Free Access
Issue
A&A
Volume 633, January 2020
Article Number L3
Number of page(s) 8
Section Letters to the Editor
DOI https://doi.org/10.1051/0004-6361/201936821
Published online 03 January 2020

© ESO 2020

1. Introduction

Faint galaxies with large effective radii have been known since the 1980s (e.g. Sandage & Binggeli 1984; Impey et al. 1988; Bothun et al. 1991; Dalcanton et al. 1997), but more recently, the name ultra-diffuse galaxies (UDGs; van Dokkum et al. 2015a) has been coined for galaxies with very similar characteristics. These are galaxies of low stellar density, defined to have low central surface brightness (μg(0) > 24 mag arcsec−2) and an effective radius (Re) of over 1.5 kpc (Re is the radius which encloses half the total flux from a galaxy; de Vaucouleurs 1948). The question of whether UDGs represent a separate class of galaxies is still under debate. Currently, known UDGs that have been discovered in clusters (Koda et al. 2015; Mihos et al. 2015; Muñoz et al. 2015; van der Burg et al. 2016; Román & Trujillo 2017a; Venhola et al. 2017; Mancera Piña et al. 2018), in groups (Román & Trujillo 2017b; Cohen et al. 2018), and in the field (Bellazzini et al. 2017; Prole et al. 2019) can have Re as large as 5 kpc which is comparable to that of large (i.e. giant) Milky Way (MW)-like galaxies. This fact has been used to suggest that UDGs are “failed” giants (van Dokkum et al. 2015a). As Re captures (at most) the central parts of giant galaxies, whether this radius can be used to fairly compare the sizes of UDGs to the more massive galaxies is questionable.

The reason why Re is incapable of reaching the boundaries of massive galaxies is that according to its definition it depends on how the light is concentrated in these objects. Therefore, if one considers that the sizes of galaxies are indicated by the location of their edges or boundaries (similar to everyday objects), then Re is undeniably a poor measurement of size. However, the idea of associating the sizes of galaxies to the location of their boundaries (or something very close to them) is not recent. Galaxy size has also been measured using limiting surface brightness isophotes, such as for example R25 (Redman 1936) or the Holmberg radius (RH; Holmberg 1958), to characterise the maximum area that galaxies spanned on the photographic plates of that era. However, similar to the effective radius, the isophotal radii were also initially defined for operational purposes and do not directly encompass any physical meaning. In spite of this, isophotal radii (and their variants; see e.g. Hall et al. 2012) as well as sizes based on light concentration, for example Re, R90 (Nair et al. 2011), and R80 (Miller et al. 2019), are being used in important scaling relations such as the fundamental plane (Djorgovski & Davis 1987), size–stellar mass (Shen et al. 2003), or the size–virial radius relation (Kravtsov 2013), to study the history and formation of galaxies.

With the aim of finding a physically motivated characterisation of galaxy size, Trujillo et al. (2020, hereafter TCK20) proposed a size parameter based on the location of the gas density threshold for star formation in galaxies. We found that the size–stellar mass relation with this measure for size has an intrinsic dispersion of only ∼0.06 dex, which is three times smaller than that of the relation with Re as galaxy size (∼0.18 dex), over five orders of magnitude in stellar mass 107M <  M <  1012M. The proposed parameter is also able to capture the boundaries of the stellar distribution of galaxies and can thus represent how large or small these objects are, in contrast to the effective radius. We refer the reader to TCK20 for an in-depth discussion on the fundamental meaning of these results and why such a size definition is different from the ones that only measure the extent of galaxies down to a given surface brightness level (e.g. RH or R25).

To complement the results presented in TCK20, in this Letter we study the implications of using the effective radius as a size measure for UDGs; and how this affects our understanding of these galaxies. We compute the physically motivated size parameter defined in TCK20 for a sample of UDGs and compare their sizes to those of dwarfs with stellar masses 107M ≤ M ≤ 108.5M as well as to the sizes of MW-like galaxies (1010M <  M <  1011M) studied in TCK20. Throughout this work we assume a standard ΛCDM cosmology with Ωm = 0.3, ΩΛ = 0.7 and H0 = 70 km s−1 Mpc−1.

2. Data and sample selection

In order to have a homogeneous dataset of dwarfs and UDGs both in depth and filter coverage, we use publicly available background-rectified imaging data in the g and r-bands of the deep IAC Stripe 82 Legacy Project1 (hereafter IAC Stripe82, Fliri & Trujillo 2016; Román & Trujillo 2018). The UDGs are taken from Román & Trujillo (2017b) and Trujillo et al. (2017). For completeness, two iconic UDGs outside the Stripe 82 footprint were also added to the sample. Imaging data for DF44 (van Dokkum et al. 2015b), a representative example of a UDG with a large Re, was obtained from the Gemini archive (GN-2016A-FT-18, PI: P. van Dokkum) and DECaLS data2 was used for [KKS2000]04 (popularized as NGC1052-DF2)3. The control sample analysed consists of 155 dwarf galaxies with stellar masses in the range 107M ≤ M ≤ 108.5M studied in TCK20. We focus on this stellar mass regime as it overlaps with the mass range of the selected UDGs for this work. Galaxies with stellar masses in the range of 1010M <  M <  1011M from the TCK20 catalogue (449 objects) are also selected to represent MW-like systems.

The IAC Stripe82 and DECaLS images are of similar depth with a limit in surface brightness of μg​​​ = ​​​29.1 mag arcsec−2 (3σ;10 × 10 arcsec2). The depth of the Gemini coaddition is μg ∼ 30 mag arcsec−2 (3σ;10 × 10 arcsec2). Only the g- and i-band data were available for [KKS2000]04.

Ultra-diffuse galaxies R2 and R3 from Román & Trujillo (2017b) were removed from the UDG sample due to light contamination in the galaxy outskirts produced by surrounding bright sources and/or stars. Therefore, our final sample includes 12 UDGs. None of the dwarf galaxies in our control sample satisfy the criteria for a UDG as in all the cases μg(0) < 24 mag arcsec−2.

3. Method

The entire analysis of this study was carried out on individual image stamps with dimensions of 100 × 100 kpc2 (for the dwarfs and UDGs) and 600 × 600 kpc2 (for the massive galaxies) in the rest frame of each galaxy. The scattered light from point sources was removed from the IAC Stripe82 image stamps using our extended (∼8 arcmin radius) point spread functions for this telescope4 (Infante-Sainz et al. 2019). All sources surrounding the galaxy of interest in the image were then masked using MTObjects (Teeninga et al. 2016), setting move_up = 0.3.

To derive the surface brightness profiles of the galaxies in our sample, the axis-ratio (q) and position angle (PA) of the galaxies were obtained by fitting an ellipse to an average isophote of 26 mag arcsec−2 in the g-band images. The centre, q, and PA of each galaxy were then visually verified prior to further analysis. These parameters were fixed and elliptical annuli were used to create the radial profile of each galaxy as well as its growth curve in flux which is needed to determine Re.

The g − r colour (g − i for DF44) profiles were derived from the surface brightness profiles and converted to mass-to-light ratio (M/L) profiles in g using the relationships from Roediger & Courteau (2015). These M/L and surface brightness profiles in the g-band for all galaxies were then converted to stellar mass density (Σ) profiles (see Eq. (1) in Bakos et al. 2008) and used to ascertain our size parameter (TCK20). The profiles were also integrated up to the μg = 29 mag arcsec−2 isophote to derive the stellar masses of the galaxies, M. Various stellar density thresholds (within the limit in depth of the images used) can easily be determined from such profiles. Here, we use R1, the radius at which Σ = 1 M pc−2, as a proxy for the location of the gas density threshold for star formation (see TCK20). For details on the background subtraction of the data, correction of the profiles due to the inclination effect and Galactic extinction, and an estimation of the uncertainties related to our measurements (stellar mass and background) we refer the reader to TCK20.

All of our measurements for the UDGs are provided in Appendix A. The measurements for the dwarf sample can be found in the online version of TCK20. For comparison, we also show the distributions of the UDGs and dwarfs using an isophotal size indicator, the Holmberg radius (RH), in Appendix B. Lacking the B-band, we used the isophote at 26 mag arcsec−2 in the g-band as a proxy for RH.

4. Results

Figure 1 shows an example of a UDG and a representative dwarf galaxy (i.e. one that lies very close to the centre and best-fit line in both the observed Re– and R1–stellar mass relations). Both galaxies have similar stellar mass (∼108M). Their corresponding surface brightness and mass density profiles are shown, and the locations of Re (dotted), R1 (solid) and the radius where Σ = 0.5 M pc−2 (called R0.5; dot-dashed, see Appendix C) are marked in the image and profiles. The reason why the effective radius of the UDG is large in comparison to that of the regular dwarf galaxy shown is because the dwarf galaxy has active star forming clumps in its central region. The presence of such clumps in these galaxies means that flux will be more concentrated in the centre which decreases the effective radius and increases their central surface brightness. Consequently, such dwarfs will not be characterised as a UDG. Similar clumps or bright regions are not usually present at the centre of UDGs which makes their effective radii larger compared to the majority of dwarfs.

thumbnail Fig. 1.

Illustration of the consequence of using Re as galaxy size for UDGs and dwarf galaxies. Here we show two galaxies of similar stellar mass at the same physical scale: UDG-B5 (top) and a representative dwarf galaxy (SDSS J224114.12−003715.0, bottom). The colour image is the gri-band composite with a grey-scaled background for contrast and contours showing Re (dotted), R1, (solid) and R0.5 (dot-dashed). The surface brightness and stellar mass density profiles derived for both galaxies are also shown.

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This last point is further demonstrated in Fig. 2 where the Σ profiles of dwarf galaxies and UDGs are over-plotted in three panels, corresponding to galaxies in three stellar mass bins: 107M ≤  M <  3 × 107M, 3 × 107M ≤ M <  108M, and 108M ≤ M ≤ 3 × 108M. Ultra-diffuse galaxies tend to be less concentrated than regular dwarfs, with lower central densities by a factor of two to three. However, the global extensions of both types of galaxy are very much alike.

thumbnail Fig. 2.

Stellar mass density profiles of dwarfs (grey) and UDGs (colours) belonging to three stellar mass bins (left to right).

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Figure 3 shows the Re–stellar mass and R1–stellar mass scaling relationships (top panels) and their distributions with respect to the best-fit lines (bottom panels). Three main features are: (1) The dispersion of the observed R1–stellar mass plane (0.086 ± 0.007 dex) is a factor of 2.5 smaller than that of the observed Re–stellar mass plane (0.213 ± 0.014 dex) for dwarf galaxies (see also TCK20); (2) UDGs populate the upper portion of the Re–stellar mass plane. A simple Kolmogrov–Smirnov (KS) test using Re/Re, fit gives an extremely small p-value of 2.1 × 10−5. After removing two galaxies that have Re <  1.5 kpc (according to our measurements) in our initial UDG sample, namely [KKS2000]04 (NGC 1052-DF2) and UDG-R1 (from Román & Trujillo 2017b), the p-value decreases to 9.3 × 10−6. Both values indicate that the null hypothesis – that dwarf galaxies and UDGs in this sample arise from the same distribution in size – can be rejected; and (3) UDGs are populated among the dwarf galaxies in the R1–stellar mass plane, showing no evidence that the distributions in R1/R1, fit of these galaxies are significantly different (p-value = 0.07). After removing the two galaxies with Re <  1.5 kpc, the p-value increases to 0.09. Therefore, the null hypothesis cannot be rejected. The UDGs shown in this work have extensions that correspond to those of dwarfs. We repeated this exercise using another popular size indicator, the Holmberg radius, and found similar results (see Appendix B).

thumbnail Fig. 3.

Comparison between Re and the physically motivated size parameter for UDGs and dwarfs. Top: Re–stellar mass relation (left) and the R1–stellar mass relation (right) for dwarfs (grey) and UDGs (colours). The best-fit line of each relation for the dwarf sample is also over-plotted. The upper left corner of each plot shows the typical uncertainty in our measurements (see TCK20). Bottom: histograms showing the distribution of Re/Re, fit (left) and R1/R1,fit (right) where “fit” refers to the best-fit line of each relation for the dwarf sample.

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On average, the location of R1 in surface brightness for galaxies in this mass range is μg(R1) ∼ 27 mag arcsec−2, but can be as faint as 28.5 mag arcsec−2. Lower stellar mass densities are even more faint (e.g. R0.5, see Appendix C), reinforcing the importance of high-quality deep images to conduct this work.

Finally, we highlight our main results in Figs. 4 and 5. Figure 4 demonstrates how using a physically motivated size parameter that captures the global extension of galaxies reveals the radical difference between the sizes of UDGs and MW-like galaxies (right panel), in contrast to the effective radius (left panel). While using Re indicates that UDGs have similar extensions to MW-like galaxies, R1 shows that the MW-like systems are, on average, ten times larger than the classical dwarfs and UDGs. In fact, the null hypothesis is completely rejected when the R1 distributions of UDGs and MW-like galaxies are compared (see also Appendix C). This result is further illustrated in Fig. 5 where an elliptical galaxy (SDSS J223954.96−005918.97) reminiscent of M 87, a MW-like spiral galaxy (SDSS J012015.34−002009.00), and the dwarf galaxy and UDG of Fig. 1 are shown to the same physical scale. The TType labels for the elliptical and spiral galaxies were taken from Nair & Abraham (2010). We see that R1 better represents the edges of galaxies compared to Re and prevents any misleading notion about the actual extension of galaxies. We emphasise the fact that the galaxies are shown to a similar depth in surface brightness. Therefore, the strikingly different sizes are not a result of the quality of the imaging data.

thumbnail Fig. 4.

Histograms showing the size distribution of UDGs, dwarfs, and MW-like galaxies. In Re (left), UDGs overlap with the dwarfs and MW-like systems in our sample and in R1 (right), the UDGs clearly separate from the MW-like galaxies and overlap with the dwarfs. These results show that UDGs have the extensions of dwarfs.

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thumbnail Fig. 5.

Top: representative galaxies of different stellar masses: a giant elliptical (M ∼ 2.5 × 1011M), a Milky Way-like galaxy (M ∼ 2.5 × 1010M), a UDG and dwarf galaxy (M ∼ 108M). All galaxies are shown to the same physical scale and a similar depth in surface brightness. Bottom: stellar mass density profiles of the same galaxies. The coloured ticks in the upper x-axis mark the location of Re. Similar ticks in the lower x-axis mark the location of R1.

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5. Discussion

The aim of this study is to investigate how the effective radius, as a size measure for UDGs, affects our understanding of these galaxies. We illustrate that the effective radii of UDGs will generally always be larger compared to that of dwarf galaxies due to the absence of luminous clumps (or substructure like bulges in the case of more massive galaxies) in their central regions. The fact that the large effective radii of UDGs can be compatible with those of MW-like galaxies has led to the interpretation that UDGs are “Milky Way-sized” (see e.g. van Dokkum et al. 2015a; Koda et al. 2015), when perhaps the more accurate statement is rather that “UDGs are less concentrated in light than dwarfs and MW-like galaxies”. For this reason, we adopted the physically motivated size measure that we developed in TCK20. As this size parameter is also better than the effective radius at representing how large or small galaxies are, we used it to fairly compare the sizes of UDGs with those of dwarfs and MW-like galaxies. Contrary to previous accounts, we demonstrate that the sizes of MW-like galaxies and UDGs are actually radically different. As a matter of fact, the sizes of UDGs with our definition (as well as with the Holmberg radius) are compatible with those of dwarfs.

However, while the KS test shows no evidence that the size distribution of the UDG and dwarf galaxy populations are different, most of the UDGs lie in the upper half of the R1–stellar mass relation (Fig. 3). This could be related to the incompleteness in our dwarf control sample arising from the spectroscopic target selection criteria of the Sloan Digital Sky Survey (SDSS), requiring that the r-band Petrosian half-light surface brightnesses of targets are at least μ50 ≤ 24.5 mag arcsec−2 (Strauss et al. 2002). The lack of faint low-mass galaxies in our dwarf sample can also be seen in the stellar mass density profiles in Fig. 2 where there are almost no dwarf galaxies with compatible central densities to UDGs. Any such bias due to spectroscopic incompleteness in our dwarf sample will also equally affect the Re–stellar mass plane. It is therefore an acceptable exercise to compare these relationships quantitatively. Were the control sample not affected by this potential incompleteness, the similarity between the dwarfs and UDGs in the distribution of their R1 as well as Re would be even greater. This should be the case when future deep spectroscopic studies (e.g. Ruiz-Lara et al. 2018; Ferré-Mateu et al. 2018) target more dwarf galaxies with lower central densities and essentially fill the upper portions of both these relations.

The use of the effective radius as a galaxy size measurement has also led to a confusion as to whether UDGs are associated to the dark matter haloes of MW-like (high-luminosity) or dwarf galaxies. Several analyses using simulations (e.g. Di Cintio et al. 2017; Chan et al. 2018) and observations (e.g. Beasley et al. 2016; Beasley & Trujillo 2016; Amorisco et al. 2018) already support the idea that UDGs reside in haloes comparable to those of dwarfs. Our results lend further support to this idea. As the size of a galaxy is believed to be proportional to the virial radius of its halo (Kravtsov 2013), the fact that our UDG sizes agree with those of dwarfs strongly suggest that both types of galaxy occupy the same dark matter haloes.

6. Conclusions

We used a proxy for the gas density threshold for star formation as a size indicator for UDGs and dwarf galaxies. We compared the size distribution of these galaxies in a physically important parameter space – the size–stellar mass plane – and show that there is no evidence that the size distributions of UDGs and dwarfs are different. The UDGs have sizes that are within the size range of dwarfs. The same result holds using the Holmberg radius as a size indicator. If low-mass, extremely diffuse Milky Way-sized galaxies exist, then in our definition of size, they need to have a radius of about 25 kpc. Such galaxies have not been found in present-day imaging surveys. Our results reinforce the importance of using physically meaningful properties in order to fairly compare classes of galaxies and draw conclusions about their nature.


3

At a distance of 13 Mpc (Trujillo et al. 2019), [KKS2000]04 no longer satisfies the criterion (Re >  1.5 kpc) to be defined as a UDG. Nevertheless, due to the popularity of this object after being reported as a “galaxy lacking dark matter” (van Dokkum et al. 2018), we include it in our sample for analysis.

Acknowledgments

We thank the anonymous referee for providing useful comments. We also thank Javier Román and Raúl Infante for providing the extended SDSS point spread functions of all filters used in this work. We acknowledge Rodrigo Carrasco for kindly allowing us to use his reduction of the galaxy DF44 from the Gemini archive. NC thanks Caroline Haigh for providing the latest version of MTObjects. She also thanks Claudio Dalla Vecchia, Jorge Sánchez Almeida and Mike Beasley for interesting conversations. We acknowledge support from the State Research Agency (AEI) of the Spanish Ministry of Science, Innovation and Universities (MCIU) and the European Regional Development Fund (FEDER) under the grants with references AYA2016-76219-P and AYA2016-77237-C3-1-P, from the European Union’s Horizon 2020 research and innovation programme under Marie Skłodowska-Curie grant agreement No. 721463 to the SUNDIAL ITN network, and from the Fundación BBVA under its 2017 programme of assistance to scientific research groups, for the project “Using machine-learning techniques to drag galaxies from the noise in deep imaging”. The IAC projects P/300624 and P/300724 is financed by the Ministry of Science, Innovation and Universities, through the State Budget and by the Canary Islands Department of Economy, Knowledge and Employment, through the Regional Budget of the Autonomous Community.

References

Appendix A: Table of measurements

Table A.1.

Measured parameters for the sample of UDGs.

Appendix B: Comparison with an isophotal size indicator: the Holmberg radius

Similar to Figs. 3 and 4, Fig. B.1 shows the distribution of UDGs, dwarfs, and MW-like galaxies using the Holmberg radius. Here we define the Holmberg radius using the isophote at 26 mag arcsec−2 in the g-band. The middle panel clearly demonstrates that UDGs have sizes that are within the size range of dwarf galaxies. The KS test using RH/RH, fit gives a p-value of 0.54. Finally, the lower panel shows that the sizes of UDGs are not compatible with those of MW-like galaxies. Therefore, our conclusions are further reinforced by taking a widely used isophotal size indicator. In other words, our conclusions regarding the sizes of UDGs are not related with our specific size definition to describe the extensions of galaxies.

thumbnail Fig. B.1.

Distribution of UDGs and dwarf galaxies using the Holmberg radius. Top: Holmberg radius (RH)–stellar mass plane for dwarfs (grey) and UDGs (colours). The best fit line of the relation for the dwarf sample is also over-plotted. The upper left corner of the plot shows the typical uncertainty in our measurements (see TCK20). Middle: histogram showing the distribution of RH/RH, fit where “fit” refers to the best-fit line of each relation for the dwarf sample. Bottom: histogram showing the RH distribution of UDGs, dwarfs, and MW-like galaxies.

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Appendix C: Other stellar mass density proxies to measure the size of dwarfs and UDGs

In TCK20, we proposed a size indicator based on the location of the gas density threshold for star formation in galaxies. On both theoretical and observational grounds, we selected the location of an isomass contour at 1 M pc−2 as a proxy for such a value. While this value is motivated by the location of the disc truncation in MW-like galaxies (see Martínez-Lombilla et al. 2019), for galaxies such as dwarfs where the level of star formation is lower, a better proxy for their gas density threshold could be given by a lower isomass contour (Leroy et al. 2008; Huang et al. 2012). Therefore, we repeated our analysis using an isomass contour at 0.5 M pc−2 instead of 1 M pc−2 as a galaxy size indicator for UDGs and dwarfs (see Fig. 1 for two examples). We call this size parameter R0.5.

Figure C.1 shows the size distribution of UDGs and dwarfs using R0.5 and that of MW-like systems using R1. Although R0.5 increases the sizes of the UDGs and dwarf galaxies as expected, their extensions never reach those of MW-like galaxies. Similar to the results shown in Fig. 4 (right panel), the null hypothesis is completely rejected upon comparing the size distributions of UDGs and MW-like galaxies. Therefore, our conclusion that UDGs do not have sizes comparable to MW-like galaxies remains unchanged even with the use of another gas density threshold for size.

thumbnail Fig. C.1.

Histograms showing the size distribution of UDGs (R0.5), dwarfs (R0.5) and MW-like galaxies (R1).

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All Tables

Table A.1.

Measured parameters for the sample of UDGs.

All Figures

thumbnail Fig. 1.

Illustration of the consequence of using Re as galaxy size for UDGs and dwarf galaxies. Here we show two galaxies of similar stellar mass at the same physical scale: UDG-B5 (top) and a representative dwarf galaxy (SDSS J224114.12−003715.0, bottom). The colour image is the gri-band composite with a grey-scaled background for contrast and contours showing Re (dotted), R1, (solid) and R0.5 (dot-dashed). The surface brightness and stellar mass density profiles derived for both galaxies are also shown.

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In the text
thumbnail Fig. 2.

Stellar mass density profiles of dwarfs (grey) and UDGs (colours) belonging to three stellar mass bins (left to right).

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In the text
thumbnail Fig. 3.

Comparison between Re and the physically motivated size parameter for UDGs and dwarfs. Top: Re–stellar mass relation (left) and the R1–stellar mass relation (right) for dwarfs (grey) and UDGs (colours). The best-fit line of each relation for the dwarf sample is also over-plotted. The upper left corner of each plot shows the typical uncertainty in our measurements (see TCK20). Bottom: histograms showing the distribution of Re/Re, fit (left) and R1/R1,fit (right) where “fit” refers to the best-fit line of each relation for the dwarf sample.

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In the text
thumbnail Fig. 4.

Histograms showing the size distribution of UDGs, dwarfs, and MW-like galaxies. In Re (left), UDGs overlap with the dwarfs and MW-like systems in our sample and in R1 (right), the UDGs clearly separate from the MW-like galaxies and overlap with the dwarfs. These results show that UDGs have the extensions of dwarfs.

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In the text
thumbnail Fig. 5.

Top: representative galaxies of different stellar masses: a giant elliptical (M ∼ 2.5 × 1011M), a Milky Way-like galaxy (M ∼ 2.5 × 1010M), a UDG and dwarf galaxy (M ∼ 108M). All galaxies are shown to the same physical scale and a similar depth in surface brightness. Bottom: stellar mass density profiles of the same galaxies. The coloured ticks in the upper x-axis mark the location of Re. Similar ticks in the lower x-axis mark the location of R1.

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In the text
thumbnail Fig. B.1.

Distribution of UDGs and dwarf galaxies using the Holmberg radius. Top: Holmberg radius (RH)–stellar mass plane for dwarfs (grey) and UDGs (colours). The best fit line of the relation for the dwarf sample is also over-plotted. The upper left corner of the plot shows the typical uncertainty in our measurements (see TCK20). Middle: histogram showing the distribution of RH/RH, fit where “fit” refers to the best-fit line of each relation for the dwarf sample. Bottom: histogram showing the RH distribution of UDGs, dwarfs, and MW-like galaxies.

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In the text
thumbnail Fig. C.1.

Histograms showing the size distribution of UDGs (R0.5), dwarfs (R0.5) and MW-like galaxies (R1).

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In the text

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