# Disks around young planetary-mass objects:
Ultradeep Spitzer imaging of NGC1333

Aleks Scholz School of Physics & Astronomy, University of St Andrews, North
Haugh, St Andrews, KY16 9SS, United Kingdom  Koraljka Muzic CENTRA, Faculdade
de Ciências, Universidade de Lisboa, Ed. C8, Campo Grande, P-1749-016 Lisboa,
Portugal  Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto
Frias, 4200-465 Porto, Portugal  Ray Jayawardhana Department of Astronomy,
Cornell University, Ithaca, NY 14853, USA  Victor Almendros-Abad  CENTRA,
Faculdade de Ciências, Universidade de Lisboa, Ed. C8, Campo Grande, 1749-016
Lisboa, Portugal  Isaac Wilson  Palomar Observatory, California Institute of
Technology, Pasadena, CA, United States

###### Abstract

We report on a sensitive infrared search for disks around isolated young
planetary-mass objects (PMOs) in the NGC1333 cluster, by stacking 70
Spitzer/IRAC frames at 3.6 and 4.5$\,\mu m$. Our co-added images go $>2.3$ mag
deeper than single-epoch frames, and cover 50 brown dwarfs, 15 of which have
M9 or later spectral types. Spectral types $>$M9 correspond to masses in the
giant planet domain, i.e., near or below the Deuterium-burning limit of
0.015$\,M_{\odot}$. Five of the 12 PMOs show definitive evidence of excess,
implying a disk fraction of 42%, albeit with a large statistical uncertainty
given the small sample. Comparing with measurements for higher-mass objects,
the disk fraction does not decline substantially with decreasing mass in the
sub-stellar domain, consistent with previous findings. Thus, free-floating
PMOs have the potential to form their own miniature planetary systems. We note
that only one of the six lowest-mass objects in NGC1333, with spectral type L0
or later, has a confirmed disk. Reviewing the literature, we find that the
lowest mass free-floating objects with firm disk detections have masses $\sim
0.01\,M_{\odot}$ (or $\sim 10\,M_{\mathrm{Jup}}$). It is not clear yet whether
even lower mass objects harbor disks. If not, it may indicate that $\sim
10\,M_{\mathrm{Jup}}$ is the lower mass limit for objects that form like
stars. Our disk detection experiment on deep Spitzer images paves the way for
studies with JWST at longer wavelengths and higher sensitivity, which will
further explore disk prevalence and formation of free-floating PMOs.

##  1 Introduction

Planets form in dusty disks surrounding newly born stars. The most
straightforward way to identify the presence of circumstellar disks is through
infrared imaging, probing for the excess emission from warm dust. The material
in the disk is a remnant from the rotating cloud core out of which the star
formed, flattened into a disk-shaped structure during the collapse, as angular
momentum is conserved. In star forming regions with ages of 1-2 Myr, more than
half of all GKM stars harbor disks. In somewhat older regions with ages of
5-10 Myr the disk fraction declines to 10-20%, as the disk material is either
accreted onto the stars, blown away by winds, or incorporated into planetary
systems (Jayawardhana et al., 1999; Haisch et al., 2001; Meyer et al., 2007).

Disks are not only found around stars, but also for young brown dwarfs,
substellar objects with masses below 0.08$\,M_{\odot}$, or
$80\,M_{\mathrm{Jup}}$, and thus unable to sustain stable Hydrogen burning.
Initial discoveries and studies of such circum-sub-stellar disks started about
two decades ago (Natta et al., 2002; Jayawardhana et al., 2003; Mohanty et
al., 2004), giving rise to the now well-established fact that most brown
dwarfs form like stars from the collapse of cloud cores (Luhman, 2012). The
Spitzer Space Telescope with its unprecendented sensitivity at 3-24 $\mu m$
was instrumental in furthering our understanding of brown dwarf disks. With
growing samples, it became clear that brown dwarf disks are common and long-
lived. The disk fractions among brown dwarfs in young star forming regions are
comparable to those of coeval stars, translating into similar disk lifetimes
(Scholz & Jayawardhana, 2008; Luhman & Mamajek, 2012). Using Spitzer, we also
learned that brown dwarf disks show evidence for the growth of dust grains
(Apai et al., 2005) and dust settling to the midplane (Scholz et al., 2007) –
prerequisites for planet formation via core accretion. Indeed, subsequent work
at longer wavelengths, in particular with ALMA, has shown the potential for
planet formation in brown dwarf disks (Testi et al., 2016). The discovery of
several planetary-mass companions orbiting brown dwarfs suggests that
substellar objects with masses of 1-8% the mass of the Sun can form their own
planetary systems (Jung et al., 2018).

In parallel to the exploration of substellar disks, deep ground-based
observing programs have established that star forming regions host objects
with masses that are an order of magnitude below the Hydrogen burning limit.
In all regions studied to sufficient depth – such as, $\sigma$ Orionis, $\rho$
Ophiuchi, NGC1333, IC348, Chamaeleon-I, Taurus, Lupus, Upper Scorpius –
surveys find objects with masses below the Deuterium burning limit of
$\sim$15$\,M_{\mathrm{Jup}}$ down to masses as low as $\sim
5\,M_{\mathrm{Jup}}$ (e.g., Zapatero Osorio et al., 2000; Lucas et al., 2001;
Scholz et al., 2012; Lodieu et al., 2018; Miret-Roig et al., 2022)

Figure 1: Comparison of a single epoch Spitzer/IRAC image of NGC1333 in
channel 2 (4.5 $\mu m$, AOR 29324032) and the stacked version of the same area
(70 frames). The marked object in the centre is the L3 brown dwarf SONYC-
NGC1333-36, one of the lowest mass spectroscopically confirmed objects in this
cluster. The image size is $2^{\prime}\times 2^{\prime}$, north is up and east
is left.

We know very little about the nature and evolution of these free-floating
planetary-mass objects (PMOs). One basic question relates to their origin:
They could be formed either like stars and brown dwarfs, from the collapse of
cloud cores (Bate, 2012). They could also be giant planets that got ejected
from their natal systems (Parker & Quanz, 2012; van Elteren et al., 2019).
From purely theoretical arguments, we expect a mix of formation scenarios in
the 1-15$\,M_{\mathrm{Jup}}$ mass domain, with a star-like origin more likely
at the upper end of the mass range, and a planetary origin dominant at the
lower (Scholz et al., 2022). The low mass limit for star formation, set by the
opacity limit of fragmentation, is expected to be in that mass range, as well
(Bate, 2012). But these notions have not been verified empirically yet.
Another fundamental question is whether or not these PMOs can form their own
miniature planetary systems.111We continue to call the potential satellites of
PMOs ’planets’, keeping in mind that if they exist, they would not be orbiting
a star, but an object itself comparable to giant planets in mass and size.

Searching for the signature of disks around free-floating PMOs will shed light
on these questions. If PMOs frequently host disks, they have at least the
potential to form their own planetary systems. Follow-up studies can then
clarify to what extent the prerequisites for planet formation (grain growth,
sufficient dust mass, longevity) are met. If free-floating PMOs form
predominantly like planets in disks around stars, we would expect their
primordial disks to be disrupted (Bowler et al., 2011). Thus, at face value,
we do not expect substantial disks to be common around PMOs if they themselves
form in circumstellar disks. Since we expect ejected planets to become more
common with decreasing mass, an important benchmark is to determine the mass
below which disks become rare or non-existent.

In this paper we present new measurements of the infrared magnitudes of PMOs
in the young cluster NGC1333, a compact active star-forming region with an age
of $\sim$1 Myr (Gutermuth et al., 2008; Scholz et al., 2009) and a distance of
300 pc, according to Gaia DR2 parallaxes of member stars (Pavlidou, 2022). In
contrast to previous infrared surveys in this cluster, we construct new,
ultradeep Spitzer images by stacking a time-series observation. In Section 2
we present the methodology and the sample. In Section 3 we derive the infrared
excess, verify the presence of disks in PMOs, and discuss PMO disks in
general. We present conclusions in Section 4.

##  2 Data analysis

###  2.1 Image stacking

NGC1333 was observed as part of the YSOVAR program during Spitzer’s Warm
Mission, under program id 61026. The field was observed in IRAC channels 1
($3.6\,\mu m$) and 2 ($4.5\,\mu m$), in the following called IRAC1 and IRAC2.
For full information on YSOVAR, see Rebull et al. (2014). In short, YSOVAR was
a large-scale monitoring program with Spitzer to study variability in young
stars, focused in particular on inner disk structure, accretion changes,
eclipsing binaries and rotation periods. The program surveyed 12 star forming
regions. The number of epochs varies from region to region, from a minimum of
about a dozen to several thousand. NGC1333 was one of the target regions for
YSOVAR, with a survey area of $2\times 2$ IRAC field of views, where a
$10^{\prime}\times 10^{\prime}$ field was covered by both channels. For more
information on the variability analysis of the NGC1333 dataset, see Rebull et
al. (2015). The fundamental reference for the IRAC instrument is Fazio et al.
(2004).

We downloaded 70 images from the YSOVAR dataset on NGC1333, for each of the
two channels, from the Spitzer Heritage Archive. The images for a given
channel all have very similar pointing, with minimal field rotation, and
similar depth. However, due to the layout of the Spitzer focal plane, the
images for channel 1 are offset compared to channel 2. Stacking was performed
using the Python reproject package (Robitaille et al., 2020), with the
function reproject_and_coadd. We stacked in two iterations, for each filter
separately: First we created 7 stacks from 10 images each, then we stacked the
resulting 7 stacks to a final deep image for each band. In Figure 1 we show a
comparison for a part of the stacked image vs. the same part of a single
image.

###  2.2 Sample selection

We created a catalogue of brown dwarfs in NGC1333 based on the recent census
by Luhman et al. (2016), which builds on several previous surveys and includes
a comprehensive literature review. Their published list comprises 203 new
members, the clear majority of which are in the area covered by our deep
Spitzer images. Most members have optical or near-infrared spectroscopic
confirmation, either by the authors of that census, or from the literature. In
addition, the authors use their own astrometry, as well as indicators of youth
(e.g., X-ray emission, infrared excess) to identify cluster members. The
Luhman census is ’nearly complete’ down to K magnitudes of 16.2 and $A_{J}$ of
3.

From that list, we selected the 65 for which the spectral type adopted by
Luhman et al. (2016) is M6 or later. A spectral type of M6 is commonly used as
the boundary between stars and brown dwarfs in star-forming regions. It
translates to a temperature of $\sim$3000 K (Mužić et al., 2014),
corresponding to an object with a mass at or slightly above the substellar
limit of 0.08$\,M_{\odot}$ for an age of 1-5 Myr, according to evolutionary
tracks (Baraffe et al., 2015). That means our sample should safely encompass
all known brown dwarfs in the NGC1333 census, but may include some very low
mass stars as well. The sample also includes 19 objects with a spectral type
of M9 or later, i.e., with estimated masses near or below the Deuterium-
burning limit. Those PMOs are the main focus of this paper.

Of the 65 likely brown dwarfs, 49 – including 14 PMOs – are covered by at
least one of our deep Spitzer/IRAC images. After the Luhman et al. (2016)
census, two more papers confirmed new substellar members for NGC1333: Esplin &
Luhman (2017) and Allers & Liu (2020). The former includes two more objects
with M9 or later spectral type, one of which is covered by our images, and was
originally reported by Oasa et al. (2008). We added it to our sample (with
spectral type M9-L4, we adopt L1.5 here). Thus our total sample of PMOs is
comprised of 15 objects. Due to the spatial offsets between the images in the
two Spitzer/IRAC channels, a few objects in our sample are only covered by one
channel. A dozen of the sources are in areas with highly variable background
or very close to the edge. Of the 65 likely brown dwarfs, 34 are in the JKS
catalogue published by Scholz et al. (2012), which combines the deep J- and
K-band photometry from Scholz et al. (2009) with the list of YSOs created from
the the single-epoch Spitzer/IRAC images taken by the C2D project (Evans et
al., 2009). For the objects covered by our deep Spitzer images, 23 (IRAC1) and
22 (IRAC2) are in the JKS catalogue.

We characterise our sample in Figure 2. In the left panel, we show spectral
type vs. K-band magnitude for the whole brown dwarf catalogue, marking those
covered by our deep images, and indicating approximate mass limits. As can be
appreciated from this figure, our sample covers the entire substellar range,
from the limit between stars and brown dwarfs to masses well below the
Deuterium burning limit. In particular, it includes all objects with spectral
types L0 or later found in this cluster thus far. In the right panel, we plot
the spatial distributions of the brown dwarfs and PMOs (marking those with
disks, see Section 3). The figure demonstrates that our images cover the
embedded cluster as well as surrounding areas.

Figure 2: Left: Spectral type vs. K-band magnitudes for brown dwarfs in
NGC1333. The dots mark all objects in this spectral type range from the list
by Luhman et al. (2016). The crosses mark the ones covered by our deep co-
added Spitzer images. Spectral types are numerically coded, M6 is 6.0, L0 is
10.0. Right: Spatial distribution of brown dwarfs (small dots), PMOs
(crosses), and PMOs with disks (empty circles) in NGC1333. We also show the
position and extent of the cluster as determined by Gutermuth et al. (2008),
as well as approximate coverage of the deep IRAC images produced here.

###  2.3 Photometry

We carried out aperture photometry for the objects in our catalogue using
tools from photutils (Bradley et al., 2021). We chose a constant aperture with
a radius of 5 pixels. We note that the native pixel size of IRAC is 1.2", but
the mosaics used here have 0.6" per pixel. The FWHM of IRAC during the Warm
Mission is 2.0", according to the IRAC Instrument Handbook. An aperture radius
of 5 pixels corresponds to a aperture diameter of 6", or approximately 3 times
the FWHM, and should capture $>90$% of the flux. For local background
estimates, we measured the median flux in an annulus with inner and outer
radius of 7 and 10 pixels centered on the object coordinates.

The magnitudes were calibrated by direct comparison with the JKS catalogue
mentioned above. The offset between the JKS IRAC magnitudes, which originally
come from the C2D source list (Evans et al., 2009), and our instrumental
measurements is constant for a wide range of magnitudes, with standard errors
of 0.03 and 0.04 mag (depending on channel). Thus, simply adding an offset
shifts our instrumental magnitudes into the standard system and makes them
comparable with the literature.

Photometric errors were estimated by adding in quadrature the Poissonian
errors in the source flux, the standard error for the background times the
number of pixels in the aperture, and the error in the calibration. The median
error, which is dominated by the calibration uncertainty, is 0.04 mag for both
channels.

All sources were checked in the stacked images. In particular, we verified
that the background determined by the photometry routine is adequate and that
the aperture is well centred on the source. As indicated earlier, some objects
are located in area of highly variable background, which may affect the
photometry.

Finally, we calculate the $K-IRAC1$ and $K-IRAC2$ color using the K-band
values listed in Luhman et al. (2016). We dereddened these colours to zero
extinction using the published $A_{J}$ extinction values for the sample, again
from Luhman et al. (2016). Those values have been determined by comparing
optical and near-infrared colours to expected photospheric values. For a
subset of our sample, Scholz et al. (2012) determined extinctions, using
photometry and spectroscopy. Comparing with the Luhman values, there is good
agreement, with a typical uncertainty $<0.5$ mag in $A_{J}$.

For the extinction correction, we convert from $A_{J}$ to $A_{K}$ using a
standard extinction law ($A_{K}=0.382A_{J}$, Mathis (1990)). For the IRAC
wavelengths, we adopt $A_{IRAC1}=0.6A_{K}$ and $A_{IRAC2}=0.5A_{K}$. These
extinction relations are in line with recently published work for star forming
regions, see Chapman et al. (2009). With very few exceptions, the brown dwarfs
in our sample have $A_{J}$ below 5. Thus, the exact choice of the extinction
law within plausible values does not change the outcomes in any significant
way. The dereddened colors vs. spectral types are plotted in Figure 3, as a
way to identify objects with infrared excess emission and thus disks. These
figures are the foundation for the discussion in Sect. 3.

For completeness, we also did full-field photometry using the SExtractor
(Bertin & Arnouts, 1996) for the stacked image and an individual image in both
channels. Comparing the peak of the histogram for the magnitudes, we find that
the stacked images are deeper by 2.4 and 2.6 mag, for IRAC1 and IRAC2
respectively. This result confirms the face-value expectation – stacking 70
images of similar quality should result in $\sqrt{70}=8.3$ better signal-to-
noise, or 2.3 magnitudes greater depth.

##  3 Results and discussion

###  3.1 Photospheric colours

To determine which objects have excess emission in their infrared colours we
first need to establish the photospheric levels in the same colours, as a
function of spectral type. We prepared a sample of stars and brown dwarfs with
spectral types from M0 to L1, all without disks (and thus without infrared
excess), that are members of nearby star-forming regions ($<10$ Myr). In this
exercise, we are building on the work in Almendros-Abad et al. (2022). The
$K-IRAC1$ and $K-IRAC2$ colours for this sample show a gradual increase from
values of 0.1 for early M dwarfs to around 1.0 for early L dwarfs. We fit
these colours for spectral types M6 or later with a linear slope, which
describes the trend seen in the data well.

$K-IRAC1=0.1007\times SpT-0.2847$ (1) $K-IRAC2=0.1179\times SpT-0.3281$ (2)

Here, the spectral type ($SpT$) is 6.0 for M6 and 11.0 for L1. We tested these
relations by doing the same procedure on only a subset of the diskless
objects, and do not find a significant change. In particular, we limited the
fit for objects with ages $<5$ Myr and the fit remains very similar. For
reference, the given relations are consistent with the ones published by
Luhman (2022) for ages $<20$ Myr.

Our photospheric colours are plotted in Figure 3 as solid lines. We also show
in the figure the typical scatter of diskless objects around this fit, as
coloured area. Objects without infrared excess in NGC1333 are expected to be
found in that area around the solid line in Figure 3.

Figure 3: Infrared colours of brown dwarfs in NGC1333 vs. spectral type, after
dereddening the colours to $A_{J}=0$. The spectral type range is M6 to L3. In
the right panel, three objects at spectral type M9 have very similar colours
and have been plotted slightly offset in spectral type for clarity. The
estimated photospheric colour is shown with a solid black line, with the
typical scatter shown as coloured area (see text for details). The dash-dotted
line is the chosen demarcation for our sample; all objects plotted on the
right side of it can be described as planetary-mass objects (PMOs).

###  3.2 Infrared excess and disk fraction

In Table 1 we list the newly derived infrared photometry and colours (before
extinction correction) for the 15 PMOs (spectral types M9 or later): 14 have
measurements in IRAC1, 13 in IRAC2. Three objects with reported magnitudes are
in areas of highly variable background, and after careful examination we deem
those measurements to be unreliable. As can be seen in Figure 3, eight objects
with good measurements have a dereddened $K-IRAC1$ colour consistent with
photospheres, six do in $K-IRAC2$. Altogether seven sources out of 12 with
robust measurements in at least one band show no evidence for infrared excess.

The remaining five objects are found to have infrared excess and are thus
likely to be PMOs with disks. Four of these are at spectral type M9, three of
them with excess in both bands (one with marginal excess in IRAC1). The fifth,
with spectral type L0, has no excess in IRAC1, but a clear excess in IRAC2.
Thus, the disk fraction from our sample is 5/12 or 42%. Due to the small
sample size, the plausible range for the disk fraction is 26-60% (assuming
1$\sigma$ Binomial confidence intervals). Visual examination of the three
objects in areas with highly variable background shows that one (SONYC-
NGC1333-9) is bright in both IRAC bands, and thus also a good candidate for
harboring a disk. The other two are faint and more likely to lack excess.
Including those three would bring the disk fraction to 6/15, or 40%. We note
that with one exception all disks in the PMO regime are found for brown dwarfs
classified as M9, at the adopted upper limit for our PMO sample. For L0 or
later objects, the disk fraction is only 1/5 or 20%, 1/6 if we add the one
with background-affected measurement.

Our analysis includes the lowest mass objects identified thus far in this
cluster, in particular, the 6 objects with spectral type L0 or later
(corresponding to masses of $\sim 0.01\,M_{\odot}$ or lower). We determine the
IR excess for five out of six, with the one exception mentioned above. It is
likely that the current optical/near-infrared surveys are not complete for
those spectral types. Therefore, the disk fractions determined here may be
affected by incompleteness at the lowest masses. If disk-bearing objects are
on average deeper embedded than those without, thus have more extinction and
are fainter, it is conceivable that we are underestimating the disk fraction
for $M<0.01\,M_{\odot}$. This, in addition to small number statistics, may
contribute to the the low fraction of disks in the L0 or later domain. Deeper
observations will be needed to clarify.

In Figure 2, right panel, we show the spatial distribution of the PMOs and
PMOs with disks relative to the cluster extent and the overall brown dwarf
sample. There is no obvious spatial bias in those samples, they are all
concentrated on the cluster itself. One of the disk-bearing PMOs is located
just outside the nominal cluster extent, but the same is true for several PMOs
without disks.

Among the brown dwarfs with valid measurements and spectral types M6 to $<$M9
in our images, 8/27 show clear infrared excess in IRAC1, 15/23 in IRAC2. This
corresponds to a disk fraction of $30\pm 10$% for IRAC1 and $65\pm 12$% for
IRAC2. Thus, based on our measurements, the disk fraction among PMOs is
consistent within the statistical uncertainties with the value for more
massive brown dwarfs. We only quote these numbers for completeness: In
contrast to the faint PMOs, our deep images do not provide a significant
benefit for the brighter M6 to $<$M9 objects.

###  3.3 Comparison with the literature

In Table 1 we also included the previous classification of the infrared excess
based on Luhman et al. (2016), in the column labelled ’IR’. This previous
paper uses mid-infrared excess measured from Spitzer images to identify
circumstellar disks. Of our list of 15, 12 have that information listed in
Luhman et al. (2016), and for all 12 our classification is in agreement with
the one in the literature. We also classify three objects for the first time
here. One of them likely has a disk: at spectral type L0, it is currently the
latest type object with a disk in NGC1333. We also classify the three objects
with spectral types later than L0 as diskless, two of them for the first time.

Luhman et al. (2016) also derived disk fractions for subsamples divided by
spectral type. They report 11/21 or 52% for spectral types later than M8 (a
slightly different subset than our sample of PMOs), 20/33 or 61% for M6-M8,
28/48 or 58% for M3.75-M5.75 and 27/39 or 69% for K6 to M3.5. Within the
statistical errorbars, these values are all consistent with each other, and
with our own measurements of the disk fractions, given above. Our measurement
is also consistent with the disk fraction of $66\pm 8$% for objects with
spectral type of M5-M9 in NGC1333, reported by Scholz et al. (2012). For
completeness, for a sample of 79 Gaia-selected low-mass stars in NGC1333,
Pavlidou (2022) measure a disk fraction of $67\pm 13$% (for the mass range 0.1
to 1.0 $M_{\odot}$, approximately).

We note here that the ‘disk fraction’ in the quoted papers and in our study is
defined as the number of Class II objects, divided over the sum of Class II
and Class III. These numbers are all based on samples from optical/near-
infrared surveys, and should not be compared directly with the higher disk
fractions derived from mid-infrared surveys (Gutermuth et al., 2008; Jørgensen
et al., 2006). One further caveat: Disk surveys based on photometry at
3-8$\,\mu m$ can by their nature not find disks with large inner holes where
the disk only causes excess emission at longer wavelength. All studies
mentioned above share this bias.

In Figure 4 we provide a summary of the currently available disk fraction
measurements for low-mass stars, brown dwarfs, and PMOs in NGC1333, including
only objects with known spectral type. As can be appreciated from this figure,
there is no significant trend in the disk fraction from late K to early L
spectral type. In that regard, our new photometry confirms results from
previous studies of the disk population for this cluster. Finding a disk
fraction largely independent of mass for low-mass stars and brown dwarfs is
also in line with what has been found for other star-forming regions with ages
between 1 and 5 Myr, including IC348 (Luhman et al., 2016), $\sigma$ Orionis
(Scholz & Jayawardhana, 2008), and Chamaeleon-I (Luhman et al., 2008). As a
sidenote, the brown dwarf disk fraction derived in the current paper from
IRAC1 excess (30%) seems to be an underestimate compared with the literature
and should be treated with caution.

Figure 4 also illustrates that there may be a decline in the disk fraction
when comparing L-type PMOs to K-M stars and brown dwarfs. As noted above,
among the six cluster members in NGC1333 with spectral type L0 or later only
one seems to have a disk, based on our measurements, translating to a disk
fraction of 20%. Given the small sample size and possible bias, as mentioned
above, it is premature to conclude that this is indeed a drop-off in the disk
fraction. If confirmed, it would constitute a very curious result, with
possible implications for the formation process. Finding more L-type objects
in this cluster would immediately give us a better handle on this issue.

We note that for somewhat older star-forming regions, like Upper Scorpius and
the TW Hydrae Association, with ages $\sim$8-10 Myr, there is some evidence in
the literature for an increased disk fraction for brown dwarfs compared to
low-mass stars (Luhman & Mamajek, 2012; Cook et al., 2017; Riaz & Gizis,
2008). For those regions, PMOs do not seem to stand out from more massive
brown dwarfs in terms of their disk fraction – but the small sample size
hampers a more definitive assessment of the longevity of disks around PMOs.

Generally speaking, the evidence is strong that free-floating PMOs can harbour
disks, and therefore, have the potential to form their own (miniature)
planetary systems. In terms of the overall mass and scale, these systems would
be more comparable to Jupiter’s Galilean moons than to the solar system.

Figure 4: Disk fraction for samples of stars and brown dwarfs with known
spectral type, from Luhman et al. (2016) (blue symbols), Scholz et al. (2012)
and this paper (orange symbols). Spectral types are numerically coded, M0 is
0.0, L0 is 10.0.

###  3.4 The lowest mass free-floating objects with disks

Our work in NGC1333 reported in this paper establishes that at least five PMOs
in NGC1333 show evidence for the presence of a circum-sub-stellar disk. The
lowest-mass among these is an object at spectral type L0 (not previously
identified as disk-bearing), corresponding to a temperature of 2200 K, which
for an age of 1-2 Myr would result in an estimated mass of $\sim$0.01
$M_{\odot}$ (Baraffe et al., 2015). The three confirmed sources with later
spectral types, and thus presumably lower masses, do not have detectable
infrared excess and are classified as diskless.

Similar studies have been carried out in other star-forming regions. Taken
together, there is now a small sample of young PMOs with securely identified
disks. This includes OTS44 (Joergens et al., 2013), the only one with an ALMA
detection at far-infrared/submm wavelengths (Bayo et al., 2017). For young
PMOs, definitively detecting an infrared excess due to a disk is challenging
work, for two reasons. One, as can be appreciated from Figure 3, the
photospheric near/mid-infrared colours increase steadily for young late M and
L dwarfs. With the typical uncertainties in the spectral typing of at least
$\pm 1$ subtype, the photospheric level for a given source is uncertain.
Second, these are faint sources, and thus, the near/mid-infrared magnitudes
come with considerable errors. Both difficulties worsen towards later spectral
types and lower masses. They can be overcome with deeper mid-infrared images
(the motivation for this study), improved spectroscopic classification, or
photometry at wavelengths where the photospheric flux is negligible compared
to the disk excess ($10\,\mu m$ or beyond).

With all that in mind, we summarize the results for regions with sufficient
survey depth.

In $\sigma$ Orionis, two planetary-mass brown dwarfs with spectral types (SOri
60 and SOri 71) have consistently been found to host disks at wavelengths up
to 8$\,\mu m$ (Scholz & Jayawardhana, 2008; Luhman et al., 2008). These two
have spectral types of L2 and L0 respectively. This cluster is slightly older
than NGC1333; for its age of 3-5 Myr, these spectral types would correspond to
$\sim$0.01 $M_{\odot}$ or above. There are some PMOs with later spectral types
and claimed excess emission (e.g., SOri 65, 66, 70), but the mid-infrared data
are inconclusive (Scholz & Jayawardhana, 2008; Luhman et al., 2008). In the
case of SOri 70, it is also not clear yet if it is in fact a young member of
the cluster and not a foreground brown dwarf (Zapatero Osorio et al., 2017).

In Chamaeleon-I, OTS44 (M9.5) is a secure detection. In addition, Luhman et
al. (2008) found Cha 1107−7626, an L0 object with a disk. Esplin et al. (2017)
report three more objects with M9-L2/L3 spectral types that may have excess
emission, but caution against interpreting it as evidence for a disk. Given
the uncertainty in their spectral type, more work needs to be done before
assigning masses or infrared excess to those.

In Taurus, the lowest mass objects with safely detected disks have late M
spectral types, according to the recent survey by Esplin & Luhman (2019). Some
L0-L1 sources also may have infrared excess. In addition, Esplin & Luhman
(2019) find one very low mass source with an L3 spectral type, which would
have a mass well below 0.01 $M_{\odot}$, with possible excess emission at
4.5$\,\mu m$. Again, this is not a robust detection yet.

$\rho$ Ophiuchi has spectroscopically confirmed brown dwarfs with disks at
spectral type M9-L1 (Testi et al., 2002; Jayawardhana & Ivanov, 2006; Alves de
Oliveira et al., 2012), which again would correspond to a mass of $\sim$0.01
$M_{\odot}$. CFHTWIR-Oph-33 has a spectral type of L3-L4 and may have excess
emission, but the mid-infrared data are inconclusive in that regard (Alves de
Oliveira et al., 2012; Esplin & Luhman, 2020). Some other very low mass
objects discussed in these papers may have infrared excess, but also very
uncertain spectral type (e.g., CFHT-WIR-Oph-58, with a range from M8.5 to L3,
see Almendros-Abad et al. (2022)).

Finally, the older Upper Scorpius association has a large population of brown
dwarfs. Two with a spectral type of L3.5 are reported to have accretion and
infrared excess (Lodieu et al., 2018). For ages of 5-10 Myr, this would
correspond to 0.01 $M_{\odot}$ or above. To our knowledge, no lower mass
objects with disks have been found here.

In summary, at this stage, the low-mass limit for objects with a safe disk
detection, across all star forming regions, is $\sim 0.01\,M_{\odot}$ (or
$\sim 10\,M_{\mathrm{Jup}}$). The surveys have found perhaps two dozen objects
in total that may have masses below that limit (depending on how the mass is
estimated) in all of the very young regions mentioned above. For the majority
of them we can already firmly rule out the presence of disks – the lowest mass
objects in NGC1333 discussed in this paper belong in that category. For about
ten of these PMOs with putative masses $<0.01\,M_{\odot}$, the currently
available data are inconclusive and insufficient for proper characterisation.
Therefore, it is certainly too early to rule out the presence of disks around
objects with masses below $0.01\,M_{\odot}$. Having said that, given that the
disk fraction among brown dwarfs is 30-60% for the young regions mentioned
above, the lack of confirmed disks below the $0.01\,M_{\odot}$ limit is
perhaps already worth noting, especially if taken together with our new
measurements for NGC1333, as illustrated in Figure 4. If the disk fraction
were similar for objects with masses $<0.01\,M_{\odot}$, essentially all
objects with tentative detections of infrared excess should have disks.
Perhaps we are beginning to see a decline of disk fractions in the planetary-
mass domain.

Sensitive mid-infrared observations of known PMOs, and deep surveys for new
objects in this mass domain are needed to verify this suspicion. JWST is
primed to help with both of these tasks. In NGC1333 specifically, a Guaranteed
Time program with NIRISS/WFSS is dedicated to a deep spectroscopic survey, and
designed to find PMOs in this cluster down to masses of 1$\,M{\mathrm{Jup}}$
(program ID 1202, see Willott et al. (2022)). Also, the unprecedented
sensitivity of the MIRI instrument will enable us to probe directly for
infrared excess in the lowest mass free-floating objects known, at wavelengths
exceeding those of Spitzer/IRAC, allowing for unambiguous disk detections.

The presence of disks is discussed in the literature as one possible signature
to distinguish between a star-like and a planet-like formation scenario
(Luhman, 2012; Testi et al., 2016). In particular, if an object gets ejected
early in the evolution from a forming planetary system, either by planet-
planet scattering or encounters with other stars, it can be expected that its
disk, if present, will be affected. Simulations on this issue, however, remain
sparse or non-existent. For the case of planet-planet-scattering, Bowler et
al. (2011) demonstrate that the disruption of a circumplanetary disk should be
common. In this context, disks detected with ALMA around young giant planets
on wide orbits are an interesting comparison. These planetary-mass companions
may have very compact disks (Wu et al., 2017), but it is unclear whether or
not their masses are lower than expected from the standard relation between
disk mass and stellar mass (Wu et al., 2020).222In the simulations by
Stamatellos & Herczeg (2015) objects formed in disks around stars are found to
harbour their own disks with substantial masses, but the authors do not model
the effect an ejection might have on those disks

Given the discussed arguments, it is plausible to assume that free-floating
planets that have been ejected from a young planetary system would not host
massive, long-lasting disks. If future observations confirm the absence of
disks for objects with masses below $0.01\,M_{\odot}$, it may indicate that
this is the low mass limit for objects to form like stars, physically set by
the opacity limit for fragmentation.

##  4 Summary

We present new Spitzer/IRAC photometry of brown dwarfs with planetary masses
in the young star-forming cluster NGC1333. To improve on previous findings, we
stack 70 images taken as a time series campaign during Spitzer’s Warm Mission
in 2011, at wavelengths of 3.6 and 4.5$\,\mu m$. Our deep images cover 50
brown dwarfs, 15 of which have spectral types of M9 or later, corresponding to
masses near or below the Deuterium-burning limit (i.e., ‘planetary-mass
objects’, PMOs). Out of 12 PMOs with good measurements, five show clear
infrared emission in excess of the estimated photospheric level. Taken
together with previous work, our results confirm that the disk fraction does
not change significantly between very low mass stars, brown dwarfs, and PMOs.
Thus, free-floating objects with masses comparable to those of giant planets
have the potential to form their own miniature planetary systems. However, we
note that among the six lowest mass objects in NGC1333, with spectral types of
L0 or later, only 1 has a clear disk detection (no. 12 in Table 1), which may
be a sign of a drop of in disk fraction at ultra-low masses, possibly
suggesting that objects at these masses are primarily ejected planets, rather
than formed from core collapse. We survey the literature for studies on disks
around PMOs and find that the lowest mass objects with a firmly detected disks
have masses around 0.01$\,M_{\odot}$. Some objects at later spectral types
have tentative detections of infrared excess, but whether or not this is
significant and due to the presence of a disk is still uncertain. Future
observations with JWST will undoubtedly probe the presence of disks around
lower mass objects, thus shedding light on the formation mechanism of free-
floating PMOs.

Table 1: Infrared photometry for planetary mass objects in NGC1333. Errors in
the IRAC magnitudes are 0.03-0.09 mag.

no | RA (deg) | DEC (deg) | SpT1 | IRAC1 | IRAC2 | K-IRAC1 | K-IRAC2 | IR1 | disk2 | first spectrum
---|---|---|---|---|---|---|---|---|---|---
1 | 52.15829 | 31.36939 | L0 | 16.72 |  | 0.98 |  | No | No | Luhman et al. (2016)
2 | 52.19013 | 31.26978 | M9 | 16.24 | 15.58 | 0.79 | 1.45 | Yes | Yes | Luhman et al. (2016)
3 | 52.21271 | 31.32450 | M9.5 | 16.84 | 16.74 | 0.85 | 0.95 | No | No | Luhman et al. (2016)
4 | 52.23542 | 31.26753 | M9.6 | 15.94 | 15.96 | 0.87 | 0.85 | No | No | Scholz et al. (2009)
5 | 52.24050 | 31.30478 | L1.5 | 16.73 | 16.74 | 1.15 | 1.14 |  | No | Esplin & Luhman (2017)
6 | 52.24683 | 31.28153 | M9 | 15.02 | 14.52 | 1.10 | 1.60 | Yes | Yes | Luhman et al. (2016)
7 | 52.25438 | 31.30956 | L0 | 16.29 | 16.11 | 0.83 | 1.01 | No | No | Luhman et al. (2016)
8 | 52.25983 | 31.37108 | M9 | 15.69 | 15.26 | 1.19 | 1.62 | Yes | Yes | Luhman et al. (2016)
9 | 52.27154 | 31.34361 | M9 | 11.64 | 10.77 |  |  | Yes | Yes? | Luhman et al. (2016)
10 | 52.27321 | 31.17053 | M9 |  | 15.12 |  | 0.77 | No | No | Scholz et al. (2009)
11 | 52.28817 | 31.36233 | M9 | 10.84 | 10.65 |  |  | No | No? | Luhman et al. (2016)
12 | 52.28858 | 31.27322 | L0 | 17.21 | 16.53 | 0.98 | 1.66 |  | Yes | Luhman et al. (2016)
13 | 52.30108 | 31.38531 | M9 | 13.77 | 13.23 | 1.39 | 1.93 | Yes | Yes | Luhman et al. (2016)
14 | 52.30775 | 31.30158 | L1 | 17.43 |  |  |  |  | No? | Scholz et al. (2012)
15 | 52.35771 | 31.27828 | L3 | 16.42 | 16.04 | 0.86 | 1.24 | No | No | Scholz et al. (2012)

1 from Luhman et al. (2016); 2 our classification, based on Figure 3

##  5 Acknowledgements

We thank the anonymous referee for a constructive and concise referee report
on this paper. This work is based on observations made with the Spitzer Space
Telescope, which was operated by the Jet Propulsion Laboratory, California
Institute of Technology under a contract with NASA. VA and KM acknowledge
funding by the Science and Technology Foundation of Portugal (FCT), grants No.
PTDC/FIS-AST/7002/2020, UIDB/00099/2020 and SFRH/BD/143433/2019. Constructive
discussions with Dimitris Stamatellos, Ken Rice, Ian Bonnell, and James
Wurster have aided the discussion in this paper. In order to meet
institutional and research funder open access requirements, any accepted
manuscript arising shall be open access under a Creative Commons Attribution
(CC BY) reuse licence with zero embargo.

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# Transposed Poisson Structures on the planar Galilean conformal algebra

Transposed Poisson Structures on the planar Galilean conformal algebra111 This
work was supported by National Natural Science Foundation grants of China
(11701345)

Henan Wu222School of Mathematical Sciences, Shanxi University, Taiyuan 030006,
China<EMAIL_ADDRESS>Wenting Zhang333 School of Mathematical Sciences,
Shanxi University, Taiyuan 030006, China<EMAIL_ADDRESS>

Abstract: Each $\frac{1}{2}$-derivation of the planar Galilean conformal
algebra is proven to be a scalar. As a corollary, all transposed Poisson
structures on the planar Galilean conformal algebra are trivial.

Keywords:  planar Galilean conformal algebra, $\frac{1}{2}$-derivation,
transposed Poisson structure.

MR(2010) Subject Classification: 17B10, 17B65, 17B68

##  1 Introduction

The origin of Poisson algebras lies in the Poisson geometry of the 1970s.
Since then, these algebras have shown their importance in several areas of
mathematics and physics, such as Poisson manifolds, algebraic geometry,
quantization theory, quantum groups, and classical and quantum mechanics. The
study of Poisson algebras also led to other algebraic structures, such as
noncommutative Poisson algebras ([1]), Jacobi algebras, Novikov-poisson
algebras ([2]). The description of all possible Poisson algebra structures
with fixed Lie or associative part is an important problem in the theory of
Poisson algebra ([3]).

Recently, the notion of transposed Poisson algebra was introduced in ([4]), by
exchanging the roles of the two binary operations in the Leibniz rule defining
the Poisson algebra. The relationship between $\frac{1}{2}$-derivations of Lie
algebras and transposed Poisson algebras was developed in ([5]). Using this
idea, possible tranposed Poisson structures with fixed Lie algebras were
studied by many authors. For example, all the possible transposed Poisson
algebra structures were described on the Witt algebra and the Virasoro algebra
in ([5]), on the twisted Heisenberg-Virasoro algebra, the
Schr$\ddot{o}$dinger-Virasoro algebra, the extended Schr$\ddot{o}$dinger-
Virasoro algebra and the twisted Schr$\ddot{o}$dinger-Virasoro algebra in
([6]), on block Lie (super) algebras in ([7]), on Galilean and solvable Lie
algebras in ([8]), on Witt type algebras in ([9]), on generalized Witt
algebras and Block Lie algebras in ([10]), respectively.

In this paper, we want to study the $\frac{1}{2}$-derivations and transposed
Poisson structures of the planar Galilean conformal algebra. This paper is
organized as follows. In Section 2, we introduce the definition of transposed
Poisson algebra, the definition of the $\frac{1}{2}$-derivations, the
definition of the planar Galilean conformal algebra. In Section 3, we compute
the $\frac{1}{2}$-derivations of the planar Galilean conformal algebra and
describe the related transposed Poisson algebra structures.

Throughout the paper, we denote by
$\mathbb{C},\,\mathbb{C}^{*},\,\mathbb{Z},\,\mathbb{Z}^{+}$ the sets of
complex numbers, nonzero complex numbers, integers, nonnegative integers,
respectively. And all vector spaces and tensor products are taken over the
complex field $\mathbb{C}$.

##  2 Preliminaries

In this section, we list some notations and results to be used in the paper.
The reader can refer to [4] for detail.

###### Definition 2.1

([4]) let $L$ be a vector space equipped with two nonzero bilinear operations
including $\cdot$ and$[,]$. The triple $(L,\cdot,[,])$ is called a transposed
Poisson algebra if $(L,\cdot)$ is a commutative associative algebra and
$(L,[,])$ is a Lie algebra such that for any $x,y,z\in L$,

$2z\cdot[x,y]=[z\cdot x,y]+[x,z\cdot y].$ (2.1)

###### Definition 2.2

([5]) Let $(L,[,])$ be an algebra with multiplication $[,]$ and $\varphi$ be a
linear transformation of $L$. Then $\varphi$ is called a
$\frac{1}{2}$-derivation if

$\varphi[x,y]=\frac{1}{2}([\varphi(x),y]+[x,\varphi(y)])$ (2.2)

for any $x,y\in L$.

###### Definition 2.3

([11]) The planar Galilean conformal algebra $\mathcal{W}$ is an infinite
dimensional Lie algebra over $\mathbb{F}$ with the basis
$\\{L_{m},H_{m},I_{m},J_{m}\,|\,m\in\Gamma\\}$ subject to the following
nontrivial relations

$\displaystyle[L_{m},L_{n}]=(m-n)L_{m+n},\ \ $
$\displaystyle[L_{m},H_{n}]=-nH_{m+n},$
$\displaystyle[L_{m},I_{n}]=(m-n)I_{m+n},\ \ $
$\displaystyle[L_{m},J_{n}]=(m-n)J_{m+n},$
$\displaystyle[H_{m},I_{n}]=J_{m+n},\ \ $
$\displaystyle[H_{m},J_{n}]=-I_{m+n}.$

##  3 $\frac{1}{2}$-derivations of $\mathcal{W}$

First, we develop a result on $\frac{1}{2}$-derivations of the general graded
Lie algebra. Assume $\mathcal{L}$ is a $G$-graded Lie algebra, i.e,

$\mathcal{L}=\bigoplus_{\alpha\in G}\mathcal{L}_{\alpha},\ \
[\mathcal{L}_{\alpha},\mathcal{L}_{\beta}]\subset\mathcal{L}_{\alpha+\beta},$

where $G$ is an abelian group. Assume $D$ is an arbitrary
$\frac{1}{2}$-derivation of $\mathcal{L}$. For any homogenous
$x_{\alpha}\in\mathcal{L}_{\alpha}$, denote $D(x_{\alpha})=\sum y_{\beta}$,
$y_{\beta}\in\mathcal{L}_{\beta}$. Define
$D_{\gamma}(x_{\alpha})=y_{\alpha+\gamma}$. By direct computation, we have
$D_{\gamma}$ is a $\frac{1}{2}$-derivation of $\mathcal{L}$ and
$D=\sum_{\gamma}D_{\gamma}$.

Note that $\mathcal{W}$ is a
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}$-graded vector space with

$\mathcal{W}_{(\overline{0},\overline{0},m)}=\mathbb{F}L_{m},\
\mathcal{W}_{(\overline{0},\overline{1},m)}=\mathbb{F}I_{m},\mathcal{W}_{(\overline{1},\overline{0},m)}=\mathbb{F}J_{m},\
\mathcal{W}_{(\overline{1},\overline{1},m)}=\mathbb{F}H_{m},\
\forall\overline{0},\overline{1}\in\mathbb{Z}_{2},\ m\in\mathbb{Z}.$

It is obvious that
$[\mathcal{W}_{(\overline{a_{1}},\overline{b_{1}},\gamma_{1})},\mathcal{W}_{(\overline{a_{2}},\overline{b_{2}},\gamma_{2})}]\subset\mathcal{W}_{(\overline{a_{1}}+\overline{a_{2}},\overline{b_{1}}+\overline{b_{2}},\gamma_{1}+\gamma_{2})}$
for any
$(\overline{a_{1}},\overline{b_{1}},\gamma_{1}),(\overline{a_{2}},\overline{b_{2}},\gamma_{2})\in\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}$.
Denote $G=\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}$. Then
$\mathcal{W}$ is a $G$-graded Lie algebra. Now we assume $D$ is an arbitrary
$\frac{1}{2}$-derivation of $\mathcal{W}$. Then $D=\sum
D_{(\overline{a},\overline{b},\gamma)}$, where
$(\overline{a},\overline{b},\gamma)\in G$. In the following, we attempt to
characterize the homogenous $\frac{1}{2}$-derivations of $\mathcal{W}$ case by
case. We can determine all $\frac{1}{2}$-derivations of $\mathcal{W}$.

Firstly we assume that $D$ is a homogenous $\frac{1}{2}$-derivation of degree
$(\overline{0},\overline{0},\gamma)$.

###  3.1 $D=D_{(\overline{0},\overline{0},\gamma)}$

In this case, denote

$D(L_{m})=a_{m}L_{m+\gamma},\ D(H_{m})=b_{m}H_{m+\gamma},\
D(I_{m})=c_{m}I_{m+\gamma},\ D(J_{m})=d_{m}J_{m+\gamma},$

where $a_{m},b_{m},c_{m},d_{m}\in\mathbb{F}$. Since
$[L_{m},L_{n}]=(m-n)L_{m+n}$, we have
$2(m-n)D(L_{m+n})=[D(L_{m}),L_{n}]+[L_{m},D(L_{n})]$. Then one gets

$2(m-n)a_{m+n}=(m+\gamma-n)a_{m}+(m-n-\gamma)a_{n},\ \ \forall
m,n\in\mathbb{Z}.$ (3.1)

###### Lemma 3.1

The solution of (3.1) is $a_{m}=a_{0}$ for any $m\in\mathbb{Z}$.

Proof. Setting $n=0$ in (3.1), we have $(m-\gamma)(a_{m}-a_{0})=0$. Then
$a_{m}=a_{0}$ for any $m\neq\gamma$. If $\gamma=0$, then $a_{m}=a_{0}$ for any
$m\in\mathbb{Z}$. If $\gamma\neq 0$, setting $m=\gamma$, $n=3\gamma$ in (3.1),
we get $a_{\gamma}=a_{0}$. Then the result follows. $\Box$

Denote $a=a_{0}$. Let $D$ act on $[L_{m},I_{n}]=(m-n)I_{m+n}$, we have

$2(m-n)c_{m+n}=(m+\gamma-n)a+(m-n-\gamma)c_{n},\ \ \forall m,n\in\mathbb{Z}.$
(3.2)

Similar to the proof of Lemma 3.1, we have $c_{m}=a$ for any $m\in\mathbb{Z}$.
Let $D$ act on $[L_{m},J_{n}]=(m-n)J_{m+n}$, we have

$2(m-n)d_{m+n}=(m+\gamma-n)a+(m-n-\gamma)d_{n},\ \ \forall m,n\in\mathbb{Z}.$
(3.3)

Similar to the proof of Lemma 3.1, we have $d_{m}=a$ for any $m\in\mathbb{Z}$.

Let $D$ act on $[L_{m},H_{n}]=-nH_{m+n}$, we have

$-2nb_{m+n}=-na+(-n-\gamma)b_{n},\ \ \forall m,n\in\mathbb{Z}.$ (3.4)

Setting $n=1$, we have $-2b_{m+1}=-a+(-1-\gamma)b_{1}$, implying $b_{m}=b_{n}$
for any $m,n\in\mathbb{Z}$. Denote $b=b_{m}$. Then we get $(\gamma-n)b=-na$
for any $n\in\mathbb{Z}$. Consequently, $a=b=0$ if $\gamma\neq 0$ and $a=b$ if
$\gamma=0$.

So we can get $D(L_{m})=aL_{m+\gamma},\ D(H_{m})=aH_{m+\gamma},\
D(I_{m})=aI_{m+\gamma},\ D(J_{m})=aJ_{m+\gamma}$ when
$D=D_{(\overline{0},\overline{0},\gamma)}$.

Then we assume that $D$ is a homogenous $\frac{1}{2}$-derivation of degree
$(\overline{0},\overline{1},\gamma)$.

###  3.2 $D=D_{(\overline{0},\overline{1},\gamma)}$

In this case, denote

$D(L_{m})=x_{m}I_{m+\gamma},\ D(H_{m})=y_{m}J_{m+\gamma},\
D(I_{m})=z_{m}L_{m+\gamma},\ D(J_{m})=w_{m}H_{m+\gamma},$

where $x_{m},y_{m},z_{m},w_{m}\in\mathbb{F}$. Let $D$ act on
$[L_{m},L_{n}]=(m-n)L_{m+n}$, we have

$2(m-n)x_{m+n}=(m+\gamma-n)x_{m}+(m-n-\gamma)x_{n},\ \ \forall
m,n\in\mathbb{Z}.$ (3.5)

Similar to the proof of Lemma 3.1. We have $x_{m}=x_{n}$ for any
$m,n\in\mathbb{Z}$. Denote $x=x_{m}$. Let $D$ act on
$[L_{m},H_{n}]=-nH_{m+n}$, we have

$-2ny_{m+n}=-x+(m-n-\gamma)y_{n},\ \ \forall m,n\in\mathbb{Z},$ (3.6)

which implies $x=0,y_{m}=0,m\in\mathbb{Z}$.

Let $D$ act on $[L_{m},I_{n}]=(m-n)I_{m+n}$, we have

$2(m-n)z_{m+n}=(m-n-\gamma)z_{n},\ \ \forall m,n\in\mathbb{Z}.$ (3.7)

###### Lemma 3.2

The solution of (3.7) is $z_{m}=0$ for any $m\in\mathbb{Z}$.

Proof. Similar to the proof of Lemma 3.1, Setting $m=0$ in (3.7), we have
$-2nz_{n}=(-n-\gamma)z_{n}$. We can get $(n-\gamma)z_{n}=0$. Then $z_{n}=0$
for any $n\neq\gamma$. If $\gamma=0$, then $z_{n}=0$ . If $\gamma\neq 0$,
setting $m=-\gamma$, $n=\gamma$ in (3.7), we get $z_{\gamma}=0$. We have
$z_{m}=0$ for any $m\in\mathbb{Z}$.

Similar to the proof of Lemma 3.1. Let $D$ act on
$[L_{m},J_{n}]=(m-n)J_{m+n}$, we have

$2(m-n)w_{m+n}=(-n-\gamma)w_{n},\ \ \forall m,n\in\mathbb{Z}.$ (3.8)

###### Lemma 3.3

The solution of (3.8) is $w_{m}=0$ for any $m\in\mathbb{Z}$.

Proof. Setting $m=0$ in (3.8), we have $(n-\gamma)w_{n}=0$ Then $w_{n}=0$ for
any $n\neq\gamma$. If $\gamma=0$, then $w_{n}=0$ . If $\gamma\neq 0$, setting
$m=-\gamma$, $n=\gamma$ in (3.8), we get $w_{\gamma}=0$, $w_{m}=0$ for any
$m\in\mathbb{Z}$.

Now we can get $D(L_{m})=0,\ D(H_{m})=0,\ D(I_{m})=0,\ D(J_{m})=0$ when
$D=D_{(\bar{0},\bar{1},\gamma)}$.

Then we assume that $D$ is a homogenous $\frac{1}{2}$-derivation of degree
$(\overline{1},\overline{0},\gamma)$.

###  3.3 $D=D_{(\overline{1},\overline{0},\gamma)}$

In this case, denote

$D(L_{m})=h_{m}J_{m+\gamma},\ D(H_{m})=e_{m}I_{m+\gamma},\
D(I_{m})=f_{m}H_{m+\gamma},\ D(J_{m})=g_{m}L_{m+\gamma},$

where $h_{m},e_{m},f_{m},g_{m}\in\mathbb{F}$.

Let $D$ act on $[L_{m},L_{n}]=(m-n)L_{m+n}$, we have

$2(m-n)h_{m+n}=(m+\gamma-n)h_{m}+(m-n-\gamma)h_{n},\ \ \forall
m,n\in\mathbb{Z}.$ (3.9)

Similar to the proof of Lemma 3.1, we have $h_{m}=h_{n}$ for any
$m,n\in\mathbb{Z}$. Denote $h=h_{m}$. Let $D$ act on
$[L_{m},H_{n}]=-nH_{m+n}$, we have

$-2ne_{m+n}=h+(m-n-\gamma)e_{n},\ \ \forall m,n\in\mathbb{Z},$ (3.10)

which implies $h=0,e_{m}=0,m\in\mathbb{Z}$. Let $D$ act on
$[L_{m},I_{n}]=(m-n)I_{m+n}$, we have

$2(m-n)f_{m+n}=(-n-\gamma)f_{n},\ \ \forall m,n\in\mathbb{Z}.$ (3.11)

Similar to the proof of Lemma 3.3, Setting $m=0$ in (3.11), we have
$(n-\gamma)f_{n}=0$. Then $f_{n}=0$ for any $n\neq\gamma$. If $\gamma=0$, then
$f_{n}=0$ . If $\gamma\neq 0$, setting $m=-\gamma$, $n=\gamma$ in (3.11), we
get $f_{\gamma}=0$. We have $f_{m}=0$ for any $m\in\mathbb{Z}$.

Let $D$ act on $[L_{m},J_{n}]=(m-n)J_{m+n}$, we have

$2(m-n)g_{m+n}=(m-n-\gamma)g_{n},\ \ \forall m,n\in\mathbb{Z}.$ (3.12)

Similar to the proof of Lemma 3.2. Setting $m=0$ in (3.12), we have
$(n-\gamma)g_{n}=0$. Then $g_{n}=0$ for any $n\neq\gamma$. If $\gamma=0$, then
$g_{n}=0$. If $\gamma\neq 0$, setting $m=-\gamma$, $n=\gamma$ in (3.12), we
get $g_{\gamma}=0$, $g_{m}=0$ for any $m\in\mathbb{Z}$.

So we can get $D(L_{m})=0,\ D(H_{m})=0,\ D(I_{m})=0,\ D(J_{m})=0$ when
$D=D_{(\overline{1},\overline{0},\gamma)}$.

Then we assume that $D$ is a homogenous $\frac{1}{2}$-derivation of degree
$(\overline{1},\overline{1},\gamma)$.

###  3.4 $D=D_{(\overline{1},\overline{1},\gamma)}$

In this case, denote

$D(L_{m})=i_{m}H_{m+\gamma},\ D(H_{m})=j_{m}L_{m+\gamma},\
D(I_{m})=k_{m}J_{m+\gamma},\ D(J_{m})=l_{m}I_{m+\gamma},$

where $i_{m},j_{m},k_{m},l_{m}\in\mathbb{F}$. $D$ restricts to a
$1/2$-derivation of the Heisenberg-Virasoro algebra,

Let $D$ act on $[L_{m},H_{n}]=-nH_{m+n}$, we have

$-2nj_{m+n}=(m-n-\gamma)j_{n},\ \ \forall m,n\in\mathbb{Z},$ (3.13)

###### Lemma 3.4

The solution of (3.13) is $j_{m}=0$ for any $m\in\mathbb{Z}$.

Proof. Setting $m=0$ in (3.13), we have $(n-\gamma)j_{n}=0$. Then $j_{n}=0$
for any $n\neq\gamma$. If $\gamma\neq 0$, then $j_{0}=0$ . If $\gamma\neq 0$,
setting $m=-\gamma$, $n=\gamma$ in (3.13), we get $j_{\gamma}=0$. If
$\gamma=0$ in (3.13) we have

$2nj_{m+n}=(n-m)j_{n},\ \ \forall m,n\in\mathbb{Z},$ (3.14)

Setting $m=0$ in (3.14), we get if $n\neq 0$, then $j_{n}=0$. Setting $n=-m=1$
in (3.14), we have $j_{1}=0$, so $j_{0}=0$. We have $j_{m}=0$ for any
$m\in\mathbb{Z}$.

Let $D$ act on $[L_{m},I_{n}]=(m-n)I_{m+n}$, we have

$2(m-n)k_{m+n}=i_{m}+k_{n}(m-n-\gamma),\ \ \forall m,n\in\mathbb{Z}.$ (3.15)

###### Lemma 3.5

The solution of (3.15) is $k_{n}=0$ for any $n\in\mathbb{Z}$.

Proof. Setting $m=0$ in (3.15), we have $(\gamma-n)k_{n}=i_{0}$. Then
$i_{0}=k_{n}=0$ for any $n\in Z$.

Let $D$ act on $[L_{m},J_{n}]=(m-n)J_{m+n}$, we have

$2(m-n)l_{m+n}=-i_{m}+(m-n-\gamma)l_{n},\ \ \forall m,n\in\mathbb{Z}.$ (3.16)

Setting $m=0$ in (3.16), we have $(n-\gamma)l_{n}=i_{0}$ Then $i_{0}=l_{n}=0$
for any $n\in Z$.

Let $D$ act on $[L_{m},L_{n}]=(m-n)L_{m+n}$, we have

$2(m-n)i_{m+n}=(m+\gamma)i_{m}+(-n-\gamma)i_{n},\ \ \forall m,n\in\mathbb{Z}.$
(3.17)

###### Lemma 3.6

The solution of (3.17) is $i_{m}=0$ for any $m\in\mathbb{Z}$.

Proof. Setting $m=0$ in (3.17),we have

$-2ni_{n}=\gamma i_{0}+(-n-\gamma)i_{n},\ \ \forall m,n\in\mathbb{Z}.$ (3.18)

We also have

$i_{n}=\frac{\gamma}{\gamma-n}i_{0},\ \ n\neq\gamma.$ (3.19)

Setting $n=-m$, taking $m\neq\\{\gamma,-\gamma,0\\}$ in (3.17), we have

$4mi_{0}=\frac{4m\gamma^{2}}{(\gamma-m)(\gamma+m)}i_{0},\ \ n\neq\gamma.$
(3.20)

By(3.20) we can get $i_{0}=0$ for any $m\in Z$. If $n\neq\gamma$ in(3.19), we
have $i_{n}=0$. setting $m=-\gamma$, $n=\gamma$ in (3.17), If $\gamma\neq 0$
we get $i_{\gamma}=0$. By (3.15) we have $(\gamma-n)k_{n}=i_{0}$, so when
$\gamma=0$, we have $i_{0}=0$ for any $n\in Z$. We have $i_{m}=0$ for any
$m\in\mathbb{Z}$.

So we can get $D(L_{m})=0,\ D(H_{m})=0,\ D(I_{m})=0,\ D(J_{m})=0$ when
$D=D_{(\overline{1},\overline{1},\gamma)}$.

As a summary, each $\frac{1}{2}$-derivation of the planar Galilean conformal
algebra is proven to be a scalar. Due to Theorem 8([5]), we get

###### Theorem 3.7

The transposed Poisson structures on $\mathcal{W}$ are trivial.

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