One of the key properties of the population of UFDs, as distinct from the properties of individual objects, is their luminosity function (LF). The relationship between the LF and the mass function of dark matter halos and subhalos encodes the physics of galaxy formation in the smallest halos and places constraints on dark matter models. Moreover, the LF provides the connection between the low luminosity galaxies observed today and their progenitor systems at high redshift, which may play a significant role in reionizing the universe (see Section 9.3).
The observed dwarf galaxy LF is only equal to the true LF if the dwarf galaxy sample is complete over the luminosity range of interest. UFDs, however, vary widely in luminosity, surface brightness, and distance, and many are close to the detection limits of the surveys in which they were discovered. The LF of such systems therefore cannot be computed until the sensitivity of dwarf galaxy searches has been accurately quantified.
Koposov et al. (2008) presented a careful analysis of the detectability of faint dwarf galaxies using an automated search algorithm in the 5th data release (DR5) of SDSS, covering 8000 deg2. They found that a significant fraction of the UFDs discovered in SDSS are close to the detection limit of their algorithm. These objects are detected in SDSS data with an efficiency of ∼ 50%, indicating that undetected dwarfs are likely to be present in the SDSS footprint. After correcting for incompleteness, Koposov et al. determined that the differential LF of Milky Way satellites can be approximated as dN / dMV = 100.1(MV+5) + 1 over the absolute magnitude range −19 < MV < −2. Translated into the Schechter form, the corresponding faint-end slope of the LF is α = −1.25. The implied total number of satellite galaxies within the virial radius of the Milky Way is 45 at MV < −5 and 85 at MV < −2. For the faintest dwarfs the incompleteness correction is very large, and it depends on the assumed radial distribution of satellites. If faint dwarfs are concentrated close to the Milky Way, then fewer such objects are expected at large distances where they are currently undetectable. Conversely, if the spatial distribution of the lowest luminosity systems is more extended then there may be enormous numbers of similar objects in the outer halo of the Galaxy. While the radial distribution of dwarfs around the Milky Way can be estimated in numerical simulations (e.g., Wang, Frenk & Cooper, 2013, Garrison-Kimmel et al., 2017), ultimately it will have to be measured observationally by deeper surveys.
A similar quantification of dwarf galaxy detectability was carried out by Walsh, Willman & Jerjen (2009) on SDSS DR6 imaging, covering 9500 deg2. Walsh, Willman & Jerjen used a more sensitive search algorithm that finds all of the SDSS satellites known at the time at ≳ 90% efficiency, but potentially with a correspondingly high false positive rate. They concluded that the transition between detectability and invisibility as a function of luminosity, surface brightness, and distance is more gradual than calculated by Koposov et al. (2008), and therefore that all of the known dwarfs should have been visible in SDSS even if they were located at significantly larger distances. According to the Walsh, Willman & Jerjen (2009) analysis, searches in the SDSS footprint are complete out to the virial radius of the Milky Way down to MV = −6.5. The extrapolated total number of Milky Way satellites is ∼ 220−340 depending on the adopted detection threshold. Many subsequent studies have used the detection sensitivity derived by Koposov et al. (2008) and/or Walsh, Willman & Jerjen (2009) to estimate the overall size of the Milky Way satellite population, generally predicting that future surveys will discover ∼ 100−300 dwarfs over the entire sky (e.g., Tollerud et al., 2008, Hargis, Willman & Peter, 2014, Newton et al., 2018, Jethwa, Erkal & Belokurov, 2018).
Unfortunately, no comparable analyses of the detectability of dwarf galaxies have been published since SDSS DR6. Consequently, the sensitivity of the final 5000 deg2 of SDSS imaging, the Dark Energy Survey, Pan-STARRS, and other smaller surveys has never been adequately quantified. Given the large number of new satellites discovered since 2009 and their apparently anisotropic distribution on the sky (Drlica-Wagner et al., 2015), updated determinations of the completeness of searches for nearby dwarf galaxies are urgently needed. Until the sensitivity of all significant surveys has been properly quantified, more detailed calculations of the total number of Milky Way satellites, their LF, and their radial and angular distributions cannot be made. What we can say at present is that the observed LF (without an incompleteness correction) peaks at MV ∼ −4, suggesting that any real turnover in the LF must be at even fainter magnitudes.
As an illustration of the discovery potential of future imaging surveys, we construct a very simple toy model of satellite detectability. Motivated by the results of Martin, de Jong & Rix (2008) for SDSS and Bechtol et al. (2015) and Drlica-Wagner et al. (2015) for DES, we assume that a satellite must contain at least 20 stars brighter than the detection limit of the survey in order to be identified. We create realizations of satellites with stellar masses corresponding to absolute magnitudes of MV = −2, −4, and −6 by randomly selecting the appropriate number of stars from an old, metal-poor mock stellar population. We then determine the median magnitude of the 20th brightest star for each absolute magnitude and calculate out to what distance that star would be detectable for a given survey depth. Note that the depth of a survey for the purpose of searching for stellar overdensities is ≳ 0.5 mag shallower than the actual 5σ detection limit because colors become uncertain and star-galaxy separation becomes unreliable at fainter magnitudes. Consistent with Koposov et al. (2008) and Walsh, Willman & Jerjen (2009), we find that SDSS should be complete at MV ≈ −6 out to beyond the virial radius of the Milky Way. Similarly, DES should be complete down to MV ≈ −4 within the virial radius. A complete search of the Milky Way's virial volume to fainter magnitudes will require full-depth LSST images.
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Figure 7. Detectability of faint stellar systems as a function of distance, absolute magnitude, and survey depth. The red curve shows the brightness of the 20th brightest star in an MV = −6 object as a function of distance. The magenta and blue curves show the brightness of the 20th brightest stars for MV = −4 and MV = −2 systems, respectively. The horizontal dashed lines indicate (from bottom to top) the limiting r magnitude for dwarf galaxy searches in SDSS, Pan-STARRS, DES, LSST single exposures, and stacked LSST images at the end of the survey. The region within the (approximate) virial radius of the Milky Way is shaded purple. |