal solid and dashed bars.

Figure 3: Background subtracted eROSITA light curves of the two samples
presented in this work, the new and the literature UCDs in the $0.2-2.0$ keV
energy band.

Figure 4: continued

##  3 Data analysis

We describe here separately the X-ray, radio and optical photometric data
analysis steps.

###  3.1 X-ray data analysis

####  3.1.1 XMM-Newton

[b]

Table 3: XMM-Newton observation log: target name (col. 1), observation ID
(col.2), date of observation (col.3), offset between proper motion corrected
expected position and X-ray source (col.4), net source counts and detection
maximum likelihood (cols.5 and 6), count rate (col.7), X-ray and fractional
X-ray luminosity (cols. 8 and 9). The last two columns refer to the energy
band of $0.2-2.0$ keV.

2MASS ID | ObsID | Obs.Date | Offset | SCTS | Det.ML | Rate | $\log(L_{\rm x})$ | $\log(L_{\rm x}/L_{\rm bol})$
---|---|---|---|---|---|---|---|---
|  |  | [arcsec] |  |  | [$\times 10^{-3}$ cnt/s] | [erg/s] |
2MJ0752 | 0690720101 | 2013-04-06 | 7.01 | 243.1$\pm$17.1 | 690 | 130.8$\pm$9.2 | 27.87$\pm$ 0.03 | -2.99
2MJ0351 | 0690720501 | 2013-03-05 | 5.63 | 310.4$\pm$21.1 | 455 | 19.1$\pm$1.3 | 26.78$\pm$0.03 | -3.72
2MJ1757 | 0743900101 | 2014-09-01 | 6.06 | 53.9$\pm$10.9 | 25 | 2.8$\pm$0.5 | 27.34$\pm$ 0.08 | -3.25
2MJ0440 | 0743900201 | 2015-03-17 | 6.03 | 333.5$\pm$20.6 | 706 | 61.8$\pm$3.8 | 26.99$\pm$ 0.03 | -3.32
2MJ0306 | 0743900301 | 2015-01-18 | $\cdot\cdot$ | $\cdot\cdot$ | $\cdot\cdot$ |  $<$17.3 |  $<$26.76 |  $<$-3.44
2MJ0838 | 0860200101 | 2020-11-18 | 6.13 | 272.9$\pm$19.5 | 361 | 26.9$\pm$1.9 | 26.70$\pm$0.03 | -3.84
2MJ0435 | 0860200201 | 2020-09-04 | 7.47 | 212.8$\pm$17.1 | 138 | 42.2$\pm$3.4 | 26.57$\pm$0.04 | -3.91
2MJ0652 | 0860200301 | 2020-09-22 | $\cdot\cdot$ | $\cdot\cdot$ | $\cdot\cdot$ |  $<$0.3 |  $<$25.90 |  $<$-4.04
2MJ0215 | 0860200401 | 2020-06-22 | $\cdot\cdot$ | $\cdot\cdot$ | $\cdot\cdot$ |  $<$2.1 |  $<$26.40 |  $<$-3.88

For nine UCDs marked with ‘X’ in Table 1 we present new observations with XMM-
Newton obtained during AO 11, AO 13, and AO 19 with exposure times ranging
from $16$ to $45$ ksec. We analyzed only the observations from the most
sensitive instrument, the EPIC/pn imaging camera, using the XMM-Newton Science
Analysis System (SAS)444SAS Data Analysis Threads:
https://www.cosmos.esa.int/web/xmm-newton/sas-threads. pipeline, version
20.0.0. We started the data processing with epproc which produces the EPIC/pn
photon events list. We extracted the high energy light curve (LC) of the whole
detector ($10.0<E<12.0$ keV) to determine the count rate above which the
particle background is too high. The corresponding time intervals are removed
for the further analysis. The remaining observing time forms the good time
intervals (GTIs). The threshold count rate in the whole-detector light curve
that defines the GTI was chosen individually for each observation based on
visual inspection, and it varies from $0.4$ cnt/s to $1.5$ cnt/s. We then
filtered the event lists applying the GTIs, pixel pattern (0 $\leq$ pattern
$\leq$ 12), and the #xmmea_ep filter.

We performed the source detection in five energy bands: $0.2-0.5$ keV,
$0.5-1.0$ keV, $1.0-2.0$ keV, $2.0-4.5$ keV, and $4.5-12.0$ keV using SAS
tools. To identify the X-ray sources associated with our sample we first
propagated the Gaia-DR3 coordinates using the Gaia-DR3 proper motions to the
epoch of the specific XMM-Newton observation and then we matched these
expected positions of the UCDs with the X-ray coordinates from the output list
of the source detection process allowing for a maximum distance of
$15^{\prime\prime}$. This way we establish that $6$ out of $9$ UCDs were
detected with EPIC/pn and that their emission is mainly within $2$ keV, with
only few and zero counts in the $2.0-4.5$ keV and $4.5-12.0$ keV energy bands,
respectively. The remaining three targets out of the $9$ observed are
undetected and they are identifiable in Table 3 where upper limits are
displayed. In this table we present all relevant X-ray parameters for the XMM-
Newton observations and the results of the source detection process, namely
target name (col.1), observation ID and date (cols.2 & 3), offset between the
optical and X-ray coordinates (col.4), source counts (col.5), detection
likelihood (col.6), net count rate (col.7), all provided by the detection
algorithm for the broad energy band ($0.2-12.0$ keV). For the three undetected
targets we calculated their upper limit count rates from the broad band
sensitivity map of the EPIC/pn observations that we extracted with the SAS
tool esensmap.

To enable a comparison with the literature, the energy band adopted hereafter
is the combination of the softest bands we used for the source detection:
$0.2-0.5$ keV, $0.5-1.0$ keV, and $1.0-2.0$ keV. Thus, the X-ray luminosity
and the fractional X-ray luminosity, given in cols.8 & 9 of Table 3, refer to
$0.2-2.0$ keV (see Sect. 3.1.4 for the calculation of these parameters).

For the spectral and temporal analysis we retained only pixels with flag $=0$.
We defined a circular photon extraction region with a radius of
$35^{\prime\prime}$ centered on the EPIC/pn source position. The background
was extracted from an adjacent source-free circular region on the same CCD
chip with a radius twice that of the source region. We grouped the spectral
channels such that each bin comprises a minimum of $10-20$ photons depending
on the brightness of the source. The response matrix and ancillary response
for the spectral analysis were created using the appropriate SAS tools, rmfgen
and arfgen.

As mentioned above, the X-ray emission of the detected UCDs is mainly within
$2$ keV, with less than $5$ % of the source counts in the harder energy band
of $2.0-12.0$ keV. This justifies our choice of the $0.2-2.0$ keV band for the
extraction of the X-ray light curves. The EPIC/pn light curves for source and
background were generated with the SAS tool evselect, and the background
subtraction was applied using the epiclccorr task, which also takes care of
corrections for instrumental effects such as vignetting. The corrected light
curves of our sample are displayed in Fig. 2 for a $1$ ks bin size and with
the background emission overlaid in green. Events that are not within the
GTIs, have been removed during the filtering process. This may lead to a
reduced number of bins in the light curve. For instance, the light curve of LP
423-31 (see left top panel in Fig. 2) does not show any “good” event during
more than half of its observation.

Visual inspection of the light curves reveals significant variability for four
UCDs: 2MASS J03510004-0052 (2MJ0351), 2MASS J17571539+7042 (2MJ1757), 2MASS
J08380224-5855 (2MJ0838), and 2MASS J04351612-1606 (2MJ0435). In Fig. 2 we
present the average count rate of the corrected source light curve and its
standard deviation with solid and dashed blue lines, respectively. The bins
located above the standard deviation upper bound can be considered to
represent possible flaring activity. The four UCDs with bins that have a count
rate above the standard deviation are, indeed, the same ones that we
identified as variable from visual inspection.

[b]

Table 4: Results from the source detection of eROSITA data: target name (col.
1), survey number and X-ray coordinates with their uncertainties (col.2 to 5),
net count rate and detection maximum likelihood in the broad band $0.2-10.0$
keV (cols.6&7), offset between proper motion corrected expected position and
eROSITA X-ray source (col.8), X-ray and fractional X-ray luminosity (cols. 9
and 10). The last two columns refer to the energy band of $0.2-2.0$ keV.

2MASS ID | eRASS | RA | DEC | RADEC_ERR | Rate | Det.ML | $\rm Sep_{\rm opt,X}$ | $\log L_{\rm x}$ | $\log(L_{\rm x}/L_{\rm bol})$
---|---|---|---|---|---|---|---|---|---
|  | [deg] | [deg] | [arcsec] | [$\times 10^{-3}$ cnt/s] |  | [arcsec] | [erg/s] |
2MJ0752 | 1 | 118.101629 | 16.202432 | 3.64 | 340.15 $\pm$ 68.44 | 61.67 | 3.13 | 28.06$\pm$0.10 | -2.79
2MJ0752 | 2 | 118.100579 | 16.202269 | 3.11 | 209.96 $\pm$ 53.05 | 49.23 | 0.84 | 27.85$\pm$0.12 | -3.00
2MJ0752 | 3 | 118.102579 | 16.202739 | 2.65 | 451.07 $\pm$ 71.40 | 125.98 | 6.54 | 28.19$\pm$0.08 | -2.67
2MJ0752 | 4 | 118.100588 | 16.202026 | 6.31 | 136.29 $\pm$ 66.28 | 9.01 | 0.86 | 27.66$\pm$0.21 | -3.19
2MJ0351 | 1 | $\cdot\cdot$ | $\cdot\cdot$ | $\cdot\cdot$ |  $<$84.60 | $\cdot\cdot$ | $\cdot\cdot$ |  $<$27.24 |  $<$-3.27
2MJ0351 | 2 | 57.751248 | -0.881977 | 4.50 | 115.89 $\pm$ 34.89 | 33.89 | 3.88 | 27.37$\pm$0.13 | -3.13
2MJ0351 | 3 | $\cdot\cdot$ | $\cdot\cdot$ | $\cdot\cdot$ |  $<$52.80 | $\cdot\cdot$ | $\cdot\cdot$ |  $<$27.03 |  $<$-3.47
2MJ0351 | 4 | $\cdot\cdot$ | $\cdot\cdot$ | $\cdot\cdot$ |  $<$36.40 | $\cdot\cdot$ | $\cdot\cdot$ |  $<$26.87 |  $<$-3.63
2MJ0440 | 1 | 70.096931 | -5.501754 | 3.96 | 81.68 $\pm$ 25.11 | 22.83 | 6.96 | 26.87$\pm$0.14 | -3.45
2MJ0440 | 2 | 70.099682 | -5.502688 | 5.87 | 102.56 $\pm$ 47.18 | 13.67 | 5.03 | 26.97$\pm$0.20 | -3.55
2MJ0440 | 3 | $\cdot\cdot$ | $\cdot\cdot$ | $\cdot\cdot$ |  $<$66.50 | $\cdot\cdot$ | $\cdot\cdot$ |  $<$26.78 |  $<$-3.54
2MJ0440 | 4 | 70.100214 | -5.501299 | 2.19 | 258.81 $\pm$ 42.59 | 128.07 | 4.38 | 27.37$\pm$0.08 | -2.95
2MJ0838 | 1 | 129.509037 | -58.936883 | 5.71 | 51.72 $\pm$ 21.96 | 12.17 | 8.07 | 26.78$\pm$0.19 | -3.77
2MJ0838 | 2 | $\cdot\cdot$ | $\cdot\cdot$ | $\cdot\cdot$ |  $<$72.06 | $\cdot\cdot$ | $\cdot\cdot$ |  $<$26.92 |  $<$-3.62
2MJ0838 | 3 | 129.507423 | -58.935126 | 5.64 | 26.98 $\pm$ 11.55 | 8.42 | 2.19 | 26.50$\pm$ 0.20 | -4.05
2MJ0838 | 4 | 129.511589 | -58.930834 | 4.84 | 35.01 $\pm$ 12.30 | 9.10 | 15.49 | 26.61$\pm$0.18 | -3.94
2MJ0435 | 1 | $\cdot\cdot$ | $\cdot\cdot$ | $\cdot\cdot$ |  $<$60.80 | $\cdot\cdot$ | $\cdot\cdot$ |  $<$26.81 |  $<$-3.67
2MJ0435 | 2 | $\cdot\cdot$ | $\cdot\cdot$ | $\cdot\cdot$ |  $<$36.70 | $\cdot\cdot$ | $\cdot\cdot$ |  $<$26.60 |  $<$-3.90
2MJ0435 | 3 | $\cdot\cdot$ | $\cdot\cdot$ | $\cdot\cdot$ |  $<$47.90 | $\cdot\cdot$ | $\cdot\cdot$ |  $<$26.71 |  $<$-3.77
2MJ0435 | 4 | 68.818034 | -16.115683 |  | 85.30 $\pm$ 0.0 | 11.07 | 6.15 | 26.96$\pm$0.04 | -3.53
2MJ0215 | 1 | $\cdot\cdot$ | $\cdot\cdot$ | $\cdot\cdot$ |  $<$16.64 | $\cdot\cdot$ | $\cdot\cdot$ |  $<$26.49 |  $<$-3.78
2MJ0215 | 2 | $\cdot\cdot$ | $\cdot\cdot$ | $\cdot\cdot$ |  $<$5.30 | $\cdot\cdot$ | $\cdot\cdot$ |  $<$25.99 |  $<$-4.28
2MJ0215 | 3 | $\cdot\cdot$ | $\cdot\cdot$ | $\cdot\cdot$ |  $<$63.80 | $\cdot\cdot$ | $\cdot\cdot$ |  $<$27.08 |  $<$-3.20
2MJ0215 | 4 | 33.791500 | -30.671689 | 6.67 | 68.35 $\pm$ 21.69 | 15.98 | 11.97 | 27.11$\pm$0.14 | -3.17
2MJ1055 | 1 | 163.829431 | -73.936249 | 7.64 | 27.99 $\pm$ 8.75 | 11.44 | 12.65 | 26.6$\pm$0.14 | -3.95
2MJ1055 | 2 | 163.817294 | -73.936770 | 4.38 | 85.32 $\pm$ 21.61 | 33.50 | 1.20 | 27.13$\pm$0.12 | -3.46
2MJ1055 | 3 | $\cdot\cdot$ | $\cdot\cdot$ | $\cdot\cdot$ |  $<$48.40 | $\cdot\cdot$ | $\cdot\cdot$ |  $<$26.88 |  $<$-3.71
2MJ1055 | 4 | $\cdot\cdot$ | $\cdot\cdot$ | $\cdot\cdot$ |  $<$22.00 | $\cdot\cdot$ | $\cdot\cdot$ |  $<$26.54 |  $<$-4.05
Literature sample
DENIS J1048 | 1 | $\cdot\cdot$ | $\cdot\cdot$ | $\cdot\cdot$ |  $<$54.24 | $\cdot\cdot$ | $\cdot\cdot$ |  $<$25.92 |  $<$-4.12
DENIS J1048 | 2 | $\cdot\cdot$ | $\cdot\cdot$ | $\cdot\cdot$ |  $<$27.52 | $\cdot\cdot$ | $\cdot\cdot$ |  $<$25.63 |  $<$-4.42
DENIS J1048 | 3 | $\cdot\cdot$ | $\cdot\cdot$ | $\cdot\cdot$ |  $<$26.46 | $\cdot\cdot$ | $\cdot\cdot$ |  $<$25.62 |  $<$-4.44
DENIS J1048 | 4 | 162.052061 | -39.939468 | 5.33 | 48.72$\pm$23.86 | 7.16 | 6.54 | 25.88$\pm$0.22 | -4.12
LHS2065 | 1 | 133.398799 | -3.494837 | 4.13 | 152.08$\pm$43.39 | 31.40 | 6.38 | 27.04$\pm$0.13 | -2.98
LHS2065 | 2 | $\cdot\cdot$ | $\cdot\cdot$ | $\cdot\cdot$ |  $<$68.83 | $\cdot\cdot$ | $\cdot\cdot$ |  $<$26.70 |  $<$-3.37
LHS2065 | 3 | $\cdot\cdot$ | $\cdot\cdot$ | $\cdot\cdot$ |  $<$28.47 | $\cdot\cdot$ | $\cdot\cdot$ |  $<$26.31 |  $<$-3.76
LHS2065 | 4 | $\cdot\cdot$ | $\cdot\cdot$ | $\cdot\cdot$ |  $<$68.65 | $\cdot\cdot$ | $\cdot\cdot$ |  $<$26.70 |  $<$-3.37

####  3.1.2 eROSITA

eROSITA data was analyzed for both the UCDs from our new dedicated X-ray/radio
campaigns and the literature sample defined in Sect. 2. Hereby, we made use of
the preliminary version of the catalog provided by the eROSITA_DE consortium
(all_s4_SourceCat3B_221031_poscorr_mpe_clean.fits) that combines the X-ray
data from all $4$ eROSITA All-sky surveys (eRASS:4). We propagated the Gaia-
DR3 coordinates of all UCDs for their proper motion (P.M.) to the mean epoch
of eRASS:4 catalog, 15 Dec 2020, and we then performed a match with the
boresight corrected coordinates (RA_CORR, DEC_CORR) of eRASS:4 within
$30^{\prime\prime}$. We found potential eROSITA matches for $2$ UCDs from the
literature and $8$ of the UCDs from our new X-ray/radio sample.

To confirm our association we performed a reverse match as in Magaudda et al.
(2022). Starting from the eRASS:4 coordinates we looked for all possible Gaia-
DR3 CTPs within $30\,^{\prime\prime}$. We found $43$ and $11$ possible Gaia-
DR3 CTPs for our new UCD and the literature samples, respectively. After
propagating the positions of all possible Gaia CTPs to the mean eRASS:4 epoch,
we selected the closest to each target of our sample we imposed that the
closest eROSITA CTP must satisfy the condition that the optical and X-ray
separation be smaller than three times the uncertainty of the eRASS:4
coordinates ($Sep_{\rm X-ray,opt}<3\times\rm{RADEC\\_ERR}$). Finally, the
match provided an eROSITA detection for all $8$ new UCDs and $2$ targets from
the literature, that is all matches achieved before the reverse match were
confirmed. We aim to compare the results from the analysis of XMM-Newton data
to those from eROSITA in the same energy bands. Since eROSITA bands used for
eRASS:4 are not the same as those of XMM-Newton that we adopted for this work,
we performed our own source detection on the eROSITA data, and we extracted
the eRASS spectra and light curves for the XMM-Newton energy bands.

From the DETUID column of eRASS:4 we found in which eROSITA sky maps our
targets are located. Then we downloaded the event files of the sky maps of our
sample for each of the four surveys. We performed the source detection using
the eSASSusers_211214 software release (Brunner et al., 2022) in the same
energy bands adopted for XMM-Newton data extraction, except for the hardest
band that is slightly narrower ($4.5-10$ keV) because eROSITA has very little
effective area above $10.0$ keV. After creating the exposure map, detection
mask and background map within the adopted energy bands, we compiled the list
of the detected sources for each of the $4$ surveys using the eSASS pipeline
ermldet for which we used a threshold detection maximum likelihood of $6.0$.
We found that $7$ of the new UCDs are detected in at least one survey, that is
one target less than according to the official eRASS:4 catalog. We recovered
the $2$ UCDs of the literature sample that are also present in eRASS:4. The
missing UCD detection can be explained by different detection likelihood
(DET_LIKE) thresholds. The value adopted for the compilation of the eRASS:4
catalog ($\rm DET\\_LIKE=5$) is smaller than ours and the missing object
(2MJ0306) shows $<20$ counts in eRASS:4.

With eROSITA data we added one more X-ray detection to our UCD sample that is
associated to 2MASS J10551532-7356 (2MJ1055), for which no time was allocated
with XMM-Newton. Moreover, we enabled a long-term variability analysis, for
instance 2MASS J02150802-3040 (2MJ0215) is an upper limit with XMM-Newton but
appears as a detection during the last eROSITA survey, eRASS4.

The UCDs without detection in any of the 4 eRASS all have a detection with
XMM-Newton. The eRASS upper limits are shallower than the XMM-Newton data and
would, therefore, not provide additional information. Therefore, we decided to
ignore them and calculate the upper limits only for those targets that are
detected during at least one eROSITA survey. The X-ray parameters of the
eROSITA sources detected during one or more eRASSs are given in Table 4 with
the results of the upper limit calculation from the extraction of the aperture
photometry with the apetool pipeline. Specifically, in Table 4 we provide the
acronym for the 2MASS name of the object and the identifying number of the
eROSITA survey (col.1&2), the X-ray coordinates with their uncertainty
(cols.3$-$5), the offset between the P.M. corrected optical position and the
X-ray coordinates (col.6), the $0.2-10.0$ keV count rate obtained from the
source detection procedure (col.7), and the detection maximum likelihood in
the same energy band (col.8). The last two columns show the $L_{\rm x}$ and
$L_{\rm x}/L_{\rm bol}$ values computed for the energy band of $0.2-2.0$ keV,
as for the XMM-Newton data (see Sect. 3.1.4).

Finally, we extracted the spectrum and light curve for each of the eROSITA
detections using the srctool pipeline with the option AUTO for the choice of
the source and background region sizes. The eROSITA light curves and spectra
are shown in Figs. 3 & 5. None of the literature targets show a bright eROSITA
detection, thus only their light curves are shown in Fig. 3. The bin size of
the light curves is $14.4$ ks corresponding to one eRODay, i.e. the time that
eROSITA takes to return to the same sky position during its scanning motion
(Predehl et al., 2021). This means there is one bin in the light curve for
each of eROSITA’s visits of the source.

####  3.1.3 X-ray spectra

We fitted all EPIC/pn and eROSITA spectra with net counts $>$200 and net
counts $>$30, respectively. These thresholds were set after several tests to
find the best-fit parameters. They are met by $6$ UCds, $5$ detected with XMM-
Newton and one from eRASS where it exceeds the above threshold in two out of 4
surveys.

We made use of XSPEC version 12.13 (Arnaud, 1996) using a one- or two-
temperature thermal APEC model. Given that nearly all observed counts are
concentrated below $2.0$ keV (see Sect. 3.1.1) we fit the apec model to the
spectrum in the $0.2-2.0$ keV energy band. Interstellar absorption is
negligible for these nearby objects. Given the low statistics of the spectra
we fixed the global abundance ($Z$) to $0.3\,Z_{\odot}$, a typical value for
stellar coronae (Maggio et al., 2007), and we adopted the solar abundance
vector set from Asplund et al. (2009). The free parameters are thus, the
temperatures ($kT$) and emission measures ($EM$). We calculated the
$1\,\sigma$ uncertainties of the fit parameters using the error command in
XSPEC. We used the best-fit parameters to compute for each spectrum the
emission measure weighted mean coronal temperature ($<kT_{\rm corona}>$)
defined as

$<kT_{\rm corona}>=\frac{\sum(\rm EM_{\rm n}\cdot kT_{\rm n})}{\sum{\rm
EM_{\rm n}}}$ (1)

where $kT_{\rm n}$ & $EM_{\rm n}$ with $n=1,2$ are the two temperatures and
two emission measures of the best-fitting model.

The results of the best-fitting models are summarized for all UCDs in Table 5,
where we show the values for the two $kT$ and $EM$ components with their
$1\,\sigma$ uncertainties (cols. 2 to 5), the reduced $\chi$ square
($\chi^{2}_{\rm red}$) and the degrees of freedom (d.o.f.) in cols. 6 & 7 and
the average coronal temperature ($<kT_{\rm corona}>$) in the last column. The
seven spectra, their best-fit models and the relative residuals between data
and model are displayed in Fig. 5.

Table 5: Best-fit X-ray spectral parameters for the five UCDs with net counts $>200$ detected with XMM-Newton and for the only object that shows two eROSITA detections with net counts $>30$. The best-fit parameters refer to the $0.2-2.0$ keV energy band. Other_Name | $kT_{\rm 1}$ | $\log(EM_{\rm 1})$ | $kT_{\rm 2}$ | $\log(EM_{\rm 2})$ | $\chi^{2}_{red}$ | d.o.f. | $<kT_{\rm corona}>$
---|---|---|---|---|---|---|---
| [keV] | [$\rm cm^{-3}$] | [keV] | [$\rm cm^{-3}$] |  |  | [keV]
XMM-Newton: Sect. 3.1.1
2MJ0752 | 0.22 $\pm$ 0.09 | 50.58 $\pm$ 0.20 | 0.86 $\pm$ 0.15 | 50.67 $\pm$ 0.12 | 0.65 | 19 | 0.57 $\pm$ 0.09
2MJ0351 | 0.28 $\pm$ 0.05 | 49.57 $\pm$ 0.11 | 1.08 $\pm$ 0.29 | 49.33 $\pm$ 0.16 | 1.25 | 25 | 0.58 $\pm$ 0.11
2MJ0440 | 0.15 $\pm$ 0.06 | 49.78 $\pm$ 0.24 | 0.75 $\pm$ 0.09 | 49.79 $\pm$ 0.08 | 1.94 | 13 | 0.46 $\pm$ 0.06
2MJ0838 | 0.22 $\pm$ 0.07 | 49.40 $\pm$ 0.18 | 1.12 $\pm$ 0.18 | 49.58 $\pm$ 0.11 | 0.85 | 21 | 0.77 $\pm$ 0.11
2MJ0435 | 0.17 $\pm$ 0.10 | 49.38 $\pm$ 0.18 | 0.84 $\pm$ 0.25 | 49.37 $\pm$ 0.12 | 1.48 | 19 | 0.50 $\pm$ 0.13
eROSITA: Sect. 3.1.2
2MJ0752 (eRASS01) | 0.48$\pm$0.28 | 51.06$\pm$0.28 | $\cdot\cdot$ | $\cdot\cdot$ | 0.22 | 2 | 0.48$\pm$0.28
2MJ0752 (eRASS03) | 0.81$\pm$0.22 | 51.18$\pm$0.13 | $\cdot\cdot$ | $\cdot\cdot$ | 0.58 | 6 | 0.81$\pm$0.22

Figure 5: X-ray spectra of UCDs associated with XMM-Newton (the first five
panels) and eROSITA (the last two panels) sources that have more than $200$
and $30$ net counts, respectively. The spectra are plotted together with the
best-fitting apec model and the residuals (both shown in green); see text in
Sect. 3.1.3 and Table 5.

We use the X-ray emission properties derived from spectral fits to the five
objects with more than 200 counts in the XMM-Newton data (Table 5) to
construct constraints on the size scale of the thermal coronal emission. We
start with the pressure equilibrium equation,

$B_{\rm eq}^{2}/8\pi=n_{e}k_{B}T$ (2)

where $B_{\rm eq}$ is the equipartition magnetic field strength that balances
thermal coronal pressure represented by electron density $n_{e}$, and coronal
temperature $k_{B}T$, where $k_{B}$ is Boltzmann’s constant. The electron
density is obtained from the volume emission measure (VEM) according to

$VEM=n_{e}^{2}V$ (3)

where $V$ is the emitting volume. Combining Eqs. 2 and 3 and assuming that the
corona is a spherical shell with height $H$ we can derive the equipartition
magnetic field strength as a function of coronal height according to

$\frac{VEM(8\pi k_{B}T)^{2}}{B_{\rm
eq}^{4}}=V=4\pi/3(R_{\star}+H)^{3}-4\pi/3R_{\star}^{3}.$ (4)

Inspection of Table 5 shows that, while there is a significant difference
between the two X-ray temperatures fitted, the volume emission measures of
each component are roughly equal within the error bars. Therefore, because the
hotter temperature plasma will provide a stronger constraint on the confining
magnetic field, we use the temperature and emission measure values of the
hotter component. The results are shown in Fig. 6 and further discussed in
Sect. 5.

Figure 6: Plot of equipartition magnetic field strength for objects with
spectral fits from XMM-Newton as a function of assumed thickness of a
spherically symmetric corona, using coronal temperatures and emission measures
from Table 5 and radius measurements from Table 2.

####  3.1.4 X-ray luminosities

For the brightest detections for which we performed the spectral analysis, we
computed the X-ray flux ($f_{\rm x}$) with the flux routine of XSPEC. For the
fainter sources, as well as for the upper limits, we converted the count rate
into flux with a conversion factor based on the results of our spectral
analysis and in the $0.2-2.0$ keV energy band. Specifically, we calculated the
CF of each bright target as the ratio of the XSPEC flux and the count rate
from the source detection. Then we calculated the mean value,

$<CF>=mean\left(\frac{f_{\rm x}}{Ct.Rate}\right)$ (5)

We adopted this approach for both instruments (EPIC/pn and eROSITA)
separately, so that the instrumental properties are implicitly considered. The
final CFs are $\rm CF_{XMM}=2.71\times 10^{-12}\pm 3.44\times 10^{-12}\,\rm
erg/cm^{2}/cts$ and $\rm CF_{eR}=8.01\times 10^{-13}\pm 7.33\times
10^{-14}\,\rm erg/cm^{2}/cts$. We then computed X-ray luminosities ($L_{\rm
x}$) and X-ray to bolometric luminosity ratios ($L_{\rm x}/L_{\rm bol}$) for
the $0.2-2.0$ keV energy band using the Gaia-DR3 distances and bolometric
luminosities from Table 2. The X-ray luminosity and $L_{\rm x}/L_{\rm bol}$
ratios as well as their upper limits for the undetected UCDs are listed in
Tables 3 & 4 for XMM-Newton and eROSITA, respectively.

To be able to compare the literature sample with that of our sample of UCDs we
converted the published X-ray luminosities to the energy band chosen for this
work. To this end, we calculated a CF for each energy band reported in the
literature that is different from our $0.2-2.0$ keV band. We used the online
count rate simulator WebPIMMS,555Count-rate simulator PIMMS:
http://heasarc.gsfc.nasa.gov/cgi-bin/Tools/w3pimms/w3pimms.pl assuming an
unabsorbed $1T$-apec model with temperature of $0.61$ keV and coronal
abundance of $0.2\,\rm Z_{\odot}$. Among the values implemented in WebPIMMS we
chose $0.61$ keV because it is the closest to the measured mean value,
computed with the emission measure weighted temperatures of the six UCDs with
spectral fits (see Sect. 3.1.3 and Table 5 for the spectral analysis). The
updated X-ray literature data are shown in Table 6 with the computed CFs, the
original values, and their references. In this table we report the results of
observations taken while the target was in a quiescent state (upper limits are
included). In cols. 8&11 we indicate if a specific object has data from the
literature of enhanced activity (X-ray flares and/or radio bursts). When the
count rate is not provided in the literature we directly converted the
literature flux from the original energy band into the $0.2-2.0$ keV band, so
no CF is provided in this case. Finally, we show the published quiescent and
bursting radio luminosities ($L_{\rm rad}$) and their references in the last
four columns of the same table.

[b]

Table 6: X-ray and Radio data of the literature sample retrieved as explained
in Sect. 2. We show the X-ray instrument, the X-ray flux and its reference
(from col. 2 to 4), the conversion factor used to calculate the flux in the
$0.2-2.0$ keV energy band used in this work (cols.5&6), the X-ray luminosity
(col.7), with the reference of detected X-ray flares (col.8) and the quiescent
and bursting radio luminosities with their reference (from col.9 to 12).

Name | Instr1 | $f_{\rm x,Lit}$ |  $\rm Ref_{xray}$2 | $\rm CF_{0.2-2.0}$ | $f_{\rm x,0.2-2.0}$ | $\log L_{\rm x}$ | X-ray2 | $\log L_{\rm R,\nu}$ |  $\rm Ref_{radio}$2 | $\log L_{\rm R,\nu}^{\rm burst}$ | Radio2
---|---|---|---|---|---|---|---|---|---|---|---
|  | [$\rm\times 10^{-14}erg/cm^{2}/s$] |  | [$\rm\times 10^{-11}erg/cm^{2}/cnt]$ | [$\rm\times 10^{-14}erg/cm^{2}/s$] | [erg/s] | flares | [erg/s/Hz] |  | [erg/s/Hz] | bursts
2MJ0036 | C | $<0.10^{0.2-8.0}$ | B05 | 3.54 |  $<$0.79 |  $<$25.86 | $\cdot\cdot$ | 13.4 | Bur05 | 13.8 | B02
2MJ0523 | C | $<0.08^{0.2-2.0}$ | B10 | 3.84 |  $<$0.90 |  $<$26.24 | $\cdot\cdot$ |  $<$13.0 | Ant07 | 13.7 | B06
2MJ0602 | C | $<0.08^{0.2-2.0}$ | B10 | 3.84 |  $<$0.90 |  $<$26.17 | $\cdot\cdot$ |  $<$12.8 | B10 | $\cdot\cdot$ | $\cdot\cdot$
2MWJ1507 | C | $<0.10^{0.2-2.0}$ | B05 | 3.84 |  $<$0.85 |  $<$25.75 | $\cdot\cdot$ |  $<$12.6 | B02 | $\cdot\cdot$ | $\cdot\cdot$
BRI0021-0214 | C | $<0.08^{0.2-2.0}$ | B10 | 3.84 |  $<$0.90 |  $<$26.22 | $\cdot\cdot$ | 12.6 | B02 | 13.8 | B02
DENIS J1048 | X | 0.660.6-10.0 | St12 | 0.18 | 0.21 | 24.62 | $\cdot\cdot$ | 12.3 | Rav11 | 14.8 | Bur05
Gl 569 B | C | 0.550.3-1.2 | St04 | 8.8 | 10.56 | 27.08 | St04 |  $<$12.5 | B06 | $\cdot\cdot$ | $\cdot\cdot$
Kelu 1 | C | 0.070.1-10.0 | Au07 | $\cdot\cdot$ | 0.06 | 25.45 | $\cdot\cdot$ |  $<$12.8 | Au07 | $\cdot\cdot$ |
LHS2065 | X | 2.920.3-8.0 | R08 | $\cdot\cdot$ | 6.46 | 26.76 | SL02 |  $<$12.8 | B02 | $\cdot\cdot$ | $\cdot\cdot$
LP 349-25 | C | 0.320.2-2.0 | W14 | $\cdot\cdot$ | 0.32 | 25.88 | W14 | 14.0 | PB07 | $\cdot\cdot$ | $\cdot\cdot$
LP 412-31 | X | 6.230.5-10.0 | St06 | 0.17 | 188.26 | 27.25 | St06 |  $<$12.8 | B10 | $\cdot\cdot$ | $\cdot\cdot$
LP 944-20 | C | $<0.03^{0.1-4.0}$ | Ru00 | 3.54 |  $<$0.42 |  $<$25.32 | Ru00 | 12.3 | B06 | 13.9 | B06
LSR J1835 | C | $<0.08^{0.2-2.0}$ | B08 | 3.84 |  $<$0.95 |  $<$25.56 | $\cdot\cdot$ | 13.3 | B06 | 14.0 | Hal08
TVLM513 | C | 0.630.3-2.0 | B08 | 4.11 | 1.11 | 26.18 | $\cdot\cdot$ | 13.4 | B02 | 14.9 | B08
vB 10 | R | 0.420.1-2.4 | F00 | 2.22 | 0.40 | 25.22 | F03 |  $<$12.6 | Kri99 | $\cdot\cdot$ | $\cdot\cdot$
vB 8 | R | 15.770.1-2.4 | F93 | 0.54 | 16.71 | 26.92 | T90 |  $<$12.1 | Kri99 | $\cdot\cdot$ | $\cdot\cdot$

  * 1

XMM-Newton: X; ROSAT: R; Chandra: C

  * 2

Ant07: Antonova et al. (2007); Au07: Audard et al. (2007); B02: Berger (2002);
B05: Berger et al. (2005); B06: Berger (2006); B08: Berger et al. (2008); B10:
Berger et al. (2010); Bur05: Burgasser & Putman (2005); F93: Fleming et al.
(1993); F95: Fleming et al. (1995); F00: Fleming et al. (2000); F03: Fleming
et al. (2003); Hal08: Hallinan et al. (2008); Kri99: Krishnamurthi et al.
(1999); PB07: Phan-Bao et al. (2007); R08: Robrade & Schmitt (2008); Rav11:
Ravi et al. (2011); Ru00: Rutledge et al. (2000); St04: Stelzer (2004); St06:
Stelzer et al. (2006); St12: Stelzer et al. (2012); SL02: Schmitt & Liefke
(2002); T90: Tagliaferri et al. (1990); W14: Williams et al. (2014b)

###  3.2 Radio data analysis

Observations were obtained as part of three NRAO/JVLA projects and one ATCA
project; observation specifics are listed in Table 8. Note that we add in the
radio flux densities for 3 of our targets which were reported previously in
the literature. Table 7 lists the specific radio frequency ranges used in pre-
upgrade VLA observations (pertinent to the literature references used in this
paper), and the JVLA and ATCA observations presented. Table 1 Our literature
references are Antonova et al. (2013) for 2M0215 and Berger (2006) for 2M0440
and 2M0435. Berger (2006) does not quantify what the quoted upper limits are,
and we are assuming that they are 3 sigma. We have recalculated the 3$\sigma$
upper limits on the radio luminosities from the literature using Gaia-DR3
distances from Table 2. These updated values are listed in 9.

####  3.2.1 JVLA data

The flux calibrator (col. 7 of Table 8) was used to establish both overall
gain solutions as well as determine bandpass calibrations, while the phase
calibrator (column 8 of Table 8) was used to correct for phase variations.
Data editing and further calibration steps were performed using standard
techniques in the Common Astronomy Software Applications (CASA) package
(McMullin et al., 2007). The calibrated visibility data for each target were
split off from the main dataset after successful calibration and imaged. The
individual epochs of observations in January 2015 were calibrated and imaged
separately, then the visibility datasets were concatenated and imaged as a
whole to increase the sensitivity. A similar approach was used for the two
epochs of observations in September 2020. The cell size for imaging the D
configuration X-band datasets was 1.5 arcsec, while for the CnB configuration
X-band datasets the cell size was 0.34 arcsec. The cell size for the B
configuration C-band datasets was 0.5 arcsec. The clean algorithm used to
create images employed natural weighting with a threshold of 10 $\mu$Jy to
discover faint radio sources.

We derived the expected position of our targets at the time of each JVLA
observation, taking into account the P.M. provided by Gaia DR3 (see Table 2).
Once images had been created for each of the four fields, we searched for any
radio sources in the vicinity of the expected target positions. For the six
epochs of 2MJ0306 and two of 2MJ0652, we searched the image for each epoch as
well as that made from combining all epochs.

The only field with a radio source near the expected target position is
2MJ0752, with a radio counterpart offset from the expected position by only
$0.27^{\prime\prime}$ (Figure 7). After deconvolving from the clean beam it
appears to be a point source with a peak flux density of 78.9$\pm$3.4
$\mu$Jy/beam in total intensity. The same field imaged in Stokes V also shows
a source at the same position, with a peak flux density of $-59.8\pm$2.6
$\mu$Jy/beam, or circular polarization fraction $V/I=-76\pm$5%. Late-type
stellar radio sources are typically the only type to experience high degrees
of circular polarization. We consider the association of this highly
circularly polarized radio point source with the brown dwarf robust,
considering the beam size of $14.45^{\prime\prime}\times 8.2^{\prime\prime}$
and the significance of the detection. For all other targets the radio sources
in the images are offset by $>30^{\prime\prime}$ from the expected target
position, i.e. they are undetected. The fields around the other four sources
are displayed in the Appendix (Figures 18 and 19). Table 9 lists the 3$\sigma$
upper limits and the detected radio flux with rms for all of the program
sources, and Table 10 describes the average, maximum and minimum values of
total and circularly polarized flux.

####  3.2.2 ATCA data

Reduction and calibration of ATCA data proceeded as noted above in Sect.
3.2.1, using Miriad software (Sault et al., 1995), with amplitude and phase
calibration using sources identified in cols. 7 and 8, respectively, of Table
8. The cell size for imaging C band observations was 0.8 arcsec, while for X
band observations it was 0.4 arcsec.

We performed similar calculations as described above in Sect.3.2.1 to
determine the expected position of the targets at the epoch of the
observation. The only object with a confident detection was 2MJ0838, with a
point source only 0.3′′ away from the expected position at both X and C bands.
Maps of the region around the expected position for this object in Stokes I
and V are shown in Fig. 8, while those for the undetected objects are
illustrated in Fig. 20. Table 9 lists the 3$\sigma$ upper limits and detected
radio luminosities; the only object with a detection is 2MJ0838. Table 10
lists the average values of the flux densities for total intensity and
circularly polarized emission averaged across the observation.

Table 7: Radio Frequency Coverage Frequency Code | Frequency Range (GHz)
---|---
AC | 4.476-6.525
AX | 7.976-10.025
VC | 3.976-8.024
VX | 7.991-10.015
LVC | 2$\times$50 MHz at 4.9 GHz
LVX | 2$\times$50 MHz at 8.46 GHz
AC=ATCA C-band, AX=ATCA X-band
VC=JVLA C-band, VX=JVLA X-band
LVC=Legacy VLA C-band, LVX=Legacy VLA X-band
Table 8: Observing log for radio data. Target | Project | Obs. Date | Obs. Start | Obs. End | Conf. | Flux Calib. | Phase Calib. | Frequency
---|---|---|---|---|---|---|---|---
2MJ0752 | 13A-360 | 2013 April 6 | 19:54:25 | 23:53:46 | D | 3C147 | J0738+1742 | VX
2MJ0351 | ” | 2013 March 5 | 00:10:49 | 03:10:20 | D | 3C147 | J0339-0146 | VX
2MJ1757 | ” | 2013 April 27 | 05:49:08 | 11:47:57 | D | 3C286 | J1716+6836 | VX
2MJ0306 | 14B-158 | 2015 Jan. 13 | 00:58:08 | 01:56:31 | CnB | 3C138 | J0334-4008 | VX
2MJ0306 | ” | 2015 Jan. 14 | 02:07:35 | 03:07:24 | ” | ” | ” | ”
2MJ0306 | ” | 2015 Jan. 14 | 03:07:27 | 04:07:15 | ” | ” | ” | ”
2MJ0306 | ” | 2015 Jan. 15 | 01:33:45 | 02:33:33 | ” | ” | ” | ”
2MJ0306 | ” | 2015 Jan. 15 | 02:33:37 | 03:33:24 | ” | ” | ” | ”
2MJ0306 | ” | 2015 Jan. 19 | 00:48:03 | 04:17:28 | ” | ” | ” | ”
2MJ0838 | C3388 | 2021 Feb. 19 | 06:30:05 | 18:28:35 | 6D | 1934-638 | J0835-5953 | AX,AC
2MJ0652 | SJ6020 | 2020 Sept. 15 | 12:35:22 | 17:34:34 | B | 3C48 | J0648-3044 | VC
2MJ0652 | ” | 2020 Sept. 24 | 11:46:27 | 16:45:39 | B | 3C48 | J0648-3044 | VC
2MJ1055 | C3388 | 2021 Feb. 20 | 00:39:35 | 12:59:05 | 6D | 1934-638 | J1057-797 | AX&AC
Figure 7: Contour image of the JVLA field around the target 2MJ0752, which has
a source only 0.27′′ from the expected position. The size is $3^{\prime}\times
3^{\prime}$ and contour levels are $3$, $5$, $10$, $15$, $20$, $30$, $50$, and
$100$ times the image rms, which is listed in Table 9.

Figure 8: Total intensity and Stokes V maps of the three arcminute region around the target 2MJ0838 observed with ATCA. Contours are shown at 3, 5, and 10 times the rms values listed in Table 9.  Table 9: Radio flux densities (with $1\,\sigma$ error) or $3\,\sigma$ upper limit for non-detections. Target | Freq. | rms | $L_{\rm R,\nu}$
---|---|---|---
| GHz | [$\mu$Jy] | [erg s-1 Hz-1]
2MJ0752 | VX | 3.3 | 1.05$\times$10${}^{13}\pm$4.5$\times$1011
2MJ0351 | VX | 8.2 |  $<$6.4$\times$1012
2MJ1757 | VX | 5.4 |  $<$7.1$\times$1012
2MJ0440 | LVX | 131 |  $<$4.5$\times$1012
2MJ0306 | VX | 2.6 |  $<$1.2$\times$1012
2MJ0838 | AC | 63 | 7.1$\times$10${}^{13}\pm$0.94$\times$1013
” | AX | 35 | 4.9$\times$10${}^{13}\pm$0.52$\times$1013
2MJ0435 | LVX | 161 |  $<$6.5$\times$1012
2MJ0652 | VC | 1.8 |  $<$1.7$\times$1012
2MJ0215 | LVC | 252 |  $<$1.8$\times$1013
2MJ1055 | AC | 36 |  $<$2.4$\times$1013
” | AX | 30 |  $<$2.0$\times$1013
1 upper limits reported in Berger (2006) are presumed
to be 3$\sigma$; 2 Antonova et al. (2013)
† separation of two UCDs is greater than the synthesized beam
width at both frequencies, so limit applies to each component
of the binary separately; see text for details.
Table 10: Total and Circularly Polarized Intensities of Radio-Detected Objects Object | Freq. | Stokes | Avg Fν | Min. Fν | Max. Fν
---|---|---|---|---|---
|  |  | ($\mu$Jy) | ($\mu$Jy) | ($\mu$Jy)
2MJ0752 | VX | I | 78.9$\pm$3.4 | 38.4$\pm$8.9 | 147$\pm$17
” | ” | V |  $-$58.9$\pm$2.6 |  $-$28$\pm$9 |  $-$130$\pm$14
2MJ0838 | AC | I | 476$\pm$63 | 231$\pm$68 | 1220$\pm$58
” | ” | V |  $-$102$\pm$18 |  $-$208$\pm$66 |  $-$704$\pm$175
” | AX | I | 330$\pm$35 | 251$\pm$60 | 485$\pm$60
” | ” | V | -39$\pm$13 |  $-$178$\pm$59 |  $-$214$\pm$51

###  3.3 TESS data

We analysed TESS light curves with the primary aim to obtain the rotation
periods of the UCDs, and secondly to identify optical flares as additional
activity diagnostics. The results of the TESS analysis and TESS identification
numbers (TIC) for all UCDs studied in this article, namely the $26$ objects
from our dedicated X-ray/radio campaigns and the literature sample, are given
in Table 11.

We uploaded the target list of the UCDs to the Barbara A. Mikulski Archive for
Space Telescopes (MAST) interface and found two-minute cadence light curves
processed by the TESS collaboration for $17$ out of the $26$ targets. Most of
them were observed in multiple sectors (see Table 11), and we downloaded $69$
individual light curves from MAST.

For our analysis we used the PDCSAP_FLUX, in which the pipeline processing has
removed some of the instrument systematics from the flux measurements. TESS
assigns a quality flag to all measurements. In the first step, we removed all
data points with a quality flag different from 0 except of ‘Impulsive outlier’
(which could be real stellar flares) and ‘Cosmic ray in collateral data’ (see
more detailed information in Magaudda et al., 2022). We then normalized the
individual sector light curves of each UCD by dividing all data points by the
median flux. The period search was performed using the method described by
Magaudda et al. (2022). In short, we used three different standard time series
analysis techniques, the generalized Lomb-Scargle periodogram (GLS;
Zechmeister & Kürster, 2009), the autocorrelation function (ACF), and sine
fitting. We then phase-folded the light curves with the best period estimate
from each method. By visual inspection we selected the methods that yielded
the period that best represents the light curves. As final period we adopted
the mean value of these methods. If light curves from more than one sector
were available for a given UCD all good period values from the different
methods in the different sectors were averaged (up to three values per
sector). Uncertainties on the $P_{\rm rot}$ values are given as the standard
deviation of those period values that were used to calculate our adopted
period. The typical results of our analysis are demonstrated on an example for
one sector of the UCD TIC 29890705 in Fig. 21.

This initial period search resulted in a significant period detection for
seven of $26$ targets. However, by eye the ACF of $3$ of the remaining UCDs
without a period estimate (i.e. TIC 70555405, TIC 117733581 and TIC 401945077)
appears to display a clear signal. Therefore, we decided to bin the original
light curves of all $19$ UCDs without a detected period with a factor of $15$
and we repeated the period search. This way, we found a reliable rotation
period for further 5 UCDs, including the three mentioned above.

Therefore, as a final result we were able to detect a period for $12$ out of
$26$ UCDs. The period for one of them, TIC 308243298, is, however, only
marginally detected in the binned data. The object was observed in $4$ sectors
but only two of them show a detection. Since this period is consistent with
the period upper limit from the $v\sin{i}$ measurement, we consider it when we
discuss the X-ray vs radio luminosity relation for UCDs with $P_{\rm rot}\leq
1$ d (see Sect. 5).

For the $14$ UCDs without a period estimate from our method or without TESS
two-minute cadence data we adopted the value listed by Crossfield (2014), who
collected published measurements of photometric rotation periods for $58$
UCDs. When no photometric period is available, we calculated its upper limit
by using $v\sin{i}$ measurements from high-resolution spectroscopic data taken
from the same reference and the computed radii listed in Table 2.

For those UCDs for which we found a valid photometric rotation period we
calculated the inclination of the systems by using the radii listed in Table 2
and the $v\sin{i}$ measurements. These data are taken from Crossfield (2014)
for all UCDs, but for one object (2MJ0752) for which no value is listed in
their work thus we adopted the one from Hughes et al. (2021). For $5$ UCDs
$v\sin{i}$ is larger than the actual rotational velocity, thus we could not
retrieve a reliable inclination. We flagged these cases with “$v\sin{i}>v_{\rm
rot}$” in Table 11 where we summarize the results of the analysis of our TESS
data. In particular, cols. 2&3 show the TESS ID of the UCD and the number of
sectors in which it was observed, cols.4 and 5 list the photometric period and
its reference, col.6 holds the literature $v\sin{i}$ value, and col. 7 the
inclination.

To detect flares we applied the method that we explained in detail in Stelzer
et al. (2022) and Magaudda et al. (2022). In short, we smoothed the light
curves and removed all data points that deviate more than 2-$\sigma$ from the
smoothed light curve. The treatment was repeated three times with decreasing
size of the boxcar width that was used for the smoothing. The final smoothed
light curve was then subtracted from the original light curve in order to
remove the rotational signal. All groups of at least three consecutive data
points that deviate more than 3-$\sigma$ from this ‘flat’ version of the light
curve were identified as potential flares. In a 5-step validation process
false positive detections were removed from this list. The validation criteria
are (1) the flare event must not occur right before or after a data point gap
in the light curve, (2) the flux ratio between flare maximum and last flare
point must be $\geq 2$, (3) the flare maximum cannot be the last flare point,
(4) the decay time has to be longer than the rise time, and (5) a fit
conducted using the flare template defined by Davenport et al. (2014) must fit
the flare better than a linear fit. The UCDs with validated flares that
fulfill all of the above mentioned criteria are marked in the last column of
Table 11.

A detailed analysis of the TESS flares is beyond the scope of this paper.
Systematic studies of large samples of flares observed with TESS on UCDs have
been published (Petrucci et al., 2023). In this work we merely us the presence
of optical flares as an additional activity indicator in the joint multi-
wavelength analysis of our sample.

[b]

Table 11: TESS observations of all UCDs studied in this article,
$v\cdot\sin{i}$ measurements retrieved from the literature and the computed
inclination. For $7$ literature targets we calculated the upper limits from
the $v\sin{i}$ measurements given by Crossfield (2014).

Object | TIC ID | $N_{\mathrm{TESS}}^{a}$ | $P_{\rm rot}$ | Refc | Flares | $v\leavevmode\nobreak\ \sin{i}^{d}$ | $i$
---|---|---|---|---|---|---|---
|  |  | [d] |  |  | [km/s] | [deg]
UCDs with new X-ray/radio campaigns
2MJ0752 | 17773418 | 3 | 0.8840 $\pm$ 0.0022 | (1) | ✓ | 9.0$\pm$2.0 | 47.3$\pm$14.7
2MJ0351 | 401945077 | 1 | 0.7288 $\pm$ 0.0002 | (1) | — | 6.5$\pm$2.0 | 38.3$\pm$14.5
2MJ1757 | 219095664 | 26 | 0.1622 $\pm$ 0.0002 | (1) | ✓ | 34.6$\pm$2.6 | 41.5$\pm$5.8
2MJ0440 | 298907057 | 2 | 0.4975 $\pm$ 0.0059 | (1) | ✓ | 17.9$\pm$1.4 | $v\sin{i}>v_{\rm rot}$
2MJ0306 | 308243298 | 4 | 0.2947 $\pm$ 0.0003b | (1) | — | 20.4$\pm$2.0 | 83.9$\pm$72.0
2MJ0838 | 44984200 | 9 | 0.1130 $\pm$ 0.0001 | (1) | ✓ | $\cdot\cdot$ | $\cdot\cdot$
2MJ0435 | 117733581 | 2 | 0.6255 $\pm$ 0.0045 | (1) | ✓ | 12.1$\pm$3.0 | $v\sin{i}>v_{\rm rot}$
2MJ0652 | 53603145 | 3 | 0.2095 $\pm$ 0.0002 | (1) | — | $\cdot\cdot$ | $\cdot\cdot$
2MJ0215 | 70555405 | 2 | 0.7025 $\pm$ 0.0016 | (1) | ✓ | $\cdot\cdot$ | $\cdot\cdot$
2MJ1055 | 277539431 | 4 | 0.1901 $\pm$ 0.0001 | (1) | ✓ | $\cdot\cdot$ | $\cdot\cdot$
Literature UCD sample
2MJ0036 | $\cdot\cdot$ | $\cdot\cdot$ | 0.12 | (2) | $\cdot\cdot$ | 40.0$\pm$2.0 | 62.2$\pm$15.0
2MJ0523 | 442929628 | 3 |  $<$0.09 | (2) | — | 18.0$\pm$2.0 | $\cdot\cdot$
2MJ0602 | $\cdot\cdot$ | $\cdot\cdot$ |  $<$0.14 | (2) | $\cdot\cdot$ | 9.0$\pm$3.0 | $\cdot\cdot$
2MWJ1507 | $\cdot\cdot$ | $\cdot\cdot$ |  $<$0.05 | (2) | $\cdot\cdot$ | 30.0$\pm$2.3 | $\cdot\cdot$
BRI0021-0214 | 244167275 | 2 | 0.17 | (2) | — | 34.2$\pm$1.6 | 76.0$\pm$25.0
DENIS J1048 | 107012050 | 3 | 0.2540 $\pm$ 0.0004 | (1) | ✓ | 18.0$\pm$2.0 | 65.0$\pm$17.9
Gl 569 B | 258105174 | 1 |  $<$0.40 | (2) | ✓ | 19.3$\pm$2.0 | $\cdot\cdot$
Kelu 1 | 229255770 | 1 | 0.07 | (2) | — | 60.9$\pm$1.9 | 41.0$\pm$6.0
LHS2065 | $\cdot\cdot$ | $\cdot\cdot$ |  $<$0.45 | (2) | $\cdot\cdot$ | 11.3$\pm$1.1 | $\cdot\cdot$
LP 349-25 | $\cdot\cdot$ | $\cdot\cdot$ | 0.07 | (2) | $\cdot\cdot$ | 55.0$\pm$2.0 | 23.9$\pm$2.4
LP 412-31 | $\cdot\cdot$ | $\cdot\cdot$ | 0.61 | (2) | $\cdot\cdot$ | 14.8$\pm$2.0 | $v\sin{i}>v_{\rm rot}$
LP 944-20 | 143029977 | 2 | 0.1598 $\pm$ 0.0003 | (1) | — | 28.9$\pm$0.8 | $v\sin{i}>v_{\rm rot}$
LSR J1835 | $\cdot\cdot$ | $\cdot\cdot$ | 0.11 | (2) | $\cdot\cdot$ | 43.9$\pm$2.2 | 71.1$\pm$17.5
TVLM513 | 311188315 | 1 | 0.08 | (2) | — | 59.2$\pm$2.0 | $v\sin{i}>v_{\rm rot}$
vB 10 | $\cdot\cdot$ | $\cdot\cdot$ |  $<$0.80 | (2) | $\cdot\cdot$ | 6.5$\pm$2.0 | $\cdot\cdot$
vB 8 | $\cdot\cdot$ | $\cdot\cdot$ |  $<$0.66 | (2) | $\cdot\cdot$ | 9.0$\pm$2.0 | $\cdot\cdot$

  * a

Number of sectors in which TESS observed the UCD

  * b

Marginal period detection (see text in Sect. 3.3)

  * c

References: (1) this work, (2) Crossfield (2014).

  * d

$v\sin{i}$ measurements from Crossfield (2014), except for the value for
2MJ0752 that comes from Hughes et al. (2021).

##  4 Objects with particular interesting radio behavior

###  4.1 2MJ0752

The source 2MJ0752 was previously reported as undetected by Berger (2006) with
an upper limit of $39\,\mu$Jy. Because of the high degree of circular
polarization in the time- and frequency-averaged data, we explored subsets to
determine whether there was any evidence of time- or frequency-dependent
variability, as an indication of the bursting behavior observed in some radio-
detected UCDs.

We initially explored the time domain only, creating a light curve by
integrating over all spectral windows, and dividing each scan in half, to
create a total of 38 time bins. The top left panel of Fig. 9 displays the time
variation of total intensity and circularly polarized flux. Upper limits in
total intensity are depicted as downward facing arrows. There are two local
peaks in the light curve of total intensity, each having an enhancement of
1.7-1.8 times the time-integrated flux density, and separated by about 1.7
hours. These however do not show up as local peaks in circularly polarized
flux.

In the bottom left panel of Fig. 9 we show the percent circular polarization,
computed only for time bins in which there was a detection in both total
intensity and circularly polarized flux. The average value of percent circular
polarization, taken from the time-averaged values of Stokes V and Stokes I
flux, is shown as a black horizontal line. The high degree of circularly
polarized flux obtained from the integrated maps does not reveal any
significant variation over the course of the observation.

We next explored dividing the data in both time and frequency to explore
spectro-temporal variations. Our initial investigation used the original 38
time bins, but subdividing the data into a few frequency bins did not reveal
any significant variations. We converged on four spectral spans and six time
intervals. These light curves are displayed in the right panels of Fig. 9.
These again did not display evidence for significant temporal or frequency-
dependent variations.

The object’s low rotational velocity of $v\sin i=9$ km/s and rotation period
of $21.1$ h measured on the TESS light curves reveal that we are seeing the
UCD at the relatively high inclination angle of $i$ $\approx$50∘ (see Table
11). The emission is highly circularly polarized ($-76\pm 5\%$, see Sect.
3.2.1), which is very suggestive of bursting behavior. While the total
intensity did show some variations, there was no apparent change in the amount
of circular polarization. Comparison with a previous epoch of observation
$\sim$ 9 years earlier, with an upper limit below the detected level of
emission, suggests that the detection spanned a long-lived burst.

The light curve of 2MJ0752 across the entire band shown in Fig. 9 reveals two
apparent double peaks, which at face value could be periodic bursts
corresponding to a short rotation period ($<2$ h). However, with a known
rotation period $\sim$21 h, perhaps this observation instead reveals the
leading and trailing edges of an auroral loss cone as it traverses our line of
sight. The time between peaks of about 1.7 h would approximately constrain the
full-width opening angle of the loss cone. And the width of each of burst,
which is roughly 0.4-0.5 h, would approximately constrain the loss-cone width.
Compared to previous radio detections of periodic bursting behavior that was
ascribed to auroral loss cones (Hallinan et al., 2006; Yu et al., 2011),
2MJ0752 has a relatively long rotation period. Additional observations are
needed to confirm that it is in fact producing periodic bursts, and to
constrain the timing of the bursts at a range of frequencies. Detailed
modelling of frequency- and time-dependence of the emission, along the lines
of Leto et al. (2016), will likewise enable constraints on the physical
properties of the emitting region.

Figure 9: Time-series of JVLA data for 2M0752. top left \- Variation of total
intensity and circularly polarized flux with time integrated over all
frequencies The black and the red line denote, respectively, the time-averaged
total intensity and the time-averaged amount of Stokes $V$ flux. bottom left
\- Variation of the percent circular polariation integrated over all
frequencies and time-average value shown as black line. top right \- Time and
frequency variations of the Stokes $I$ flux density, broken up into six time
bins and four frequency spans. bottom right \- Variations of circularly
polarized flux with the same time and frequency bins as for total flux
density.

###  4.2 2MJ0838

The source 2MJ0838 was detected at both C- and X-bands. In addition to a
detection in total intensity (Stokes I), the source was evident in Stokes V
images as well. In the C-band Stokes I image there was a strong source
detected in addition to the target. We performed image fits to the sources to
determine their positions and average characteristics. The average flux
density values for the target, 2MJ0838, are listed in Table 10. Light curve
creation proceeded in Miriad with the task uvfit. For each invocation of the
task, all source parameters for the fit to the visibility data for the
specified time bin were fixed except the flux density of the target. This
process was repeated individually for both Stokes I and V, and for each
frequency band, to create light curves with approximately 7.5 minute time bins
(roughly half of each scan length). The light curves of 2MJ0838 in both radio
bands as well as the time evolution of the polarization are shown in Figs. 10
and 11.

Inspection of the temporal behavior of circularly polarized and total
intensity flux densities reveals interesting behavior. The C-band light curve
shows evidence of total intensity variability at a factor of six at least,
with additional periods where the source is undetected. The X-band light curve
displays far less evidence for variability during the times when the source
was detected. The average flux densities over the whole observation, derived
from image fits, are overlaid as horizontal lines in Figs. 10 and 11. For both
bands these averages are lower than the flux densities in time intervals when
the source is detected, which reflects the significant fraction of the
observing time when the source is undetected. In general, when there is a
detection in the Stokes V flux density, there is little variability in these
detected flux density levels. We note though that there are relatively few
intervals when there is a Stokes V detection. The average values of circularly
polarized flux density derived from the entire observation are smaller in an
absolute value sense than the time intervals when it is detected; this mirrors
the behavior noted for the average total intensity. Table 10 delineates these
average, minimum, and maximum detected flux densities.

The periods of time with elevated C-band flux densities correspond in the
X-band to either upper limits or to no evidence for variability. The time
period 07:20:35-09:52:55 is indicative of the first type, variability at
C-band with undetected flux densities at X-band, while the time period
10:59:45-12:58:25 shows no indication of variability at X-band and factors of
$\sim$three variability at C-band. We term these ”C1” and ”C2”, respectively.

We investigated the frequency dependence of the large and varying C-band flux
during the decay phase of C2 by repeating the above calculations in each time
bin but splitting the 2.048 GHz bandpass into four 512 MHz spectral intervals.
We determined the flux of the bright object noted above in each spectral
interval, and repeated the uvfit calculations, allowing only the peak flux
density of the target to vary. In order to investigate spectral variations, we
filtered on time bins where the signal to noise ratio of the time bin across
the entire C-band bandpass was larger than 8, and the signal-to-noise ratio of
each of the four subbands was larger than 3. This resulted in five time bins.
These times are indicated in Figure 10 and the spectral variations, including
the flux density measurement at X-band, are shown in Figure 12.

Figure 10:  Total intensity and circular polarization behavior of 2MJ0838 with
ATCA at C band. Light curve bin size is 7.5 minutes. Open squares indicate
detections of total intensity, and filled symbols indicate detections of
circularly polarized flux density. For upper limits in total intensity, an
arrow from 1$\sigma$ to 3$\sigma$ is shown. The dashed lines and forward
(backward) hatched regions indicate the average flux density and standard
deviation for total intensity (circularly polarized flux) determined from
image analysis spanning the entire time. Approximate extent of events ’C1’ and
’C2’ are indicated. Specific time bins in the decay phase of event C2 are
denoted, and their spectral energy distributions are plotted in Figure 12.
Figure 11:  Total intensity and circular polarization behavior for 2MJ0838 at
X band. Figure labels are the same as for Figure 10.  Figure 12: Spectra of
individual time bins during the decay phase of event C2 on SCR 0838 noted in
Figure 10. The C-band was divided into four spectral bins for which the flux
was computed, while the X-band is represented by a single flux measurement.
The spectral index calculated from the two lowest frequency bins is shown for
the first three time bins in the decay of C2.

Figure 13: Radio vs. X-ray luminosity. Left panel: the lowest measured X-ray
luminosity in the $0.2-2.0$ keV energy band, including upper limits. The black
line represents the relation from Güdel & Benz (1993). Right panel: same as
left panel with the radio bursts and X-ray flares presented as open squares
and stars, respectively. See Sect. 5 for more details. Figure 14: Radio to
X-ray luminosity ratio vs rotation period. TESS & X-ray flares are shown with
open filled circles & stars, respectively, while radio bursts with open filled
squares. UCDs that are placed on the x-axis are undetected with both X-ray &
radio instruments. The solid horizontal line at $\log(L_{\rm R,\nu}/L_{\rm
x})\sim-15.5$ represents the relation of Güdel & Benz (1993) as a function of
$P_{\rm rot}$, while the vertical line at $P_{\rm rot}=0.2$ d is from Pineda
et al. (2017) to separate the radio-loud from the radio-quiet regime.

Figure 15: The activity-rotation relation in terms of $L_{\rm x}/L_{\rm bol}$
vs $P_{\rm rot}$ and based on SpT-color code. As in Figs.13 & 14, we present
the deepest X-ray luminosity, including the upper limits (upside-down open
triagles) as a function of rotation periods extracted from TESS light curves
or calculated from $v\cdot\sin{i}$ measurements when the rotational modulation
do not provide a clear value (left-oriented arrows). The UCD sample of this
work (open circles) is compared with the M dwarfs studied in Magaudda et al.
(2020, 2022), see Sect. 5

.

We speculate that events C1 and C2 on 2MJ0838 are gyrosynchrotron emission
from flares, since we have not detected circular polarization associated with
these events as is typical for coherent bursty emission. The initial stages of
the decay of C2 shown in Fig. 12 reveal a very steep frequency dependence of
the flux in the first two frequency bins. We constrain the spectral index from
these two bins for the first three time intervals, and overplot the fit
results. This reveals that the emission from 4.5-5.5 GHz at least in these
three time bins during the decay phase of C2 is consistent with a falling
spectrum, possibly indicating optically thin emission. The flux densities at
higher frequencies, 5.5 -10 GHz, show very little variation with frequency.
This constrains the peak frequency of the presumed gyrosynchrotron emission to
be at a value below $4.5$ GHz. We note that this value is lower than what is
typically seen from solar and stellar gyrosynchrotron tlfares, which have peak
frequencies 5-10 GHz or higher (Shaik & Gary, 2021; Dulk, 1985).

Using equation 39 in Dulk (1985), we explore for the C2 event of 2MJ0838 the
dependence of the peak frequency on the magnetic field strength in the radio-
emitting source as well as the product of the total number density of non-
thermal electrons times the length scale of the radio-emitting region, denoted
$NL$ by Dulk (1985). This relationship also depends on the power-law index of
the accelerated particle distribution determined from radio observations
($\delta_{\rm r}$) as well as an assumption about the angle of propagation
with respect to the line of sight, $\theta$ (here assumed to be 45∘. We use
the values of spectral index $\alpha$ constrained from Fig. 12 to infer the
index of the accelerated electron distribution $\delta_{r}$, where
$\alpha=1.22-0.9\delta_{r}$ under optically thin conditions for
gyrosynchrotron emission (Equation 35 in Dulk, 1985). Time bins C2D1 and C2D2
in Fig. 12 provide bounds for the values of $\alpha$ and hence $\delta_{r}$.
For these time bins we find $\delta_{r}=8.8\pm 2.6$ and $\delta_{r}=5.2\pm
2.4$, respectively. Figure 16 shows the relationships among these parameters
for a wide range of B and NL; we have assumed that the angle of propagation
$\theta$ is 45∘ for these parameter space calculations. Based on recent
studies of solar flares the nonthermal electron density appears to range
between 105 and 1011 cm-3 (Fleishman et al., 2022), and loop sizes from solar
to stellar flares can range from $\leq$ 108 cm (Cargill & Klimchuk, 2006) to
the size of the stellar diameter (Stelzer et al., 2006). We use a wide range
of parameter space for $N$ and $L$ to investigate trends. Inspection of Figure
16 shows that in combination with the peak frequency constraints, the lower
value of $\delta_{r}$ is more constraining on allowed regions of magnetic
field strength. In particular, magnetic field strengths of only a few tens of
Gauss are compatible with a nonthermal electron density near the maximum found
in Fleishman et al. (2022) (1011 cm-3) and a length scale of order the radius
of 2MJ0838 (1010cm). This is similar to the equipartition field strength
derived from X-ray observations in Sect. 3.1.3, but whereas the X-ray
constraints are determined from the average coronal properties, the current
constraints derive from a specific event corresponding to elevated flux levels
and are not necessarily characteristic. Additional modelling will be needed to
constrain these parameters further.

The circular polarization properties of 2MJ0838 are intriguing. They do not
appear to be periodic; the detections at both X and C bands are just slightly
above 3$\sigma$. Additionally, they indicate moderate degrees of circular
polarization (40-50%) when there is a detection, which is different from the
highly circularly polarized bursts seen in 2MJ0752 and other UCDs which
approach 100% circular polarization. This is also a much larger value of
circular polarization fraction than is typically seen from quiescent
gyrosynchrotron emission, generally $\sim$20% or less (White et al., 1989). If
we identify the frequencies at which the circular polarization is detected
with the electron gyro-frequency, then we can place a constraint on the
magnetic field strength in the steady radio-emitting region, with $\nu_{\rm
obs}=s\cdot\nu_{B}$, where $s$ is the harmonic number of the emission being
observed, and $\nu_{B}$ is the gyrofrequency, $\nu_{B}\approx 2.8\times
10^{6}B[G]$ Hz. The large frequencies at which we detected the circular
polarization, namely 9000 MHz, imply field strengths of one to several
kilogauss.

We use the method described by Osten et al. (2016); Smith et al. (2005) to
convert the radio light curve into a constraint on the non-thermal energy as a
function of magnetic field strength in the radio-emitting source and the
distribution of accelerated particles $\delta_{r}$ as derived above. This
contour plot is displayed in Fig. 17. While we do not have simultaneous
measures of radiated flare energy, we note that this range spans 1030-1034 erg
for values of $\delta_{r}$ consistent with the observational constraints;
these are flare energy ranges noted from a recent analysis of TESS white light
flares on ultracool dwarfs (Petrucci et al., 2023). For realistic flare
energies (Fig. 17) the magnetic field in the flaring source is on the order of
a few $100$ G, or less.

The equipartition field, derived from average X-ray measurements of this
object in Section 3.1.3, is considerably smaller than what is derived from the
flare analysis. On its face, this implies that these coronae could be
extremely compact. An apt comparison is that of Ness et al. (2004), whose
analysis of X-ray grating data of late-type stars combined measurements of
electron density and volume emission measures, to suggest “pathologically
small” coronal scale heights of 107 cm or less ($H/R_{\star}<0.001$ for the M
dwarfs considered in their sample). They invoke a filling factor less than
unity to overcome this situation. Constraints on the sizes of coronae of other
late-type stars indicate larger extents: X-ray eclipses of the G dwarf
component in the $\alpha$ CrB system by its X-ray dark A star companion reveal
a coronal height above the limb less than a solar radius (Schmitt & Kurster,
1994). Similarly, imaging observations of the Sun reveal a uniform typical
height of the bright portion of the solar corona to $\sim$0.3R⊙ (DeForest,
2007).

Figure 16: Contour plot of dependence of the peak frequency of gyrosynchrotron
emission as a function of magnetic field strength in the radio-emitting source
and the product of total number density of accelerated electrons and size
scale of the emission $NL$. Two different sets of contours are shown, which
bracket the inferred values of $\delta_{r}$ seen in the frequency-dependent
decay of event C2 in SCR0838 shown in Figure 12. The larger values of $NL$
provide more constraints on the field strength in the radio-emitting region,
in order to be consistent with the low peak frequency and the smaller value of
$\delta_{r}$.  Figure 17: Contour plot of estimated non-thermal energy in
radio flare C2 seen on SCR0838, as a function of the magnetic field strength
($B$) in the radio emitting region and the power-law index of the distribution
of accelerated particles, $\delta_{\rm r}$. Horizontal lines indicate the
range of $\delta_{\rm r}$ from the frequency dependence in Fig. 12.

##  5 Discussion

With our increased database of UCDs with sensitive X-ray and radio data we
present an updated view of the $L_{\rm R,\nu}-L_{\rm x}$ relation (Fig. 13).
If multiple observations were obtained in either waveband we show the most
sensitive one (left panel), including upper limits in case of non-detections.
Radio and X-ray upper limits are displayed with filled upside-down and left-
oriented triangles, respectively, while upper limits in both bands are
presented with smaller filled diamonds and full detections with filled
circles. In the right panel of Fig. 13, we highlight objects for which X-ray
flares and radio bursts have been observed in at least one instance. Among all
objects of this work, $26$ UCDs including the literature sample, we detected
optical flares for $9$ objects, among which $4$ show X-ray flaring activity,
$3$ emitted radio bursts, and only $2$ present activity along all three
wavelength bands. For the UCDs from our dedicated campaigns whether there is
X-ray flaring activity or not is inferred from the variability in the XMM-
Newton light curves (Fig. 2). This choice is justified by the fact that the
time resolution of XMM-Newton is higher than the one of eROSITA, that we
recall being only $4$ h and, thus, longer than typical X-ray flares on UCDs
(see e.g. Stelzer et al., 2006). Since we have one radio observation for each
target, we account for variability simply if there is a burst in their
emission. For the literature sample, instead, we adopted the notion of
flaring/bursting versus quiescent X-ray and radio activity as presented in the
references. In both panels of Fig. 13 we also show the GB-relation (see Sect.
1 for more details).

We confirm previous findings that UCDs are drastically offset from the GB-
relation. With our enhanced sample statistics we overall confirm the picture
of two groups of UCDs with different activity characteristics outlaid by
Stelzer et al. (2012), with X-ray flaring objects being closer to the GB-
relation and radio bursting ones often associated with faint or no measurable
X-ray emission. The majority of the joint X-ray/radio sample is detected in
X-rays and at levels of $L_{\rm x}$ fainter than the stars on which the GB-
relation was defined (the dashed part of the line in Fig. 14). This
(apparently) leftwards displacement with respect to the GB-relation could be
explained by the lack of sensitivity at radio wavelengths. The color code of
this figure shows that radio bursts are mostly detected from fast-rotating
UCDs in agreement with the results of Pineda et al. (2017). Their two regimes
of fast-rotating “radio-loud” and slow-rotating “radio-quiet” UCDs were the
basis of our target selection (see Sect. 1). Since we work with photometric
period while Pineda et al. (2017) used $v\sin{i}$ measurements, we had
translated their boundary between the two regimes ($v\sin{i}\approx 38\,\rm
km/s$) to a value of $P_{\rm rot}\approx 0.2$ d (assuming a typical late-M
dwarf radius of $R_{\star}\sim 0.15\,R_{\odot}$ and $i\approx 90^{\circ}$. In
contrast, X-ray flares are found on UCDs that are distributed throughout the
range of rotation periods considered in Fig. 13, implying no evident relation
for UCDs between their X-ray activity and rotation.

The X-ray scale height/equipartition magnetic field analysis in Sect. 3.1.3
reveals either a very compact corona for values of magnetic field strength of
a few tens of Gauss, or extremely low confining magnetic field strength for a
more extended corona. This could originate from the assumptions of homogeneity
and spherical symmetry.

One UCD of our sample (2MJ0752) is simultaneously detected along the two
wavebands adopted for this work and its X-ray emission is one of the
brightest. We do not infer X-ray variability from the XMM-Newton light curve
(see Fig. 2), although we clearly see variable X-ray emission from eROSITA
detections (see Fig. 3). In Sect. 4.1 and Fig. 9 we presented the radio light
curve of this UCD that shows two apparent double peaks. This brightness
modulation suggests the presence of a periodic burst corresponding to a
shorter rotation period ($<2$ h) than the one of 2MJ0752 (listed in Table 11).
We linked this radio emission to a possible auroral electron precipitation, as
the one typical of giant planets of the solar system. The X-ray brightest and
radio simultaneous detections of this UCD are showing signatures expected from
higher mass M dwarfs along with emerging evidence of radio auroral emission,
reinforcing the hypothesis of a bimodal dynamo mechanism ruling the
X-ray/radio activity along the ultracool dwarf regime.

For a direct investigation of the dependence of radio and X-ray activity on
rotation we show in Fig. 14 the relation between $L_{\rm R,\nu}/L_{\rm x}$ and
$P_{\rm rot}$, emphasizing objects for which an X-ray flare (open star),
optical flare (open circle) or a radio burst (open square) is present in our
dataset. UCDs that are upper limits in both X-ray and radio bands are placed
on the bottom of the plotting window, and we indicate with left-oriented
arrows the rotation periods that are upper limits, calculated from $v\sin{i}$
measurements. In this $L_{\rm R,\nu}/L_{\rm x}-P_{\rm rot}$ space the GB-
relation is a horizontal line, and we placed a vertical solid line at $P_{\rm
rot}=0.2$ d to separate the radio-loud and radio-quiet regimes from Pineda et
al. (2017).

Since only $2$ of the $26$ UCDs experienced activity along all three
wavelenghts, searching for a relation between the mechanism that generates a
radio bursts rather than an optical or X-ray flare remains a difficult task.
Thus, a larger sample and $B$ measurements are required and in order to
investigate such dichotomy that might be caused by the structure of the
magnetic field.

Given the fast rotation of UCDs and their peculiar X-ray/radio behavior it is
interesting to study their position in the rotation-activity relation compared
to earlier M dwarfs. We use the representation in terms of fractional
luminosity, $L_{\rm x}/L_{\rm bol}$ versus $P_{\rm rot}$ (see Fig. 15). The
$L_{\rm x}/L_{\rm bol}$ ratio has the advantage of removing most of the mass
dependence of X-ray brightness in the saturated regime. In most studies of the
activity-rotation relation this parameter is combined with the Rossby number
($R_{0}=\tau_{\rm conv}/P_{\rm rot}$, where $\tau_{\rm conv}$ is the
convective timescale). For late-type stars activity-rotation relations that
make use of the Rossby number display a smaller horizontal spread than those
involving only $P_{\rm rot}$ (Wright et al., 2011; Magaudda et al., 2020).
However, since the convective timescale is notoriously difficult to constrain
and no parametrizations exist for UCDs later than SpT = M9 we work with the
rotation period. In Fig. 15 the SpT-color code applies to both the UCDs from
this article and the comparison sample of early- to mid-M dwarfs from Magaudda
et al. (2020, 2022), but for clarity we additionally highlight the UCDs with
open black circles. As in the other figures, we display for the UCDs the most
sensitive X-ray measurement, including the upper limits that are plotted as
upside-down open triangles. The left-oriented arrows indicate for which object
we calculated the rotation period from $v\sin{i}$ measurements. As stated in
many previous works (Wright et al., 2011; Wright & Drake, 2016; Stelzer et
al., 2016; Wright et al., 2018; González-Álvarez et al., 2019; Magaudda et
al., 2020, 2022), the activity rotation relation of early- to mid- M dwarfs
splits into two regimes: for $P_{\rm rot}\lessapprox 10$ d the X-ray activity
does not depend on the rotation (saturated regime) and at longer periods the
X-ray activity declines with increasing rotational period (unsaturated or
‘correlated’ regime).

For our analysis of the $L_{\rm x}/L_{\rm bol}-P_{\rm rot}$ relation we only
consider the quiescent state of our sample. Our sample of late-M dwarfs spans
the entire range of $L_{\rm x}/L_{\rm bol}$ that covers the typical activity
level of both the saturated & unsaturated regime of the earlier SpTs from
Magaudda et al. (2022). This spread and consequent low activity of UCDs
evidently show a lack of correlation between $L_{\rm x}$ and $P_{\rm rot}$,
and it is usually ascribed to their mostly neutral atmospheres, that $-$ even
if the magnetic field is present $-$ do not allow coupling with matter and
thus outer magnetic heated atmospheres are not prominent (Mohanty et al.,
2002). Previous works already found a significant drop in X-ray activity
beyond spectral type $\sim$M7 (Cook et al., 2014; Berger et al., 2010).
Specifically, Berger et al. (2010) showed $L_{\rm x}/L_{\rm bol}\approx
10^{-4}$ for M7–M9 and $L_{\rm x}/L_{\rm bol}\approx 10^{-5}$ as deepest
limits for L dwarfs, while Cook et al. (2014) found a $L_{\rm x}/L_{\rm
bol}-$spread of $2$ dex for UCDs located in the saturated regime, i.e. $P_{\rm
rot}\leq 10$ d. We found a $L_{\rm x}/L_{\rm bol}-$spread for UCDs with
$P_{\rm rot}\leq 1$ d larger than the one presented in Cook et al. (2014) only
for quiescent X-ray emissions (see Fig. 15). In fact, we observe a declining
$L_{\rm x}/L_{\rm bol}$ for later SpT than M7 down to $L_{\rm x}/L_{\rm
bol}\approx 3\times 10^{-6}$. This detection refers to a M9 dwarf from the
literature, DENIS J1048, for which the deepest $L_{\rm x}$ is from XMM-Newton
observation (see Tables 6 & 4). Moreover, with our larger and updated sample
of UCDs we confirm the results of Cook et al. (2014), for which the authors
inferred the decrease of the X-ray activity of fast rotating UCDs caused by a
decrease in the effectiveness of the magnetic dynamo. Furthermore, with our
larger and updated sample of UCDs we confirm the results of Cook et al.
(2014), whereby the authors coupled the decrease in X-ray activity of rapidly
rotating UCDs with a decrease in the effectiveness of the magnetic dynamo.
Cook et al. (2014) suggested that the X-ray activity decrease for UCDs is due
to the presence of a bimodal dynamo across late-type dwarfs that produces
distinct magnetic field topologies. In other words, UCDs with large-scale
fields may spin down more efficiently than those with weaker and tangled
fields.

##  6 Conclusions

We present a collection of simultaneous X-ray/radio observations of a sample
of $10$ UCDs that we enlarge with archival X-ray and radio data from the
literature. The final sample counts $26$ late-M dwarfs. We added two more
radio-detected dwarfs to the list of known emitters: 2MJ0752 and 2MJ0838. The
first object is a slow rotating ($v\sin{i}=9$ km/s) and one of the X-ray
brightest radio-detected UCDs. It exhibits a possible loss cone along the line
of sight, suggested by the presence of two double peaks in its radio light
curve that correspond to a rotation period shorter that the one extracted from
TESS data ($P_{\rm rot}=0.88$ d). Given that there is a bias in radio
detection fraction against the slowest rotating UCDs (McLean et al., 2012;
Stelzer et al., 2012), it becomes an even more intriguing object to follow up.
The detection of this target during what looks like a long-lasting radio
burst, after a previous epoch in which a lower upper limit was obtained,
suggests that the actual fraction of radio-emitting objects amongst UCDs is
much higher than one would estimate based on single epochs alone. The other
radio-detected object (2MJ0838) displays evidence for X-ray emission and
incoherent radio flares, along with a moderate circularly polarized flux which
does not vary appreciably. We speculate that this object is experiencing a
transition of its magnetic behavior, producing signatures typical of higher
mass M dwarfs along with emerging evidence of auroral emission.

We updated the $L_{\rm R,\nu}-L_{\rm x}$ relation for UCDs with rotation
periods lower than $1$ d. While comparing our results with those from Güdel &
Benz (1993) for earlier-type stars we confirmed that rapid rotators lie the
furthest away from the GB-relation, and are more likely to exhibit radio
bursts indicative of loss cone-auroral emission. Paudel et al. (2018) noted
the ubiquity of white-light flaring in UCDs. Our results support and extend
this conclusion to shorter wavebands by noting that X-ray flaring occurs
throughout the region of radio and X-ray luminosities in Fig. 13, while
objects with radio bursts tend to be experienced more by fast rotating UCDs.
Based on our examination of radio, X-ray and rotation period results we
observed and confirmed the presence of a radio-loud regime for UCDs with
$P_{\rm rot}\leq 0.2$ d, already proposed by Pineda et al. (2017), and a
radio-quiet regime where most of the X-ray flares take place for larger
$P_{\rm rot}$.

Finally, we examined $L_{\rm x}/L_{\rm bol}$ as a function of $P_{\rm rot}$
for our sample of UCDs in comparison with the results of Magaudda et al.
(2022) for earlier-type M dwarfs. We observed that despite their fast rotation
the X-ray emission of UCDs covers the whole $L_{\rm x}/L_{\rm bol}$ range
($\approx-3.0...-5.5$) that comprises the saturated and the unsaturated
regimes of earlier M dwarfs. This shows that no evident relation seems to
emerge between the X-ray emission of UCDs with respect to their rotation. The
low activity detected for our sample of UCDs can be ascribed to their mostly
neutral atmospheres, that do not allow coupling with matter (Mohanty et al.,
2002) and to a decrease of dynamo efficiency that suggests the presence of a
bimodal dynamo across late-type dwarfs that changes the magnetic field
topology (Cook et al., 2014).

###### Acknowledgements.

EM is supported by Deutsche Forschungsgemeinschaft under grant STE 1068/8-1.
This work is based on data from eROSITA, the soft X-ray instrument aboard SRG,
a joint Russian-German science mission supported by the Russian Space Agency
(Roskosmos), in the interests of the Russian Academy of Sciences represented
by its Space Research Institute (IKI), and the Deutsches Zentrum für Luft- und
Raumfahrt (DLR). The SRG spacecraft was built by Lavochkin Association (NPOL)
and its subcontractors, and is operated by NPOL with support from the Max
Planck Institute for Extraterrestrial Physics (MPE). The development and
construction of the eROSITA X-ray instrument was led by MPE, with
contributions from the Dr. Karl Remeis Observatory Bamberg & ECAP (FAU
Erlangen-Nuernberg), the University of Hamburg Observatory, the Leibniz
Institute for Astrophysics Potsdam (AIP), and the Institute for Astronomy and
Astrophysics of the University of Tübingen, with the support of DLR and the
Max Planck Society. The Argelander Institute for Astronomy of the University
of Bonn and the Ludwig Maximilians Universität Munich also participated in the
science preparation for eROSITA. The eROSITA data shown here were processed
using the eSASS/NRTA software system developed by the German eROSITA
consortium. This paper includes data collected with the TESS mission, obtained
from the MAST data archive at the Space Telescope Science Institute (STScI).
Funding for the TESS mission is provided by the NASA Explorer Program. STScI
is operated by the Association of Universities for Research in Astronomy,
Inc., under NASA contract NAS 5–26555.

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## Appendix A

The Appendix displays radio maps of the field around objects which did not
produce a detection at radio wavelengths.

Figure 18: Contour images of the JVLA field around the three JVLA targets in D
and CnB configuration as observed at X band which were not detected. The size
is $3^{\prime}\times 3^{\prime}$ for the two first images taken in D
configuration and $2.6^{\prime}\times 2.6^{\prime}$ for the last one observed
in CnB configuration. Contour levels are $3$, $5$, $10$, $15$, $20$, $30$,
$50$, and $100$ times the image rms, which is listed in Table 9. The right
ascension and declination of the phase center for the observation for each
target is listed in Table 8. Figure 19: Zoom in of region around 2MJ0652 with
the JVLA as observed at C band. Contours are 5, 8, and 10 times the rms of
1.8$\mu$Jy determined from a blank region of the field. Figure 20: Maps of the
three-arcminute region around the lower frequency of the targets 2MJ1055
observed with ATCA, which was not detected at either frequency. Contours are
shown at 3, 5, and 10 times the rms values listed in Table 9.

## Appendix B

We present an example of our TESS rotation analysis showing the results we
obtained for the UCD TIC 29890705.

Figure 21: Result of the period search for TIC 298907057 observed with TESS in
sector 32: from left to right results refer to Lomb-Scargle periodogram, auto-
correlation and sine-fitting. top panels \- light curves phase-folded with the
periods obtained with each of the different methods; bottom \- periodogram,
autocorrelation function and the original light curve with the sine fit.