B Famaey

S Mcgaugh

M Milgrom

MOND and the dynamics of NGC1052-DF2

Keywords:

The dwarf galaxy NGC1052-DF2 has recently been identified as potentially lacking dark matter. If correct, this could be a challenge for MOND, which predicts that low surface brightness galaxies should evince large mass discrepancies. However, the correct prediction of MOND depends on both the internal field of the dwarf and the external field caused by its proximity to the giant elliptical NGC1052. Taking both into consideration under plausible assumptions, we find σ MOND = 13.4 +4.8 -3.7 km s -1 . This is only marginally higher than the claimed 90% upper limit on the velocity dispersion (σ < 10.5 km s -1 ), and compares well with the observed root mean square velocity dispersion (σ = 14.3 km s -1 ). We also discuss a few caveats on both the observational and theoretical side. On the theory side, the internal virialization time in this dwarf may be longer that the time scale of variation of the external field. On the observational side, the paucity of data and their large uncertainties call for further analysis of the velocity dispersion of NGC1052-DF2, to check whether it poses a challenge to MOND or is a success thereof.

INTRODUCTION

In a recent paper, van Dokkum et al. (2018) report the lineof-sight velocities of ten globular clusters around the dwarf galaxy NGC1052-DF2. They infer from these a surprisingly low velocity dispersion for such a galaxy: σ < 10.5 km s -1 at 90% confidence. They then deduce that the total mass within the outermost radius of 7.6 kpc would be < 3.4 × 10 8 M ⊙ . The stellar mass which van Dokkum et al. (2018) assume for this galaxy, M * ≈ 2 × 10 8 M ⊙ , is based on a stellar mass-to-light ratio of Υ * ≈ 2 M ⊙ /L ⊙ . This stellar mass is close to the dynamical mass, implying no need for dark matter. If all these deductions are correct, this would in principle contradict MOND.

MOND (Milgrom 1983;Famaey & McGaugh 2012;Milgrom 2014) is a paradigm predicting the dynamics of galaxies directly from their baryonic mass distribution. The postulate is that, for gravitational accelerations below a 0 ≈ 10 -10 m s -2 the actual gravitational attraction approaches (g N a 0 ) 1/2 where g N is the usual Newtonian gravitational attraction.

The successes of MOND are best known for galaxies that are isolated, axisymmetric, and rotationally supported (e.g., Sanders & McGaugh 2002;Gentile et al. 2011).

⋆ E-mail: benoit.famaey@astro.unistra.fr stacy.mcgaugh@case.edu moti.milgrom@weizmann.ac.il

In such systems, there is a clear and direct connection between the baryonic mass distribution and the rotation curve (McGaugh et al. 2016;Lelli et al. 2017). This empirical relation is indistinguishable from MOND (Li et al. 2018).

MOND also has a good track record of predictive success for pressure supported systems like NGC1052-DF2 (McGaugh & Milgrom 2013a,b;Pawlowski & McGaugh 2014;McGaugh 2016). The analysis in such cases is complicated by the same uncertainties as in Newtonian analyses, such as that in the stellar mass-to-light ratio and the unknown anisotropy in the velocity tracers. Unique to MOND is the external field effect (EFE, Milgrom 1983;Bekenstein & Milgrom 1984;Famaey & McGaugh 2012;Haghi et al. 2016;Hees et al. 2016). Because of the nonlinearity of MOND, the internal dynamics of a system can be affected by the external gravitational field in which it is immersed. When the external field dominates over the internal one, the amplitude of the MOND effect, and the corresponding amount of dark matter inferred, is reduced. Some interesting effects unique to MOND are related to the EFE, such as the prediction of asymmetric tidal streams of globular clusters (Thomas et al. 2018).

An essential consequence of the EFE in MOND is that the predicted velocity dispersion of a dwarf galaxy depends on its environment. An object in isolation is expected to have a higher velocity dispersion than the same object in orbit around a massive host. This difference is perceptible in pairs of photometrically indistinguishable dwarf satellites of Andromeda (McGaugh & Milgrom 2013b). Indeed, the EFE was essential to the correct a priori prediction (McGaugh & Milgrom 2013a) of the velocity dispersions of the dwarfs And XIX, And XXI, and And XXV. These cases are notable for their large scale lengths and low velocity dispersions (Collins et al. 2013) -properties that were surprising in the context of dark matter but are natural in MOND. A further example is provided by the recently discovered Milky Way satellite Crater 2 (Torrealba et al. 2016). McGaugh (2016) predicted that this object would have a velocity dispersion of 2.1 +0.9 -0.6 km s -1 , much lower than the nominal expectation in the context of dark matter. Caldwell et al. (2017) subsequently observed 2.7±0.3 km s -1 .

Here we take the external field of the host galaxy NGC1052 into account to predict the expected velocity dispersion of NGC1052-DF2 in MOND. In §2 we adopt a nominal mass-to-light ratio of Υ * = 2 M ⊙ /L ⊙ to obtain σ MOND ≈ 13.4 km s -1 . For a factor of two uncertainty in the V-band stellar mass-to-light ratio, the allowed range is 9.7 ≤ σ MOND ≤ 18.2 km s -1 . This range of variation follows simply from plausible uncertainty in the conversion of luminosity to stellar mass; other potentially relevant uncertainties are discussed further in §3. Brief conclusions are given in §4.

PREDICTING THE VELOCITY DISPERSION OF NGC1052-DFIN MOND

Isolated prediction

McGaugh & Milgrom (2013a) outlined the procedure by which the velocity dispersions of satellite galaxies like NGC1052-DF2 can be predicted. For a spherical, isotropic, isolated system,

σ iso = 4 81 GMa 0 1/4 ≃ Υ * 2 1/4
20 km s -1 for NGC1052-DF2.

(1) This is the same result obtained by van Dokkum et al. (2018) for the same assumptions. However, this is not the correct MOND prediction, as this dwarf is not isolated: the external field due to the giant host NGC1052 is not negligible.

Instantaneous external field effect prediction

The isolated MOND boost to Newtonian gravity is predicted to be observed only in systems where the absolute value of the gravity both internal, g, and external, g e (from a host galaxy, or large scale structure), is less than a 0 . If g e < g < a 0 then we have isolated MOND effects. If instead g < g e < a 0 , then the system is quasi-Newtonian. The usual Newtonian formula applies, but with a renormalized gravitational constant. When g e ≪ a 0 , the renormalizing factor is simply a 0 /g e . The velocity dispersion estimator is simple (McGaugh & Milgrom 2013a) when g ≫ g e (isolated) or g ≪ g e (EFE) in that it depends only on the dominant acceleration (g or g e ). When g ≈ g e , both must be taken into consideration. This turns out to be the case for NGC1052DF2.

To estimate these quantities, we make the same assumptions as van Dokkum et al. (2018). For a V-band stellar mass-to-light ratio of Υ * ≃ 2, the baryonic mass is 2 × 10 8 M ⊙ . At the deprojected 3D half-light radius r 1/2 = (4/3)R e = 2.9 kpc, the internal Newtonian acceleration is g N = 1.34 × 10 -2 a 0 . This corresponds to g i ∼ 0.12a 0 for an isolated object in MOND.

For the external field, we assume g e = V 2 /D assuming a flat rotation curve for the host NGC1052 of V = 210 km s -1 , in accordance with the stellar mass of the host estimated by Bellstedt et al. (2018) and consistent with the kinematic observations of van Gorkom et al. (1986). At the projected distance of NGC1052-DF2 from its host, this yields an external field g e = 0.15a 0 . This estimate of the external acceleration is rather uncertain for observational, not theoretical, reasons. For specificity, we adopt g e = 0.15a 0 here, and discuss the uncertainties further in §3.

The internal isolated MOND acceleration g i ∼ 0.12a 0 and the external one from the host g e ∼ 0.15a 0 are thus of the same order of magnitude. The exact calculation should then be made with a numerical Poisson solver. Nevertheless, a good ansatz in such a case is to consider the net MOND effect in one dimension (eq. 59 of Famaey & McGaugh 2012):

(g + g e ) µ g + g e a 0 = g N + g e µ g e a 0 , (2)
Here, µ is the MOND interpolating function, g the norm of the internal gravitational field one is looking for, g e the external field from the host, and g N the Newtonian internal gravitational field. Adopting here the 'simple' interpolating function 1 of MOND which is known to provide good fits to galaxy rotation curves (Famaey & Binney 2005;Gentile et al. 2011), this equation can easily be solved for g. Wolf et al. (2010) provides us with a mass estimator at the deprojected 3D half-light radius r 1/2 , where we can consider that the system feels the effect of the renormalized gravitational constant in MOND,

G eff = G × [g(r 1/2 )/g N (r 1/2 )].
Solving Eq. ( 2) yields G eff = 3.64G at the half-light radius. Using this and solving the mass estimator of Wolf et al. (2010) for the velocity dispersion leads to σ MOND ≈ 13.4 km s -1 .

(3) This is lower than the isolated case (20 km s -1 ) considered by van Dokkum et al. (2018), as is always the case when the EFE is important. This analytic estimate of the predicted velocity dispersion in MOND is in good agreement with an independent estimate by Kroupa et al. (2018), whose work is based on fits to numerical simulations by Haghi et al. (2009) of globular clusters embedded in an external field.

To estimate an uncertainty on this prediction, we consider a factor of two variation in the V-band mass-to-light ratio. That is, we obtain σ MOND = 9.7 km s -1 for Υ * = 1 M ⊙ /L ⊙ , and σ MOND = 18.2 km s -1 for Υ * = 4 M ⊙ /L ⊙ . Whether this range is appropriate for this particular dwarf is pure supposition that we must make in any theory. Note also that if we apply the same range to the mass of the host galaxy, this largely compensates for possible variations in distance of NGC1052-DF2 with respect to the host.

The MOND velocity dispersion discussed by van Dokkum et al. ( 2018), 20 km s -1 , only applies in isolation. It simply is not the correct prediction for MOND in the case of NGC1052-DF2 given its proximity to the giant elliptical NGC1052. The velocity dispersion we predict for MOND with proper consideration of the EFE disagrees far less with the data of van Dokkum et al. (2018) than the isolated case. Rather than being removed from the 90% c.l. upper limit of 10.5 km s -1 by a factor of two, the MOND prediction differs by only a few km s -1 and overlaps with it at the 1σ level. The correct MOND prediction is very close to the raw r.m.s. velocity dispersion (14.3 km s -1 ) measured by van Dokkum et al. (2018). Clearly we cannot exclude the theory when one realization of the data is entirely consistent with it.

CAVEATS

The prediction of velocity dispersions in MOND is most clean and simple in the isolated, low acceleration case, depending only on the stellar mass given the usual assumptions of spherical symmetry, isotropy, and dynamical equilibrium. Examples of isolated dwarfs are provided by And XXVIII (McGaugh & Milgrom 2013a,b) and Cetus (Pawlowski & McGaugh 2014). The problem becomes more involved when the EFE is relevant. In addition to the stellar mass of the dwarf, the extent of the stellar distribution matters to the determination of the internal acceleration. Diffuse objects2 like NGC1052-DF2 or Crater 2 are more subject to the EFE than compact objects like globular clusters. In addition to the internal field, we also need to measure the external field. This depends on the mass of the host and the 3D distance separating the host from its dwarf satellite. In dense environments where there may be multiple massive systems, matters obviously become more complicated still.

Estimating the external field can be challenging. We discuss here a few of the issues that arise, both observational (how well do the data constrain g e ) and theoretical (is it fair to assume the instantaneous value of the EFE at this moment in the dwarf's orbit). The estimates made above adopt reasonable assumptions consistent with those of van Dokkum et al. ( 2018), but these need not be correct.

Observational caveats

We have already noted that the r.m.s velocity dispersion (14.3 km s -1 ) is higher than the 90% c.l. upper limit (10.5 km s -1 ). This odd situation stems entirely from the omission of a single velocity tracer by van Dokkum et al. (2018, the object labeled 98 in their Fig. 2). While this is an outlier statistically, its exclusion is a choice that leads to a perverse physical situation. Without this one globular cluster, the mass of the object is too low for it to remain gravitationally bound. This raises the question as to why it is present at all, which is a essentially the original puzzle pointed out by Zwicky (1937). Stranger still, once excluded, van Dokkum et al. (2018) argue that the intrinsic velocity dispersion is as low as 3.2 km s -1 , at which points there is not enough dynamical mass to explain the observed stars. It is beyond the scope of this work to reanalyze these data; hopefully others will comment independently on the probability distribution of the velocity dispersion when only ten velocity tracers are available. One might also be concerned whether this tracer population is representative of the massweighted velocity dispersion (see discussion in Milgrom 2010, and references therein), or if nearly face-on rotation of the globular cluster system could play a role.

A key uncertainty in our analysis is the distance, which is required to determine the absolute acceleration scale. Distances are notoriously difficult to constrain for individual galaxies. We largely agree with the assessment of the distance laid out by van Dokkum et al. ( 2018), but for completeness point out where this matters most.

The absolute distance to NGC1052-DF2 matters insofar as this determines its association with the putative giant host NGC1052. If the two are not associated, then the object is likely isolated. Even that is not guaranteed given the proximity of NGC1042 along the line of sight. Setting that aside, a closer distance would be more consistent with the isolated MOND case as the stellar mass shrinks as D 2 and the predicted velocity dispersion along with it. Conversely, a significantly larger distance (beyond NGC1052) would be problematic for MOND as the object would then be more massive and should have a higher velocity dispersion.

The absolute 3D distance relative to the host NGC1052 also matters, as it affects the strength of the external field. While the projected separation is well determined for the assumed distance, the additional distance along the line of sight is not. As this increases, the EFE weakens, tweaking the calculation above. However, this is largely degenerate with uncertainties in the mass of the host, so it is difficult to provide a sensible error estimate that does not look like hedging. These are observational uncertainties, not theoretical hedges.

Theoretical caveats

MOND is a nonlinear theory. The approximation we have adopted to predict the velocity dispersion implicitly assumes that the external field is effectively constant. In reality, the amplitude and vector direction of the external field varies as the dwarf orbits its host. The issue then becomes whether the internal dynamics of the dwarf have time to come into equilibrium with the continually changing external field (Brada & Milgrom 2000). If so, the prediction we obtain is valid. If not, it is no more valid than if the condition of dynamical equilibrium is not obtained in Newtonian dynamics, as would be the case if the dwarf were being tidally stripped of its dark matter.

In MOND, the internal dynamics of an object depends on the external field in which it is immersed as soon as the external field dominates over the internal one. If the external field does not vary quickly in time with respect to the time it takes for the object to settle back to virial equilibrium, the "instantaneous" external field approximation that we have applied is appropriate. This works well for quasi-circular orbits, as the amplitude of the external field is steady even though the orientation vector varies. In general, the external field is expected to change over time scales D/V, where D is the distance to NGC1052 and V its circular velocity. This timescale is of the order of 3 × 10 8 years. The relevant internal time is the time it takes for the system (in the present case the system of globular clusters being utilized as velocity tracers) to come back to virial equilibrium. If this would be just half a revolution around NGC1052-DF2 at a typical radius of 5 kpc for the globular clusters, and for a typical velocity of 10 km s -1 , this time would be 1.6 × 10 9 years. So the dynamical state of the system of globular clusters has not obviously had time to equilibrate. Consequently, it might retain a memory of a previous value of the external field, and the system would then basically be out-of-equilibrium. Much depends on the details of its orbit: whether it is on a radial or circular orbit, and when its last pericenter passage was. One should bear in mind that the same issues pertain to tidal disruption in a Newtonian analysis, albeit with a different amplitude: one obvious interpretation for a dwarf to be devoid of dark matter is that it has been tidally stripped thereof, in which case it is not in equilibrium and any velocity tracers must be interpreted with a grain of salt.

The stability of dwarfs orbiting massive hosts has been investigated numerically by Brada & Milgrom (2000). Applying their criteria as in McGaugh & Wolf (2010), it indeed appears that NGC1052-DF2 may be on the fringe where dynamical equilibrium ceases to apply for the ultrafaint dwarfs of the Milky Way. More generally, Brada & Milgrom (2000) describe how an orbit dwarf may experience oscillations in velocity dispersion and scale length as it orbits from pericenter to apocenter. It is easy to imagine that NGC1052-DF2 is in a phase where its velocity dispersion is minimized while the scale length is maximized. While there is no way to ascertain if this is the case, this is an anticipated effect in MOND, and illustrates some of the subtle nonlinear effects that can arise.

We may also estimate the tidal radius in MOND. At the projected distance from the host, this is ∼ 8 kpc. Intriguingly, this is just beyond the outermost globular cluster. Consequently, the limit on the distance from the host obtained from the Jacobi radius by van Dokkum et al. (2018) does not apply in MOND.

The caveats discussed in this subsection are theoretical: nonlinear theories can be sensitive, and great care must be taken to extract the correct prediction. That said, we see no reason to expect the bulk velocity dispersion to vary outside the bounds stipulated by the plausible range in the stellar mass-to-light ratio, at least for our calculations referenced to the half-light radius. It would be interesting to compute the detailed radial variation of the velocity dispersion (Alexander et al. 2017), but such a computation is well beyond the scope of this letter, and indeed, beyond the scope of foreseeable observations to test for this remote dwarf galaxy.

CONCLUSIONS

We have predicted the velocity dispersion of the dwarf galaxy NGC1052-DF2 in MOND. We find σ MOND ≈ 13.4 +4.8 -3.7 km s -1 , not the ∼ 20 km s -1 discussed by van Dokkum et al. (2018). The difference arises because of the external field effect. The acceleration of stars within the dwarf is comparable to the acceleration of the dwarf in its orbit around its host galaxy, NGC1052. Consequently, neither field may be ignored, and the net MOND effect is reduced from the isolated case. This, in turn, reduces the predicted velocity dispersion, which consequently is not in conflict with the measurements of van Dokkum et al. (2018).

© 2018 The Authors

The result is not particularly sensitive to the choice of interpolation function since the object in question is deep in the regime of low acceleration where all interpolation functions converge.MNRAS 000, 1-4 (2018)

The quantity of merit in MOND is the surface density, not the mass. Lower surface density means lower acceleration. That NGC1052-DF2 is massive for a dwarf does not shield it from the EFE as it is a diffuse object.

MNRAS 000, 1-4 (2018)


Oliver Müller

Benoit Famaey

Hongsheng Zhao

Predicted MOND velocity dispersions for a catalog of ultra-diffuse galaxies in group environments

Keywords: Galaxies: kinematics and dynamics, Galaxies: dwarf, Cosmology: theory

Predicted MOND velocity dispersions for a catalog of ultra-diffuse galaxies in group environments

Oliver Müller, Benoit Famaey,

Introduction

In recent years, the number of newly discovered ultra-diffuse galaxies (UDGs) in the nearby Universe has exploded. While it has long been known that this special type of galaxy exists in galaxy clusters environments (e.g., Sandage & Binggeli 1984;Impey et al. 1988), the discovery of UDGs in nearby galaxy groups (e.g., Crnojević et al. 2014;[START_REF] Müller | Predicted MOND velocity dispersions for a catalog of ultra-diffuse galaxies in group environments[END_REF]Cohen et al. 2018) has brought new insights, as well as opportunities to better study these systems. Today, UDGs have been defined by their low surface brightness of > 25 mag arcsec -2 in the V-band, and large radial extent, with projected half-light radii r eff > 1.5 kpc (van Dokkum et al. 2015). In morphology, they are not distinguishable from normal dwarf galaxies and appear both as early and late type galaxies (Sandage & Binggeli 1984). Their formation is still under intense debate with different proposed formation mechanisms. For example it has been argued that these galaxies are failed Milky Way type galaxies (van Dokkum et al. 2015;Yozin & Bekki 2015), just normal dwarf galaxies affected by stellar feedback, quenching, and outflows (Amorisco & Loeb 2016;Di Cintio et al. 2017;Chan et al. 2018), or rather a mixed bag of objects (Toloba et al. 2018).

Recently, the UDG NGC 1052-DF2, a putative member of the NGC 1052 galaxy group (Cohen et al. 2018), located at 20 Mpc, has caught some attention, following the announcement by van Dokkum et al. (2018) that this galaxy is apparently lacking dark matter. This result was inferred from the velocity dispersion of ten bright globular clusters associated with this galaxy. Using a biweight estimator, van Dokkum et al. (2018) derived a velocity dispersion of only σ vel = 3.2 km s -1 with a 90% upper limit of σ vel = 10.5 km s -1 , which is consistent with the baryonic matter of this galaxy alone. Following a more conservative approach, Martin et al. (2018) obtained a value of σ vel = 9.2 km s -1 , with a 90% upper limit of σ vel = 17.2 km s -1 . While the globular cluster population can be used as a tracer for the mass of the system (Toloba et al. 2018), ultimately, the velocity dispersion of the stellar body should be used. For example, in the Fornax dwarf spheroidal, where both estimates are available, there is a strong discrepancy between the two values (Laporte et al. 2018). We note that, for NGC 1052-DF2 a stellar velocity dispersion has recently been measured using IFU observations with VLT+MUSE (Emsellem et al. 2018;Fensch et al. 2018).

The apparent lack of dark matter in a dwarf galaxy could have some challenging consequences for alternative gravity scenarios such as Modified Newton Dynamics (MOND, Milgrom 1983), as previously investigated by, e.g., Gentile et al. (2007Gentile et al. ( , 2012)); Lelli et al. (2015). The argument is that, in MOND, the baryonic matter alone gives the high observed velocities in galaxies, and notably the flat rotation curves in spirals. This argument was indeed brought up by van Dokkum et al. (2018) in the case of NGC 1052-DF2. This simple assessment, however, is only true if the galaxy resides in relative isolation, that is, if no other galaxy is nearby. In the case that there is another gravitational potential close by, its environmental effect has to be taken into account when estimating the internal velocity dispersion, a phenomenon exclusively appearing in MOND. This so-called external field effect (EFE) arises from the non-linearity of the MONDian Poisson equation (Bekenstein & Milgrom 1984).

The EFE can ultimately lower the velocity dispersion of a system, or in other words, make it Newtonian. Because this EFE appears only in MOND, and not within the standard model of cosmology, it is a highly intriguing feature to test and distinguish these paradigms. For a thorough discussion of the EFE and its consequences, see e.g. Haghi et al. (2016); Thomas et al. (2018); the Supplementary Materials of Kroupa et al. (2018); or the extensive review of MOND (Famaey & McGaugh 2012).

Article number, page 1 of 9 arXiv:1901.02679v1 [astro-ph.GA] 9 Jan 2019

For the Andromeda dwarf galaxies in the Local Group, it has been realized that this EFE plays a crucial role, allowing MOND to successfully predict the velocity dispersion of those dwarfs (McGaugh & Milgrom 2013a,b). For the "feeble-giant" Crater-II (Torrealba et al. 2016), an ultra-faint dwarf galaxy satellite of the Milky Way, a very low velocity dispersion has been predicted due to the EFE (McGaugh 2016) and subsequently measured (Caldwell et al. 2017). Several more predictions of velocity dispersions for the ultra-faint dwarf galaxies around the Milky Way (Pawlowski & McGaugh 2014;Pawlowski et al. 2015;Cortés & Hernandez 2017) now await follow-up measurements. Taking the EFE into account for NGC 1052-DF2, the expected velocity dispersion was estimated to be σ vel 14 km s -1 , consistently estimated by two independent teams (Famaey et al. 2018;Kroupa et al. 2018). While the former used an analytic expression to get this results, the latter invoked an N-body integrator (Londrillo & Nipoti 2009) to get this estimate, thereby validating the analytic estimate. This result alleviates the claim that NGC 1052-DF2 could falsify MOND, but shows that UDGs are an excellent testbed to study the impact of the EFE. Following up on this previous work, we hereby make a priori predictions for the velocity dispersion of two dozen UDGs in the nearby Universe, thus making MOND and its EFE falsifiable in a currently less-explored regime than that of spiral galaxies. Interestingly, Milgrom (2018a,b) has recently extended the succesful predictions of MOND to the regime of the internal dynamics of galaxy groups themselves.

The paper is organized as follows: In Section 2 we discuss how the velocity dispersion can be predicted in MOND, in Section 3 we present a catalog of nearby UDGs in galaxy group environments, in Section 4 we predict the velocity dispersion for the UDG catalog, and in Section 5 we present a brief discussion and estimation on how these values can be measured. Finally, we give our summary and conclusions in Section 6.

The External Field Effect calculation

The Poisson equation of MOND reads, with Famaey & Binney (2005) interpolating function:

∇ • g µ g a 0 = 4πGρ, µ(x) = x 1 + x , ( 1
)
where g is the gravitational acceleration vector, ρ the baryonic density, and a 0 = 1.2 × 10 -13 km s -2 (Begeman et al. 1991) is the MOND acceleration constant marking the transition from the Newtonian to the deep-MOND regime. The fact that µ < 1 when in the regime of low acceleration (g < a 0 ), in which NGC 1052-DF2 clearly is, would suggest that such galaxies should always display a 'phantom' dark matter behaviour, i.e. a dark matter behavior when interpreted in the context of Newtonian dynamics. However, when the galaxy is not isolated, one should take into account the EFE, such that

g = g ex -∇φ, (2)
where g ex is the gravitational field from the neighbouring galaxies, and φ the internal MOND potential. In a MONDian universe, the velocity dispersion of an isolated spherical and isotropic system in the low acceleration regime is easily calculated from its baryonic mass M alone (e.g., McGaugh & Milgrom 2013a):

σ iso = 4 81 GMa 0 1/4 , ( 3
)
where M is the total baryonic mass. This formula applies when the total acceleration (internal plus external acceleration) is well bellow a 0 , i.e. g + g ex < a 0 . When the internal acceleration is larger than the external, g ex < g < a 0 , then we can apply the given formula for the isolated case. However, if g < g ex < a 0 the object is quasi-Newtonian, where Newton's law of Gravity applies with a renormalization of the gravitational constant G (McGaugh & Milgrom 2013a;Famaey et al. 2018). The external acceleration can be approximated by g e = v 2 rot /D, where v rot is the rotation curve of the external galaxy and D is the separation between the two objects. Using Eq. 59 of Famaey & McGaugh (2012), one can estimate the MOND acceleration g at the half-light radius with:

(g + g ex ) µ g + g ex a o = g N,1/2 + g ex µ g ex a 0 , (4)
with

g N,1/2 = GM 2r 2 1/2 , ( 5
)
where µ(x) = x/(1 + x) is the simple interpolation function (Famaey & Binney 2005), and g N,1/2 is the Newtonian internal field for the mass embedded within the 3D deprojected half light radius (r 1/2 = 4/3 r eff ). The expression solved for g is rather long, so we refrain from showing it here and refer to the footnote 34 of Famaey & McGaugh (2012), where it is explicitly given. We can now estimate the true velocity dispersion of the system, corrected for the external field. Having estimated g, we can calculate the renormalization of the gravitational constant with

G renorm = G • (g/g N,1/2 ). (6)
Finally, we can use the mass estimator in Wolf et al. (2010) Eq. 2 to calculate the velocity dispersion:

σ = 0.5M G renorm 3 r 1/2 , ( 7
)
where 0.5M corresponds to the mass embedded within r 1/2 . This cooking recipe to calculate the velocity dispersion in the quasi-Newtonian regime is discussed in more detail in Famaey et al. (2018), who used it to successfully derive the MONDian value for NGC 1052-DF2 to be between 8.9 and 19.0 km s -1 depending on the interpolating function, stellar mass-to-light ratio, and three-dimensional distance to the host. Note that the latest measured velocity dispersion value, using deep MUSE IFU observations (Emsellem et al. 2018) for the stellar body of NGC 1052-DF2 is σ stellar = 18.8 km s -1 , and for the globular cluster and planetary nebulae population associated to this galaxy is σ GC,PN = 10.5 km s -1 -which are well within the allowed MONDian range for this system.

A catalog of nearby ultra-diffuse galaxies

Data

Several independent teams have taken up the effort to search for nearby low-surface brightness galaxies in group (e.g., Merritt et al. 2014;Carlin et al. 2016;Javanmardi et al. 2016;Bennet et al. 2017;Müller et al. 2015Müller et al. , 2017a;;Mihos et al. 2018) and cluster (e.g. Mihos et al. 2015;van der Burg et al. 2016;Mihos et al. 2017;Venhola et al. 2017;Eigenthaler et al. 2018) environments. While most detections in these surveys correspond to the regime of the normal dwarf galaxies, also several UDGs have been discovered, based on their integrated light profiles. Up to now, no compilation of these objects exists. Therefore we have collected UDGs from the literature -mainly selecting objects with half-light radius r eff estimates larger than 1.5 kpc (including those objects with very close values to 1.5 kpc, e.g. NGC 7814-DGSAT-7 with r eff = 1.49 kpc) and residing in a galaxy group environment based on their projected position in the sky. Therefore we exclude UDGs residing in clusters, because in these environments too many things can affect the velocity dispersion of the UDGs. Also, we only consider galaxies in the nearby Universe (z < 0.01). While going further would indeed increase the sample (Greco et al. 2018), the galaxy's association to a certain galaxy group becomes more difficult to estimate.

In Table 1, we present our sample of nearby UDGs within galaxy group environments. We have compiled the relevant data needed to estimate the velocity dispersion in MOND (see Section 2). For this, we need the total luminosity of the galaxy L , the half-light radius r eff , and the projected separation to the host galaxy ∆ pro j . When missing, we have derived values from the published data, e.g. the mean surface brightness from the apparent magnitude and the half-light radius, the luminosity in solar units from the absolute magnitude, and the projected separation simply from the coordinates. We have transformed the apparent magnitudes into solar luminosities by using L = 10 0.4 (m -m+DM) , where m = 4.83 mag is the solar apparent magnitude in the V band, m is the apparent magnitude of the UDG, and DM is the distance modulus to the host.

We also need the rotation curve of the host galaxies to estimate their gravitational influence. In our sample, galaxies with an asymptotic circular velocity measurement are: NGC 1052 (v rot = 210 km s -1 , van Gorkom et al. 1986), M96 (v rot =240 km s -1 , Moiseev et al. 2004) Fraternali et al. 2011). From there, the external field at the position of the UDG can be estimated directly from the centripetal acceleration.

For elliptical galaxies where no rotation curve data is available, we converted the K band luminosity using a Mass-to-Light (M/L) ratio of 0.8 (de Blok et al. 2008) to get the total stellar mass. From there, the gravitational field at the distance of the UDG can be estimated in the context of MOND. This was the case for Cen A (K = -23.9 mag), NGC 3625 (K = -21.9 mag), NGC 3669 (K = -22.3 mag), NGC 7814 (K = -24.0 mag), NGC 5485 (K = -23.8 mag), NGC 3384 (K = -23.4 mag), M 105 (K = -23.9 mag), NGC 5475 (K = -22.8 mag), and NGC 3619 (K = -24.0 mag) using the photometry by the Two Micron All Sky Survey (2MASS, Skrutskie et al. 2006) and a distance modulus given in Table 1.

Finally, for one face-on spiral galaxy (NGC 1042), no rotation curve was available. Therefore we first converted again its K band luminosity using a Mass-to-Light (M/L) ratio of 0.8 to get the stellar mass of the system. Then, we estimated the additional total gas mass with Eq. 2 from Di Cintio & Lelli (2016) and added this to the stellar mass to get the total baryonic mass, and estimate the MOND external field at the position of the UDG from there.

Because galaxy group environments can host several large galaxies (M V < -19 mag) we have to be careful to select the influencing galaxy when calculating the MONDian velocity dis-persion of the UDG. If there is a bright galaxy apart from the dominant galaxy in the group -which is mostly indicated by the group's name -close to the UDG we have to re-evaluate the MONDian velocity dispersion for those systems. A precise evaluation of the EFE as a function of the 3D position of the UDG in the group should then be performed using a numerical Poisson solver. In the present paper, we present an order of magnitude of the effect on the velocity dispersion by considering the EFE associated to each bright galaxy separately.

Individual galaxy groups

We now have a closer look at the different galaxy groups and evaluate the impact of their different bright galaxies onto the UDGs.

NGC 1052 group: This galaxy group consists of three bright galaxies, the dominant elliptical galaxy NGC 1052, the faceon spiral galaxy NGC 1042, and the edge-on spiral galaxy NGC 1035. While NGC 1052-DF2 is only affected by the dominant elliptical, we note that NGC 1052-DF1 is at a separation of only 28 kpc from NGC 1042, which indeed has a major impact on the predicted velocity dispersion. On the other hand, NGC 1052-DF4 is separated by only 21 kpc from NGC 1035.

NGC 1084 group: To the north-east of the NGC 1052 group, a single spiral galaxy -NGC 1084 -resides well in isolation, bringing no additional complications to our calculations for NGC 1084-DF1.

M 96/Leo-I group: This group consists of 7 bright galaxies (see e.g. Leo-Triplet: The Leo-Triplet is sometimes considered as part of the Leo-I group, based on their same velocity and distance measurements. However, they are well separated on-sky into two distinct associations (Müller et al. 2018). The dominant edge-on spiral galaxy NGC 3628 is also the closest galaxy to the UDG dw1117+15, which is almost 2 degrees to the north of the system.

Cen A group: Cen A is the closest elliptical galaxy in the nearby universe and is the dominant galaxy in the Cen A group (Müller et al. 2017a). Because the distances to Cen A and Cen A-MM-dw1 are well established with high-precision tip of the red giant branch measurements (Rejkuba 2004;Crnojević et al. 2018), no confusion is expected for their association.

NGC 5485 group: Several surveys have targeted the nearby M 101 group of galaxies and announced the discovery of new dwarf galaxy members (Merritt et al. 2014;Müller et al. 2017b;Bennet et al. 2017). Within its projected virial radius, reside the two massive elliptical galaxies -NGC 5485 and NGC 5473 with similar mass estimates -at a distance of 27 Mpc. Some of the putative M 101 members have recently been identified as background UDG members of the NGC 5485 group by their non-detected red giant branch star population in HST observations [START_REF] Müller | Predicted MOND velocity dispersions for a catalog of ultra-diffuse galaxies in group environments[END_REF] These two galaxies have the same systemic velocity measurement as the NGC 5485 group, which could indicate that M 101-DF5 could be associated to these these two galaxies, and not to NGC 5485 itself. We therefore consider two cases for M 101-DF5, a) one case where it resides within NGC 5485 external field, and b) another where it resides within NGC 5475 external field.

NGC 2683 group: This galaxy group is made up of one large spiral galaxy and resides in isolation, making the predictions for NGC 2683-DGSAT-1 and NGC 2683-DGSAT-2 straightforward.

NGC 4594/M 104 group: This famous galaxy group with the M 104 (the Sombrero galaxy) in its heart is again, very isolated, making the prediction for NGC 4594-DGSAT-1 straightforward.

NGC 3625 group: The isolated spiral galaxy NGC 3625 is in projection very close to the shell galaxy NGC 3619, and the elliptical galaxy NGC 3613 which are well separated in velocity space from NGC 3625. It is therefore difficult to disentangle the membership of the UDGs between those groups. If NGC 3625-DGSAT-2 is a NGC 3625 member, the external field of NGC 3625 is almost negligible, however, if it is associated to NGC 3619 (D = 25±1.8 Mpc), it is deep in the quasi-Newtonian regime. For the other two UDGs in this region (NGC 3625-DGSAT-3, NGC 3625-DGSAT-4), the separation to NGC 3619 is too large to be strongly affect by the EFE.

NGC 3669 group: The isolated spiral galaxy NGC 3669 is to the east of the NGC 3625 and NGC 3619 groups. However, their separation is too large for major confusions, allowing straight-forward predictions for NGC 3669-DGSAT-2 and NGC 3669-DGSAT-3.

NGC 7184 group: This group again is well isolated, with the spiral galaxy NGC 7184 at its center, making the predictions for NGC 7814-DGSAT-2 and .NGC 7814-DGSAT-7 straightforward.

Predicted velocity dispersions

For all galaxies of Table 1 we have first calculated whether the external field dominates the dynamics or not. To make a distinction between the isolated and the quasi-Newtonian regimes we estimate the fraction g/g ex as a function of the three dimensional separation of the UDG to its main host (or to various possible hosts), up to one Megaparsec. We can safely assume that, when the internal acceleration is less than twice the external one, the EFE will play a role. In this case, we thus use the σ as previously derived by including the EFE in Eqs. 4-7. Above this value of the internal to external field ratio, the galaxy can be assumed to be isolated, therefore σ iso applies as in Eq. 3.

In Figure 1 we show the predictions for the 18 UDGs in our sample which have one clear host. The curves start at the projected separation of the UDG to its host. We derive upper and lower limits for the curves as follows: for every UDG, we vary the V-band M/L ratio between 1 and 4, as well as take into account the distance uncertainty to the host. We also allow the mass of the host to vary by ±20 percent. The lower limit is therefore given by a V band M/L = 1, a distance to the host at its lowest limit (making the UDG less luminous), and the +20 percent upper limit for the host mass, whilst the upper limit is given by a V band M/L = 4, the upper limit for the host galaxy system distance (making the UDG more luminous), and the -20 percent lower limit for the host mass. We also give a typical estimate in between those two curves, with M/L = 2 and the literature values for the distance and mass (or rotational velocity) of the host. We have simplified the Figures by using the transition g/g ex = 2 for the typical case also for the transition for the lower and upper limits. Correctly we would need to take into account that in the lower case, the EFE will have a larger range until the UDG reaches the regime of isolation. We also left a gap between the σ estimation including the EFE and the isolated case. This is caused by the fact that we would need to carefully interpolate between the two regimes, which is not the focus of this work.

For 4 UDGs in our catalog, the host dominating the EFE is ambiguous. For NGC 3625-DGSAT-2 the two possible host galaxies are not even within the same group and have therefore different nominal distances. For the others, to make a precise prediction, one would need to use a numerical Poisson solver as a function of the full 3D configuration of the galaxy group.

To give an idea of the range of possible predictions, we provide in Fig. 2 the predictions as a function of the separation to each potential host galaxy separately.

In Table 2 we present the minimal and maximal allowed values for the velocity dispersion, as well as a typical value for the UDG, if it has a V band M/L ratio of 2 (which is typical for old dwarf galaxies) and has a physical distance of √ 3/2 • ∆ pro j . This distance simply assumes that the depth along the line-of-sight is the same as the on-sky separation.

For 17 of our 22 UDGs, we note that the EFE was indeed important to consider and will effectively lower the stellar velocity dispersion of the galaxy for most of its possible 3D separations.

Discussion

The discovery of the UDG NGC 1052-DF2 with its spectroscopic follow-up observations opened up a novel opportunity to study MOND and the EFE in this class of galaxies. In Figure 3 we present the MONDian calculation for this galaxy, this time including the σ measurements by van Dokkum et al. ( 2018) and Emsellem et al. (2018). The former gave a velocity dispersion of σ = 3.2 +5.5 -3.2 km s -1 (shown in red in Figure 3), using globular clusters as tracer of the stellar body of the system. It is apparent that such a low value for the velocity dispersion would indeed be an outlier in MOND (to be more precise, a 2σ outlier, see Figure 1 in Kroupa et al. 2018). Emsellem et al. (2018), using precise IFU spectroscopy -obtained with MUSE mounted at the Very Large Telescope -directly derived a stellar velocity dispersion of σ = 16.3 +5.0 -5.0 km s -1 (shown in grey in Figure 3). This value is in excellent agreement with the MONDian estimates made here and in previous studies (Kroupa et al. 2018;Famaey et al. 2018).

Additionally, Emsellem et al. (2018) found two more globular clusters and three Planetary Nebulae. Using them, and updated velocities for five previously known globular clusters, they derive a velocity dispersion of σ = 10.5 +4.0 -2.2 km s -1 , which is at the lower limit of allowed values in MOND (see Table 4).

Assuming that the stellar velocity dispersion σ = 16.3 km s -1 is the true value for this system, and that we live in a MON-Dian universe, NGC 1052-DF2 is then most likely at a physical separation of 124 kpc to its host NGC 1052, considering the external field. This does make insofar sense, that this is a typical radial separation of a satellite to its host. However, NGC 1052-DF2's systemic velocity is off by 300 km s -1 , which could indicate that this galaxy is not a bound member of the NGC 1052 group. Only precise distance measurements can tell whether this galaxy indeed belongs to the group, or whether it is a field galaxy in the fore-or background. In the latter case, an estimate with the corresponding distance would be needed to check the validity of MOND.

The MUSE observations are indeed an encouraging prospect for future studies of these UDGs. At these low-surface brightness levels, it is possible to get an estimate of the velocity dispersion under ten hours of observation time, which is expensive but certainly doable. Other today's available facilities which potentially can conduct such studies are KCWI installed at Keck and Megara at the Gran Canaria Telescope. In the future, the 1 The values here are slightly higher than in Famaey et al. (2018) because of very slightly different assumptions for the baryonic mass of the UDG and host, as well as in the definition of the upper limit.

2 NGC 3625-DGSAT-2 could also be associated to NGC 3619 (second shown entry) instead of NGC 3625 (first shown entry), which is closer to us, therefore we provide the predictions for both cases.

next-generation telescopes like the ELT will certainly be able to conduct such measurements in short times for a large sample of galaxies.

We refrain here from making any predictions of the velocity dispersion of the UDGs in terms of the standard ΛCDM framework. In principle, we could collect the dark matter halos in high-resolution simulations for the satellite galaxies, given the observed luminosity and construct a range of possible values for the velocity dispersion. However, the origin of UDGs is still not understood. If they are failed Milky Way type galaxies, they will possess a large dark matter halo to begin with, and on the other hand, if they origin from dwarf galaxies, they will reside in smaller dark matter halos. This huge range of possibilities makes it unfeasible to make a prediction of the velocity dispersion in the standard context.

We note that two of the UDGs within our sample -Cen A-MM-Dw1 and M 96-DF6 -already have precise distance measurements available (Crnojević et al. 2018;Cohen et al. 2018). The three dimensional separation between Cen A and Cen A-MM-dw1 is 249 kpc with a distance uncertainty for Cen A-MM-dw1 of ±120 kpc. This galaxy resides at a separation where the EFE plays a role. In Figure 1 limit is given by the isolated case. For M 96-DF6, the three dimensional separation between M 96 and M 96-DF6 is 276 kpc, with a distance uncertainty for M 96-DF6 of ±300 kpc. This gives a typical MONDian velocity dispersion of σ = 9.5 +1.0 -1.0 km s -1 . For these estimates, we have not varied the distance to the host. Once taken into account, this will additionally increase the allowed interval for the velocity dispersion.

With the updated velocity dispersion for NGC 1052-DF2 at hand, within the ΛCDM paradigm it is possible to estimate the total mass (dark matter + baryonic mass) with equation 2 given in Wolf et al. (2010). With a stellar velocity dispersion of σ = 16.3 km s -1 we get a mass within r 1/2 of 5.08×10 8 M . With an absolute magnitude of -15.3 mag in the V band the galaxy has a total baryonic mass of 2.27 × 10 8 M (using a M/L = 2 and a V band magnitude of 4.83 for the Sun), leading to a dynamical to visible mass ratio of 4.5 within the half-light radius. This rough estimate is well in agreement with Local Group dwarfs in the corresponding luminosity regime (see McConnachie 2012). Therefore, the observed velocity dispersion of NGC 1052-DF2 is not peculiar, and hence comfortable in both paradigms, the ΛCDM standard model of cosmology and the alternative gravity paradigm (MOND).

If NGC 1052-DF2 cannot help us to distinguish between MOND and ΛCDM, are there other galaxies in our sample which do? The most affected galaxy in terms of the EFE in our sample -max(σ iso /σ typical ) -is NGC 4594-DGSAT-1. Let us make a gedankenexperiment, where we live in a MONDian universe, but fail to realize that, and thus describe it in terms of Newtonian dynamics. Assume that the calculated typical velocity dispersion correctly describes the UDGs, and thus that NGC 4594-DGSAT-1 has a typical velocity dispersion of σ = 5.8 km s -1 . Now our Newtonian observers have exactly measured this value to infinite precision, and using it with Wolf et al. (2010) they will derive a mass within r 1/2 of 5.0 × 10 7 M . The mass of the baryonic content of the galaxy inferred from the light is 6 × 10 7 M , giving a dynamical to visible mass ratio within r 1/2 of only 1.7, and hence a dynamical mass-to-light ratio M dyn /L of 3.4. For a galaxy with M V = -13.9 mag this is similar to Milky Way dwarf satellite Leo-I (Mateo et al. 2008), sitting at the lower end of the luminosity-M dyn /L relation (see Figure 11 in McConnachie 2012). However, if the lowest possible value in MOND was measured, i.e. σ min = 3.5 km s -1 , the Newtonian observers would derive a M dyn /L ratio of only 1.2, clearly too small for primordial dwarf galaxies in ΛCDM. This simple gedankenexperiment shows that we can in principle distinguish between primordial dwarf galaxies in the ΛCDM framework and the MOND predictions for the cases where the UDG is strongly affected by the EFE and high-precision distance and velocity dispersions measurements are available. Let us note that such a situation has already happened for some Andromeda dwarfs (McGaugh & Milgrom 2013a,b) and Crater-II (McGaugh 2016). As these previous examples show, such an observation would of course not falsify ΛCDM, but would pose interesting challenges to it. In terms of falsifying MOND with such objects, as discussed in Famaey et al. (2018), an issue could always be whether the internal dynamics of a UDG on an eccentric orbit have time to come into equilibrium with the continually changing external field. Hence, once an interesting object challenging MOND is found, elucidating this issue would necessitate more involved simulations.

On the other hand, to tentatively falsify MOND, it would be well worth to follow-up some of the UDGs for which we have shown that the EFE plays a minor role, or no role at all. In that case, when their velocity dispersion is measured to be lower than the MOND prediction for the isolated case, then there is not much room in MOND for those velocity dispersions to vary (assuming that the values used for the predictions like the distance and luminosity are correct). This is in contrast to the ultrafaint dwarf galaxies in the Milky Way, which MOND tidal effects would certainly bring out of equilibrium, complicating any comparison between observations and predictions (McGaugh & Wolf 2010;Fattahi et al. 2018;Read et al. 2018).

Summary and conclusion

Ultra-diffuse galaxies are puzzling objects for cosmology. While they resemble galaxies like the Milky Way in size, they are as faint and feeble as dwarf galaxies. In MOND -when they are close to a large galaxy -they can be affected by its external field effect, making them quasi-Newtonian. This EFE -unique to MOND -will lower the velocity dispersion of these systems, when compared to isolation. We have compiled a catalog of UDGs in group environments in the nearby universe to predict their velocity dispersions. We used an analytic expression to include the external field in our calculations and derived velocity dispersions as function of the true three dimensional separation of the UDGs from their host. When the internal accelerations of the UDGs overrule the external effects, we present the prediction based on simple MOND formula for isolated objects. By varying the Mass-to-Light ratio of the UDG, as well as the distance to its putative host, we give upper and lower limits allowed from MOND. Future observations of these UDGs can therefore test the validity of MOND and test the external field effect.

Two of our UDGs -Cen A-MM-Dw1 and M 96-DF6 -have high-precision distance measurements with the HST. In these cases, we have calculated the three dimensional distances to their host and made a more narrow prediction of the expected velocity dispersion in MOND, assuming that they follow a typical M/L ratio of 2. For Cen A-MM-Dw1 we expect a velocity dispersion of σ = 11.8 +2.2 -1.5 km s -1 , and for M 96-DF6 a velocity dispersion of σ = 9.5 +1.0 -1.0 km s -1 . For NGC 1052-DF2 high-precision IFU data have become available, taken with VLT+MUSE. This is the first time the stellar velocity dispersion itself of an UDG has been measured. We compared this new measurement to the MOND prediction, already made in Famaey et al. (2018); Kroupa et al. (2018), and find excellent agreement. This is an encouraging prospect, as it shows that testing our predictions is feasible with today's instruments.

As a last caveat, we should mention that the predictions made here make rather simple assumptions, such as no strong anisotropy, no rotation on the sky, as well as an analytic MOND estimate with only one interpolating function and with an implicitly small eccentricity for the UDG orbit. As discussed in Famaey et al. (2018), a UDG on a very eccentric orbit with a continually changing external field could display a lower velocity dispersion than the prediction made here. Hence, once an interesting object challenging MOND is found, elucidating this issue would necessitate more involved simulations.

  , NGC 3628 (v rot =213 km s -1 , Wilding et al. 1993) NGC 1084 (v rot = 140 km s -1 , Burbidge et al. 1963), NGC 2683 (v rot = 215 km s -1 , Casertano & van Gorkom 1991), NGC 4594/M 104 (v rot = 350 km s -1 , Faber et al. 1977), NGC 7814 (v rot = 215 km s -1 ,

  Fig. 1 in Müller et al. 2018) at a mean distance of 10.7 Mpc. Of those, the two elliptical galaxies M 105 and NGC 3384 have separations of 53 kpc and 67 kpc from M 96-DF6, respectively. The large spiral galaxy M 96 has a projected separation of 172 kpc. This UDG sits almost at the center of this massive group, complicating accurate predictions. Fortunately, for M 96-DF6, HST distance measurements are available (D = 10.2 ± 0.3 Mpc Cohen et al. 2018), indicating that this UDG is at the nominal distance of M 96 (D = 10.4 ± 0.3 Mpc Karachentsev et al. 2013), and farther separated from M 105 (D = 11.3 ± 0.1 Mpc Karachentsev et al. 2013) and NGC 3384 (D = 9.4 ± 0.1 Mpc Karachentsev et al. 2013). Farther outside of this galaxy assembly resides dw1055+11.

Fig. 1 .

 1 Fig.1. The predicted velocity dispersions including the EFE (blue lines) and for the isolated case (dotted lines) as function of the three dimensional separation up to 1 Mpc. The inclusion of the EFE is stopped when g/g ex > 2. The vertical dotted line corresponds to the on-sky separation (and thereby lower limit of the 3D distance) between the host and the UDG. The upper and lower limits of the curves are derived by varying the M/L and the distance to the host, see text. The last two entries -M 96-DF6 and Cen A-MM-Dw1 -have high-precision distance measurements available, we therefore indicate the three dimensional separation (grey line) as well as the associated 1σ uncertainties (grey area).

Fig. 2 .

 2 Fig. 2. Continuation of Figure1, in the cases where multiple external fields have potentially a strong influence on the ultra-diffuse galaxies. For NGC 3625-DGSAT-2 the two host galaxies are not within the same group and have therefore different nominal distances.

Fig. 3 .

 3 Fig. 3. The velocity dispersions for the ultra-diffuse galaxy NGC 1052-DF2. The blue and black lines correspond to the MOND predictions as in Figure 1. The red area denotes the 1σ range of the velocity dispersion measurement of van Dokkum et al. (2018), the gray are the stellar velocity dispersion measurement ofEmsellem et al. (2018).

Table 1 .

 1 . Many more candidates await dis-A&A proofs: manuscript no. aanda A list of nearby ultra-diffuse galaxies within galaxy groups.

	Name	RA	dec	m V µ eff,V	r eff ∆ pro j (m -M) L	ref
		J2000	J2000	mag mag arcsec -2 kpc kpc mag	10 6
	NGC 7814-DGSAT-2 00:03:06.9 +16:18:30.8 17.9 26.9	2.29 47	31.02	30.1 Henkel et al. (2017)
	NGC 7814-DGSAT-7 00:00:44.0 +15:27:14.3 18.7 27.0	1.49 258 31.02	14.4 Henkel et al. (2017)
	NGC 1052-DF4	02:39:15.1 -08:06:58.6 16.5 25.1	1.60 165 31.51	171.8 Cohen et al. (2018)
	NGC 1052-DF1	02:40:04.6 -08:26:44.4 18.2 27.4	2.51 109 31.51	35.9 Cohen et al. (2018)
	NGC 1052-DF2	02:41:46.8 -08:24:09.3 16.2 25.1	2.06 79	31.51	226.5 Cohen et al. (2018)
	NGC 1084-DF1	02:42:38.0 -07:20:16.3 16.2 24.7	1.63 280 31.34	193.8 Cohen et al. (2018)
	NGC 2683-DGSAT-1 08:52:47.8 +33:47:33.1 14.7 26.5	4.10 68	30.05	234.5 Javanmardi et al. (2016)
	NGC 2683-DGSAT-2 08:55:23.3 +33:33:32.4 16.3 25.8	1.39 105 30.05	53.7 Javanmardi et al. (2016)
	M96-DF6	10:46:53.1 +12:44:33.5 16.6 27.0	2.11 172 30.15	44.9 Cohen et al. (2018)
	dw1055+11	10:55:43.5 +11:58:05.0 16.9 26.7	1.78 411 30.15	34.0 Müller et al. (2018)
	dw1117+15	11:17:02.1 +15:10:17.0 17.4 27.4	2.04 398 30.15	21.5 Müller et al. (2018)
	NGC 3625-DGSAT-2 11:21:22.9 +57:34:50.1 20.3 26.9	1.54 151 32.86	18.0 Henkel et al. (2017)
	NGC 3625-DGSAT-4 11:21:40.8 +57 24 37.0 18.5 26.3	2.17 263 32.86	94.6 Henkel et al. (2017)
	NGC 3625-DGSAT-3 11:22:12.0 +58:02:11.9 19.0 25.5	1.48 221 32.86	59.7 Henkel et al. (2017)
	NGC 3669-DGSAT-2 11:24:48.3 +57:37:58.0 20.0 26.2	1.86 93	33.17	31.4 Henkel et al. (2017)
	NGC 3669-DGSAT-3 11:26:38.8 +57:41:19.1 18.3 26.9	3.91 123 33.17	150.4 Henkel et al. (2017)
	NGC 4594-DGSAT-1 12:39:55.1 -11:44:38.4 16.2 26.1	1.67 24	30.12	62.8 Javanmardi et al. (2016)
	CenA-MM-Dw1	13:30:14.3 -41:53:34.8 14.2 25.4	1.82 92	27.84	48.9 Crnojević et al. (2018)
	M101-DF5	14:04:28.1 +55:37:00.0 17.7 27.7	4.90 342 32.16	103.0 Merritt et al. (2016)
	M101-DF7	14:05:47.5 +55:07:57.3 20.1 28.4	2.60 112 32.16	11.3 Merritt et al. (2016)
	M101-DF4	14:07:33.8 +54:42:39.2 18.5 27.6	3.60 140 32.16	49.3 Merritt et al. (2016)
	M101-DF6	14:08:18.7 +55:11:30.6 19.8 27.5	2.90 117 32.16	14.9 Merritt et al. (2016)

tance measurements and can potentially be UDG members of the NGC 5485 group as well. While M 101-DF4, M 101-DF6, and M 101-DF7 are in projection closer to NGC 5485 such that it is reasonable to use this galaxy as the major external field provider, the case of M 101-DF5 is less clear. It has an on-sky separation of 44 arcmin to NGC 5485, and only 10 arcmin and 23 arcmin to the other nearby galaxies NGC 5475 and NGC 5443, respectively.

Table 2 .

 2 The predicted velocity dispersions.Nameσ typical σ min σ max km s -1 km s -1 km s -1Notes. * denotes that potentially the ultra-diffuse galaxy is embedded in multiple strong external fields, in which case we present the smallest velocity dispersion value from all of them.

	NGC 7814-DGSAT-2	4.6 3.0	16.2
	NGC 7814-DGSAT-7	7.4 4.9	13.5
	NGC 1052-DF4	17.3 9.2 *	23.4
	NGC 1052-DF1	6.8 3.7 *	15.8
	NGC 1052-DF2 1	15.3 10.2 25.1
	NGC 1084-DF1	19.8 16.6 24.1
	NGC 2683-DGSAT-1 10.9 7.0	27.4
	NGC 2683-DGSAT-2 10.1 6.7	19.0
	M 96-DF6	8.7 5.6	16.5
	dw1055+11	10.5 7.4	15.4
	dw1117+15	8.5 5.7	13.7
	NGC 3625-DGSAT-2 2 9.6 7.0	13.7
		6.1 4.0	11.0
	NGC 3625-DGSAT-4 16.5 13.9 20.7
	NGC 3625-DGSAT-3 14.7 12.4 18.5
	NGC 3669-DGSAT-2 11.7 9.8	16.0
	NGC 3669-DGSAT-3 17.3 14.5 23.7
	NGC 4594-DGSAT-1	5.1 3.5	21.5
	Cen A-MM-Dw1	9.6 6.3	17.1
	M 101-DF5	13.9 6.6 *	20.8
	M 101-DF6	5.1 3.2	12.8
	M 101-DF4	8.4 5.6	17.3
	M 101-DF7	4.6 2.9	12.0

Article number, page 2 of 9 Oliver Müller et al.: Predicted MOND velocity dispersions for a catalog of ultra-diffuse galaxies in group environments

Acknowledgements. O.M. thanks the Swiss National Science Foundation for financial support. We thank the referee for their useful and constructive report.