# A disk and no signatures of tidal distortion in the galaxy “lacking” dark
matter NGC 1052–DF2

Mireia Montes STScI Prize Fellow  Space Telescope Science Institute, 3700 San
Martin Drive, Baltimore, MD 21218, USA  Ignacio Trujillo Instituto de
Astrofísica de Canarias, c/ Vía Láctea s/n, E38205 - La Laguna, Tenerife,
Spain  Departamento de Astrofísica, Universidad de La Laguna, E-38205 - La
Laguna, Tenerife, Spain  Raúl Infante-Sainz Instituto de Astrofísica de
Canarias, c/ Vía Láctea s/n, E38205 - La Laguna, Tenerife, Spain  Departamento
de Astrofísica, Universidad de La Laguna, E-38205 - La Laguna, Tenerife, Spain
Matteo Monelli Instituto de Astrofísica de Canarias, c/ Vía Láctea s/n, E38205
- La Laguna, Tenerife, Spain  Departamento de Astrofísica, Universidad de La
Laguna, E-38205 - La Laguna, Tenerife, Spain  Alejandro S. Borlaff NASA Ames
Research Center, Moffett Field, CA 94035, USA

(Received January 1, 2018; Revised January 7, 2018)

###### Abstract

Using ultra-deep imaging ($\mu_{g}=30.4$ mag/arcsec2; 3$\sigma$,
10$\arcsec\times$10″), we probed the surroundings of the first galaxy
“lacking” dark matter KKS2000[04] (NGC 1052–DF2). Signs of tidal stripping in
this galaxy would explain its claimed low content of dark matter. However, we
find no evidence of tidal tails. In fact, the galaxy remains undisturbed down
to a radial distance of $80$ arcsec. This radial distance triples previous
spatial explorations of the stellar distribution of this galaxy. In addition,
the distribution of its globular clusters (GCs) is not extended in relation to
the bulk of the galaxy (the radius containing half of the GCs is $21$ arcsec).
We also found that the surface brightness radial profiles of this galaxy in
the $g$ and $r$ bands decline exponentially from 35 to 80 arcsec. That,
together with a constant ellipticity and position angle in the outer parts of
the galaxy strongly suggests the presence of a low-inclination disk. This is
consistent with the evidence of rotation found for this object. This finding
implies that the dynamical mass of this galaxy is a factor of 2 higher than
previously reported, bringing the dark matter content of this galaxy in line
with galaxies of similar stellar mass.

editorials, notices — miscellaneous — catalogs — surveys

††journal: ApJ††facilities: HST(ACS), INT, GTC††software: Astropy (The Astropy
Collaboration et al., 2018), SExtractor (Bertin & Arnouts, 1996), SCAMP
(Bertin, 2006), SWarp (Bertin, 2010), Gnuastro (Akhlaghi & Ichikawa, 2015),
Maneage (Akhlaghi et al., 2021), photutils v0.7.2 (Bradley et al., 2019),
pillow (van Kemenade et al., 2020), ellipse (Jedrzejewski, 1987), numpy
(Oliphant, 2006), scipy (Virtanen et al., 2020), Astrodrizzle (Gonzaga et al.,
2012), Astrometry.net (Lang et al., 2010)

##  1 Introduction

Dark matter (DM) is a key constituent in the current favored models of galaxy
formation. For this reason, the claimed existence of two old and long-lived
galaxies “lacking” DM (van Dokkum et al., 2018b, 2019) in the field of view
(FOV) of the massive elliptical galaxy NGC 1052 sparked a lot of controversy
(Martin et al., 2018; Trujillo et al., 2019; Laporte et al., 2019; Monelli &
Trujillo, 2019; Nusser, 2019, to name a few). These galaxies have been argued
to have a stellar vs. dark matter mass ratio of $\sim 1$, while for similar
galaxies the observed ratios are $\lesssim 0.1$ (e.g., Mancera Piña et al.,
2019; Müller et al., 2020). Their existence could imply a challenge to the
current galaxy formation paradigm, as without DM the gas would not have
collapsed and formed stars.

A growing number of theoretical studies suggest an explanation for the
existence of these galaxies within the current cosmological paradigm: tidal
stripping of the DM of these galaxies (Ogiya, 2018; Nusser, 2020; Yang et al.,
2020; Jackson et al., 2021; Macciò et al., 2021; Doppel et al., 2021). In a
recent paper, Montes et al. (2020, hereafter, M20) found that the second
galaxy “lacking” DM, NGC 1052–DF4, is showing tidal tails pointing to an
interaction with its nearby neighbor, NGC 1035. As stars are more centrally
concentrated than the DM, this means that tidal stripping has removed a
significant fraction of the DM before starting to affect the stars of the
galaxy.

The current cosmological paradigm, Lambda Cold Dark Matter ($\Lambda$CDM),
predicts significant numbers of faint stellar streams in the outskirts of the
majority of nearby massive galaxies (e.g., Bullock & Johnston, 2005; Cooper et
al., 2010). All this structure starts to show up at surface brightness fainter
than $30$ mag/arcsec2. However, the detection and characterization of faint
substructures in stellar halos is still a work in progress as their intrinsic
low surface brightness prevents a complete census of these structures.

In this paper, we take advantage of ultra-deep observations and recent
advances in low surface brightness data processing to explore the tidal
stripping scenario in KKS2000[04], better known as NGC 1052–DF2 (for
convenience, this is the name we will use throughout this paper). The presence
of tidal tails surrounding NGC 1052–DF2 will serve as the smoking gun of
ongoing tidal disruption and therefore, explain the “lack” of DM observed in
the first galaxy of this kind. We follow the same steps than in M20 to search
for signs of interaction in this galaxy. Sec. 2 presents the data used in this
work. Then, we explore the distribution of GCs in Sec. 3 as tidal stripping
will place them along the orbit of the galaxy. Secondly, we explore the
stellar light of the galaxy, deriving its surface brightness profiles in Sec.
4.1 and its shape in Sec. 4.2. Finally, we discuss the implications of our
findings for the claimed rotation of the galaxy (Sec. 5.1) and its association
with its closest (in projection) neighbors (Sec. 5.2). All magnitudes in this
work are given in the AB magnitude system.

##  2 Data

The data used in this paper come from three different facilities: the _Hubble
Space Telescope_ (_HST_) and two ground based telescopes: the 10.4m Gran
Telescopio Canarias (GTC) and the 2.5m Isaac Newton Telescope (INT). We
describe the details of each observation on what follows.

###  2.1 Hubble Space Telescope imaging

NGC 1052–DF2 was observed with _HST_ ACS Wide-Field Channel (WFC) as part of
the program GO-14644 (PI: van Dokkum) available through the MAST
archive111https://mast.stsci.edu/portal/Mashup/Clients/Mast/Portal.html. The
data consists of one orbit in the $F606W$ and one in the $F814W$ band. The
total exposure time is $545$ s in $F606W$ and $580$ s in $F814W$.

The data reduction of the _HST_ imaging of NGC 1052–DF2 is described in detail
in Trujillo et al. (2019) and summarized below. The charge-transfer efficiency
(CTE) corrected data were downloaded from MAST and flc files were used to
build the final drizzled mosaics with Astrodrizzle (Gonzaga et al., 2012). To
estimate the sky, we masked thoroughly the images using a combination of
NoiseChisel (Akhlaghi & Ichikawa, 2015) and manual masking of the galaxy. The
centroid of the sky distribution derived using the masked images was
calculated using bootstrapping222Using the code Bootmedian, available at
https://github.com/Borlaff/bootmedian. Finally, to mitigate the gradients
present in the images, NoiseChisel was used to generate a final sky background
model of the final co-added images that was subtracted from the images.

Following M20, we use these high-resolution _HST_ observations together with
photometry from the GTC HiPERCAM deep data to characterize and potentially
find new GCs around NGC 1052–DF2.

###  2.2 Deep ground-based imaging

One of the biggest challenges in low surface brightness imaging is to process
ultra-deep observations in a manner that preserves the faint light of the
object of interest while correcting for all systematic effects introduced by
the instrument/telescope. In this section, we describe the data reduction
performed for our ground-based ultra-deep imaging. Both the data acquisition
and data processing are aimed at preserving the low surface brightness light
around NGC 1052–DF2. Most of the reduction steps are common for both datasets
and described in the following. Specific steps for each of the telescopes are
discussed later.

A key step for obtaining ultra-deep images is the dithering pattern. NGC
1052–DF2 was observed by the GTC and INT following the dithering strategy
outlined in Trujillo & Fliri (2016) aimed at reducing scattered light from the
telescope structure as much as possible. In short, this consists in a
dithering pattern with large steps (typically the size of the source under
investigation) and whenever possible rotation of the camera. This ensures we
achieve the flat background required for our goals.

Bias frames were taken the same night of the observations as part of the
different observing programs. The master bias frames were created as the
$3\sigma$-clipped mean of the individual frames and subtracted from the
original images. The flat-field frames are derived using the same science
exposures obtained for this work333Dome or twilight flats can introduce
gradients in the data due to inhomogeneities in illumination. Using the same
science images to derive the flat-field frames ensures that we are not
introducing gradients (e.g., Trujillo & Fliri, 2016).. The bias corrected CCD
frames present steep gradients making the proper masking of the sources
difficult in those science frames, a crucial step to achieve an accurate flat-
field correction. For this reason, the flat-field frames were built following
a two-step process. First, we derive a preliminary flat by stacking each bias-
corrected frame. This preliminary flat is used to correct each CCD frame in
order to flatten and, consequently, properly mask the science exposures.
Second, we combine these masked, and normalized, images to create the final
master flat-field frames.

We, then, performed the astrometric calibration of the different frames. We
used the astrometry.net software (Lang et al., 2010) to produce an approximate
astrometric solution, later refining it with SCAMP (Bertin, 2006).

Once all frames were corrected of all systematic effects and in a common
astrometric solution, they were resampled into a common grid using SWarp
(Bertin, 2010) and stacked using a $3\sigma$-clipped mean into a final image,
one per filter.

####  2.2.1 GTC HiPERCAM images

HiPERCAM (Dhillon et al., 2018) is a quintuple-beam, high-speed astronomical
imager able to obtain images of celestial objects in five different filters
($u$, $g$, $r$, $i$, $z$) simultaneously. The image area of each of the five
CCDs is $2048\times 1024$ pixels ($2.7\arcmin\times 1.4\arcmin$; 1
pixel$=0\farcs 08$) divided into four channels of $1024\times 512$ pixels. NGC
1052–DF2 was observed on the 2019-01-11. The whole reduction process was done
within a controlled and enclosed software environment as described in Akhlaghi
et al. (2021).

The data reduction of the NGC 1052–DF2 images follows the steps detailed in
M20 for NGC 1052–DF4. After the standard calibration per CCD channel (bias and
flat-field), each set of four channels were assembled into a single image. The
different exposures that went into the final images were visually inspected
and those with low quality or strong gradients were rejected. The photometric
calibration of these images was done using SDSS DR12 (Alam et al., 2015). The
final exposure time is 0.6h for each band, corresponding to point-like source
depths ($5\sigma$ in $2\arcsec$ diameter apertures) of: $25.32\pm 0.05$,
$24.69\pm 0.05$, $24.63\pm 0.07$, $24.18\pm 0.06$, $23.71\pm 0.06$, in the
$u$, $g$, $r$, $i$, $z$ bands, respectively. The sky subtraction was done by
subtracting a constant value computed from the masked image.

Unfortunately, the FOV of HiPERCAM is comparable to the size of the galaxy
making it impossible to reliably measure the sky. This results in over-
subtraction in the outer parts of the galaxy. Therefore, we will use these
images to study only the photometry of the GC system around NGC 1052–DF2.

####  2.2.2 INT images

The Isaac Newton Telescope (INT) is a 2.5-m telescope situated in the Roque de
los Muchachos Observatory in the island of La Palma. This telescope is
equipped with a wide-field camera (WFC), mounted at the prime focus, providing
imaging over a 33-arcmin FOV with a wide variety of broad- and narrow-band
filters. The WFC consists of 4 thinned EEV 2k$\times$4k CCDs. The pixel scale
of the camera is $0\farcs 33$. The data were taken between 2020-10-12 and
2020-10-16 in two different filters $g$ and $r$ for a total exposure time of
4.4h and 3.2h, respectively.

In this case, after following the data reduction outlined above, the
photometric calibration of the images was done using the SDSS DR15 (Aguado et
al., 2019). Thanks to the high quality of the flat-field correction, the sky
subtraction was done by only using an average constant applied to the entire
image. No gradients were present in the final co-adds of each band.

The nominal depth of the final images measured in $10\arcsec\times 10\arcsec$
boxes is $30.4$ and $29.5$ mag/arcsec2 in the $g$ and $r$ bands ($3\sigma$
above the background), respectively, measured using the method in appendix A
in Román et al. (2020). For comparison, the limiting surface brightness
(3$\sigma$; 12″$\times$12″) of the Dragonfly images in the field of NGC1052
are between $\mu_{g/r}=$ 27.4 and 28.0 mag/arcsec2 (Merritt et al., 2016) and
$\mu_{r}=$ 28.5 mag/arcsec2 in Müller et al. (2019) ($3\sigma$;
$10\arcsec\times 10\arcsec$).

###  2.3 Scattered light removal from nearby stars

The careful modeling and removal of stars in deep images is now a common
technique in low surface brightness science (e.g, Slater et al., 2009;
Trujillo & Fliri, 2016; Infante-Sainz et al., 2020; Montes et al., 2021). In
order to reliably explore low surface brightness features around NGC 1052–DF2,
it is key to remove the light scattered by the brightest objects in the image.
In fact, there is the bright star HD16873 (R.A. = $2^{h}42^{m}9.3^{s}$, Dec =
$-8^{d}22\arcmin 32\farcs 218$, V= 8.34 mag), $\sim 350$ arcsec to the east of
NGC 1052–DF2, creating a source of extra light, contamination, that can bias
our results. The removal of this scattered light contamination is particularly
imperative for the images where we are exploring the distribution of light
around NGC 1052–DF2: the INT $g$ and $r$ images.

To correct for this scattered light, we first need to model the 2D profile of
the star. We use the software ellipse in IRAF. ellipse fits elliptical
isophotes to the 2-D images of sources using the method described in
Jedrzejewski (1987).

Prior to modelling the star, we need to carefully mask foreground and
background sources as their presence can affect the fitting. For this, we used
a two-step SExtractor setting; a “hot+cold” mode (Rix et al., 2004; Montes et
al., 2021). The “cold” mode masks extended sources while the “hot” mode is
optimized to detect the more compact and faint sources that are embedded
within the galaxy light. We run SExtractor in a deep combined INT $g+r$ image.
The “cold” mask was further expanded 8 pixels while the “hot” was expanded 2
pixels, leaving the bright star we want to model unmasked. Because the PSF in
our INT images is highly asymmetric, especially noticeably in bright stars, we
only modeled the section of the star that is close to NGC 1052–DF2, masking
the rest manually. This way we ensure that the output model is suitable for
our purposes. In addition, we manually masked NGC 1052–DF2 to cover any
diffuse light that has remained unmasked from the above procedure that could
contaminate the flux of the star at large radius.

Once ellipse was run in the masked image, we created the model of the star
with bmodel and subtracted it from the images. In addition, we also modelled
and subtracted two fainter stars $\sim 150$ arcsec to the east and west of NGC
1052–DF2 (R.A. = $2^{h}41^{m}36.88^{s}$ Dec = $-8^{d}24\arcmin 56\farcs 80$,
and R.A. = $2^{h}41^{m}54.47^{s}$ Dec = $-8^{d}24\arcmin 20\farcs 80$) .

##  3 The spatial distribution of the globular cluster system of NGC 1052–DF2

Due to the compact sizes (half-light radii of a few parsecs) and masses of
globular clusters (GCs, $10^{4}-10^{6}$ M⊙; Harris, 1996), they appear as good
tracers of the properties of their host galaxy as they are easily observable
in distant system even at large radial distances (see Brodie & Strader, 2006).

The spatial distribution of the GC system of a galaxy reveals clues about its
present state. In an accretion event, the stripped material is placed along
the orbit of the parent satellite galaxy, forming tidal streams (e.g., Toomre
& Toomre, 1972; Johnston et al., 1996). In the same way, the GCs will also be
placed along the orbit and exploring their observed spatial distribution can
indicate if this process is happening and the direction of the orbit.

One of the goals of this work is to explore whether the GCs align through a
particular direction, suggesting that NGC 1052–DF2 is undergoing tidal
stripping or, on the contrary, the GCs are more spherically distributed around
NGC 1052–DF2, which is expected if the galaxy is not experiencing any
interaction.

###  3.1 GC selection

Figure 1: Histograms of FWHM (left), ellipticity (middle) and $F606W$
magnitude mF606W (right) for all the sources simultaneously detected sources
in the HST and HiPERCAM images (grey). The magenta histograms show the
distribution of the spectroscopically confirmed GCs in Shen et al. (2021a).
The blue vertical lines mark the range in each parameter of the initial
selection. The mint green histograms show the sources that simultaneously
satisfy the conditions of FWHM, ellipticity and magnitude enclosed by the
vertical blue lines.

In this section, we combine the high resolution of _HST_ with the multicolor
information from HiPERCAM to identify new GCs for NGC 1052–DF2. Recent works
have shown the increased effectiveness of identifying GCs using more
information from their spectral energy distributions (Montes et al., 2014;
Muñoz et al., 2014).

van Dokkum et al. (2018b) and Shen et al. (2021a) confirmed spectroscopically
12 GCs in NGC 1052–DF2. Here, we try to identify more GCs following the steps
in M20. First, we run SExtractor (Bertin & Arnouts, 1996) on the _HST_ images
in dual-mode using F814W as the detection image. In the same way, SExtractor
was also run in the HiPERCAM images using $r$ as our detection image. Both
catalogues were matched based on the position of the sources in the sky.

We pre-selected GC candidates thanks to the high-resolution of the _HST_
data444We have rerun the analysis with the images of NGC 1052–DF2 from
GO-15851 (PI: van Dokkum) and the results obtained do not change with this
deeper dataset.. This selection is based upon the properties of the confirmed
GCs in Shen et al. (2021a). In addition to the morphology selection with _HST_
similar to that in M20, we also used the measured $F606W-F814W$ color to
eliminate sources that are clearly too red to be a GC candidate.

The pre-selection criteria is as follows.

  * •

$0.1\arcsec<$ FWHM $<0.23\arcsec$

  * •

$0<$ ellipticity $<0.23$

  * •

$21<$ mF606W $<24$ mag

  * •

mF606W \- mF814W $<0.8$ mag

Figure 2: The ($u$-$r$)-($r$-$z$) color-color diagram of the initial sample of
candidate GCs (mint green). The spectroscopically confirmed GCs of Shen et al.
(2021a) are highlighted in magenta. The blue box indicates the color-color
region we have selected to create our final sample of GC candidates. The inset
shows a zoom-in into the selection box for ease of viewing.

The aim of this selection is to find GCs that are similar to the confirmed GCs
while allowing for sources that might have been too faint for spectroscopic
follow-up. Fig. 1 shows the histograms of FWHM, ellipticity and mF606W of all
the sources detected with SExtractor (filled in grey), the confirmed GCs in
Shen et al. (2021a) (dashed in magenta) and the GC candidates based in our
preliminary selection (filled in mint green). Nineteen objects fall within
this pre-selection, including eleven objects in Shen et al. (2021a) (see
below).

The next step is using the colors of the GCs from HiPERCAM to narrow down the
selection. Fig. 2 shows the ($u$-$r$) vs. ($r$-$z$) color-color diagram for
the pre-selected GC candidates (mint green) from our preliminary selection. We
have marked 11 GCs in Shen et al. (2021a) as the magenta circles555One of the
GCs identified in Shen et al. (2021a) (GC-39) is outside the FOV of HiPERCAM,
and has been added manually after this step.. As seen in Fig. 2, the confirmed
GCs fill a very narrow region in the color-color space and, therefore, we
further narrowed down the initial selection to objects within $1.4<u-r<2.2$
mag and $0.24<r-z<0.51$ mag (blue box). These ranges in color are based on the
$\pm 2\sigma$ around the median colors of the confirmed GCs (magenta circles),
with $\sigma$ being their dispersion in color. One of the spectroscopically
confirmed GCs (C-14) fall outside this selection box as it appears to be
blended with another source and therefore its photometry might be
contaminated, but was not discarded from the final selection.

We identified fourteen GCs, two in addition to the ones confirmed
spectroscopically. The coordinates and magnitudes of all the objects are
listed in Table 1. The magnitudes are corrected by the extinction of our
Galaxy (Schlafly & Finkbeiner, 2011): A${}_{u}=0.10$, A${}_{g}=0.08$,
A${}_{r}=0.06$, A${}_{z}=0.03$, A${}_{F606W}=0.06$ and A${}_{F814W}=0.04$. Two
of the GCS (C-6 and C-10) have ground-based magnitudes that are brighter than
their _HST_ magnitudes suggesting that they are affected by blending.

Table 1:  Confirmed and candidate GCs of NGC 1052–DF2. ID | Other IDa | RA | DEC | $m_{u}$ | $m_{g}$ | $m_{r}$ | $m_{z}$ | $m_{\rm F606W}$ | $m_{\rm F814W}$
---|---|---|---|---|---|---|---|---|---
|  |  |  | $\mathrm{mag}$ | $\mathrm{mag}$ | $\mathrm{mag}$ | $\mathrm{mag}$ | $\mathrm{mag}$ | $\mathrm{mag}$
C-1∗ | GC-59 | 2h41m48.09s | $-8\arcdeg 24\arcmin 56.82\arcsec$ | $24.23\pm 0.28$ | $22.86\pm 0.04$ | $22.58\pm 0.03$ | $22.22\pm 0.07$ | $22.82\pm 0.01$ | $22.38\pm 0.01$
C-2∗ | GC-71 | 2h41m45.14s | $-8\arcdeg 24\arcmin 22.31\arcsec$ | $24.51\pm 0.20$ | $22.79\pm 0.03$ | $22.47\pm 0.03$ | $22.14\pm 0.06$ | $22.66\pm 0.01$ | $22.26\pm 0.01$
C-3∗ | GC-73 | 2h41m48.22s | $-8\arcdeg 24\arcmin 17.49\arcsec$ | $22.96\pm 0.06$ | $21.55\pm 0.01$ | $21.31\pm 0.01$ | $20.96\pm 0.02$ | $21.49\pm 0.01$ | $21.15\pm 0.01$
C-4 |  | 2h41m47.61s | $-8\arcdeg 24\arcmin 18.21\arcsec$ | $25.39\pm 0.61$ | $23.87\pm 0.08$ | $23.63\pm 0.07$ | $23.12\pm 0.15$ | $23.73\pm 0.03$ | $23.33\pm 0.03$
C-5∗ | GC-77 | 2h41m46.55s | $-8\arcdeg 24\arcmin 13.36\arcsec$ | $23.54\pm 0.09$ | $21.96\pm 0.02$ | $21.78\pm 0.01$ | $21.30\pm 0.03$ | $21.98\pm 0.01$ | $21.61\pm 0.01$
C-6∗ | GC-85 | 2h41m47.75s | $-8\arcdeg 24\arcmin 05.30\arcsec$ | $20.89\pm 0.13$ | $19.42\pm 0.02$ | $19.10\pm 0.02$ | $18.73\pm 0.04$ | $22.39\pm 0.01$ | $21.97\pm 0.01$
C-7 |  | 2h41m45.91s | $-8\arcdeg 23\arcmin 56.43\arcsec$ | $25.77\pm 0.73$ | $24.09\pm 0.09$ | $23.67\pm 0.08$ | $23.20\pm 0.18$ | $23.82\pm 0.03$ | $23.39\pm 0.03$
C-8∗ | GC-91 | 2h41m42.17s | $-8\arcdeg 23\arcmin 53.27\arcsec$ | $24.20\pm 0.25$ | $22.43\pm 0.02$ | $22.16\pm 0.02$ | $21.72\pm 0.05$ | $22.44\pm 0.01$ | $22.04\pm 0.01$
C-9∗ | GC-92 | 2h41m46.89s | $-8\arcdeg 23\arcmin 50.69\arcsec$ | $23.32\pm 0.12$ | $21.96\pm 0.02$ | $21.61\pm 0.02$ | $21.24\pm 0.04$ | $22.35\pm 0.01$ | $21.83\pm 0.01$
C-10∗ | GC-93 | 2h41m46.72s | $-8\arcdeg 23\arcmin 50.75\arcsec$ | $21.87\pm 0.26$ | $20.08\pm 0.04$ | $19.78\pm 0.03$ | $19.36\pm 0.07$ | $22.91\pm 0.02$ | $22.53\pm 0.02$
C-11∗ | GC-98 | 2h41m47.34s | $-8\arcdeg 23\arcmin 34.54\arcsec$ | $24.44\pm 0.24$ | $22.90\pm 0.03$ | $22.65\pm 0.03$ | $22.27\pm 0.07$ | $22.85\pm 0.01$ | $22.44\pm 0.01$
C-12∗ | GC-101 | 2h41m45.20s | $-8\arcdeg 23\arcmin 27.73\arcsec$ | $24.56\pm 0.28$ | $22.95\pm 0.04$ | $22.66\pm 0.04$ | $22.39\pm 0.09$ | $22.98\pm 0.02$ | $22.54\pm 0.01$
C-13∗ | GC-39 | 2h41m45.07s | $-8\arcdeg 25\arcmin 24.87\arcsec$ | … | … | … | … | $22.31\pm 0.01$ | $21.93\pm 0.01$
C-14∗ | GC-80 | 2h41m46.23s | $-8\arcdeg 24\arcmin 07.32\arcsec$ | $23.54\pm 0.12$ | $22.35\pm 0.03$ | $22.22\pm 0.02$ | $21.70\pm 0.05$ | $22.56\pm 0.01$ | $22.17\pm 0.01$
∗GC spectroscopically confirmed in Shen et al. (2021a).
a ID in Shen et al. (2021a).

###  3.2 Spatial distribution of the GCs

In this section, we address the spatial distribution of the GCs in NGC
1052–DF2 in order to obtain clues about the interaction state of the galaxy.
As a satellite galaxy is interacting with a massive galaxy, the stripped
material (both stars and GCs) of the satellite will be deposited along its
orbit. This is the case for NGC 1052–DF4 where its GC system align in a
particular axis along the galaxy (see Fig. 4 in M20) and the spatial
distribution of GCs is more extended than that of the stellar body of the
galaxy.

Fig. 3 shows an RGB image created using the HiPERCAM $g$, $r$ and $z$ and a
black and white INT $g$ image for the background. The highlighted sources are
the 12 confirmed GCs from Shen et al. (2021a, magenta) and the two new
candidates obtained in the previous section (mint green). From Figure 3, it is
evident that the distribution of GCs of NGC 1052–DF2 is remarkably different
from that of NGC 1052–DF4; the spatial distribution of the GCs in NGC 1052–DF2
is more concentrated. The radius containing half of the GCs (Re,GC) is
$21.0\pm 0.1$ arcsec. This is similar to the value of the effective radius
published for the stellar light distribution of this galaxy using HST data
($R_{e}=22.6$ arcsec; van Dokkum et al., 2018b). In contrast, the radius
containing half of the GCs in NGC 1052–DF4 was 9 times larger than the $R_{e}$
of the stellar light (M20). Furthermore, all but two of the GCs are contained
in the stellar body of the galaxy ($<3R_{e}$). This concentrated distribution
of GCs with respect to the galaxy body (i.e., Re,GC/R${}_{e}\sim$1) has been
found in other UDGs that present no signatures of tidal distortions (see
Saifollahi et al., 2021). There is not preferred direction, or axis, in the
spatial distribution of GCs of this galaxy.

However, from Fig. 3, there is a hint that a larger number of GCs are located
towards the northern side of the galaxy. To explore whether this excess is
statistically significant, we computed its significance by deriving the
probability the globular clusters distribute in that way using a binomial
distribution. We divided the GCs North and South from the minor axis of the
galaxy (PA = $-31$ deg, defined in Sec. 4.1). 9 (8) clusters are North of the
minor axis and 5 (4) are South, for the total (spectroscopically confirmed)
sample of GCs. We find a probability of $0.64\pm 0.13$ for the total sample
and $0.66\pm 0.14$ for the spectroscopically confirmed (Shen et al., 2021a)
sample. Within the errors ($\sim 1\sigma$), the spatial distribution of the
GCs is consistent with the null hypothesis (i.e., the GCs are distributed with
similar probability in the two sides of the galaxy, probability of 0.5).
Therefore, we can not reject the idea that the observed distribution of GCs in
this galaxy is compatible with a random distribution.

Figure 3: Color composite of a postage stamp of a region $240\arcsec\times
240\arcsec$ around NGC 1052–DF2. The black and white background is the
$g$-band INT image. The mint green circles highlight the candidate GCs found
in this work while the magenta circles are the GCs spectroscopically confirmed
in Shen et al. (2021a). The blue ellipses indicate $1R_{e}$, $2R_{e}$ and
$3R_{e}$, respectively, of the stellar light of the galaxy ($R_{e}=22.6$
arcsec, as reported in van Dokkum et al. 2018b).

##  4 The stellar component of NGC 1052–DF2

In this section, we use ultra-deep INT imaging to address whether this galaxy
presents tidal features that can point to tidal interactions with one of the
nearest massive galaxies and explain its seemingly low content of DM.

###  4.1 Surface brightness profiles

Figure 4: Output from ellipse for the INT $g$ and $r$ images of NGC 1052–DF2.
The left panel shows the $200\arcsec\times 200\arcsec$ region around NGC
1052–DF2 with the fitted ellipses overplotted. The middle panel shows the
surface brightness radial profiles as a function of the semi-major axis for
the $g$ (mint green) and $r$ (teal). The rightmost panel presents the
ellipticity (top) and PA (bottom) radial profiles. The profiles in the middle
panel have been derived by fixing the ellipticity and PA to the values
indicated as a pink solid line in the rightmost panels.

The goal of this paper is to investigate whether NGC 1052–DF2 shows any signs
of interaction with a massive nearby neighbor. Note that this galaxy’s
distance has remained controversial and therefore, which massive galaxy it is
associated with is unclear. van Dokkum et al. (2018b) and van Dokkum et al.
(2018a) claimed a distance of $\sim 20$ Mpc which connects it with NGC
1052666This same group has claimed a distance of $22.1\pm 1.2$ Mpc using
deeper _HST_ data (Shen et al., 2021b).. On the contrary, Trujillo et al.
(2019) and Monelli & Trujillo (2019) place the galaxy at $\sim 13.5$ Mpc,
associating it with the spiral galaxy NGC 1042777The revised surface
brightness fluctuation estimation in Zonoozi et al. (2021) places an upper
limit to the distance to NGC 1052–DF2 of 17 Mpc..

In surface brightness profiles, signs of interactions will appear as an excess
of light at large radius and deviations from the morphology of the inner parts
of the galaxy (e.g., Johnston et al., 2002). To explore whether this is the
case for NGC 1052–DF2, we derived the radial profiles for the INT images using
ellipse. It provides the median intensity, ellipticity and position angle (PA)
for each of the fitted isophotes.

Beforehand, we need to mask thoroughly the image. For this, we used the same
procedure described in Sec. 2.3. In this case, the “cold” mask was further
expanded 6 pixels while the “hot” was expanded 2 pixels, leaving NGC 1052–DF2
unmasked in the “cold” mask. That leaves the diffuse light of the galaxy
unmasked while all compact sources like background galaxies and GCs are
masked. The mask is shown in Appendix A.

Once we have all contaminant sources masked, we derived the radial profiles
for the INT $g$ and $r$ images with ellipse. In this case, we run ellipse in
three steps, similar to the approach followed in Huang et al. (2018). We first
ran the task allowing all parameters to vary freely. In the second run, we
fixed the centers to the median centers of the isophotes returned by ellipse
in the first iteration. In the third iteration, we hold the ellipticity and PA
fixed to derive the surface brightness profiles in both $g$ and $r$ bands.
This last step ensures that we reach regions where the signal-to-noise ratio
is low.

Fig. 4 shows the output of ellipse for the $g$ and $r$ bands. In the leftmost
panel, a postage stamp of a $200\arcsec\times 200\arcsec$ region around NGC
1052–DF2 is shown with the fitted ellipses overplotted. The right panel shows
the ellipticity (top) and PA (bottom) as a function of the semi-major axis
(SMA).

The 1-D radial surface brightness profiles as a function of the SMA are shown
in the middle panel, to a radius of $\sim 80\arcsec$. The shaded regions
represent the error in each elliptical isophote. The error is computed as the
combination of the error in the intensity of each isophote, as provided by
ellipse, and the error of the sky given by the distribution of background
pixels in each image. The 1-D surface brightness profiles in both bands are
derived with a fixed ellipticity and PA ($0.17$ and $-31$ deg, respectively,
shown as the pink lines in the right panel).

Figure 5: Observed 1-D surface brightness profiles of NGC 1052–DF2 in the
g-band (mint green line). The shaded region represents the error in each
elliptical isophote computed as described in Sec. 4.1. The pink solid line
shows the Sérsic fit with n=0.6 and $R_{e}=22.6$ arcsec as reported in van
Dokkum et al. (2018b). The dashed black line correspond to an exponential fit
to the profile for $r>20$″. It is evident that neither the Sérsic nor an
exponential can fully describe the profile of this galaxy.

These profiles derived for NGC 1052–DF2 show that the structure of the galaxy
is more complex than previously reported. Fig. 5 shows that the observed
profile in the $g$-band is well described with a Sérsic model in the inner
$\sim 30$ arcsec with n = 0.6. However, this cannot provide a faithful
description of the galaxy at larger radius. On the other hand, an exponential
fit ($R_{e}=24.4\pm 0.1$ arcsec) cannot reproduce the galaxy in its inner
regions. Interestingly, the fact that the outer exponential can not describe
the inner regions suggests that if the outer part of the galaxy is a disk
structure it is truncated in the central region of the galaxy. Exponentially
truncated disks in their inner regions have been proposed by Breda et al.
(2020) to explain the structure of many spiral galaxies.

The exponential decline seen in Fig. 4 is reinforced by the behaviour of the
ellipticity and PA with radius. Once they reach $r\sim 35$ arcsec, they become
roughly constant. Signs of interaction or a warp in the galaxy will show up as
an increase in PA and ellipticity with radius, that are not seen here. The
change in the PA with radius seen within $35$ arcsec was also reported in
Müller et al. (2019) for this galaxy. From the profiles in Fig. 4, we computed
the new effective radius of the galaxy to be $R_{e}=28.4\pm 0.3$ arcsec.

In addition, we derived the radial $g-r$ profile and the stellar mass density
profile of NGC 1052–DF2. This is shown in Fig. 6. In the left and middle
panels, we plot the $g-r$ color profile to a radius of $70$ arcsec (beyond
this radius the errors are larger than 0.2 mag and therefore not reliable).
The leftmost panel shows the color estimate for different metallicities
derived from the Vazdekis et al. (2016) models for a fixed age of $8.5$ Gyr
(i.e., the age of NGC 1052–DF2 estimated spectroscopically, Ruiz-Lara et al.,
2019) as horizontal dashed lines. The black horizontal dashed line corresponds
to the metallicity derived for the central part of this galaxy in Ruiz-Lara et
al. (2019) using spectroscopy.Similarly, the dashed lines in the middle panel
are the different colors of the models for a fixed metallicity, [Fe/H] =
-1.18, but varying age. The rightmost panel of Fig. 6 shows the stellar mass
density profile for this galaxy. To derive it, we apply equation 1 and 2 in
Montes & Trujillo (2014) (see also Bakos et al. 2008) to link the observed
surface brightness in the $g$-band to a stellar mass to light (M/L) ratio. The
M/L ratio was derived from the prescriptions given in Roediger & Courteau
(2015). For this, we assumed a Chabrier (2003) initial mass function.

The color profile shows a clear negative gradient with radius. Two different
scenarios could explain why NGC 1052–DF2 becomes bluer with radius. In Fig. 6,
we interpret the color gradient as an age and/or a metallicity gradient. We
can not rule out any of these two alternatives as it is not possible to
disentangle between age and metallicity with only one color. However, the lack
of HI detection in this galaxy (Chowdhury, 2019) suggests that it is similarly
old everywhere and that the color trend could be explained as a metallicity
gradient along its radial profile. In fact, there is a hint at a metallicity
gradient, but no age gradient, in the MUSE data of NGC 1052–DF2 in Fensch et
al. (2019). Assuming that the age of the galaxy is the same along its entire
structure, the color gradient would indicate that the stellar population is
getting more metal-poor towards the outskirts. If that was the case, the
derived metallicity using spectroscopy would only be representative of the
innermost parts of the galaxy ($<1R_{e}$).

The stellar mass density profile also exhibits the two components, Sérsic and
exponential, present in the surface brightness profiles. There is a shallow
core down to $\sim 23$ arcsec followed by an exponential decline down to the
limit where the data allows to trace reliably the galaxy (i.e. down to $0.07$
M${}_{\odot}/\mathrm{pc}^{2}$).

Figure 6: Left and middle panel: the INT $g-r$ color profile of NGC 1052–DF2
shows a clear negative gradient. The horizontal dashed lines in the left panel
indicate the $g-r$ color of the Vazdekis et al. (2016) models at a fixed age
of $8.5$ Gyr and different [Fe/H] (as labelled), while in the middle panel
they indicate different ages at a fixed metallicity of [Fe/H] = -1.18. Right
panel: Stellar mass density profile of NGC 1052–DF2.  Figure 7: HiPERCAM RGB
color image of the $240\arcsec\times 240\arcsec$ around NGC 1052–DF2 with the
INT $g$ image in black and white for the background. The left panel shows the
contours at four different surface brightness levels. There is no evidence of
tidal features in our ultra-deep imaging of NGC 1052–DF2. The right panel
shows the rotation map from Lewis et al. (2020) with its PA (pink area) and
the stellar PA from the outer parts. Both PA are in agreement, suggesting that
the galaxy is indeed rotating.

###  4.2 The shape of NGC 1052–DF2

Matter stripped from satellite galaxies forms tidal tails which will
eventually become completely unbound and form larger tidal features. In N-body
simulations (e.g., Klimentowski et al., 2009), dwarf galaxies orbiting a
massive host galaxy potential show two tidal tails emanating from the two
opposite sides of the dwarf as the particles are seen to move in the direction
of the tidal forces, creating a shape reminiscent of an S.

To explore whether we can rule out the presence of tidal features up to the
depth of our data, we derive the isocontours of this galaxy at four different
surface brightness limits. For this we use our deepest image: the INT
$g$-band. As in M20, we used matplotlib’s contour to draw the contours in the
left panel of Fig. 7. We use the same mask derived in Sec. 4.1 and convolve
the image with a Gaussian of $\sigma$ = 5 pix (1 pix = $0\farcs 33$) to
further enhance structures that might be buried in the noise of the outer
parts of the galaxy. The different surface brightness levels we explore are:
$26.0$ (darker teal shade), $27.0$, $28.0$ and $29.5$ mag/arcsec2 (lightest
teal shade).

From the contours in Fig. 7 we can confirm the results in Sec. 4.1: the outer
parts of NGC 1052–DF2 appear symmetric not showing any sign of tidal stripping
down to a surface brightness of 29.5 mag/arcsec2. This image supports what we
found for the surface brightness profiles in the previous section: a structure
compatible with a low inclination disk.

The limiting surface brightness of the image in the $g$-band (30.4
mag/arcsec2; 3$\sigma$ in $10\arcsec\times 10\arcsec$ boxes) can be used to
make a rough estimation (upper limit) of the amount of stellar mass that could
be hidden under our detection limit. The outer parts of the galaxies have a
color $g-r\sim 0.4$. At 30 mag/arcsec2, this is equivalent to 0.03 M⊙/pc2.
Therefore, if there were any stellar mass which such density in the radial
range between 80 to 100 arcsec (i.e. $\sim$9000 arcsec2), that will correspond
to a total stellar mass in that region of $\sim 1.1\times 10^{6}$ M⊙ (at 13
Mpc) or $\sim 2.5\times 10^{6}$ M⊙ (at 20 Mpc). That is, the amount of
potentially stripped stellar mass of DF2 is $\sim 2$% or less of its total
stellar mass.

###  4.3 Tidal radius of NGC 1052–DF2

Tidal forces are dependent on the gradient of a gravitational field and so
tidal effects are usually limited to the immediate surroundings of a galaxy.
The two most prominent galaxies in the FOV, and therefore the ones that could
produce tidal stripping in NGC 1052–DF2, are NGC 1052 and NGC 1042. van Dokkum
et al. (2018b) used the vicinity of NGC 1052–DF2 to NGC 1052 ($13\farcm 7$ in
projection) to support its distance of 20 Mpc. However, Monelli & Trujillo
(2019) finds two groups in the same line of sight, one at 13.5 Mpc, containing
NGC 1042, and one at $\sim 20$ Mpc, containing NGC 1052. They associated NGC
1052–DF2 to the former (NGC 1042), at a projected separation of $20\farcm 5$.

Even though the projected physical distance of NGC 1052–DF2 to both galaxies
would be equivalent to $80$ kpc, the total mass of NGC 1052
(M${}_{1052,dyn}(<40$ kpc) = $4.14\pm 0.71\times 10^{11}$M⊙, Forbes et al.,
2019) is more than 10 times larger than that of NGC 1042 (M1042,dyn($<8$
kpc)$=3.0^{+3.0}_{-1.5}\times 10^{10}$ M⊙, Trujillo et al. in prep.). In order
to estimate the dynamical mass of NGC 1042 to the same radial distance than
NGC 1052 (i.e., 40 kpc), we can assume that the rotational velocity of NGC
1042 is constant. Using this, then the dynamical mass is proportional to the
radius. Therefore, the dynamical mass of NGC 1042 (very likely an upper limit
due to our assumption) is M${}_{1042,dyn}(<40$ kpc$)=1.5^{+0.3}_{-0.2}\times
10^{11}$ M⊙. Given this mass difference between the two galaxies, we would
expect to see a different tidal effect on NGC 1052–DF2 depending on its
associated host.

To explore where we expect to see the tidal features for both NGC 1052 and NGC
1042 we calculated the tidal radius of NGC 1052–DF2 produced by each galaxy.
The tidal radius identifies the radius where the tidal effects are expected to
be important in the satellite and where we should be looking for tidal
distortions. To derive the tidal radius ($r_{t}$) for NGC 1052–DF2, we follow
equation 5 in Johnston et al. (2002) but for the instantaneous $r_{tidal}$;
that is the tidal radius at the current position of the satellite from the
parent galaxy assuming a circular orbit. This is given by:

$r_{tidal,inst}=D\times\left(\frac{m_{dyn}}{M_{host,dyn}}\right)^{1/3}$ (1)

Figure 8: Expected tidal regions on NGC 1052–DF2 produced by NGC 1042 (mint
green elliptical ring, bigger) and NGC 1052 (pink elliptical ring, smaller).
The background is a SDSS image of the FOV around NGC 1052–DF2. The nearby
massive galaxies, the elliptical NGC 1052 and the spiral NGC 1042, are
labelled. For ease of view, we show a zoom-in of the $240\arcsec\times
240\arcsec$ region around NGC 1052–DF2. The lack of tidal features around NGC
1052–DF2 does not favour the hypothesis that this UDG is physically bound to
NGC 1052.

Fig. 8 shows the expected tidal radii derived for NGC 1052 (in pink) and for
NGC 1042 (in mint green) as elliptical rings. The elliptical rings encompass
the range of radius from the uncertainties in the total masses of the
galaxies. It is evident from the Figure that the difference in mass between
the two potential hosts will have a different impact in the satellite. The
tidal radius produced by NGC 1042 is $r_{t}^{1042}$ = $148_{-48}^{+39}$
arcsec. On the other hand, the tidal radius if the UDG was associated to NGC
1052 is $r_{t}^{1052}$ = $55_{-17}^{+14}$ arcsec, well within the observed
extension of the galaxy in this work.

##  5 Discussion

In this paper, we study in detail the first galaxy “lacking” DM, NGC 1052–DF2,
using ultra-deep images to assess if tidal stripping could explain its claimed
low content of DM. In M20, we explored tidal stripping as an alternative
scenario to explain the low DM content in NGC 1052–DF4. Our aim in this paper
was to repeat the same analysis for NGC 1052–DF2. However, the distribution of
GCs (Fig. 3), the surface brightness profiles (Fig. 4) and the contours (left
panel of Fig. 7) of this galaxy show no signs of interaction. The lack of
tidal features implies that currently there is no significant tidal stripping
in this galaxy.

###  5.1 Evidence of rotation in NGC 1052–DF2

The dynamical analyses of NGC 1052–DF2 have assumed that this galaxy is
pressure supported (e.g., van Dokkum et al., 2018b; Martin et al., 2018;
Laporte et al., 2019). However, if a significant rotation component is
present, it will influence the current dynamical mass estimate of this galaxy.

Dynamical studies of the stellar light of the innermost regions of this galaxy
have found contradictory results. The lack of rotation reported in Danieli et
al. (2019) is likely caused by the small FOV of the KCWI instrument ($20\farcs
4\times 16\farcs 5$) that coincides with the presence of the bulge-like
structure seen in the radial profiles of Fig. 4 at $R<35$ arcsec. In the other
hand, evidence for rotation of NGC 1052–DF2 has been presented in Emsellem et
al. (2019) and Lewis et al. (2020). Note that the rotation axis found in the
stellar component by Emsellem et al. (2019) is almost perpendicular to the
axis found in Lewis et al. (2020) using the GC kinematics. While it is not
clear the reason of this disagreement, we speculate that it might be caused by
the limited FOV of MUSE ($60\arcsec\times 60\arcsec$) imaging only one side of
the galaxy compared with the larger radial distances probed with the GCs888We
suggest that the MUSE results should be revisited taking into account the PA
of the external parts of the galaxy found in this work..

The profiles in Fig. 4 show that at $R>35$ arcsec the galaxy presents a
component that is best described with an exponential. This suggests the
presence of a disk in NGC 1052–DF2 that only shows up in deep images (see also
Müller et al., 2019). In the right panel of Fig. 7, we show the inner
$240\arcsec\times 240\arcsec$ region of NGC 1052–DF2 with the rotation map
from Lewis et al. (2020) overplotted. The rotation map obtained from re-
analyzing the dynamics of the GC system is in nice agreement with the disk
structure at large radius. We indicated with the mint line the PA of the outer
parts ($R>35$ arcsec) of the galaxy derived from the ellipse fits in Sec. 4.1.
This stellar PA seems to be perpendicular to the projected rotation axis of
the rotation map. In order to assess that, we have also plotted the PA of the
rotation map and its errors as reported in Lewis et al. (2020) (pink area).
Both angles agree within the errors.

The presence of a disk-like structure in the outer parts of the galaxy
provides further evidence of the rotation of NGC 1052–DF2. We used eq. 9 in
Lewis et al. (2020) to calculate a rough estimate of the dynamical mass of the
galaxy considering the rotation component measured in that work. Using the
axis ratio of the disk measured with ellipse in Sec. 4.1, we can derive a
rough estimate of the inclination angle $i$ of this galaxy using $b/a=cos(i)$
(here we do not consider any thickness for the disk). That gives us an
inclination angle of $34^{+6}_{-7}$ degrees. The rotation parameters were
taken from the analysis in Lewis et al. (2020):
$v_{rot}=12.44^{+4.40}_{-5.16}$ km/s and $\sigma=3.88^{+4.42}_{-2.72}$ km/s.

Under the assumption that the galaxy is at 20 Mpc, we found that the dynamical
mass enclosed by the GCs is M($<7.6$ kpc) = $9.0^{+17.1}_{-6.8}\times 10^{8}$
M⊙. Assuming a stellar mass of $\sim 2\times 10^{8}$ M⊙, this corresponds to
Mdyn/M∗ $=4.5^{+8.6}_{-3.4}$. If a distance of 13.5 Mpc is assumed (Trujillo
et al., 2019; Zonoozi et al., 2021), then the estimates are : M($<5.1$ kpc) =
$6.1^{+11.5}_{-4.5}\times 10^{8}$ M⊙ and Mdyn/M∗ $=12.1^{+23.0}_{-9.1}$,
assuming a stellar mass of $5\times 10^{7}$ M⊙ (Trujillo et al., 2019). The
dynamical mass derived here is a factor of 2.6 larger than that derived in van
Dokkum et al. (2018b) for a distance of 20 Mpc, and a factor of 1.5 larger
than in Trujillo et al. (2019) at 13.5 Mpc.

As discussed in Lewis et al. (2020), the presence of rotation in this galaxy
evidenced also by the presence of a disk component at large radius, affects
the inferred M/L ratio bringing the DM content of NGC 1052–DF2 more in line
with galaxies of similar stellar masses (e.g., Mancera Piña et al., 2019;
Müller et al., 2020). Furthermore, a closer distance of $\sim 13.5$ Mpc would
make even more room for DM in this galaxy, and far from “lacking” dark matter.

###  5.2 The tidal radius of NGC 1052–DF2 and its distance

In Sec. 4.3, we discuss the estimated tidal radii of NGC 1052–DF2 depending on
the two potential host galaxies: NGC 1042 at 13.5 Mpc (Trujillo et al., 2019)
or NGC 1052 at 20 Mpc (van Dokkum et al., 2018b). Both radii were visually
represented in Fig. 8.

Our results suggest that if NGC 1052–DF2 was gravitationally bound to NGC
1052, we would have already seen tidal features around this UDG. However, the
lack of any sign of tidal stripping is not in contradiction with a potential
association of NGC 1052–DF2 with NGC 1042 and, therefore, it is consistent
with the distance of $13.5$ Mpc found in Trujillo et al. (2019) and Monelli &
Trujillo (2019). One can argue that we are observing the projected distance
between NGC 1052–DF2 and its host, which would be the minimum distance between
both galaxies. The fact that we are not detecting any feature within a radius
of $80$ arcsec means that this would be the minimum tidal radius. That is a
factor of $\sim 2$ larger than the $r_{t}^{1052}$ calculated in Sec. 4.3 and
therefore we can infer that the distance between both galaxies has to be
larger than $\sim 168$ kpc. The virial radius of NGC 1052 is $390$ kpc (Forbes
et al., 2019), and consequently, we cannot fully reject the scenario where NGC
1052–DF2 and NGC 1052 are associated only on the basis of the absence of tidal
features. However, the lack of tidal distortion is indicative that if they are
associated, NGC 1052–DF2 is in its initial infall into the group. If this was
not the case, NGC 1052–DF2 would have already experienced the tidal forces of
the massive galaxy or even been disrupted. This is in agreement with Forbes et
al. (2019). The idea that the absence of tidal features indicates an initial
infall of the UDG also holds if we use the dynamical mass derived in the
previous section (Sec. 5.1). A higher dynamical mass (and therefore a larger
amount of dark matter) will “shield” the UDG from the tidal forces produced by
NGC 1052. The new tidal radius using the new dynamical mass estimate would be
$(2.6)^{1/3}\times r_{t}^{1052}=75^{+19}_{-24}$ arcsec, at the limit of our
ultra-deep imaging.

##  6 Conclusions

The existence of long-lived ($\sim$9 Gyr) and stable (without signatures of
tidal distortion) galaxies “lacking” DM present a challenge in our
understanding of galaxy formation. This is especially problematic if the
galaxy inhabits a dense environment. Here, we explore the tidal stripping
scenario to explain the claimed low content of DM in the first of this kind of
galaxies: NGC 1052–DF2. We use ultra-deep data to image the outer parts of the
galaxy down to $R\sim 80$ arcsec (i.e., three times larger than previous
imaging of this galaxy) finding no signs of tidal features. However, we see a
disk-like structure from $35$ to $80$ arcsec in agreement with the rotation
previously reported for this galaxy. The dynamical mass derived taking into
account the rotation component is a factor 2.6 larger than in van Dokkum et
al. (2018b) and 1.5 larger than the one measured in Trujillo et al. (2019).
This translates into a dark matter vs. stellar mass ratio of
$4.5^{+8.6}_{-3.4}$ at $20$ Mpc ($R<7.6$ kpc) or $12.1^{+23.0}_{-9.1}$ at
$13.5$ Mpc ($R<5.1$ kpc), bringing the DM content of this galaxy more in line
with other galaxies of similar stellar masses ( $\gtrsim 10$, Mancera Piña et
al., 2019; Müller et al., 2020).

We thank the referee for their useful comments that helped improve the
original manuscript. We would like to thank Javier Román for his useful
comments on the paper and his reduction of the INT images. We also thank
Geraint Lewis for providing the rotation map of NGC 1052–DF2 and helpful
comments about the rotation of this system. We acknowledge support from the
grant PID2019-107427GB-C32 from the Spanish Ministry of Science and
Innovation. We acknowledge financial support from the European Union’s Horizon
2020 research and innovation programme under Marie Skłodowska-Curie grant
agreement No. 721463 to the SUNDIAL ITN network, and the European Regional
Development Fund (FEDER), from IAC project P/300624, financed by the Spanish
Ministry of Science, Innovation and Universities (MCIU), through the State
Budget and by the Canary Islands Department of Economy, Knowledge and
Employment, through the Regional Budget of the Autonomous Community. A. B. was
supported by an appointment to the NASA Postdoctoral Program at the NASA Ames
Research Center, administered by Universities Space Research Association under
contract with NASA. Based on observations made with the GTC telescope, in the
Spanish Observatorio del Roque de los Muchachos of the Instituto de
Astrofísica de Canarias, under Director’s Discretionary Time. Based on
observations made with the Isaac Newton Telescope, in the Spanish Observatorio
del Roque de los Muchachos of the Instituto de Astrofísica de Canarias.

##  Appendix A Masking of NGC 1052–DF2

Sec. 4.1 describes the procedure to create the mask that is used to derive the
surface brightness profiles. Fig. 9 shows this mask applied to a composite of
an RGB HiPERCAM images and a black and white background from the $g$-band INT
image.

Figure 9: Mask (teal regions) applied to a HiPERCAM color composite of a
postage stamp of a region $140\arcsec\times 240\arcsec$ around NGC 1052–DF2.
The black and white background is the $g$-band INT image. This shows the
thorough masking necessary in these deep images.

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