# A Tip of the Red Giant Branch Distance of $22.1\pm 1.2$ Mpc to the Dark
Matter Deficient Galaxy NGC 1052–DF2 from 40 Orbits of Hubble Space Telescope
Imaging

Zili Shen Astronomy Department, Yale University, 52 Hillhouse Ave, New Haven,
CT 06511, USA  Shany Danieli NASA Hubble Fellow  Astronomy Department, Yale
University, 52 Hillhouse Ave, New Haven, CT 06511, USA  Department of Physics,
Yale University, New Haven, CT 06520, USA  Yale Center for Astronomy and
Astrophysics, Yale University, New Haven, CT 06511, USA  Institute for
Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA  Pieter van Dokkum
Astronomy Department, Yale University, 52 Hillhouse Ave, New Haven, CT 06511,
USA  Roberto Abraham David A. Dunlap Department of Astronomy and Astrophysics,
University of Toronto, Toronto ON, M5S 3H4, Canada  Dunlap Institute for
Astronomy and Astrophysics, University of Toronto, Toronto ON, M5S 3H4, Canada
Jean P. Brodie University of California Observatories, 1156 High Street, Santa
Cruz, CA 95064, USA  Charlie Conroy Harvard-Smithsonian Center for
Astrophysics, 60 Garden Street, Cambridge, MA, USA  Andrew E. Dolphin Raytheon
Technologies, 1151 E. Hermans Road, Tucson, AZ 85756, USA  Steward
Observatory, University of Arizona, 933 North Cherry Avenue, Tucson, AZ
85721-0065 USA  Aaron J. Romanowsky Department of Physics and Astronomy, San
Jose State University, San Jose, CA 95192, USA  University of California
Observatories, 1156 High Street, Santa Cruz, CA 95064, USA  J. M. Diederik
Kruijssen Astronomisches Rechen-Institut, Zentrum für Astronomie der
Universität Heidelberg, Mönchhofstraße 12-14, D-69120 Heidelberg, Germany
Dhruba Dutta Chowdhury Astronomy Department, Yale University, 52 Hillhouse
Ave, New Haven, CT 06511, USA

(Received March 17, 2021; Revised April 7, 2021)

###### Abstract

The large and diffuse galaxies NGC 1052–DF2 and NGC 1052–DF4 have been found
to have very low dark matter content and a population of luminous globular
clusters. Accurate distance measurements are key to interpreting these
observations. Recently, the distance to NGC 1052–DF4 was found to be $20.0\pm
1.6$ Mpc by identifying the tip of the red giant branch (TRGB) in 12 orbits of
Hubble Space Telescope (HST) Advanced Camera for Surveys (ACS) imaging. Here
we present 40 orbits of HST ACS data for NGC 1052–DF2 and use these data to
measure its TRGB. The TRGB is readily apparent in the color-magnitude diagram.
Using a forward model that incorporates photometric uncertainties, we find a
TRGB magnitude of $m_{\rm F814W,TRGB}=27.67\pm 0.10$ mag. The inferred
distance is $D_{\rm TRGB}=22.1\pm 1.2$ Mpc, consistent with the to the bright
elliptical galaxy NGC 1052 . The new HST distance rules out the idea that some
of NGC 1052–DF2’s unusual properties can be explained if it were at $\sim 13$
Mpc; instead, it implies that the galaxy’s globular clusters are even more
luminous than had been derived using the previous distance . The distance from
NGC 1052–DF2 to NGC 1052–DF4 is well-determined at $2.1\pm 0.5$ Mpc,
significantly larger than the virial diameter of NGC 1052. We discuss the
implications for formation scenarios of the galaxies and for the external
field effect, which has been invoked to explain the intrinsic dynamics of
these objects in the context of modified Newtonian dynamics.

photometry — galaxies: distances and redshift — cosmology: dark matter —
galaxies: individual(NGC 1052–DF2)

††journal: ApJL††facilities: HST(ACS)††software: astropy (Price-Whelan et al.,
2018), Dolphot (Dolphin, 2000), Tiny Tim (Krist, 1995)

##  1 Introduction

First identified by Karachentsev et al. (2000), NGC 1052–DF2 is a galaxy in
one of the fields of the Dragonfly Nearby Galaxies Survey (Merritt et al.,
2016) that was found to be unusual in follow-up observations (van Dokkum et
al., 2018b; Cohen et al., 2018). At the distance of the bright elliptical
galaxy NGC 1052 (D = $19.4-21.4$ Mpc; Blakeslee et al., 2001; Tonry et al.,
2001; Tully et al., 2013), NGC 1052–DF2 has the stellar mass ($\sim 2\times
10^{8}M_{\odot}$) and metallicity of typical dwarf galaxies (Fensch et al.,
2019) but a large size ($R_{\rm e}=2.2$ kpc), placing it in the regime of
ultra-diffuse galaxies (van Dokkum et al., 2015). Furthermore, it has a large
population of luminous globular clusters (GCs) and a very low velocity
dispersion of $8.5^{+2.3}_{-3.1}$ km s-1 (see van Dokkum et al., 2018b;
Danieli et al., 2019; Emsellem et al., 2019; Shen et al., 2020). In 2019, a
second galaxy in the same field, NGC 1052–DF4, was found to have properties
almost identical to those of NGC 1052–DF2 (van Dokkum et al., 2019; Danieli et
al., 2020; Shen et al., 2020). The velocity dispersions of both galaxies
($4-10$ km s-1; van Dokkum et al., 2018b, 2019; Danieli et al., 2019; Emsellem
et al., 2019) are consistent with expectations from the stellar mass alone
($\approx 7$ km s-1), suggesting that the galaxies have no or very little dark
matter (van Dokkum et al., 2018b). van Dokkum et al. (2018b) claimed that NGC
1052–DF2 invalidates alternative gravity theories such as Modified Newtonian
Dynamics (MOND; Milgrom, 1983), as the (apparent) effects of dark matter
should be observed in all galaxies in these models.

The unusual nature of NGC 1052–DF2 and NGC 1052–DF4 sparked a vigorous debate
in the community. Whereas initial concerns focused on the velocity dispersion
measurements (see, e.g., Martin et al., 2018), more recently the distance to
the galaxies has become a point of contention. Their radial velocities of
$1803$ km s-1 and $1445$ km s-1, their projected proximity to the elliptical
galaxy NGC 1052, and surface brightness fluctuations in 1+1 orbit F814W and
F606W HST images all indicate a distance of $\approx 20$ Mpc for the galaxies
(van Dokkum et al., 2018b; Blakeslee & Cantiello, 2018; Cohen et al., 2018;
van Dokkum et al., 2018a). However, Trujillo et al. (2019) and Monelli &
Trujillo (2019) derived distances of 13–14 Mpc to both NGC 1052–DF2 and NGC
1052–DF4 from the same 1+1 orbit HST images, claiming that individual red
giant stars were detected and associating a sharp increase in the number of
detections around $m_{F814W}\approx 26.5$ mag to be the tip of the red giant
branch (TRGB). A distance of 13 Mpc is one way to resolve or alleviate the
unusual properties of NGC 1052–DF2 and NGC 1052–DF4: for a smaller distance,
the galaxies are no longer “ultra-diffuse”, the GCs are smaller and less
luminous, and the dark matter fraction increases.

Accurate distances to both galaxies are important for deriving accurate
luminosities, sizes, and masses – of both the galaxies themselves and their
GCs – and are also key to placing constraints on alternative gravity models
such as MOND. As pointed out by Kroupa et al. (2018), Müller et al. (2019a),
Haghi et al. (2019), and Famaey et al. (2018), van Dokkum et al. (2018b) had
neglected an essential element of MOND, namely the external field effect
(EFE). The EFE, unique to MOND, causes a low mass galaxy in orbit around a
massive galaxy to have a lower velocity dispersion than the same object in
isolation. Thus, the low velocity dispersions of NGC 1052–DF2 and NGC 1052–DF4
may be consistent with the expectations from MOND if both galaxies are in
close proximity to NGC 1052 (Kroupa et al., 2018).

Accurate distances can be obtained using HST data that are much deeper than
the 1+1 orbits in the Cohen et al. (2018) study, as the TRGB measurement is
sensitive to the depth and quality cuts used in the analysis. In a single
F814W orbit, individual red giants can only be detected out to $\approx 14$
Mpc, right where Trujillo et al. (2019) and Monelli & Trujillo (2019) claimed
to see the onset of the giant branch. With $\approx 10$ orbits, the TRGB can
be detected out to 20 Mpc. Danieli et al. (2020) obtained the needed data for
NGC 1052–DF4, and using a total of 8 orbits in F814W and 4 orbits in F606W
measured a TRGB distance to that galaxy of $20.0\pm 1.6$ Mpc. Here, we present
even deeper HST imaging (20+20 orbits in F814W and F606W) for NGC 1052–DF2. We
unambiguously identify the TRGB and derive a distance to the galaxy that
should be definitive. Furthermore, we derive an accurate and (as it turns out)
interesting relative distance between NGC 1052–DF2 and NGC 1052–DF4, making
use of the fact that nearly all systematic uncertainties cancel in the
comparison. This relative distance provides important context for the presence
of tidal features in the outskirts of the galaxies (Montes et al., 2020) and
constrains the EFE. Vega magnitudes are used throughout this paper.

##  2 HST/ACS Data

Figure 1: Deep image of NGC 1052–DF2 obtained with HST/ACS. The total exposure
time is 41840s in F814W and 40560s in F606W, 20 orbits each. The length of
each arrow is 5 arcsec for scale ($\sim 0.5$ kpc at the distance obtained in
this work). The shows resolved red giant stars in a background of unresolved
blue stars.

NGC 1052–DF2 was observed with HST ACS in Cycle 27 (program 15851), split in
three epochs from June 2020 to September 2020. We obtained 19 orbits each in
the F814W and F606W filters. Adding these data to the previous 1 orbit each in
the same filters obtained in 2017 (program 14644), the total exposure time is
41840s in F814W and 40560s in F606W. NGC 1052–DF2 was placed near the center
of the field of view, varying the exact location between visits to ensure
relatively uniform coverage despite the chip gap.

Within each orbit, four exposures were taken in a four-point dither pattern to
remove cosmic rays, identify hot pixels, and dither over the chip gap. The
STScI standard pipeline was used to perform bias and dark current subtraction,
flat-fielding, and CTE correction on individual exposures and produces
calibrated flc files. We used the TweakReg utility in DrizzlePac to align the
160 flc files. We used AstroDrizzle to remove astrometric distortion, correct
sky background, and flag cosmic-rays in the aligned images, resulting in a
combined (drc) image in the F814W and F606W filters.

Figure 1 shows a color image of NGC 1052–DF2 generated from the drizzled
images. In these deep data, the galaxy is well-resolved. The yellow points are
resolved red giants which appear among a background of bluer unresolved
subgiants and main sequence stars. We do not confirm the “Maybe Stream”
proposed in Abraham et al. (2018). The clumps of luminous giants that make up
the apparent feature are detected but we do not find an accompanying excess of
fainter stars that would be present if it were a tidal feature. We conclude
that the apparent feature is a chance alignment of background galaxies and
Poisson fluctuations in the number of bright stars.

##  3 Photometry

The software package DOLPHOT111http://americano.dolphinsim.com/dolphot/
(Dolphin, 2000) was used to carry out photometry on individual flc files from
the STScI ACS pipeline. Before running the photometry detection step, all 160
flc files as well as the deepest drizzled image (which combines 80 stacked
F814W flc images with a total exposure time of 41840s) were ran through
several preparation steps using the DOLPHOT/ACS pre-processing tools. We used
the acsmask routine to mask bad columns and hot pixels, the splitgroups
routine to split each flc into two chips, and the calcsky routine to determine
the background value in each image.

DOLPHOT analyzed all 160 flc images simultaneously, using the deep combined
drc image as reference. Stars were detected in each image by fitting Tiny Tim
PSFs (Krist, 1995) and the fluxes were measured separately in the F814W and
F606W frames. Although this approach is memory-intensive for a deep data set
like ours, the fact that no resampling is required has the advantages of
preserving the noise properties and optimally using the information in the flc
files. The key DOLPHOT parameters were: the sky fitting parameter FitSky=2,
the aperture radius RAper=3, and Force1=1 which forces the sources to be
fitted as stars. These parameters were identical to the values used for the
TRGB analysis on NGC 1052–DF4 (Danieli et al., 2020), and we refer to that
paper for further information.

Stars were selected from the raw DOLPHOT output catalog with a set of quality
cuts. The photometry catalog was corrected for Galactic extinction: 0.04 mag
in F814W and 0.06 mag in F606W (Schlafly & Finkbeiner, 2011). Objects were
considered to be reliable stars only if they pass all of the following
criteria: signal-to-noise ratio $>4$ in F814W and $>3$ in F606W; object-type
$\leq 2$, which corresponds to good star or faint star); sharpness parameter
$|\text{sharp}_{F814W}|\leq 0.5$ and $|\text{sharp}_{F606W}|\leq 0.5$;
crowding parameter $|\text{crowd}_{F814W}|\leq 0.5$ and
$|\text{crowd}_{F606W}|\leq 0.5$. Compared to the criteria in Danieli et al.
(2020), we increased the signal-to-noise threshold in F606W to match our
deeper data.

Characterization of the systematic and random uncertainties in the photometry
is an important component of the analysis, as they have an impact on the
apparent location of the TRGB (see, e.g. Makarov et al., 2006). In DOLPHOT,
stellar PSFs (using models appropriate for their individual $x,y$ positions)
can be placed in the HST data, and then analyzed alongside the actual stars.
The scatter and systematic offset in the magnitudes of injected and recovered
stars can then be taken into account in the measurement of the TRGB location.
We ran DOLPHOT photometry on 180,000 artificial stars. The stars uniformly
sampled the magnitude range $25<F814W<29$ and the color range
$1.0<F606W-F814W<1.5$, and were distributed between $R_{\text{eff}}$ $(\approx
20\arcsec)$ and $4R_{\text{eff}}$. The artificial stars were injected into
individual flc images and analyzed with the same methodology as the real
stars.

Figure 2 shows examples of stellar detections in two regions of NGC 1052–DF2.
There is a sharp increase in the number of sources fainter than 27.5 mag,
$\sim 3$ times more sources detected with $27.5<m_{\rm F814W}<28.0$ than with
$27.0<m_{\rm F814W}<27.5$. The sudden increase at 27.5 mag is especially
visible in the outer regions of the galaxy, where stellar density is lower and
crowding is less severe.

Figure 2: Stars detected after quality cuts in NGC 1052–DF2, shown in small
regions in the ) and ). RGB stars with $27.5<m_{\text{F814W}}<28.0$ are
circled in orange and AGB stars with $27.0<m_{\text{F814W}}<27.5$ in green.
The number of sources brighter than 27.5 mag decrease visibly on the image.
Blended objects (especially in the center) do not pass the quality cuts. Stars
in the central part of the galaxy ($R<R_{\text{eff}}$) are excluded from the
analysis.

##  4 TRGB Distance

###  4.1 Color-magnitude Diagrams

Figure 3: TRGB edge detection for NGC 1052–DF2. Top panels: the
color–magnitude diagram (left), the binned and smoothed F814W luminosity
function (middle), and its response function to a sobel kernel (right) are
shown for stars located at $R_{\text{eff}}<R<4R_{\text{eff}}$. Black points
are stars that pass quality cuts, and the luminosity function is calculated
for stars that pass an additional color cut to select the RGB (the diagonal
dashes lines). Crosses indicate photometric scatter. measured by fitting a
Gaussian, $m_{\text{TRGB,F814W}}=27.52\pm 0.17$ mag, while the dotted orange
line marks the TRGB for 13 Mpc (Trujillo et al., 2019). Bottom panels: the
color-magnitude diagrams of stars in radial bins. The dashed line indicates
the respective TRGB measurement.

Color–magnitude diagrams (CMDs) were constructed for different radial regions
in NGC 1052–DF2 and shown in Figure 3. The main panel and the bottom panels of
Figure 3 all show the characteristic sharp decrease in the number of stars
around $m_{F814W}\sim 27.5$ mag, indicating the onset of the red giant branch
(RGB)222The photometry catalog is available at
https://github.com/zilishen/NGC1052-DF2-public.. The F814W luminosity is shown
in the middle panel, as measured within the following color range:
$2<(F814W-28)+2.3\times(F606W-F814W)<3.2$. This luminosity function is fitted
in § 4.2 and § 4.3 to identify the location of the TRGB, which is then used to
derive the distance.

###  4.2 Edge Detection

The first method, edge detection, measures the TRGB from the first–derivative
of the binned and smoothed luminosity function of the RGB and asymptotic giant
branch (AGB) stars. This method was previously used for NGC 1052–DF4 (Danieli
et al., 2020) and numerous other galaxies (e.g., Lee et al., 1993). Advantages
of the method include that it is straightforward to compare results to other
studies and other galaxies, and that it provides a robust measurement of the
approximate location of the TRGB. We caution, however, that it does not take
measurement uncertainties into account.333In practice, the systematic bias
(see § 4.3) causes the TRGB to appear fainter than the true value, whereas
photometric scatter causes it to appear brighter. The F814W-band luminosity
function was smoothed with a Gaussian kernel with a standard deviation of 0.4
mag and then filtered with a Sobel kernel. The response to the Sobel kernel
peaks where the slope of the luminosity function reaches a maximum, and this
is the location of the TRGB. The peak location was identified by performing a
Gaussian fit to the response function to determine its center and width.

The TRGB measurement we obtained with the edge-detection method is shown in
Figure 3. The main result came from a sample in the radial region
$R_{\text{eff}}<R<4R_{\text{eff}}$ (upper left panel). We measure an
extinction–corrected $F814W$ TRGB magnitude of $m_{\text{TRGB,F814W}}=27.51\pm
0.22$ mag for NGC 1052–DF2, where the uncertainty is the width of the Gaussian
fit to the peak. The CMD in each radial bin from $R_{\text{eff}}$ to
$4R_{\text{eff}}$ and the corresponding measured TRGB magnitude is shown in
the bottom row of Figure 3. The measurement is consistent within error bars
across all radial bins, even in the innermost region where crowding is most
severe and many initial source detections are rejected by the quality cuts. We
infer that the TRGB is located near $m_{\rm F814W}\sim 27.5$, and not near
$m_{\rm F814W}\sim 26.5$ as was found by Trujillo et al. (2019).

###  4.3 Forward Modeling

Figure 4: TRGB forward modeling results for NGC 1052–DF2. Top left: the F814W
luminosity function of the intrinsic model without contamination (grey) and
with contamination added (red). Top right: the systematic offset (bias) and
the scatter in the recovered magnitudes using the artificial star test Main
panel: the best-fit model (red) of the observed F814W luminosity function
(black), with a TRGB magnitude of $m_{\text{TRGB,F814W}}=27.67$ . The reduced
$\chi^{2}$ value for this fit is 1.8. For comparison, the observed luminosity
function of NGC 1052–DF4 (Danieli et al., 2020) is scaled and plotted in the
dashed yellow line.

The second method to measure the TRGB is fitting the luminosity functions of
the RGB and AGB. We derived a precise TRGB location with the same forward
modeling procedure as Danieli et al. (2020). Briefly summarized, this approach
generates an intrinsic luminosity function of a stellar population by drawing
stars from an isochrone according to a initial mass function, then perturbs
the photometry of individual stars according to the (magnitude-dependent)
photometric errors to obtain an observed luminosity function, and finally adds
contaminating stars according to a simple linear distribution in magnitude. As
the of the is not very sensitive to age or metallicity, the results are not
sensitive to the details of the stellar population synthesis model; following
Danieli et al. (2020) we used an old (10 Gyr), metal-poor ([Fe/H] $=-1$)
stellar population, The two parameters, the TRGB magnitude and the
contamination, were varied to search for a best fit to the F814W luminosity
function.

The photometric errors were characterized using artificial star tests (see§
3). Compared to the input magnitudes of injected artificial stars, the
recovered magnitudes show a systematic offset (“bias”), scatter, and
incompleteness. These results are shown in the right panels of Figure 4. The
bias and scatter increase towards fainter magnitudes and completeness
decreases as expected for fainter stars. At $m_{\text{F814W}}=27.67$, the
recovered magnitudes are 0.21 mag fainter than the input magnitudes, with a
$1\sigma$ scatter of 0.13 mag, and the completeness fraction is 80%. Compared
to the photometric errors of Danieli et al. (2020), our deeper data show a
similar level of bias and smaller scatter. We find a $\sim 0.5$ mag fainter
50%-completeness limit, as expected from the increase from 8 orbits to 20
orbits in the F814W filter.

Starting from $10^{9}$ stars drawn from a MIST isochrone (Dotter, 2016; Choi
et al., 2016), the magnitude of each star was shifted by the parameterized
bias and perturbed by a random sample of the scatter. Possible contamination
was parametrized by a linear function

$N_{contam}=20c\times(m-26)$ (1)

for stars fainter than 26 mag, and no contamination for stars brighter than 26
mag. After adding the contamination, the luminosity function was multiplied by
the completeness fraction.

This model was fitted to the stars that satisfy the quality and color cuts and
are located between $R_{\text{eff}}$ and $4R_{\text{eff}}$ from the center.
The observed F814W luminosity function had 50 magnitude bins from 25.50 to
27.85 mag, and the best fit was found by minimizing $\chi^{2}$ while varying
$m_{TRGB}$ and $c$. We obtained a best-fit TRGB magnitude of $27.67\pm 0.02$,
where the error is the statistical uncertainty that encompasses 68% of the
marginal likelihood. The reduced $\chi^{2}$ of the best fit was 1.8. The best-
fit model is shown in the main panel of Figure 4. The fit is excellent, as it
reproduces not only the overall increase in the number of stars at faint
magnitudes but also several of the individual small “steps” in the luminosity
function. To our knowledge, our previous paper on NGC 1052–DF4 was the first
application of this method; the fact that it is able to fit small scale
features in the luminosity function may warrant wider application of this
technique.

The TRGB magnitude derived with forward modeling is fainter than that
determined from edge detection. This difference of $0.15$ mag, previously seen
in Danieli et al. (2020), is due to the effect of photometric scatter on the
apparent location of the TRGB. Many more stars scatter from faint to bright
magnitudes than the other way, due to the steep increase in the luminosity
function around the TRGB. We adopt the forward modeling value as the best
measurement but add 0.1 mag systematic uncertainty 50 % of the 0.20 mag bias
at the location of the best-fit TRGB magnitude. The final result is therefore
$m_{\rm TRGB}=27.67\pm 0.02$ (random) $\pm 0.10$ (systematic).

From the TRGB measurement, we derive the distance to NGC 1052–DF2. The zero-
point calibration of the absolute magnitude of the TRGB depends on the color
(Rizzi et al., 2007):

$M_{F814W}^{ACS}=-4.06+0.20[(F606W-F814W)-1.23]$ (2)

For a TRGB median color of 1.25 mag, as measured from the color-magnitude
diagram, the absolute magnitude of the TRGB is $M_{F814W}=-4.056$ mag, with a
standard uncertainty of 0.07 mag (McQuinn et al., 2017).

The errors on the final distance measurement have three components: the
statistical uncertainty from the fitting procedure (0.02 mag), the systematic
uncertainty from the artificial star tests (bias of 0.1 mag), and finally the
zero-point calibration uncertainty of 0.07 mag. Adding these in quadrature
leads to a final distance modulus of $31.72\pm 0.12$ mag and a distance of
$D=22.1\pm 0.2$ (statistical) $\pm 1.02$ (systematic) $\pm 0.7$ (calibration)
$=22.1\pm 1.2$ Mpc.

##  5 Discussion

Figure 5: Top panel: Dragonfly r-band image of the NGC 1052 field. Bottom
panels: Four possible arrangements of NGC 1052–DF2, NGC 1052–DF4, NGC 1052,
and NGC 1035. The fiducial panel shows the measured radial and projected
distances of the four galaxies. The radial distances of NGC 1052–DF2 and NGC
1052–DF4 in other panels move within 1.5$\sigma$ error. The second panel shows
that NGC 1052–DF2 could be within range of EFE in MOND, but NGC 1052–DF4 would
be outside due to their relative distance of $2.1$ Mpc. The third panel is
most consistent with MOND, where NGC 1052–DF2 and NGC 1052–DF4 are each within
the EFE radius of a massive galaxy, but NGC 1035 is 1.3$\sigma$ away from its
fiducial distance. The rightmost panel shows a scenario where NGC 1052–DF2 and
NGC 1052–DF4 are equidistant from NGC 1052, possibly resulting from a high-
speed encounter.

In this study, we used extremely deep HST/ACS data to measure the TRGB
distance to NGC 1052–DF2. Our best fit TRGB distance is $22.1\pm 1.2$ Mpc.
This distance to NGC 1052–DF2 is just consistent with, but slightly larger
than, two previous independent measurements of surface brightness fluctuations
(SBF): $D_{\rm SBF}=18.7\pm 1.7$ Mpc (van Dokkum et al., 2018a) and $D_{\rm
SBF}=20.4\pm 2.0$ Mpc (Blakeslee & Cantiello, 2018). The TRGB distance should
be more reliable than the SBF distances, as SBF absolute magnitudes are more
model-dependent (see, e.g., Greco et al., 2021). The new distance measurement
to NGC 1052–DF2 is consistent with the distance to the elliptical galaxy NGC
1052 within the uncertainties (19.4 – 21.4 Mpc; Blakeslee et al., 2001; Tonry
et al., 2001; Tully et al., 2013), . The new HST data rule out a distance to
the galaxy of 13.6 Mpc, which had been proposed by Trujillo et al. (2019). The
original single orbit HST data were not deep enough to identify individual RGB
stars at a distance of 20 Mpc, and as shown in van Dokkum et al. (2018a), the
apparent feature at $m_{\rm F814W}\approx 26.5$ in those data can be
reproduced by a combination of blends and AGB stars.

Our new distance confirms the unusual dark matter and GC properties of NGC
1052–DF2, and the slight upward revision of the distance to NGC 1052–DF2 makes
its luminous GCs even more spectacular. The previously reported luminosities
of the GCs in van Dokkum et al. (2018c) and Shen et al. (2020) assumed a
distance of 20 Mpc. With the revised distance, all of the GCs in NGC 1052–DF2
are 0.2 mag more luminous than previously reported. The GC luminosity function
now peaks at $M_{v}\approx-9.3$, even further from the canonical value of
$-7.5$ (see Rejkuba, 2012).

Interestingly, the TRGB distance to NGC 1052–DF2 is 2.1 Mpc larger than the
TRGB distance to NGC 1052–DF4 (20.0 $\pm$ 1.6 Mpc, Danieli et al., 2020) and
this difference is statistically significant. Data for these two galaxies both
come from HST/ACS and are analyzed with the same methods, using the same
forward modeling assumptions and the same absolute magnitude calibration. The
only difference between the two measurements is the depth of the data: in the
luminosity function filter (F814W), there are 8 orbits for NGC 1052–DF4 and 20
orbits for NGC 1052–DF2. This translates into a difference in the amplitude of
the photometric errors of $\approx 0.05$ mag. We conservatively assume that
the uncertainty in the relative TRGB magnitudes of the two galaxies is the
quadratic sum of the forward modeling fit uncertainties and this 0.05 mag
difference. This leads to a distance difference of $\Delta D=2.1\pm 0.5$ Mpc.
Figure 5 displays a visual summary of these findings. The relative distance
from NGC 1052–DF2 to NGC 1052–DF4 is kept fixed in the bottom panels.

As shown in the bottom panels of Figure 5, the relative distance between NGC
1052–DF2 and NGC 1052–DF4 is significantly larger than the virial diameter of
NGC 1052, which means that at least one of the galaxies is not bound to the
group. The virial diameter is $780\pm 54$ kpc (Forbes et al., 2019), the error
comes from their reported velocity dispersion uncertainty. NGC 1052–DF2 is
perhaps the most likely candidate for being unbound, given its large radial
velocity compared to other galaxies in the group (van Dokkum et al., 2018b),
and the fact that its distance is only just consistent with that of NGC 1052.

Previously proposed formation scenarios for dark matter deficient galaxies
fall broadly into two categories: formation within the group and tidal
stripping. The first group of theories includes formation in the chaotic gas-
rich environment of the assembling central galaxy (van Dokkum et al., 2018b)
and formation in a QSO outflow from the central black hole in NGC 1052. In
particular, it has been proposed that a high-velocity collision between two
gas-rich dwarfs occurring in the protogroup environment could explain both the
unusual GCs and the lack of dark matter in these galaxies (Silk, 2019). The
large relative distance and the large relative velocity could be consistent
with such scenarios. The second category of scenarios involves a progenitor
falling into a Milky-Way sized halo, losing gas due to supernova feedback and
ram pressure and losing dark matter by tidal stripping (Nusser, 2020). Tidal
stripping of dark matter is possible if the progenitor is on a tightly bound
and quite radial orbit, has a cored density structure for the dark halo
(Ogiya, 2018), and if the accretion happened early on, $z>1.5$ . These
scenarios are – at least at face value – more difficult to reconcile with at
least one of the galaxies being no longer bound to the group.

The relative distance also places a new constraint on the interpretation of
NGC 1052–DF2 and NGC 1052–DF4 in the context of MOND (see § 1). van Dokkum et
al. (2018b) argued that the low velocity dispersion of NGC 1052–DF2 could
falsify MOND, but they neglected the external field effect (EFE). Kroupa et
al. (2018), Haghi et al. (2019), and Müller et al. (2019a) found that the
observed velocity dispersions of both NGC 1052–DF2 and NGC 1052–DF4 are
consistent with MOND if both galaxies are physically close ($<300$ kpc; Kroupa
et al., 2018) to a massive galaxy. The most obvious candidate for this massive
galaxy is NGC 1052 itself, but our relative distance measurement of $\Delta
D=2.1\pm 0.5$ Mpc rules out NGC 1052–DF2 and NGC 1052–DF4 both being within
300 kpc of NGC 1052. We cannot exclude that two different galaxies
independently produce the EFE, as NGC 1052–DF4 is close in projection to the
low luminosity spiral galaxy NGC1035 and it has been suggested that it is
interacting with it (Montes et al., 2020). The Tully-Fisher distance of
NGC1035 is consistent with it being in the foreground of the group ($15.6\pm
2.2$ Mpc, $v=1392$ km s-1; Kourkchi & Tully, 2017). The issue with this
scenario is that NGC 1052–DF2 and NGC 1052–DF4 would have formed completely
independently, one in a massive group and the other as a satellite of an
isolated low luminosity galaxy in the foreground, despite their near-identical
morphologies, kinematics, and extreme globular cluster populations. Moreover,
deep Dragonfly imaging does not confirm the bridge of stars between NGC
1052–DF4 and NGC1035 that was found by Montes et al. (2020) (M. Keim et al.,
in preparation).

Looking ahead, further constraints on the physical processes that led to the
formation of NGC 1052–DF2 and NGC 1052–DF4 can be obtained by combining the
new distance information with the GC luminosity functions (Leigh & Fragione,
2020; Trujillo-Gomez et al., 2020), GC dynamics (e.g. orbital decay timescale;
Nusser, 2018; Dutta Chowdhury et al., 2019), and tidal features (Müller et
al., 2019b).

We thank Stacy McGaugh, Pavel Kroupa, Oliver Müller, Hosein Haghi, and
Indranil Banik for insightful discussions on MOND and we thank John Blakeslee
for comments on SBF distances. Support from STScI grant HST GO-15851,14644,
and 15695 is gratefully acknowledged. ZS is supported by the Gruber Science
Fellowship. S.D. is supported by NASA through Hubble Fellowship grant HST-
HF2-51454.001-A awarded by the Space Telescope Science Institute, which is
operated by the Association of Universities for Research in Astronomy,
Incorporated, under NASA contract NAS5-26555. AJR is supported by National
Science Foundation grant AST-1616710 and as a Research Corporation for Science
Advancement Cottrell Scholar. J.M.D.K. gratefully acknowledges funding from
the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through
an Emmy Noether Research Group (grant number KR4801/1-1) and the DFG
Sachbeihilfe (grant number KR4801/2-1), as well as from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme via the ERC Starting Grant MUSTANG (grant agreement number 714907).

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