Detection of UFDGs with Gaia

Detection of satellite remnants in the Galactic Halo with Gaia – III. Detection limits for Ultra Faint Dwarf Galaxies

Abstract

We present a method to identify Ultra Faint Dwarf Galaxies (UFDGs) candidates in the halo of the Milky Way using the future Gaia catalogue and we explore its detection limits and completeness. The method is based on the Wavelet Transform and searches for over-densities in the combined space of sky coordinates and proper motions, using kinematics in the search for the first time. We test the method with a Gaia mock catalogue that has the Gaia Universe Model Snapshot (GUMS) as a background, and use a library of around 30 000 UFDGs simulated as Plummer spheres with a single stellar population. For the UFDGs we use a wide range of structural and orbital parameters that go beyond the range spanned by real systems, where some UFDGs may remain undetected. We characterize the detection limits as function of the number of observable stars by Gaia in the UFDGs with respect to that of the background and their apparent sizes in the sky and proper motion planes. We find that the addition of proper motions in the search improves considerably the detections compared to a photometric survey at the same magnitude limit. Our experiments suggest that Gaia will be able to detect UFDGs that are similar to some of the known UFDGs even if the limit of Gaia is around 2 magnitudes brighter than that of SDSS, with the advantage of having a full-sky catalogue. We also see that Gaia could even find some UFDGs that have lower surface brightness than the SDSS limit.

keywords:
The Galaxy: halo, formation - galaxies: dwarf - dark matter - methods: data analysis - Astronomy & Celestial Mechanics: astrometry
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1 Introduction

The current cosmological cold dark matter paradigm posits the assemblage of large structures in the Universe from smaller ones (Press & Schechter, 1974; White & Rees, 1978; Springel, Frenk & White, 2006). A galaxy like ours must have formed by the merger of a large number of smaller systems, that even today, must be still in the process of being accreted. A discrepancy between the predicted and observed number of galaxy satellites has given rise to the so called “missing satellite problem” (Klypin et al., 1999; Moore et al., 1999). However, in recent years, an entirely new population of hitherto unknown systems with very low luminosity and surface brightness, dominated by dark matter, the so called “Ultra Faint Dwarf Galaxies” (UFDGs), has been discovered, opening up the possibility of resolving this problem (e.g. Simon & Geha, 2007; Bullock, 2010). The knowledge of their structural properties, chemical abundances and stellar populations is also key to understanding fundamental issues (see review by Belokurov, 2013) like the process of star formation and the role of feedback in these relatively low-mass environments (Brown et al., 2014); how to distinguish between a dwarf galaxy and a globular cluster in some extreme cases (e.g. Segue 1 and 2, Willman 1, Boo II and CmB, Forbes & Kroupa, 2011); or to what extent UFDGs could have contributed to the stellar population found in the Galactic halo today (e.g. Kirby et al., 2008).

So far, all known UFDGs were discovered as over-densities in deep large-area photometric surveys, the vast majority in SDSS (e.g. Willman et al., 2005a, b; Zucker et al., 2006a, b; Belokurov et al., 2006, 2007, 2008, 2009, 2010), together with recent findings in Pan-STARRS (Laevens et al., 2015) and the Dark Energy Survey (DES, Koposov et al., 2015; Bechtol et al., 2015).

The ESA Gaia mission, launched in December 2013, offers excellent prospects for the discovery of new members of the UFDG population. Gaia (de Bruijne, 2012; Perryman et al., 2001) will measure accurate positions, parallaxes and proper motions for all stars out to its survey limit of (, depending on the colour of the source), where is the white light photometric pass-band of Gaia (Jordi et al., 2010). Multi-colour photometry will be obtained for all stars and radial velocities will be collected for stars at mag, where indicates the pass-band of the Radial Velocity Spectrograph on-board. Gaia will also provide astrophysical information on all the sources observed, primarily through multi-colour photometry. The astrophysical parameters of all Gaia sources will be provided as part of the survey data products (Bailer-Jones et al., 2013).

Although the Gaia survey is not as deep as SDSS, Pan-STARRS or DES, it is all sky at a spatial resolution comparable to that of Hubble Space Telescope (HST), and will deliver high accuracy astrometry (positions and proper motions) for all sources. The combination of these unique features is what makes the comparatively shallow survey of Gaia potentially powerful in the search for UFDGs. Here we aim to exploit this in a technique to identify UFDGs. The combination of positions and kinematics has proven to be most efficient in the search for dark matter subhalos in cosmological simulations (e.g. Behroozi, Wechsler & Wu 2013; Onions et al. 2012). But, to our knowledge, this is the first time that both configuration space and kinematics are included in the search of UFDGs. As we will show, Gaia will enable us to probe parts of the UFDG parameter space which have not been covered before, and will allow for a comprehensive study of the spatial distribution around the Milky Way (MW) of this faint galaxy population.

The present work continues the series (Brown, Velázquez & Aguilar, 2005; Mateu et al., 2011), in which we have assumed the task of building ever more realistic Gaia mock catalogues, and used them to test tools that we have introduced to detect and characterize substructure in the stellar halo of our Galaxy.

In Section 2 we introduce our Gaia mock catalogue, which serves as our laboratory to study the detectability of UFDGs. This includes a stellar background and our synthetic UFDGs. The details of the Gaia selection function and error model used to generate the Gaia observables are described as well. In Section 3, we present our detection tool, which consists of a peak identifier that is applied in the sky and proper motion planes, a cross-matcher that identifies peaks with common members in both planes, and a procedure to evaluate the statistical significance of the matched peaks. Section 4 presents our results. Detection limits are shown as a function of astrophysical parameters and of “effective parameters”, namely a combination of the former on which our detection method depends directly. In Section 5 we summarize the limits of our method as well as the assumptions that it is based on. Our conclusions are presented in Section 6.

2 The Gaia mock catalogue

Our Gaia mock catalogue is the stage where we assess the success, efficiency and limits of our UFDGs detection technique. As such, it represents a controlled, but realistic environment. There are several elements that compose it. First, we need a model of the Galaxy, from which suitable stellar backgrounds can be extracted (Section 2.1). We also need a mass model for our synthetic UFDGs and a stellar population model (Section 2.2). The latter is because our UFDGs are not merely ensembles of particles, but stellar properties must be assigned to them, as they impinge on the value and quality of their Gaia observables. The previous elements allow us to assemble an extensive library of UFDGs at various distances and with a wide range of intrinsic parameters, projected against stellar backgrounds at several Galactic latitudes. We then use a Gaia selection function and error model to transform the theoretical quantities into realistic Gaia observables, as our detection method should work based on them only (Section 2.3). In Section 2.4 we present the filtering method that we use to eliminate foreground stars. Finally, we examine the nature of the UFDGs projections in the sky and proper motion planes, as these are the basic input variables that our method works on (Section 2.5).

2.1 The Galactic Background Model

We use as a Galactic background the Gaia Universe Model Snapshot (GUMS) from Robin et al. (2012), which is a simulated catalogue of the sources expected to be observed by Gaia, at a fixed epoch. It includes the simulation of Galactic sources, Solar system and extragalactic objects.

We note here that Gaia will observed large numbers (potentially millions) of galaxies and about half a million QSOs (de Bruijne et al., 2015; de Souza et al., 2014), which will all appear as faint point sources and could thus complicate the search for UFDGs. However, discrete source classification will be part of the published data (Bailer-Jones et al., 2013) and in this work we assume that we can rely on this to filter out galaxies and QSOs (but see Sect. 5). Therefore, we use only Galactic sources, and restrict the catalogue to a range in latitude of , to avoid the crowding and high extinction expected near the Galactic plane.

The Galactic sources in GUMS are generated based on the Besançon Galactic Model, which includes the Galactic Thin and Thick Disks, Bulge and Halo, based on appropriate density laws, kinematics, star formation histories, enrichment laws, initial mass function (IMF) and total luminosities for each of the component populations, described in detail in Robin et al. (2012). Objects are simulated with masses down to the hydrogen burning limit, corresponding to spectral types down to . Binary and multiple star systems are also simulated (see details in Arenou, 2011), introduced with a probability that depends on the mass and evolutionary state of the primary star. The probability distribution for the separations is assumed to be a log-normal with the parameters reported by Duquennoy & Mayor (1991) for (primary) stars down to solar masses, and from Close et al. (2003) for low mass stars.

2.2 The UFDG Model

The Dynamical Model

For our basic synthetic UFDG dynamical model we use a simple Plummer sphere with isotropic velocity distribution. A particular realization of this model is uniquely defined by two of the following parameters:

  • : Total mass

  • : Core radius

  • : Half–mass radius

  • : Velocity dispersion (3-D)

The total gravitational binding energy of a Plummer sphere is:

By virial equilibrium, we can establish a relation between the total mass, core radius and velocity dispersion:

In appropriate astronomical units, the previous relation is:

Observationally, the scale–length usually reported is the half–light radius. Under the assumption of a position independent mass–to–light ratio (i.e. well mixed), the half–mass radius and its light counterpart coincide. We assume this and use indistinctly as the half–light, or the half–mass radius. For a Plummer sphere, the relation between the core and half–mass radius is: . Then, the previous relation between velocity dispersion, mass and radius can be written as

(1)

For a model with a given and , the mass-to-light ratio of the UFDG is given by the ratio of its total mass , derived from Eq. 1, and the chosen total V-band stellar luminosity . The number of particles in the realization () and the total stellar mass () are a consequence of the assumed total luminosity, star formation history and stellar mass function.

The stellar population model

Figure 1: Hess diagram vs intrinsic colour for the UFDG’s stellar population model (see text for details). The colour scale is proportional to the logarithmic number of stars in each bin. The right -axis indicates the maximum distance up to which a star with a given will be observable by Gaia, given the expected magnitude limit of (assuming ). For the gray bands see discussion at end the of Section 2.3

We simulate the stellar population of the UFDGs as a single star formation burst with an age of Gyr and metallicity , assuming a Chabrier (2003) IMF. We use the HB13 Stellar Population Synthesis code from Hernández-Pérez & Bruzual (2013), which allows for a consistent treatment of isolated and binary stars. The prescriptions assumed in this code are similar, yet not identical, to those used for the statistical orbital properties of binaries in GUMS (see Section 2.1). In HB13, the binary probabilities and orbital parameters are randomly drawn and assigned to each primary star in the population at age zero and the evolution is followed using the Hurley, Tout & Pols (2002) binary evolution code. Binary probabilities are assumed to depend on the mass of the primary using the prescription from Lada (2006), and the distribution of periods, and thus, separations, of Duquennoy & Mayor (1991). The resulting versus intrinsic colour Hess diagram for the stellar population used for the UFDGs is shown in Fig. 1.

Parameters of the simulated UFDGs

Each of the simulated UFDG has 9 free parameters:

  1. intrinsic parameters:

    • Total V-band luminosity:

    • Half–light radius:

    • Velocity dispersion:

  2. extrinsic parameters:

    • Heliocentric distance: D

    • Position in the sky (, )

    • Galactocentric velocity vector modulus:

    • Azimuthal and latitudinal orientation angles of the galactocentric velocity vector: ,

() () () () () () () () () () ()
ranges - 5- 1-500 10-250 0-180 0-90 24-550 0-360 -90 0.04- 10- 10-
fiducial 80 10 20 90 30 453 0 0 902 94
Table 1: Ranges of parameters in the simulated UFDGs (first row) and parameters of the UFDG used as our fiducial case. See Sections 2.2.3 and 2.3 for definitions. and can be obtained from the other parameters.
Figure 2: Left: vs. . Right: vs. . The yellow dots correspond to members of our UFDG library. Note that this library contains only systems with at least 10 stars observable by Gaia. Solid lines indicate constant total-mass models. Known UFDGs and classical dwarf spheroidal galaxies are shown with green diamonds and blue squares respectively (data from McConnachie, 2012). The labels correspond to: Sgr (1), For (2), LeoI (3), Scl (4), LeoII (5), Sex (6), Car (7), UMi (8), Dra (9) for blue squares and CVnI (1), Her (2), Boo (3), UMa (4), LeoIV (5), CVnII (6), UMaII (7), CmB (8), BooII (9), WilI (10), SegI (11), SegII (12), LeoV (13) for green diamonds. We do not include LeoT, which would not be observable with Gaia, and PscII that lacks measurements on some parameters.

We have generated a set of libraries with a total of 30 000 UFDGs covering large ranges of the 9 parameters (see Table 1). Our main library is generated with the following parameters drawn at random: i) the number of stars that would be observable by Gaia , ii) the heliocentric distance , iii) the apparent size of the UFDGs in the sky and iv) in the proper motion plane , and v) the center-of-mass velocity. These quantities are described in detail in Sections 2.3 and 2.5. The first four parameters are generated from a uniform distribution in a logarithmic scale. For the last one, the angles and and the modulus are generated following a uniform distribution, with between zero and the local escape velocity for the Galaxy3. The remaining parameters (namely , and ) are obtained from the ones above. We also require to be at least 10 in this library. This library is designed with particular goals described in detail in Sections 4.2 and 4.3.

Fig. 2 illustrates the range explored in half-light radii and velocity dispersion (left panel), as well as in mass–to–light ratio and total V-band luminosity (right). The observed values of these parameters for known UFDGs and classical dSph galaxies are shown with green diamonds and blue squares, respectively. Note that the range explored by our synthetic library (yellow dots) is much larger than the observed one for , and . In particular, the large range covered in results in a very large range of (the library spans an even larger range of than shown in Fig. 2). We are pushing the limits of the parameter space explored, towards regions where the detection would be observationally more difficult, i.e. towards larger and (top and right areas of left panel), and lower luminosity and high (top and left areas of right panel). The fact that is generated uniformly, together with the large scatter in luminosity for small due to stochastic effects, produces the diffuse boundary in in the right panel.

2.3 The Gaia selection function and error model

Here we present our model for the Gaia observations that includes the selection function and the Gaia error model that we apply to the GUMS model and the simulated UFDGs. The Gaia observables are the 5 astrometric parameters (, , , , ), the radial velocity, the Gaia photometry (including the Gaia magnitude and the two broad band magnitudes and ). The final Gaia catalogue will also provide three atmospheric parameters (metallicity, surface gravity and effective temperature) and extinction. The true values for these observables and parameters are obtained directly from the models. The conversion from the Johnson-Cousins photometric system to Gaia magnitudes is done following the transformation given in table 3 from Jordi et al. (2010). We do not consider extinction because all fields used in our study are at relatively high latitudes (at least ).

The GUMS model and the simulated UFDGs include binary and multiple systems. To determine which ones will be resolved by Gaia, we use a prescription used within the Data Processing and Analysis Consortium4. In this model the minimum angular separation on the sky that Gaia can resolve depends on the apparent magnitudes of the stars in the system, with the minimum separation being mas. For the unresolved cases, a single detection is considered by computing the total integrated magnitude, averaging positions and taking the atmospheric parameters (such as surface gravity) of the primary star in the system.

As an example, if we take a field5 of centered at and , there are initially 25 521 objects, from which 57 per cent are single stars, 13 per cent are stars of resolved multiple systems and 30 per cent are unresolved systems6. For a simulated UFDG at kpc, these fractions are 63, 6 and 31 per cent, respectively.

To simulate Gaia -like errors for the GUMS catalogue and the simulated UFDGs, we use the code presented in Romero-Gómez et al. (2015), updated to the post-launch performance7 as described in de Bruijne, Rygl & Antoja (2015). Up to date information is available from the Gaia web pages8. The uncertainties on the astrometry, photometry and spectroscopy are mainly functions of the magnitude and colour. The geometrical factors and the effect of the number of passages due to the scanning law are also taken into account9. For the surface gravity we take a constant error of 0.25 dex, based on table 4 of Bailer-Jones et al. (2013). Lacking a model of the Gaia performances for unresolved systems, we use the same prescriptions as for single or resolved stars. Only stars with magnitude (the Gaia magnitude limit) are considered.

From all the Gaia astrometric observables, we can not make use of parallaxes to infer distances to UFDGs stars. The median relative error in parallax of the stars in the UFDGs in the range of distances considered here is at least of 70 per cent and on average 170 per cent, since they can be very faint and distant objects. Besides, radial velocities are not available for most of the cases as 90 per cent of the UFDGs in the range of distances explored here have at most 10 per cent of stars that are brighter than the magnitude limit of the Gaia spectrograph (). Therefore, we use as our observables only the two angular positions in the sky ( and ) and the two proper motions ( and ). The errors in the angular coordinates in the sky are of the order of 0.05–0.4 whereas in proper motion these are about 0.03–.

Figure 3: Number of stars observed by Gaia , as a function of the total V-band luminosity and distance of the UFDG.

The number of UFDG stars seen by Gaia (not the same as the total number of stars in the realization ) is determined by the total stellar luminosity of the system and the distance of the UFDG (given an assumed stellar population model). On the right axis of Fig. 1 we indicate the distance limit associated to the Gaia deepest magnitude . Note that at distances larger than only giant stars are observed. In Fig. 3 we show the number of UFDG stars observed by Gaia as function of luminosity and distance. For instance, UFDGs of luminosity around have no stars bright enough to be observed by Gaia beyond but they will have around 15 observable stars around . The oscillations with distance present at around 25, 60 and are because the type of stars of the UFDG population that Gaia can detect changes as it is observed at different distances, depending on whether or not features like the main sequence turn-off are observable. These distances have been marked in the Hess diagram of Fig.1 and they correspond to the main sequence turn-off, the extreme horizontal branch and the horizontal branch, respectively. Note also the stochasticity around small numbers of observable stars.

Figure 4: Left: vs. . Right: Apparent size in the proper motion plane vs. apparent size in the sky. The yellow dots show the range spanned by the synthetic UFDG library. Known UFDGs and classical dwarf spheroidal galaxies are shown with green diamonds and blue squares respectively (data from McConnachie, 2012). The labels are as in Fig 2.

The left panel of Fig. 4 shows the range covered by our library in number of stars observable by Gaia and distance to the Sun. To include the known UFDGs and classical dSphs in this plot we have computed assuming the stellar population model described in Section 2.2.2, and the total luminosity and distance reported by McConnachie (2012) for these systems10. Here we can see that there are real systems that go beyond the range covered by our library towards small number of observed stars. We must remember that is the number of stars that would be seen by Gaia, which has a mag brighter limit than the SDSS 11 used to identify those systems (e.g. Belokurov et al., 2007). This is a limitation imposed by Gaia that we cannot get around. Note also that the boundaries of the regions spanned by the library in this panel are sharp by construction (see Section 2.2.3).

2.4 Filtering the foreground

Along a given line-of-sight (LOS), it is important to minimize the number of background stars12 with respect to the number of stars in the UFDG. We use a parallax cut to filter out foreground disk stars, which have large parallaxes with small errors. Thus, we discard data for stars with mas, i.e. an observed parallax which, within the errors, corresponds to distances smaller than . We also filter out foreground disc dwarfs with the implementation of a surface gravity cut: we discard stars with , where is the atmospheric parameter derived from the Gaia  observables. With these two cuts, we reduce typically by an order of magnitude. For instance, there were 25 521 stars in the GUMS model in our fiducial field ( and ) and with the cuts we reduce this number to =1 413.

We stress that these cuts have been designed to have minimal loss of observable stars from the UFDGs, particularly at relatively large distances (kpc), for which dwarf stars will not be observable by Gaia. The fraction of lost stars is up to 70 per cent for nearby UFDGs at . However, it goes down to 30 per cent at and is less than 10 per cent for distances larger than . Nevertheless, with the cuts we are maximizing the relative fraction of UFDG stars with respect to the background in all cases, given that the fraction of stars lost from the background is larger than that of the UFDGs. It is our experience that it is difficult to devise a unique algorithm that can identify our target systems at all distances, and so, limits are introduced as a necessary compromise. It is clear that specifically tailored algorithms could be used for nearby cases.

2.5 The sky and proper motion planes

Figure 5: Sky (top) and proper motion plane (bottom) for the field of our fiducial UFDG in Table 1. The stars belonging to the system are shown as green dots while the background stars are in black.

The starting point of our detection procedure (Section 3) is the UFDGs projections in the sky and proper motion planes, thus it is essential to understand the behaviour of these projections of the systems and the background.

Fig. 5 shows the stars in the customary field of view of our fiducial simulated system in the sky (top) and proper motion planes (bottom). The parameters of this system are listed in Table 1. The stars belonging to the UFDG are coloured in green, while the background stars are in black. This system is hardly seen in the sky plane because it is very diffuse. But note how it is much more compact in the other plane. The compactness of the UFDGs in the proper motion plane is a general characteristic of most of our simulated UFDGs that improves considerably our search (Section 4), being a fundamental advantage of the Gaia data.

Note also the very different nature of the background in these planes. In the sky the background is roughly constant, but becomes markedly non–uniform in the proper motion plane. This requires a special treatment when assigning significance to the peaks (Section 3.3).

The apparent sizes of an UFDG in the sky and proper motion planes are set by its intrinsic size and velocity dispersion, combined with its distance from the Sun. These sizes can span a wide range in both planes. The half-light angular size is given by

(2)

In our synthetic library, varies between (see Table 1) and and between 10 and (this is the approximate distance limit to detect at least 10 stars with Gaia, for a luminous UFDG with ). This implies a range of apparent angular sizes of , ].

In the proper motion plane, the apparent size is

(3)

For values of in the range and again between 10 and , we end up with a range of apparent sizes of (but see below).

The right panel of Fig. 4 shows the sizes spanned in the sky and proper motion planes by the UFDGs in our library. As we will see in Section 3, these two parameters are the most important, together with the number of visible stars in the UFDG, in determining the detectability of the system. We can see here that our library extends well beyond the spread covered by real systems. Note again that the boundaries of the library are sharp in this panel, as it is generated with apparent parameters drawn at random from a uniform distribution in a logarithmic scale.

Figure 6: Top: median error in for synthetic UFDG with velocity dispersions around () at different heliocentric distance. Bottom: dispersion in the proper motion of the same set of synthetic UFDG (red dots). The black curve shows the median error in calculated in logarithmic bins from the top panel. The error bars correspond to the standard deviation. The blue line is the expected size according to Eq. 3 for .

It is also important to note that the apparent size of an UFDG in the proper motion space is greatly influenced by the observational errors. To illustrate this, we use a set of simulated UFDGs located at different distances and with velocity dispersions between and . Each black dot in the top panel of Fig. 6 shows the error in (similar for ) in each simulated UFDG computed as the median of all the individual stars errors in each UFDG. The proper motion error slightly increases with distance, as one would naively expect due to the fainter magnitudes. But the error also oscillates with distance. This is because the Gaia performances depend on the magnitude and colour of the star and the type of stars in the UFDG that Gaia can detect, and thus the fraction of stars with certain magnitudes and colours, changes as it is observed at different distances (as seen in Section 2.3). One can see that the error has minima around 30, 70 and . These are distances slightly larger than the ones at which a sudden increase in the number of stars of certain types occurs. They correspond to the main sequence turn-off, the extreme horizontal branch and the horizontal branch, as discussed previously (gray shaded stripes in Fig. 1).

The bottom panel of Fig. 6 shows the real size in the proper motion plane (red dots) computed as the standard deviation of the proper motion coordinate () of the stars in each UFDG. We also overplot the error in () at each distance (black curve) taken as the median error in logarithmic bins from the top panel. The blue line in this plot shows the expected size according to Eq. 3 for a velocity dispersion of . We see that the sizes of the UFDGs decrease up to and for larger distances they follow the oscillations due to the Gaia errors. Once the size of the UFDGs is dominated by the observational error, the apparent size oscillates between 0.1 and . For smaller velocity dispersions, e.g. , the errors dominate already at a distance of . Therefore, the minimum apparent size of the UFDGs is set by the observational errors. For the range of parameters explored here, this is above in per cent of the cases. In what follows we take instead of as a better measure of the apparent size of the UFDG in proper motion space.

As seen in Section 2.3, the errors in the angular coordinates in the sky are around 0.05-0.4 , which is negligible compared to the apparent sizes of the UFDGs. For this reason, we do not observe a similar effect in the sky plane.

3 The detection tools

In this section, we present all the different elements that compose our detection method to identify UFDG candidates against the background.

Our strategy is as follows. We consider fields of view of in the sky. We first detect over-densities independently in the sky and the corresponding proper motion planes. For this we use the Wavelet Transform (WT, see Section 3.1). We do this for over-densities of different amplitudes and sizes, and keep the most significant ones. After this, we perform what we call the cross-match of peaks (Section 3.2). This consists in counting how many stars belong simultaneously to a certain peak in the sky and to a certain peak in proper motion space. We do this for all pairs of peaks of any size between both spaces. For each cross-match, we finally compute the probability that the observed number of common stars is just a coincidence (Section 3.3 to 3.5). Cross-matches with a low probability are selected as possible UFDG candidates. Below we detail each of the steps of our method.

One may wonder why this separate treatment for the sky and proper motion planes. After all, what we are looking for is a single peak in the 4-D space of positions in the sky and proper motion planes. This is because of the very different nature of these two planes, which results in the impossibility of having a natural metric in the combined space. Any metric will imply the introduction of an arbitrary dimensional scale which will limit the nature of the systems found. This is why we have preferred to work on the sky and proper motion planes separately, and then use the cross match procedure to relate peaks. The peaks that we do identify correspond to single peaks in the combined 4-D space, but not necessarily using a unique metric, as our combination of different wavelet scales in both planes allows for a larger range of identified peaks than if using a single metric.

Although the whole detection process might seem complex, it is quite straighforward from the computational point of view. The entire algorithm takes a total of 40 s to run for our fiducial field of in a single Intel(R) Core(TM) i7-3770 CPU @3.40GHz. This might change depending on the LOS but, as a first approximation, the celestial sphere above would require 86 h of CPU time, which in fact can be spread into several CPU for different LOS.

3.1 Wavelet analysis

Figure 7: Wavelet Transform (WT) at different scales for our fiducial UFDG for the sky (top) and proper motion planes (bottom). The black dashed circle shows the true position and size of the system in each plane. The circle in the proper motion plane is very small but can be seen better in the smaller scales. The position is calculated as the median of the coordinates (positions and proper motions) of the stars in the UFDG that are observed by Gaia, while the size of the circle is taken as the maximum between the standard deviation of the coordinates. Red, orange and green crosses indicate peaks at between 1 and 2, between 2 and 3, and significance, respectively.

To detect over-densities in the sky and proper motion planes like the ones of Fig. 5, we use the WT (Starck & Murtagh, 2002). This can be thought of as a “localized” Fourier transform that gives information about certain frequencies and where in the image these frequencies are located. Due to the wide range of apparent sizes of our simulated UFDGs (see Section 2.5), our method needs to be able to detect over-densities of different sizes. In the application here a discrete set of frequencies (i.e. scales) are probed and we get information about the localization of those particular structures. We use here the à trous (“with holes”) variant of the WT (Starck & Murtagh, 2002) which computes a discrete set of scale-related “views” of a 2-D function or image. We have previously used this technique to detect moving groups in the stellar velocity distribution of the Solar Neighborhood and surroundings (Antoja et al., 2008, 2012). To perform the calculations we use the MR software developed by CEA (Saclay, France) and Nice Observatory.

Although the WT works at a specific scale, it can identify over-densities within some range in size. Nevertheless, it is important to realize that we are probing a discrete set of scales in the images, and therefore, it is critical to choose those scales wisely. We explore 4 logarithmically spaced scales in each plane within the ranges found in Section 2.5. For the sky, as we are dealing with fields of , we have chosen the scales , , and . Even though the higher scale puts a limit on the maximum size of an UFDG that can be detected in principle, the innermost parts of the more luminous UFDGs can still be detected, even if they have larger angular sizes. For the proper motion plane, we use scales of , , and . Here, what we are missing are exceptional cases with extremely high velocity dispersion which are very close.

An example of the WT planes in the sky for our fiducial UFDG in Table 1 is shown at the top part of Fig. 7, while the bottom part shows the proper motion plane. In each case, the four scales mentioned are shown. The blue colours are proportional to the values of the WT.

After the WT, we search for relative maxima to detect the over-densities. The algorithm computes the Wavelet Probability (WP), that is the probability that the detected over-densities in the wavelet space are not due to Poisson noise. For this it uses a model for this type of noise in wavelet space. This is done by first using the Anscombe transform (Anscombe, 1948) that converts a signal with Poisson noise into Gaussian noise, for which the treatment in the WT planes is more straightforward (see Starck & Murtagh 2002 and references therein). Here we will consider only WT peaks that have a WP of being real detections of per cent (green crosses in Fig. 7), (orange crosses) and (red crosses), similar to 3, 2–3 and 1–2 significance levels in the Gaussian case, respectively. The size of the crosses in this figure indicates the size or scale that is being probed in each WT plane (also indicated in the top of the plots). There is an additional condition on the over-densities: they should have at least 5 stars to be considered a peak.

The UFDGs are optimally detected (i.e. with higher WP) when the scales probed are similar to their apparent sizes. In the example of Fig. 7 a black circle indicates the position and extension of the UFDG in the sky plane and in the proper motion plane. In the sky images, for scales that are smaller than the apparent size of the UFDG (two first panels), some peaks are detected inside the region occupied by the UFDG but with low WP (1 or 2, red and orange crosses). For larger scales, which in this case are similar to the apparent size of the UFDG, the detection is above 3 (green crosses). In other cases, the detection is always below 3 or below 2 because the UFDGs can be very diffuse in this plane, as already highlighted in Section 2.5.

On the other hand, the UFDG is very compact in proper motion space, and it stands out as an over-density for all scales studied in the left part of the panels. In our example, the fiducial UFDG is always detected above 3 (green crosses). In other cases, the best detection is for a particular scale that is close to the apparent size of the UFDG.

Note also how in both planes, a number of low–WP random detections appear (most of red crosses in Fig. 7). Because of this, we need to discard false over-densities and keep only good UFDG candidates (Section 3.2). Besides, in the proper motion case, two over-densities with high WP are also detected in the center of the distribution for the two largest scales. These correspond to the peaks of the background distribution, which as we have seen in Fig. 5, is not uniform. Note that the distribution of background stars in the proper motion plane will be different for each LOS and therefore, its centroid will shift to different positions in this plane. This does not occur for the sky plane which presents a uniform background.

3.2 Cross-matching peaks in the two planes

So far, we have detected peaks in the sky and proper motion planes, separately. However, contrary to false detections, an UFDG is an over-density in the 4-D combined space ---. This is precisely the feature that we need to exploit to identify UFDGs, beyond what has been currently achieved.

To do this, we list the stars contributing to the peaks identified separately in the sky and proper motion planes. By stars belonging to a certain peak, we mean those that are enclosed in a circle around the peak with the radius of the WT scale in the considered plane. In practice, because Gaia is a point source catalogue, we can identify the stars by their id number. Then we see whether a large fraction of these stars belong simultaneously to a certain peak in the sky and a peak in the proper motion plane. We call this “cross-match of peaks”. This cross-match is done for every peak and at every scale in the sky, compared to every peak at every scale in the proper motion plane.

The computation of the probability of having this cluster of common stars occurring by chance is computed as described in Section 3.3. In Section 3.4 we explain how we filter out false detections. Because each UFDG can be detected in more than one scale, we also need to keep only independent detections. This is explained in Section 3.5.

3.3 Assessing the probability of the detections

Here we describe the statistics machinery that we devised to assess the probability of detection, i.e. compute which detections have a very low probability of occurring by chance.

We are interested in , i.e. the probability of observing a certain number of common stars in a peak in the sky and a peak in the proper motion plane, given the expected number of common stars . This probability13 is simply given by the Poisson probability distribution function

(4)

An estimate of is given by:

(5)

where is the expected number of stars in the peak and describes the (normalized) number density of stars in the proper motion plane, both under the assumption that no UFDG is present. indicates the area of the peak over which we are integrating, which is a circle with a radius given by the WT scale, centered on the coordinates of the peak in question. For convenience, we use hereafter the logarithm of the probability, .

For simplicity, we assume that the background density in the plane is uniform, which is reasonable for the field size used, and therefore , where is the total area of the field in the sky plane (in our case 4 ), is the wavelet scale in the plane of the sky and is the number of background stars in the field. The latter is computed from the observed data itself, by taking the 8 fields adjacent to our problem field, with the same total area. For each of these fields we compute the total number of stars and we take the median. This is a better estimation of the number of background stars than the total number of stars in the considered field, specially in cases of luminous UFDGs that have a number of observed stars that is not negligible compared to the number of background stars.

In the proper motion plane, however, it is crucial to account for the fact that the density is not constant and this is achieved by the integral term in Eq. 5, which, multiplied by gives the number of common stars expected to lie within the area of the detected peak.

The distribution of stars in the proper motion plane is different depending upon the direction on the sky, so it must be computed independently for each field. We do this from the observed data itself, by taking the mentioned 8 adjacent fields with the same total area. For each of these fields we compute the density as a (normalised) 2-D histogram in the plane with a pixel size of . Taking these eight 2-D histograms, we compute the pixel-by-pixel median density to obtain a statistically reliable estimate in each of the matrix cells, minimizing the effect of outliers. With this, we are assuming that the distribution of background proper motions remains similar among these adjacent fields. This is indeed the case. For instance, variations in the median proper motion in longitude and latitude among the adjacent fields are in general smaller than the pixel size. We numerically evaluate the integral in Eq. 5 using the trapezoid rule in 2-D and bi-linear interpolation on the median density matrix.

It could happen that one or various of the adjacent fields contains UFDGs. This would yield a wrong estimation of and . The fact that we use the median of the 8 fields helps to alleviate this issue. However, in case very luminous UFDGs are present, our algorithm checks if the number of stars in one of the adjacent fields is significantly larger than in the others. This is done by checking that the dispersion in the number of stars in the 8 fields is not larger than 2.5 times the square root of the median. In this case, the algorithm could be re-run without the field in question.

Instead of using the probability of Eq. 4, we can also use the significance , defined as the number of times above the expected value of the distribution, scaled to the dispersion of the distribution

(6)

The advantage of using instead of is that is positive and it increases for more relevant detections.

3.4 Setting a threshold probability for detection

l b
90 30 86.8 150.2 86.7 105.0 85.4 50.3 83.4 0.3 81.7 0.0
90 42 85.7 44.9 85.4 38.7 84.3 1.2 83.0 0.0 81.6 0.0
90 55 84.6 37.4 84.4 35.9 83.6 1.8 82.6 1.1 81.8 0.3
90 68 81.4 39.0 81.2 38.9 80.8 27.8 80.2 0.3 79.3 0.0
90 80 84.6 7.8 84.2 0.5 83.4 0.1 82.8 0.0 82.2 0.0
170 30 87.7 68.6 87.4 67.5 85.9 0.4 84.6 0.1 83.4 0.0
170 42 84.7 6.3 84.4 0.6 83.8 0.2 83.2 0.0 82.6 0.0
170 55 79.9 47.6 79.7 43.8 79.3 0.4 78.8 0.0 77.8 0.0
170 68 82.2 1.7 82.1 0.5 81.7 0.0 81.5 0.0 80.6 0.0
170 80 82.9 3.2 82.6 0.4 81.5 0.0 81.1 0.0 80.8 0.0
Table 2: Detections, real and false, as percentage of the number of UFDGs used in each LOS.

We are only interested in those detections which have very low probability (very negative ) and also a number of common stars14 . However, the central peak(s) of the background in the proper motion plane can appear also as a detection. This is because we estimate its expected number of stars using the adjacent fields and any small fluctuation above this value can give a significant detection, though with a larger . To filter peaks that correspond to the background and not to the UFDGs, we can conservatively select a relatively low (very negative) threshold value for , below which we consider detections to be relevant. However, we must realize that, as we lower the threshold value , although we eliminate spurious peaks, we start losing relevant detections, so this is a compromise between false positives and losing bona fide peaks.

We have explored this compromise on various LOS’s. In Table 2 we list the percentage of recovered UFDGs  and of false detections from the total number of tested UFDGs , as a function of 5 different values for the threshold and 10 different LOS’s. For values above -9.0 there are several fields where the percentage of false detections is above 20 per cent. For a threshold of -12.0, all false detections are at most 1.1 per cent. Although we could choose a in between those two to make it the most optimal, we conservatively choose .

3.5 Independent Detections

As explained before, the cross-match of peaks is done for all peaks at all scales and a given UFDG can be detected in more than one scale. This means that we need to select which of the many detections made in a given field, are in fact independent detections, i.e. different objects.

We first organize all detections by increasing , choose the detection with the lowest and compare its and coordinates with the remaining detections. Now, we choose the next independent detection as the one with the lowest that lies, both in the sky and proper motion planes, at a distance larger than the sum of the WT scales of the two detections (i.e. they do not overlap). We repeat this procedure until we have gone through all the available (relevant) detections.

Figure 8: as a function of the number of common stars for our fiducial field and UFDG. The color scale is proportional to , i.e. the fraction of recovered stars from each UFDG. The dashed horizontal line mark the line below which we consider detections as relevant. The detection with the lowest (marked with a cross) is what we take as the best independent detection (see text for details).

To illustrate the behaviour of and the selection of independent detections, in Fig. 8, we plot for our fiducial UFDG, as a function of the number of common stars for all detections in this field, that is the results of cross-matching all peaks at all scales in the sky and proper motion plane. In the plot we use a color scale proportional to , defined as the number of stars from those that truly belong to the UFDG divided by the total number of stars originally in the UFDG. In other words, is the fraction of recovered stars from each UFDG. Dots below the horizontal dashed line are relevant detections, i.e. with below (Section 3.4).

There is a correlation of with . As expected, detections with larger numbers of common stars have on average lower . There is also a sequence that moves across the plot above the threshold. This corresponds to peaks in the proper motion background (note that they are black points, i.e. with no stars belonging to the UFDG). As explained before, these detections are filtered by our threshold.

Also, detections with the largest values of have low , i.e. they are significant detections. However, as the number of stars in common increases, the value of decreases, reaches a minimum and then increases again. The minimum value occurs for detections at the optimum scales in the sky and proper motion planes. It is in this case that a large fraction of UFDG stars lie in the detected peak inside the WT scale, and the background is sufficiently low, so that the difference between the observed and expected number of common stars in the peak is maximal. Increasing the WT scale past the optimum values causes the inclusion of more stars of the UFDG in the peak but also more stars of the background that might not necessarily belong to both peaks in the sky and proper motion plane simultaneously, and hence, this causes to go back to larger values. This is a very convenient behaviour which allows us to select detections at the optimum WT scales. The detection with the lowest (marked with a cross) is what we take as the best (and in this case, only) independent detection. Finally, notice that in this example no false positives are picked up.

4 Results

As we have seen in Section 2.2.3, we face a 9-D parameter space. Even with our library of more than 30 000 different synthetic UFDGs, it is clear that we can cover only a limited amount of this vast hypervolume.

To explore this space with some order, we will rely first on a series of carefully curated ensembles of cases. In each one, all parameters, except two, will be kept fixed (Section 4.1). This allows us to take 2-D sections of the original parameter space. Then we will identify in Section 4.2, a reduced number of combinations of the original parameters that our detection procedure depends on directly, and which we call “efective parameters”. In Section 4.3, we explore the limits and completeness of our method in the space of effective parameters, as well as in some of the most relevant original parameters. The effect introduced by changing the background level as we look at different LOS’s is discussed in Section 4.4.

4.1 The physical parameter space

Figure 9: Detectability tests run with several ensembles of UFDGs with only two varying parameters: vs top), vs (middle), vs distance D (bottom). The color scale indicates the detection significance . UFDGs with significance over 200 have been plotted with a colour saturated at this value. Black crosses indicate UFDGs that were not detected.
() () () () () () () () ()
80 5 20 90 30 300. 0. 0.
Table 3: Values for the fixed parameters in the ensembles of UFDGs shown in Fig. 9.

In the ensembles of tests presented here, we vary only 2 parameters, keeping the other 7 parameters constant. The values for the fixed parameters are shown in Table 3.

The vs plane

In the first test we use synthetic UFDGs with varying and . As indicated in Table 3, all UFDGs are located at a fixed position in the sky at , , at a heliocentric distance of and have a luminosity of . These are approximately the mean values for the observed UFDGs. The velocity dispersion varies logarithmically between 3 and and the half-light radius between and . Due to stochastic variations in each realization, despite the luminosity and distance being constant, the number of detectable stars varies between 38 and 207. The results of the test are shown in the top panel of Fig. 9. Each symbol (squares and crosses) in this plot corresponds to one simulated UFDG. The color scale in the panels is proportional to the detection significance (Eq. 6). Black crosses indicate UFDGs that were not detected. From this plot we can evaluate the detection limits as a function of and .

UFDGs with larger than are not detected (for this fixed distance and luminosity). This is because their apparent size in the sky is very big, making them extremely diffuse. In fact, results in an angular size equal to the sky fields that we are using for our analysis (). We also notice that for velocity dispersions below the detection significance depends mainly on the half-light radius (vertical contours). This is because in this regime the apparent size of the UFDG in proper motion is in fact constant and set by the observational errors (Section 2.5). Above this velocity dispersion, the contours bend slightly to the left, meaning that for a given size in the sky, the detection is more significant for lower velocity dispersions. Note that we are exploring velocity dispersions up to , i.e. significantly larger than the typical velocity dispersion of of known UFDG and classical dSph galaxies (McConnachie, 2012).

The vs plane

The middle panel of Fig. 9 is a test with 625 UFDGs, where the velocity dispersion and the modulus of the velocity vector are varied. In this case, is varied linearly instead of logarithmically. Notice that in this ensemble, we only change the position and spread of the UFDG peak in the proper motion plane. In particular, we have chosen the values for the fixed velocity angles ( and ), so that the UFDG peak position moves horizontally across the proper motion plane as we vary , covering all possible contrasts between background and UFDG, and coinciding with the background peak for .

Note how for a fixed value of , the best detections are the ones for lower velocity dispersions, which produce more concentrated peaks. Besides, something that immediately stands out from this plote, compared with the others shown in Fig. 9, is the shallow variation in the detection significance across the entire part of this plane. At the horizontal region around , we see that the significance is the lowest, as we expected, but this is a very subtle effect. This lack of sensitivity indicates that, although they play a role, these two parameters (and specially ) have little effect on the detectability of the UFDG.

The vs plane

The bottom panel of Fig. 9 shows the results of the experiment where luminosity and distance were varied between and , and between 10 and , respectively. There are 509 UFDGs in this test. Their mass–to–light ratio is between and . Their observable number of stars varies between 2 and . Note that here we consider a lower value for the minimum than 10 as indicated in Table 1 to sample in detail the detection limit. We find, however, that the minimum number of that gives a positive detection is 5 for this particular example.

In this test, there are several competing effects. For a fixed luminosity, as we increase the distance, the size in the sky and proper motion planes decreases, which favours identification, but on the other hand, the number of visible stars also decreases, which makes identification harder. The first effect scales as , while the second, being an individual star luminosity problem, scales as . So, at large distances the latter dominates and we lose the UFDGs, as seen in this panel. For instance, UFDGs with luminosities around are not detected beyond . Also, given a fixed distance, more luminous objects are detected with higher significance.

There is an interesting modulation in the colour contours in this panel. There are three leftward indentations of better detections at around 10, 25 and , which are better seen in the red and light-blue colours. These features are not statistical fluctuations, but the result of the effect of the Gaia observational errors in measured proper motions. As explained in Sect 2.5 (Fig. 6), the size of the UFDGs in proper motion plane changes with distance in a peculiar way, presenting several minima at around the mentioned distances. At these distances, therefore, the UFDGs are slightly more concentrated in proper motion and, hence, easier to detect. The upper left part of the panel, which does not contain any coloured squares or black crosses, is the region where systems do not have stars that can be observed by Gaia.

From this simple tests, one can see that some properties of the UFDGs are more relevant for the detections. In particular, the luminosity and distance, which set the number of observable stars together with the apparent size in the sky, seem to have a larger impact on the significance of our detections, than the size in proper motion space and the position of the peak with respect to the background.

4.2 The “effective” parameter space

If we look at the essence of our problem devoid of its astronomical context, our task is to identify common peaks in two different planes, subject to a noisy and not necessarily uniform background. Seen as such, the key parameters upon which a successful detection depends are: the height of the peaks compared with the background level and the spread of the peaks, that is the apparent sizes of the UFDGs in the sky and proper motion planes , and the number of observed stars that they are composed of , with respect to the background. The probability will also depend on the projection of the center of mass velocity in the proper motion plane, since this determines how close the UFDG peak appears to the center of the proper motions distribution, where the majority of the background contaminants lie.

One can also see that for a certain UFDG that has been detected in the optimal scales (that is almost all stars in the UFDG are enclosed inside the joint peak detection), , where is the number of background stars that fall inside the joint peak detection. Assuming that the number of background stars inside the joint peak is similar to the expected one, that is , the significance of Eq. 6 is equivalent to

(7)

As , it follows from Eq. 7 that UFDGs that have the same ratio and all the rest of the parameters equal (including the LOS), have also approximately the same significance15. The same applies to UFDGs with the same ratio . For this reason, given a certain LOS (we deal with different LOS in Section 4.4) we can describe our detection problem based on these two quantities and , together with the position of the peak in proper motion space. We call these the “effective parameters”. The latter has, however, less relevance compared to other properties, as already seen.

These quantities depend in turn on other astronomical parameters that characterize the system and its position with respect to the observer. But the successful detection of an UFDG depends only on a limited number of combinations of them, that is on the effective parameters. The importance of the effective parameters is that they reduce the dimensionality of the parameter space where we need to determine the boundaries of successful detection of our procedure. In particular, while we describe our UFDGs by using 9 physical parameters, the detection of these systems depends only on 3 (and mainly 2) effective parameters.

Here we test that this concept is indeed correct. To evaluate the dependency of the detection limits and significance on these effective parameters, we have built a library of UFDGs with varying , but keeping constant the apparent sizes in the sky and proper motion planes, as well as projected center of mass velocity. Note that there is no straightforward way of generating UFDGs with the same real size in proper motion space because of the effects of observational errors in proper motion (Section 2.5), and therefore we do it approximately by generating UFDGs with constant , assuming that . Thus in this exercise we vary the physical parameters , and in a way that their combination (Eqs. 2 and 3) result in constant and . We also change with distance in order to obtain the same proper motion peak for these UFDGs.

Figure 10: Detectability tests run with a library of synthetic UFDGs built with constant apparent size in the sky and proper motion plane , as well as projected center of mass velocity. Regions of constant effective parameters and are vertical lines in this plot. The panel show the significance as function of the physical parameters distance (left axis) and half-light radii (right axis) as a function of . UFDGs with significance over 200 have been plotted with a colour saturated at this value. Black crosses indicate UFDGs that were not detected.

Fig. 10 shows the significance of the UFDGs of this experiment as function of and distance (left axis). The right axis shows the half-light radii which is related to through Eq. 2 to produce the same . Although we do not include them in this plot, one could also draw other axes for and , which are also related to to produce the same and the same peak position in proper motion space. As and are constant for all these UFDGs, the effective parameters and are constant along vertical lines in this plot. We see that equal significance contours are approximately vertical (but see discussion below), which illustrates that indeed, for constant effective parameters the significance does not depend on the physical parameters (, , ) for these experiments (contrast this with the case of Fig. 9) but only on and .

However, the colours do not follow exactly vertical contours. This is due to the effects of observational errors in proper motion which make the apparent size of the UFDGs in the proper motion plane oscillate with as already explained in Section 2.5 (Fig. 6). This is the same effect as in Fig. 9 (bottom panel). Here we generated these UFDGs with constant, but not . In Fig. 10 we can see how UFDGs at around are better detected because at this distance the apparent size decreases. Although there is a similar effect around and , these are not so clearly seen here.

4.3 Limits of detection and completeness

Having seen that UFDGs with the same effective parameters have the same significance, we can explore the detection limits and completeness as function of them alone. In the following we show the significance of a large synthetic library (15 000 UFDGs) that spans a large range in the effective parameters. All of the physical parameters of the UFDGs in this test are varied, except the LOS ( and ).

Figure 11: Top: Detectability tests run with a large library of synthetic UFDGs as function of two effective parameters: and . The color scale indicates the significance . UFDGs with significance over 300 have been plotted with a colour saturated at this value. Black crosses indicate UFDGs that were not detected. Bottom: Significance of the detections of the same library but being these only detections in the sky plane (see text for details).

Fig. 11 (top panel) shows the results of this test as function of the two effective parameters and . The colours are well separated, i.e. not strongly mixed, in this plot, showing that despite the physical properties of the UFDGs being very different16, the significance of the detections depends mainly of these two effective parameters. Though the third effective parameter (position of the peak in proper motion space) changes in the UFDGs of this test, the uniformity of the colours in a certain region of this plot indicates that its influence is not as relevant as that of the other parameters, as already shown.

More in detail we can also see that for a higher fraction of the plot and specially the upper half, the colours follow a approximately vertical structure, i.e. the significance is mainly given by . For the lower part, the contours are more curved. Finally, note also how the undetected objects lie in the regions of low and/or low , i.e. most diffuse objects.

We find that the minimum significance of our positive detections is . This is because of the threshold imposed to in order to filter false detections (see Section 3.4).

Figure 12: Fraction of detected UFDGs as function of effective parameters. Black squares with a central white dot are regions where the fraction recovered is exactly equal to 0 (this is to differentiate from regions with small recovered fraction). The blue squares and the green diamonds show the estimated positions of classical dSphs and UFDGs, respectively, with labels as in Fig. 2. White contours indicate the detection significance from Fig. 11 (top).

In Fig. 12 we use the same test described above to estimate the fraction of detected UFDGs in each region of the effective parameter space. To do this, we have binned logarithmically this space and computed how many of the generated UFDGs in each bin are successfully detected. We only plot bins with at least 4 simulated UFDGs. The median number of UFDGs in each cell is 14. Note how for most of the space explored this fraction is close to 1 (ocher colours). There is a transition zone where fractions go from to . To differentiate regions with small recovered fraction from regions with this fraction equal to 0 (all of them with dark colours), we have marked the latter with a white central dot. The region where our method is not able to detect objects is the low and/or low , as expected.

In this plot we superpose white contours indicating the significance of the detections from Fig. 11 (top), computed as the median significance in the same grid used in this plot, including the cases that were not detected, that is with , in the computation. Note that these contours are just approximate, given that there is some dispersion in the significance. For instance, around the contours of and , the dispersion in is of 5-10. We see how the transition zone corresponds to the region of significance roughly around . For significance higher than 10, the recovered fractions is between 0.7 and 1.0.

We also plot in Fig. 12 the estimated values of effective parameters for the known MW satellites (classical dSph and UFDGs, blue squares and green diamonds, respectively). The values of for these know systems have been estimated by interpolating in a plot similar to Fig. 6 (bottom) but for velocity dispersions between 5 and . As expected, all of the classical satellites of the MW lie in a region of effective parameters where our algorithm applied to the mock Gaia data successfully detects all simulated systems. Note that some of the classical satellites lie outside the higher limits of the plot. Out of the 13 UFDGs that would have observable stars by Gaia (Leo T would not be observable), 1 of them lies in a region where is 1.0 (Boo (3)) and 4 of them lie in regions with recovery fraction of 0.9 (CVnI (1), Her (2), UMaII (7), CmB (8)). Besides, WilI (10), UMa (4), SegII (12) and BooII (9) are recovered with fractions of 0.8, 0.7, 0.6, 0.5, respectively. SegI (11) and LeoIV (5) are in regions with of 0.3 and 0.1, respectively. Finally, CVnII (6) and LeoV (5) are outside the limits of detection. Note that in this plot we see how systems with effective parameters similar to the known UFDG would be detected by our algorithm. But some of the known UFDG have a small number of observables stars (e.g. LeoIV (5) and SegII (12) have and , respectively) and our tests are done for a minimum of . Nevertheless, for these low luminosity known cases, the estimated is very uncertain.

It is outstanding that our tests indicate that it is possible to detect with Gaia UFDGs similar to some of the ones detected by SDSS which is mag deeper. This is because, whereas the UFDGs of SDSS were detected with photometry alone, our search is done also in the proper motion plane. The bottom panel of Fig. 11 is the same as the top panel, but plotting the significance that would correspond to these detections if the search had been made only in the sky plane, that is not including proper motion data in the detection algorithm. The significance is now calculated through , where is the number of stars in the detected peak in the sky. We only plot the significance for the detections that had at least17 . The remaining simulated UFDGs are plotted with black crosses. The colours follow now vertical contours as the vertical axis plays no role. Comparing this plot with the top panel we see how much the significance increases when the proper motions are included in the search. Red colours (maximum significance) are only achieved for higher , that is for more densely populated objects in the sky (right part). Also the limits of detection are now located at larger .

Figure 13: Surface brightness of the detected UFDGs as function of the effective parameters. Black dots indicate UFDGs that were not detected. Detected objects with surface brightness larger than 30 mag/arsec are highlighted in red colours. The blue squares and the green diamonds show the estimated positions of classical dSphs and UFDGs, respectively, with labels as in Fig. 2.

Fig. 13 shows the surface brightness of the detected UFDGs as function of the effective parameters. Contours of similar surface brightness are approximately diagonal in this parameter space. We have marked with red the detected synthetic UFDGs with surface brightness dimmer than 30 mag/arsec which is the global SDSS surface brightness limit as found in Koposov et al. (2008). Very interestingly, the red squares mark out an area in the parameter space of UFDGs less bright than the SDSS limit and that would be possible to explore with Gaia. Note, nonetheless, that this region has a recovery fraction smaller than 0.8.

In Fig. 13 (and in Fig. 12) all known satellites of the MW lie in an approximate diagonal line in the effective parameter space. We believe that this is a projection of the fundamental curve mass-radius-luminosity studied in e.g. Tollerud et al. (2011), or more in detail, a consequence of the Faber-Jackson and the - scaling relations. The plot shows that our algorithm would be able to detect objects that are outside this diagonal. However, UFDGs that lie below the diagonal have surface brightness brighter than the SDSS limit and they would have already been detected, unless they are all located outside the SDSS footprint. On the other hand, part of the red region with surface brightness dimmer than the SDSS limit but that Gaia could probe lies outside the diagonal and, therefore, the detection of objects in it relies on the existence of them. Note, however, that the scatter across the diagonal is large.

Figure 14: Fraction of detected UFDGs (colour-scale) as a function of (top), (middle) and (bottom) versus . Black squares with a central white dot are regions where the fraction recovered is exactly equal to 0. The blue squares and the green diamonds show the estimated positions of classical dSphs and UFDGs, respectively. The black diagonal line in the bottom panel shows the SDSS surface brightness limit 30 mag/arsec.
Figure 15: Same as low panel of Fig. 14 but only considering UFDGs with velocity dispersion .

Fig. 14 illustrates the recovered fraction (colour-scale) of UFDGs but now in terms of the physical parameters , and as a function of . In these panels, the region with fractions between and (ocher colours) occupies a much smaller portion of the explored ranges. This is because, in these plots, in any given bin only two physical parameters are fixed while the remaining are varying in the entire explored range, which can result in a very different detection significance. This has the effect of lowering on average, while also resulting in more diffuse boundaries between areas with different , as opposed to the sharp boundaries seen in the effective parameter plane of Fig. 12, which corroborates the fact that our detection scheme does depend mainly on the effective parameters. In this figure we also show the positions of known UFDGs and classical dSph galaxies, though without the number labels. These are shown to illustrate the typical recovery fraction one would expect for a galaxy with, e.g. a given , if its velocity dispersion and other parameters are unknown but restricted to the range spanned by our library.

This implicit dependence on the other physical parameters means that the behaviour of in these plots will change depending on the assumed distributions for the different parameters, and so, strictly speaking the reported is only valid under the assumed log-uniform distributions. For instance, if we consider only UFDGs with small velocity dispersion (; Fig. 15), the boundaries of detection improve significantly, i.e. the algorithm could detect larger UFDGs at the same given . The simple distributions assumed for the physical parameters do allow us, however, to illustrate the limits of our method. Finally, it is worth noticing that the recovered fractions shown in Figs. 12 and 14 can be interpreted in a probabilistic sense as the probability that any individual galaxy is detected by our method, given two of its physical or effective parameters.

The diagonal line in the bottom panel of Fig. 14 shows the SDSS surface brightness limit of 30 mag/arcsec. Lines of equal surface brightness are diagonal lines with slope of 5 in this plot. The detection limits of our algorithm (for example considering the line delineated by the red or blue coloured bins) have a similar slope at a slightly lower surface brightness with an additional vertical limit at (but note that all these depends on the underlying distribution of physical parameters).

The shape of the detection frontiers in the lower panel of Fig. 14 is similar to the ones of Koposov et al. (2008) in their Figs. 10 and 11. For brighter systems the detection limits follows a diagonal line with the slope of a constant surface brightness line, followed by a vertical cut at certain absolute magnitude. In the case of Koposov et al. (2008), the surface brightness and absolute magnitude limits vary as function of distance. However, the detection limits of the two studies are not directly comparable because our method is based on different information, as it includes kinematics. Our effective parameter space, where the detection limits are defined, is essentially different (with more dimensions).

4.4 Changing the background

The results of the previous section correspond to our fiducial field . We now explore how these results change for different LOS’s that have a different number of stars and a different distribution of proper motions.

Let be the significance of a detection in the fiducial field, in which we have an UFDG with and a background with observed stars. Because , from Eq. 7, it follows that . Then, given an UFDG with the same effective parameters but in an arbitrary LOS, for which the number of background stars has changed in a ratio , its significance is

(8)

This is a useful relation that allows us to establish the significance and detection limits in the effective parameter space for different LOS’s without running additional experiments.

For instance, the number of background stars in the fiducial field is and for two different LOS’s at and this is and , respectively. Therefore the background has decreased by factors and , respectively, with respect to the fiducial case. Thus, we expect the significance of UFDGs with the same effective parameters to increase by and , respectively.

In the following we check that this relation is correct. We use a library of 1000 UFDGs (randomly extracted from the library of Section 4.3), and locate copies of it in different LOS’s. We then compare the one-by-one significance for different pairs of LOS’s. Note, however, that because we keep the proper motion of each UFDG constant, its relative position with respect to the centroid will change depending on the LOS (because the underlying distribution changes), thus changing one of the effective parameters. For this reason, and also because of the approximations used to derive Eq. 8 and that Gaia errors change with LOS, we expect a certain dispersion around the values predicted by Eq. 8.

field A field B expected median AMD
90 30 90 42 1.4 1.2 0.2 0.8 1.9
90 30 90 55 1.6 1.4 0.3 0.9 2.4
90 30 90 68 1.9 1.7 0.4 1.0 3.0
90 30 90 80 1.9 1.9 0.4 1.0 3.2
90 42 90 55 1.2 1.2 0.2 0.7 1.8
170 30 170 42 1.3 1.2 0.2 0.8 1.8
170 30 170 55 1.3 1.4 0.2 0.9 2.2
170 30 170 68 1.4 1.4 0.2 0.9 2.1
170 30 170 80 1.4 1.5 0.3 1.0 2.4
90 30 170 30 1.4 1.4 0.5 0.6 3.3
90 30 170 42 1.8 1.7 0.6 0.7 4.1
90 30 170 55 1.9 1.9 0.7 0.8 4.4
90 30 170 68 2.0 1.8 0.7 0.8 4.5
90 30 170 80 2.0 2.0 0.8 0.9 4.5
Table 4: Comparison of the significance of an ensemble of 1000 UFDGs located at different fields.

Table 4 compares the expected value of with the median observed values computed with all 1000 UFDGs at different pairs of LOS A and B. We also give the Absolute Median Deviation (AMD), and the 10 per cent () and 90 per cent () percentiles. The median ratios differ at most 0.2 from the expected values. We see also that there is some expected dispersion with respect to this value. The cases where more dispersion is observed are when we compare fields at different longitudes (last rows). This is because in these cases the distribution of background proper motion changes the most. However, by looking at the percentiles we see that most of the dispersion comes from values that are higher than the expected value (thus improving the significance). The is always around 1. This means that the significance of all the fields B is smaller than predicted, but larger than the significance of the fields A in per cent of the cases. For per cent of the cases the significance of all the fields B is larger than expected.

In conclusion, the significance of the detections of our fiducial field at and are maintained or improved in per cent of the simulated cases in the other fields where was smaller. The boundaries of the detection will also improve for these fields. But the scaling between significance and fraction of recovery is not straightforward and one would need to evaluate this in each particular field. Eq. 8 offers, however, a fast approximate way of comparing the success of the detections in different LOS’s.

5 Caveats

The method that we have introduced here has its limitations and assumptions, which we will review here.

First of all, we emphasize that our method does not aim to characterize and study UFDGs but it is a probabilistic method to identify possible candidates. When applied to real data, it will provide us with a list of candidates that will need to be studied in detail. The colour-magnitude diagrams of Gaia photometry can be used for this, as well as to derive morphological properties, kinematics, distances, etc, once a proper filter to select the UFDG population is designed, as has been done with SDSS (Willman et al., 2002). Also, a follow up using ground-based facilities will be required to obtain radial velocities and detailed chemical abundances.

Our procedure has been tested for Galactic latitudes above in fields of and has been designed for UFDGs at distances larger than 10 kpc. If we want to apply it to search for nearer systems, different parallax and cuts would be needed. However, larger scales in the sky would have to be probed, increasing the background level. Another limitation of the method is that it is optimized for UFDGs with angular sizes smaller than the fields, and the detection of larger systems would require, again, to probe larger scales in the sky and perhaps a different strategy.

Likewise, it has been tested for UFDGs modeled as Plummer spheres with isotropic velocity distributions where light follows mass. A change in these assumptions that results on a variation in the footprints in the sky or proper motion planes, will change the effectiveness of our method, although the limits we have encountered should remain the same, when expressed in terms of the effective parameters. It is the mapping from structural to effective parameters that would need to be established for the new UFDG models. Similarly, the boundaries that define the limits of our detection method in will simply map into different boundaries in stellar luminosity, if the stellar population content is changed.

Our background clearly comes from smooth distributions without streams or clouds. A clumpy halo may affect the number of background stars compared to our estimations with GUMS. However, our algorithm will also detect other systems that are not necessarily UFDGs as long as they present some coherence in the 4-dimensional space that we use. These detections, rather than being considered as additional false positives, will be interesting systems to be followed-up.

Previous studies to detect UFDGs with photometric surveys apply an isocrhone masking or a probabilistic modelling in the colour-magnitude or colour-colour diagram in order to filter out field stars (e.g. Koposov et al., 2008). Instead, here we do cuts using parallax and surface gravity. Some preliminary tests show that the addition of an isocrhone masking in the Gaia vs. plane in our algorithm may be beneficial in particular cases. This merits a separate investigation that we aim to undertake in the future.

Although we have not included unresolved galaxies and quasars in our simulated background, we have checked that these will have a minor effect in our results. According to Bailer-Jones et al. (2013) (their table 3) the fraction of misclassified galaxies and quasars is 2.5% (2% misclassified as stars and 0.5% as binary systems) and 8.9% (5.9% as stars and 0.1% as binary systems), respectively. From GUMS simulations we have estimated that the number of galaxies and quasars in our fiducial field would be 4051 and 120, respectively. Therefore, there will be objects (mainly galaxies) misclassified as stars, corresponding to an increase of 7% of the back/foreground population in our fiducial field, and up to % for other LOS (). However, this increase in the number of field stars will imply approximately a change in the significance of the detections only of a factor 0.96 and 0.88 (Eq. 8), respectively, in the two LOS described.

One important aspect of the Gaia astrometric data that we have not taken into account in this work is the issue of covariances in the estimated astrometric parameters. As explained in Lindegren et al. (2012), the statistical correlation between the different astrometric parameters will occur between the parameters of the same source and also between the parameters of different sources. The within-source error covariances can be similar for collections of sources in small areas of the sky, as can be seen for example in the statistical plots in Volume 1 to the Hipparcos Catalogue documentation (ESA, 1997). In the proper motion plane for small areas on the sky (such as used in this study) this can lead to apparent structure in the proper motion distribution (caused by elongated and preferentially oriented error-ellipses).The between-source covariances will have a similar effect and are estimated in the case of Gaia to be most pronounced over areas of order radius on the sky (the value of the correlation half-length estimated in Holl, Hobbs & Lindegren, 2010). This means that for a large fraction of UFDGs the between-source correlations will be important in addition to the within-source correlations. To first order the main effect will be that the interpretation of the WT maps will be more involved, where a distinction will have to be made between real and spurious structure in the proper motion plane.

The within-source covariance matrix of the astrometric parameters will be provided as part of the Gaia data releases. The covariance matrix of the astrometric parameters of different sources cannot be calculated for the full Gaia catalogue but it is feasible to do so for limited groups of sources as demonstrated in Holl & Lindegren (2012) and Holl, Lindegren & Hobbs (2012). Hence we will be able to account for the error covariances but we defer to a future study the details of how to implement this in practice.

6 Discussion and conclusions

We have introduced an automatic procedure to identify UFDG candidates in the future Gaia database and charted its detection limits. The main advantages of using Gaia data on the search of UFDGs are, first, the inclusion of kinematics (proper motions) in the detection algorithm for the first time; and second, the Gaia full sky coverage, being the first unbiased homogeneous survey to be used for this purpose.

Our procedure identifies significant overdense peaks in the planes of the sky and of proper motions that share common stars. Then the probability of this occurring by chance is assessed and used to discard spurious detections. We have used a library of synthetic UFDGs to probe the 9-D space of intrinsic (, , ) and extrinsic (, , , , , ) UFDG parameters, spanning ranges that extend well beyond those occupied by currently known systems.

We have identified the “effective parameters” that our algorithm depends mainly on. The main two are the ratios of the number of observable stars by Gaia in the UFDGs to their apparent sizes in the sky () and proper motion planes (). The position of the peak in proper motion with respect to the background also influences the detection, but is not as relevant. These parameters reduce the dimensionality of our problem to 3, mainly 2, parameters.

We have charted the limits of detectability and completeness (recovery fraction) of our search in the effective parameter space (Fig. 12) for a LOS at and . Detections can be made with high significance over most of the explored region, which includes the majority of the currently known UFDGs, with a recovered fraction that remains above 70 per cent over most of it. It is only in the corner of small effective parameters that the efficacy of our method decreases abruptly. On the other hand, the limits of our detection procedure can not be described in terms of a limiting surface brightness alone (Fig. 13), because of the inclusion of kinematics in the search.

We have derived a relation that allows us to know the approximate detection significance of the synthetic UFDGs at LOSs with a different number of background stars. The translation from significance to recovery fraction is not straightforward and one would need a more thorough characterization per LOS. However, most of the results presented here are for a pessimistic case compared to higher latitudes, or to the outer galaxy (), where we expect less field contamination.

Furthermore, we have explored the extent to which current detectability limits can be pushed forward, opening the possibility of detecting real systems hitherto not found. We have found that there is a region in the effective parameter space where there are currently no observed systems. Part of this region corresponds to UFDGs with surface brightness brighter than the SDSS limit and, therefore, they would have already been detected, unless they are all located outside of the SDSS footprint. But more interestingly, we have seen that Gaia will be able to probe a region of the effective parameter space of surface brightness dimmer than the SDSS limit, if such objects exist, albeit with a recovery fraction smaller than 0.8. Note that the recent UFDG discoveries made with DECam have similar surface brightness to the ones detected by SDSS (see Fig. 17 in Koposov et al., 2015). Also because of the different detection methodologies followed by SDSS and DECam compared to Gaia, the nature of the detection limits is completely different, thus offering the possibility to explore uncovered regions of the parameter space (both with respect other surveys in the north and in the south) and for all sky.

We can make a very rough estimation of the number of UFDGs that Gaia will detect from the recovery fractions that we have found for our synthetic search (Fig. 12), assuming isotropy on the distribution of satellites in the MW halo, and considering only SDSS UFDGs. There is 1 known object (Boo) that would be detected with a recovery fraction of 1.0, 4 objects (CVnI, Her, UMaII, CmB) with a fraction of 0.9, and 4 objects (WilI, UMa, SegII, BooII) with fractions of 0.8, 0.7, 0.6 and 0.5, respectively. We do not count objects with a recovery fraction below 0.5. This makes a total of 7.2 UFDGs in a sky area equivalent to SDSS ( of the sky, Koposov et al. 2008). If we assume that Gaia will detect UFDGs only above , which corresponds to 1/2 of the sky, there should be of the order of new UFDGs (i.e. currently not known) over the 1/2-1/5=3/10 of the sky that remains unexplored, that is subtracting the area already covered by SDSS. These calculations are based on the field at and but could be slightly better for higher latitudes.

However, by the arrival time of the Gaia catalogue (see below) other surveys such as ATLAS (Shanks et al., 2013), Pan-STARRS (Kaiser et al., 2010) and DES (Diehl et al., 2014) will have covered great fraction of this area. But a part of the South Galactic cap will still remain completely unexplored (at declinations below stripe SPT of DES). We estimate this to be a fraction of of the whole sphere (by taking the part of the spherical cap in equatorial coordinates below that lies in the range ). Therefore, there should be of the order of new UFDG in this unexplored area. However, we emphasize that our method uses information not used in other searches, namely proper motions, and thus, it could lead to new discoveries, made possible, not by the covered region in the sky or depth of probing, but by their motion in the sky. As such, our method complements present searches.

Moreover, the number of discovered new candidates could be higher because as discussed above, Gaia could also detect more UFDGs with lower surface brightness than the SDSS limit. Besides, under the assumption of anisotropy in the spatial distribution of satellites, this number could be larger if the Gaia footprint happens to cover preferential directions. In fact, the importance of having a full-sky catalogue in this type of search for the first time is that it will allow to put constraints on the isotropic distribution of the satellites and, therefore, their origin.

The known UFDGs with high recovery fraction mentioned above could be seen as standard systems for future Gaia discoveries but only in terms of effective parameters. Thus, one can not interpret this as if, for instance, all objects with the same half-light radius and the same distance as Boo will be detected, but rather as systems for which the combination of all physical parameters produce similar effective parameters will be detected with high probability. Note also that we have not considered in this calculation the influence of the third effective parameter, which we have shown to be less important.

Our proposed method can be applied fully to the third Gaia data release scheduled for18 2017/2018. This release will include the five-parameter astrometric solutions as well as the object classification (necessary to eliminate contaminant extra-Galactic objects) and astrophysical parameters such as , necessary for filtering out foreground dwarfs. Preliminary searches could be conducted using earlier releases; e.g. with the first data release in summer 2016, using only on sky coordinates; or with the second data release in early 2017, using full sky and proper motion information, yet without the possibility of using the foreground filters as explained here, since astrophysical parameters will not yet be available.

Finally, there is the future possibility that the Gaia magnitude limit will be pushed down to . This will obviously be positive in terms of the number of observable stars in each UFDG, but will also increase the foreground/background contamination, so the effect in the detection probabilities will have to be assessed.

Acknowledgments

We thank V. Belokurov, the referee, for a thorough review and insightful comments on the original manuscript. TA is supported by an ESA Research Fellowship in Space Science. CM acknowledges support from a post-doctoral fellowship of DGAPA-UNAM, Mexico. CM and LA acknowledge support from DGAPA/UNAM grant IG100115 and the hospitality and support of the University of Barcelona during part of this investigation. FF acknowledge support from MINECO (Spanish Ministry of Economy) - FEDER through grant AYA2012-39551-C02-01 and ESP2013-48318-C2-1-R, and from the European Community’s Seventh Framework Programme (FP7/2007-2013) under grant agreement GREAT-ITN FP7 264895. We thank the CEA and Nice Observatory for the MR software. We thank Merce Romero-Gomez and GaiaUB team for the code to simulate Gaia errors.

Footnotes

  1. pagerange: Detection of satellite remnants in the Galactic Halo with Gaia – III. Detection limits for Ultra Faint Dwarf GalaxiesDetection of satellite remnants in the Galactic Halo with Gaia – III. Detection limits for Ultra Faint Dwarf Galaxies
  2. pubyear: 2007
  3. We compute the escape velocity as , with km s and kpc.
  4. http://www.cosmos.esa.int/web/gaia/dpac (Mignard et al. 2008)
  5. In what follows, we always work with fields. To cover the same solid angle, regardless of latitude, we have converted the Galactic longitude of the stars , to , where and are the longitude and latitude of the center of the field, respectively. For simplicity, we use instead of hereafter.
  6. After the cuts in parallax and surface gravity (see Section 2.4), these fractions become 54, 5 and 41 per cent, respectively. The relative increase of unresolved systems is because we are selecting large distances and giant stars (dwarf stars have been removed), which have higher binary fractions.
  7. The code was released at the Gaia Challenge Workshop and is publicly available at https://github.com/mromerog/Gaia-errors
  8. http://www.cosmos.esa.int/web/gaia/science-performance
  9. http://www.cosmos.esa.int/web/gaia/table-6
  10. The SFH assumed in the stellar population model is reasonably representative of the SFH of known UFDGs. For simplicity we assume the same model for the classical dwarfs to get a rough estimate of , although these have very different SFHs.
  11. The SDSS survey is mag deeper comparing the and bands, or between 1 and mag deeper comparing the and bands. This is estimated by taking the two extreme colours of the stars in our simulated UFDGs, that is 0.25 and 1.5, and convert these to and colours following Jordi et al. (2010).
  12. The Galactic sources in GUMS are actually foreground and background stars. We use hereafter “background” for simplicity.
  13. Do not confuse this probability for the combined sky and proper motion planes , with the wavelet probability WP used in Section 3.1
  14. The last condition is required in order to select only over-densities but not under-densities.
  15. Note also that UFDGs with the same or (apparent “surface” density) do not have the same significance.
  16. Remember that this is not a cross section as in Fig. 9.
  17. Note that in the top panel the minimum found was 5.
  18. http://www.cosmos.esa.int/web/gaia/release

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