Hot White Dwarfs: UV catalogs

Catalogs of Hot White Dwarfs in the Milky Way from GALEX’s Ultraviolet Sky Surveys. Constraining Stellar Evolution.

Luciana Bianchi, Boryana Efremova, James Herald, Léo Girardi, Alexandre Zabot, Paola Marigo and Christopher Martin
Department of Physics and Astronomy, Johns Hopkins University, 3400 N.Charles St., Baltimore, MD 21218, USA; bianchi@pha.jhu.edu
Astronomical Observatory of Padova, INAF, Vicolo dell’Osservatorio 5, I-35122 Padova, Italy
Universidade Federal da Fronteira Sul, Campus de Laranjeiras do Sul, Brazil
Dept. of Astronomy, University of Padua, Vicolo dell’Osservatorio 3, I-35122, Padova. Italy
CA Institute of Technology, Pasadena, USA
E-mail: bianchi@pha.jhu.edu
Accepted —- Received September 23, 2019
Abstract

We present comprehensive catalogs of hot star candidates in the Milky Way, selected from GALEX far-UV (FUV, 1344-1786Å) and near-UV (NUV, 1771-2831Å) imaging. The FUV and NUV photometry allows us to extract the hottest stellar objects, in particular hot white dwarfs (WD), which are elusive at other wavelengths because of their high temperatures and faint optical luminosities. We generated catalogs of UV sources from two GALEX’s surveys: AIS (All-Sky Imaging Survey, depth ABmag19.9/20.8 in FUV/NUV) and MIS (Medium-depth Imaging Survey, depth 22.6/22.7mag). The two catalogs (from GALEX fifth data release) contain 65.3/12.6 million (AIS/MIS) unique UV sources with error0.5mag, over 21,435/1,579 square degrees. We also constructed subcatalogs of the UV sources with matched optical photometry from SDSS (seventh data release): these contain 0.6/0.9million (AIS/MIS) sources with errors 0.3mag in both FUV and NUV, excluding sources with multiple optical counterparts, over an area of 7,325/1,103 square degrees. All catalogs are available online. We then selected 28,319 (AIS) / 9,028 (MIS) matched sources with FUV-NUV-0.13; this color cut corresponds to stellar  hotter than 18,000 K (the exact value varying with gravity). An additional color cut of NUV-r0.1 isolates binaries with largely differing  ’s, and some intruding QSOs (more numerous at faint magnitudes). Available spectroscopy for a subsample indicates that hot-star candidates with NUV-r0.1 (mostly “single” hot stars) have negligible contamination by non-stellar objects. We discuss the distribution of sources in the catalogs, and the effects of error and color cuts on the samples. The density of hot-star candidates increases from high to low Galactic latitudes, but drops on the MW plane due to dust extinction. Our hot-star counts at all latitudes are better matched by Milky Way models computed with an initial-final mass relation (IFMR) that favours lower final masses. The model analysis indicates that the brightest sample is likely composed of WDs located in the thin disk, at typical distances between 0.15-1kpc, while the fainter sample comprises also a fraction of thick disk and halo stars. Proper motion distributions, available only for the bright sample (NUV18 mag), are consistent with the kinematics of a thin-disk population.

keywords:
Astronomical Data Bases: catalogues — stars: white dwarfs — stars: evolution — Galaxy: stellar content — ultraviolet: stars — galaxies: Milky Way

1 Introduction

The vast majority of stars (initial mass 8)  end their lives as white dwarfs (WD), after passing through the asymptotic giant branch (AGB) and planetary nebula (PN) phases, in which they shed much of their mass. The ejected material enriches the interstellar medium (ISM) with newly synthesized nuclear products (mainly He, C, N, and possibly O), to different extents, depending on the initial stellar mass and exact evolutionary path (e.g. Marigo 2001, Karakas 2010). Intermediate mass stars are the main providers of carbon and nitrogen, whereas low-mass stars are the most relevant component for the mass budget of stellar remnants in galaxies. Most of the stellar mass is shed in the AGB and PN phases, but the evolution through these phases is still subject to considerable uncertainties, in particular regarding mass loss and the efficiency of the third dredge-up. Stars within an intial mass range of 0.8 to 8 end as WDs with a narrow mass range, mostly below 0.8. It is important to understand how the mass of their precursors relates to the final WD mass, in order to understand the relative contribution of different stars to the chemical enrichment of elements such as He, C, N and O.

While the evolution of the WD progenitors in the main sequence phase is fairly well understood and observationally constrained, the hot-WD population is hitherto quite elusive, owing to their small radius, hence low optical luminosity, and extremely hot temperatures, to which optical colors are insensitive (see e.g. Bianchi (2007), Bianchi et al. (2007a), Bianchi et al. (2007b)) as well as to their very short lifetimes on the constant-luminosity post-AGB phase. To make matters more difficult, the post-AGB luminosity at a given stellar temperature varies significantly according to the stellar mass, making it impossible to infer absolute luminosity from other physical parameters. The evolutionary time spent on the constant-luminosity post-AGB phase and on the cooling track is also a strong function of the mass (Vassiliadis & Woods 1994). Therefore, the exact relation between progenitor’s initial mass and WD mass (initial-final mass relation, IFMR), remains to date a crucial missing link in our understanding of stellar evolution and chemical enrichment of the Galaxy.

A characterization of the population of hot WDs in the Milky Way can reduce these uncertainties, and lead to a better understanding of processes that drive the chemical evolution of galaxies like the Milky Way. UV photometry combined with optical measurements significantly increases the sensitivity to the hottest temperatures. For example, the color difference between a  =50,000K and 20,000K star is about 1.5mag in FUV-g, but 0.4mag in U-B, and 0.15 mag in g-r which are comparable to photometric errors when large surveys are considered. The sensitivity gained by extending the measurements to UV wavelengths is more critical for discerning the hottest stars (see e.g. Fig.s 5-7 of Bianchi et al. (2007a) and Bianchi 2009).

The census and characterization of cool compact objects has significantly improved in recent years thanks to optical and IR surveys. The Eisenstein et al. (2006) catalog from the SDSS fourth data release (DR4) contains 9316 spectroscopically confirmed WDs and 928 subdwarfs over an area of 4783deg; about one fourth (2741) have  18,000K as estimated by the SDSS pipeline automated spectral analysis. An additional 5-6000 WD are expected from DR7 (Kleinman, Nitta, & Koester (2009)). Seven ultracool WD were added to the census by Harris et al. (2008). Gontcharov et al (2010) extracted from 2MASS, Tycho-2, XPM and UCAC3 catalogs combined 34 WDs, 1996 evolved (11,000 60,000K) and 7769 unevolved ( 7,000K) subdwarfs, using multicolor photometry and proper motions of stars with 6K14 mag. The current version (2008) of the McCook & Sion (1999) catalog of spectroscopically confirmed WDs includes 10132 entries (all types), a factor of five increase over a decade, with respect to the original (1999) version listing 2249 WDs. Vennes et al. (2002) give a catalog of 201 DA WDs, spectroscopically confirmed from the 1,000 H-rich DA WDs discovered in the 2dF QSO redshift survey; Croom et al. (2004) 2dF catalog includes 2071 WDs over 2000 deg. WDs in binaries from the SDSS were catalogued and studied by several authors, see e.g. Rebassa-Mansergas et al.(2010) and references therein, Silvestri et al (2007), Heller et al. (2009, WD-M star binaries) from spectroscopy; other authors used optical-IR photometry to search for binaries among the known WDs (e.g. Tremblay & Bergeron 2007, Watcher et al. 2003). Special classes such as CVs are addressed by other works (e.g. Szkody et al. 2009 and references therein, Gaensike et al. 2009), while others studied in detail the very local population (e.g. Holberg et al (2002) discuss 122 objects within 20 pc from the Sun, which they estimate to be an 80% complete sample within this distance, and which include some double-degenerate systems).

Finding the hottest, smallest stars, however, remained a challenge prior to the GALEX UV sky surveys, which provide deep sensitivity and large area coverage. For example, 105 were found in the original EUVE whole-sky survey, with small subsequent additions obtained by combining EUVE and Rosat: see e.g. Dupuis (2002) for a review and discussion. The Rosat whole-sky survey produced 175 WD in X-rays, mostly DAs (Fleming et al. 1996, from Rosat PSPC).

Such catalogs of confirmed or candidate WDs enable the study of these objects as astrophysical probes, of stellar evolution, of MW structure, of the local neighborhood, etc. Most importantly, comprehensive catalogs enable selection of targets for follow-up spectroscopy, which then provides the detailed physics of these objects, especially when extended to the UV and far-UV wavelengths, as proven by the score of results enabled by IUE, FUSE, and HST spectrographs.

Bianchi (2009, 2007), Bianchi et al. (2009a, 2007a, 2005) have demonstrated the power of far-UV and near-UV measurements, afforded for the first time over large areas of the sky by GALEX, to unambiguously detect and characterize extremely hot stellar sources. Not only the UV wavelengths are more sensitive to the temperatures of hottest stars, but the combination of UV and optical colors allows a better separation of different classes of astrophysical objects (e.g. Bianchi 2009 and references therein).

Our present work provides a selection of hot-star candidates from UV photometry; about 40,000 hot-star candidates with FUV, NUV photometric errors0.3mag (about 74,000 with photometric errors0.5mag) have also SDSS optical photometry. The majority are likley hot WD with log(g) between 7 and 9. The catalog covers different latitudes and enables a first analysis of this stellar population with Milky Way models. Larger catalogs of UV sources with and without optical SDSS match are also constructed, and made available as online products.

In Section 2 we describe the method used to construct “clean” catalogs of unique UV sources from two GALEX surveys with different depths and coverage, and subsets of these catalogs with matched optical photometry, and we present the catalogs’ characteristics. In Section 3 we extract samples of hot-star candidates, and we analyze them with Milky Way models in Section 4. Discussion and conclusions are given in Section 5.

Figure 1: TOP: Sky coverage (in Galactic coordinates) of the GALEX data release GR5 showing the major surveys: AIS (orange), MIS (green), DIS (blue). BOTTOM: the portion of the GALEX GR5 sky coverage for AIS and MIS overlapping with SDSS seventh data realease (DR7)

2 The UV Sky Surveys and the source catalogs

2.1 The sky survey data

The Galaxy Evolution Explorer (GALEX) is imaging the sky in far-UV (FUV, 1344-1786Å, = 1538.6Å) and near-UV (NUV, 1771-2831Å, = 2315.7Å) simultaneously, with a field-of-view of 1.2 diameter and a resolution of 4.2/5.3 [FUV/NUV] (Morrissey et al. (2007)). The images are sampled with 1.5   pixels. Nested surveys with different depth and coverage are in progress. The widest sky coverage is provided by the All Sky Imaging Survey (AIS) and the Medium [depth] Imaging Survey (MIS), that reach typical depths of 19.9/20.8mag (FUV/NUV) and 22.6/22.7mag (FUV/NUV) respectively, in the AB magnitude system. The Nearby Galaxy Survey (NGS, Bianchi et al. 2003, Bianchi 2009, Gil de Paz et al. 2007), with over 300 fields at MIS depth, targeted nearby, hence fairly extended, galaxies, therefore it has been excluded in the present work, lest some sources from galaxies “shredded” by the pipeline intrude on our catalog. See also Bianchi (2009), Bianchi (2011), Bianchi et al (2011), Bianchi et al. (2007a), for a general discussion of the content of the UV sky surveys.

In this paper we use data from the GALEX fifth data release (GR5) AIS and MIS surveys, which include a total of 28,269 and 2,161 fields respectively. The data are taken from the MAST archive. We restrict the catalogs to sources within the central 1 diameter of the field (for good photometry and astrometry, and to exclude edge artifacts). With such restriction, and eliminating overlaps, these surveys cover a total unique area of 21,434.8 (AIS) and 1,578.6 (MIS) square degrees (Section 2.4). Section 2.2 describes the construction of the catalog of unique GALEX sources. In order to separate the UV sources by astrophysical classes, we examine in this work the portions of the GALEX GR5 AIS and MIS surveys that are also included in the footprint of the Sloan Digital Sky Survey seventh data release (DR7), which provides five optical magnitudes: u g r i z in addition to the GALEX FUV, NUV magnitudes. The overlap between GALEX GR5 and SDSS DR7 includes 10,316 / 1,655 (AIS/MIS) GALEX fields, and the area coverage of the overlap is 7,325 (AIS) and 1,103 (MIS) square degrees. Details of the area calculation are provided in Section 2.4. The sky coverage of AIS and MIS in the GALEX data release GR5 and its overlap with SDSS DR7 are shown in Fig. 1.

Figure 2: TOP: distribution of FUV exposure times for AIS (10sec bins) and MIS surveys (50sec bins); a few fields have longer exposure times, off the scale of the plots.    BOTTOM: Number of UV sources (all UV detections, not just our selected hot stars) in each GALEX field (1 diameter), before merging the catalogs and removing overlaps. Three shades of blue (light/medium/dark) for AIS indicate exposure times of 120 / 120-220 / 220 sec respectively, and lighter / darker green for MIS fields indicate FUV exposures less/more than 1200 sec. Generally the number of UV source detections in a field increases with depth of exposure, as expected, and as seen more distinctly for the AIS at the high latitudes, uncomplicated by dust extinction. A few fields, including some with short exposures, have overdensities. While the number of MW stars increases towards the MW disk, as shown by the AIS sources, at MIS depth most UV sources are extragalactic, hence show no correlation with Galactic latitude except for the foreground extinction. The sharp drop in the center, due to dust extinction, defines the MW dust disk.

2.2 The catalog of unique UV sources

Here we describe the procedure used to construct the catalog of unique GALEX sources (i.e. eliminating repeated measurements). All catalogs described in this paper are made publicly available from our website http://dolomiti.pha.jhu.edu/uvsky , and from MAST at http://galex.stsci.edu , and as High-Level Science Products (http://archive.stsci.edu/hlsp/ , in the “Catalogs” section). Therefore, we provide here information on how they were constructed that will be relevant for potential users, as well as to others interested in constructing future versions of similar samples.

We extracted catalogs of GALEX sources from STScI MAST (www.mast.stsci.edu), at the CASJobs SQL interface (www.galex.stsci.edu). Sources were extracted from the table “photoobjall”, from the MIS and AIS surveys separately, with the criteria that the source distance from the field center had to be 0.5 and the photometric error less than 0.5 mag in NUV. In other words, we initially included in our general GALEX source catalog all NUV reliable detections, regardless of whether they have also an FUV detection. For our selection of hot star candidates, we will eventually impose the additional criterion of good FUV photometry. The additional restriction of err0.5 mag significanly reduces the number of sources (by a factor of up to 10, see Table 2), with respect to the total number of NUV detections, and of course introduces a bias in the source catalog, the hottest and bluest sources (the subject of this paper) being not affected but the redder sources being progressively eliminated, as discussed in sections 2.5 and 3.3. For more discussions about statistical properties, and biases inherent to sample selections, see Bianchi et al (2011).

A few observations planned as part of the MIS or AIS surveys actually partly failed, and resulted in one of the detectors (most often FUV) not being exposed (such observations are typically repeated later). The observations with one of the two bands having zero exposure time were eliminated from our catalog, otherwise they will bias the statistics of FUV-detection over number of total NUV detections, and the corresponding fields were not counted in the area calculation.

Fig. 2 shows the distribution of exposure times (FUV is shown, but NUV is generally equal or larger), for the AIS and MIS fields. The typical exposure time for MIS is 1500 sec, which is met or exceeded by the majority of fields. The AIS survey aims at exposure times of the order of 100 sec. We retained also fields with exposures shorter than typical. Therefore, while the typical depth of the two surveys (AIS and MIS) differ by 2 mag, the exposure level is not strictly homogeneous across each catalog.

Fig. 2-bottom panel shows the number of GALEX sources in each GALEX field. We color-coded the fields by three ranges of exposure time for the AIS (two for MIS), since the number of sources detected above a given error-cut generally increases with exposure time. This plot is useful to check for fields with overdensities, since the surveys, the broad AIS in particular, include also some stellar clusters. In general the spread in number of sources per field is just about a factor of two for MIS. A few AIS fields around latitude -30 have an overdensity of almost one dex, however the total number of AIS fields in each 10-wide latitude bin (used in our analysis, see next section) is very large, and a few overdensities do not affect our analysis of stellar counts with Milky Way models over wide areas. Fields with high density of sources, and in particular the stellar clusters included in the surveys, will be separately analyzed elsewhere. They are included in our catalog for completeness and for possible use by others, although customized photometry is desirable in very crowded fields (e.g. de Martino et al. 2008).

Nearby galaxies are generally observed as part of the ’NGS’ survey which is excluded from our catalog, but a few large galaxies are also in the footprint of AIS and MIS and bright knots of galaxies “shredded” by the pipeline may enter the catalog as separate sources (see Bianchi et al. (2007a)). A few such sources may have FUV-NUV-0.13 and appear pointlike in the SDSS catalog, and therefore may enter our catalog of hot-star candidates (section 3), but they would not affect the statistical results. For other, more specific uses of our catalog, they may be removed by checking against a list of nearby galaxies, as we did in Bianchi et al. (2007a). Using the “child” flags from the SDSS pipeline, that track deblended sources, proved not to be useful in indentifying and weeding out sources from “shredded galaxies”.

The searches and download of GALEX sources with the above criteria were done using the java CASJobs command-line tool (casjobs.jar). One problem we often encountered when using this tool to extract and download query results from the database is that sometimes the download pipe gets broken prematurely, and the results are truncated without any warning being given. In order to verify that the output file contained all the results, a separate count query was run on the SQL server and compared to the number of output sources in the downloaded file each time. It was necessary to subdivide the search in small latitude strips due to various limits set by the CASjobs interface (e.g. query length, output file size), as well as to avoid the frequent problem of the long searches being interrupted.

The GALEX archive contains multiple observations of the same source, when some fields overlap or are repeated. Having all measurements of each source is useful for variability studies (which will be addressed in a forthcoming work), and for chosing the best measurement when several are available. For our present purpose we constructed from the total output a unique-source catalog, in the following manner. GALEX sources were considered possible duplicates if they lied within 2.5   of each other. If two GALEX sources were within this distance, but had the same “photoextractid” (i.e., they are both from the same observation) they were both considered unique. Otherwise, they were assumed to be multiple observations of the same source. We choose, to represent the unique source, the measurement with the longest NUV exposure time. In the case of equal exposure times, the observation where the source was closer to the field center was chosen (i.e. the source with the smallest “fov_radius” value from the “photoobjall” table), as photometric quality is usually better in the central part of the field.

2.3 The matched UV-optical source catalog

A portion of the GALEX survey areas is in the footprint of the SDSS DR7 (see Fig.1 - bottom panel), and for the UV sources in this area we constructed a catalog of matched optical sources. We uploaded the coordinates of our GALEX unique sources into the SDSS SQL interface (version 3_5_16 rev 1.70, at casjobs.sdss.org) and queried for SDSS matches against the “photoprimary” table (SDSS source catalog that includes only unique sources) using an inital search radius of 4.2. A match radius of 3 was eventually used in the final catalog, as we shall see.

As with the GALEX searches, the SDSS searches were done in an automated way on 1-wide Galactic latitude strips using the casjobs command-line tool. We found that there is currently a bug in the SDSS version of this tool (v0.03) that causes the last character of a file to be dropped when extracting query results. We compensated for this by writing our query to pad the end of the file with an additional character.

2.3.1 Multiple matches.

A given GALEX source may have multiple SDSS matches within the search radius, given the higher spatial resolution of the SDSS (1.4 ). In such cases the CASjob search returns multiple lines with the same GALEX source and the various SDSS matches. We ”ranked” the multiple SDSS matched sources based on the distance, with the closest SDSS source being retained as the ”match” for the GALEX source and the additional SDSS sources being noted. When using UV-optical colors for source classification, UV sources with multiple optical matches must be excluded, because, even though the closest optical source may be the actual counterpart, the UV photometry at the GALEX spatial resolution of 5 may be a composite measure of two optical sources, and therefore the color would not be meaningful. The fraction of UV sources with multiple optical matches, eliminated from the analysis sample, is then taken into account when estimating density of astrophysical sources (# per square degree). This fraction is given in Table 2. Sources with multiple optical matches are listed separately in our hot-star catalogs, and are included with “rank”0 in the total matched-source catalog. The relative number of UV sources with multiple optical matches, shown in Figure 3, increases towards low Galactic latitudes, as expected. The figure refers to our final catalog with match radius of 3, and pointlike sources only: approximately 10% of the UV sources have more than one optical counterpart, at intermediate and high Galactic latitudes, in agreement with our previous work on earlier data releases which covered mainly high Galactic latitudes (Bianchi et al. 2005, 2007a). The fraction increses to slightly over 20% for latitudes b 10-20, reflecting the higher density of stars in the MW disk, and is still very uncertain at latitudes b10, where we have little area coverage. For a larger match radius (4.2), the fraction of multiple matches increases significantly, relative to the total number of matches, and so does the incidence of spurious matches (Sect. 2.3.3), therefore we adopted a final match radius of 3  in our catalog. The fraction is also higher if exended sources are included.

Figure 3: Fraction of UV sources having multiple optical matches within a 3 match radius, as a function of Galactic latitude. Green dots show MIS data, blue dots show AIS data, divided in strips of 1 (small dots), as well as averaged over 5 (blue circles). The increase towards the Galactic plane reflects the higher density of optical sources at low MW latitudes. The plot also shows a slight North-South asymmetry. The fractions shown are for point-like sources, with error cuts of 0.3 mag. The fraction increases if a larger match radius is used. The red dots are the incidence of spurious matches (see text).

2.3.2 Other caveats.

In the interest of others who may want to apply the same procedures, we also mention that, due to the large current area coverage and consequent large number of sources, when it is necessary to perform the searches on small contiguous portions of the sky (e.g. 1 strips), it may happen that the coordinates of a GALEX UV source fall in one strip (close to the latitude limit), and the SDSS match falls in the next latitude range, due to a small difference in coordinates. Such cases are included in the final catalogs in the latitude range which is appropriate according to the position of the GALEX source. Care should also be taken when using the Galactic coordinates as returned by the GALEX database, as those coordinates differ by 0.5 from those derived by converting the GALEX RA, Dec coordinates to Galactic coordinates using standard astronomical packages (e.g., WCSTools from the Smithsonian Astrophysical Observatory). These differences probably arise because the GALEX database stores Galactic coordinates using real data types, while using double precision data types to store RA and Dec values. Therefore, sorting GALEX search results based on the Galactic coordinates returned by the database may give different results than using WCSTools to derive those coordinates from the GALEX RA and Dec.

Another contingency that must be tested for, in the above procedures, is whether a given SDSS source matches multiple GALEX sources. This can occur if a SDSS source lies somewhere in between two GALEX sources which were deemed unique. In this case, the match query returns a match of each of the GALEX sources with the same SDSS object. We retain the match with the GALEX source which is the closest to the SDSS source, and eliminate the other. This occurs rarely.

2.3.3 Spurious matches.

The probability of spurious matches was estimated as follows. We randomly selected 30% of the coordinates from our matched GALEX-SDSS catalogs (in separate 5 bins of Galatic latitude), and searched the SDSS database against coordinates offset by 0.5 from those of the real sources. The spurious match rate, shown in Fig. 3 (red dots), is the number of incidences where one or more matches were found (within a search radius of 3”) divided by the number of uploaded coordinates. As for the multiple matches, it is higher towards the Galactic plane, where the stellar density increases and therefore also the probability of random matches within a given radius. The rate is of the order of a few% at high and intermediate latitudes. However, matches that are positional coincidences are likely to have odd colors and the fraction may be lower in selected analysis samples.

2.3.4 Comparison with other catalogs

We point out, in the interest of users of our catalogs, that there are several differences from other basic matched source catalogs posted on MAST. Most notably, we include only the central 1 diameter portion of the field, therefore we conservatively eliminate all edge sources (mostly defects or bad measurements but also some good sources), but we retain all good portions. Second, we consider all existing measurements for each source, from which the best is chosen. For example, the matched catalog described by Budavari et al. (2009) instead, eliminates overlap by reducing each GALEX field to an hexagon (selection of primary). In this procedure, the vertices of the hexagons include sources near the edge (i.e. possibly also sources with poor photometry, and artifacts). Moreover, since the hexagons are fixed at nominal positions, but the actual pointings may differ, unnecessary gaps between fields (which may not exist in the actual observations) are introduced, as the authors point out. Our partition, described in section 2.4, and following Bianchi et al. (2007b), avoids such problems and enables better and more homogeneous photometry quality, as well as easy and precise calculations of the unique-area coverage for our GALEX surveys, and of overlap area with any other survey, consistent with the actual source catalog rather than with nominal field centers. This is preferable for our analysis. Such differences may cause samples selected from different matched catalogs, to differ slightly.

2.4 Area coverage

In order to derive the density of sources extracted from our catalogs, we computed the total areas of unique coverage, taking into account overlap between fields, in the GALEX GR5, and then the area overlapping with SDSS DR7, following the method of Bianchi et al. (2007b). Since we restricted the catalogs to sources within the central 1 of the GALEX field, for each field we considered an effective radius of 0.5. Our code scans the entire sky and calculates the unique area covered by the GALEX fields, and the portion of this area covered also by the SDSS. First, the whole sky is divided in small, approximately square, tassels, along Galactic longitude and latitude. We used steps in l,b of 0.05/0.1 for MIS/AIS respectively. Then, we find the distance between the center of each tassel in our whole-sky grid and the center of each GALEX field, and sum the areas of the tassels that are within half degree of a field center, avoiding to count the same tassel twice, which eliminates overlap between fields.

The error in the estimated area depends on the step used to calculate the grid of tassels (tassel size), and the number of tassels. The area of each tassel decreases at high latitudes (the step in l,b is kept constant over the whole sky), hence the statistical error due to the tassels along the field’s edge (i.e. tassels that fall partly inside and partly outside the 1 circle of a GALEX field) varies with the location of the fields, both with latitude and with the relative location of the field center to a grid step. Such errors cancel out statistically for a large number of fields. For a very small area however, i.e. for a few fields, it is desirable to use smaller tassels. We estimate the uncertainty by computing the areas several times, each time offsetting the positions of the GALEX field centers by about half the size of a tassel, in both latitude and longitude directions. The resulting uncertainty is 1 square degree, for our total area and the areas of each latitude bin.

In order to calculate the area of overlap between the GALEX MIS and AIS surveys and SDSS DR7, we initially matched the centers of all tassels included in our GALEX coverage (MIS and AIS) to the SDSS DR7 footprint: the areas of tassels deemed within the SDSS footprint were summed to obtain the area coverage of our matched source catalogs. However, we discovered an issue in the SDSS database DR7 footprint that prevented its use for calculating the overlap areas. The output returned from the SDSS database footprint query is currently somewhat incorrect, perhaps due to some SDSS fields which appear included in the DR7 footprint in the database (therefore counting towards area coverage in any estimate), however do not have sources in the ”photoprimary” table or corresponding images because the observation failed or was marked as bad for some reason. We have consulted several experts of the SDSS database, but we found no way so far to identify such fields specifically. When we plotted on the sky the SDSS matched sources (obtained from the SDSS “photoprimary” catalog), and the sky-tassels deemed by the footprint-query to be within the footprint, we noted some not irrelevant areas of mismatch (most notably, in the latitude strip 50-60South we found the largest “false positive” area, about 17.5deg in total; the discrepancies are much smaller at other latitudes). We attempted to perform the footprint query on the SDSS database using different syntaxes. Specifically, a basic all-inclusive footprint SQL search returns several “false-positive” answers (i.e. several contiguous areas that the database considers within the footprint, have no SDSS sources), hence the area based on such outputs would be overestimated with respect to the source catalog; a more detailed footprint query, confining the search to ”PRIMARY” footprint area, returns both false-positives and false-negatives. We finally could not use the SDSS footprint query, and wrote a separate code to independently search, for our entire GALEX sky coverage, sub-areas with and without SDSS sources; we considered the areas devoided of any SDSS sources as outside the SDSS footprint111These regions may be ’in’ the footprint for SDSS database purposes, but they contribute no sources to the catalog, hence counting them towards the area coverage would be inconsistent. This procedure was complicated because (1) in order to decide whether a tassel contained SDSS sources, we had to consider SDSS sources of any color, not just our hot-star catalog (which includes the rarest stars in the sky, hence some tassels may actually not have any hot star even though they are located within the footprint), and (2) we had to consider much larger sky-tassels than our original grid, to avoid false-negatives, and then iterate within the ”positive” tassels with a progressively finer grid (down to 0.05deg, about 1/16th of a GALEX field), to confine the uncertainty of the area estimate. It is important for our analysis, which compares density of sources (#/square degree) with model predictions, to estimate areas consistent with the source catalog used, in this case the PRIMARY catalog from the SDSS. We used our area estimates in the following analysis. We stress, for future reference, that there was no way to detect the inconsistency between catalog and database footprint other than by plotting the distribution of sources and footprint-tassels on sky coordinates in various ways. This is a desirable test on any such work.

The GALEX GR5 AIS and MIS sky coverage, and the GALEX-SDSS overlap area in the matched GR5xDR7 releases, restricting the GALEX field to 1 diameter and eliminating overlaps between GALEX fields, are given in Table 1, for the whole coverage and for separate ranges of Galactic latitude.

2.5 Catalog statistics and effect of error cuts on the sample

Figure 4: Distribution of UV sources in NUV magnitudes, and effects of error cuts and color cuts. LEFT: the whole catalog of GALEX unique sources (MIS), described in Section 2.2, with progressive error cuts in NUV, and in both FUV and NUV. A more stringent NUV error cut simply causes a brighter cutoff at the faint end, as expected, while the requirement of significant detection (or good photometry) also in FUV modifies the overall distribution (“redder” sources being eliminated), which becomes double-peaked. The distribution of the “bluest” sources (FUV-NUV -0.13), which include hot stars (this paper) and some QSOs (Bianchi et al. 2009a) is also shown. RIGHT: NUV magnitude distribution of the matched UV-optical sources. Here we also use the spatial information from the optical data, to separate point-sources (eliminating most galaxies, but not QSOs with a high contrast between the central source and the underlying galaxy). The point-like sources with FUV-NUV-0.13 and FUV, NUV error 0.3mag (solid orange line) are the hot star candidate sample analyzed in this paper, and the green histogram are the “single” hot stars. The figure, with the analysis in section 4, shows that a potential halo component (NUV24-25mag) is eliminated by our error cut at the current survey’s depth; on the other hand, including objects with larger errors would cause our color selection to include low z QSOs.

Figure 4 shows the UV magnitude distribution of our GALEX unique-source catalog, and the effect of progressive error cuts in NUV, and in both NUV and FUV. While a progressively stringent error cut in one band (NUV) simply truncates the sample to a brighter magnitude limit, an additional error cut in FUV, which effectively raises the faint limit of the sample in this band, causes a relatively higher decrease of redder sources, and the histogram of sources distribution becomes double-peaked. The same effects are seen (right panel) for the matched GALEX-SDSS sources catalog. This catalog can also be separated in pointlike and extended sources using the SDSS spatial information (1.4   resolution). Such distinction shows that most of the faint-magnitude peak of the source distribution is due to extended sources, which are likely galaxies, as can be expected (e.g. Bianchi 2009). The most restricted sample in the right-panel histograms, the point-like matched sources with error 0.3mag in both FUV and NUV, and FUV-NUV-0.13, is the subject of this paper; it contains mostly hot star candidates (  approximately 18000K), with some contamination by QSOs at faint magnitudes and red optical colors, discussed in section 3.2 (see also Bianchi et al. (2009a)).

The analysis of the hot-star sample with Milky Way models (see later) shows that the magnitude limit introduced by our error cut of 0.3 mag eliminates mostly halo and thick-disk MW stars from the sample (between the green solid- and dashed-lines on the right-hand panel). On the other hand, including sources with larger photometric errors would introduce significant contamination of the sample by non-stellar objects (see Fig 5).

More statistical analysis of the catalogs, and discussion of potential biases in flux-limited sample selection, are given by Bianchi et al (2011).

Figure 5: Color-color diagrams for the GALEX UV sources matched to SDSS sources, at AIS and MIS depth. Blue/black densities are pointlike/extended sources respectively. Model colors for stars (T sequence, for different gravities), QSO (redshift sequence, cyan) and galaxies (age sequences, green) are shown. The two purple stellar sequences (label “WD”) in the right-side panel are for logg=9.0 and 7.0. Although gravity cannot be separated well photometrically at high  ’s, the vast majority of hot stellar sources are clustered in between these two model sequences, and much fewer are seen along the logg=5 and 3 (red and yellow) sequences at hot temperatures, as expected. In the 2.5mag shallower AIS survey the relative number of stars and extragalactic objects is comparable, while at MIS depth the latter dominate. From such diagrams the hot star samples are selected (see text), with color cuts shown by the green lines in the left-hand plot.

3 Selection of hot stars

The matched GALEX-SDSS sources in our catalogs are shown in color-color diagrams in Figure 5 as density plots. Model colors for major classes of astrophysical objects are also shown to guide the eye in interpreting the distribution of sources. We refer to Bianchi et al. (2009a) and references therein for other similar figures and description of the model colors.

We selected hot stellar candidates by extracting the matched pointlike sources with FUV-NUV  -0.13 (in the AB magnitude system), which corresponds to stars hotter than 18,000 K, the precise  depending on the stellar gravity, and on the model atmospheres adopted (discussed later).222The measured colors also depend on extinction, however for a Milky Way-type dust, with R=3.1, and moderate reddening amounts, the GALEX FUV-NUV color is basically reddening-free. In fact, the higher selective extinction A/  in the FUV range is approximately equalled by the effect of the broad 2175Å  feature in the NUV range, see Fig. 2, and Bianchi (2009). The absorption in each UV band, however, is much higher than in optical bands. This FUV-NUV limit was chosen to largely exclude all extragalactic objects, based on colors derived from canonical templates of QSOs and galaxies (e.g. Bianchi 2009 and references therein, see also Fig. 5).

As shown by Figure 5 in this work, and by Fig. 7 (lower panels) Bianchi et al. (2007a), at AIS depth (21th mag) sources with FUV-NUV  -0.13 have very little contamination by extragalactic objects, but at fainter magnitudes a number of QSOs and galaxies with extremely “blue” FUV-NUV colors is found. Bianchi et al. (2009a) have examined the nature of such uncommon QSOs, and their Figure 1 and 2 show the location of these objects in the color-color diagrams. These QSOs have extremely “blue” FUV-NUV (more negative than typical QSOs at any redshift), but optical colors typical of QSOs and galaxies. Therefore they overlap, in UV–optical color-color diagrams, with the locus of hot stars with a cool companion, i.e. hot stars having very blue (i.e. negative) FUV-NUV color but optical colors redder than what their FUV-NUV would imply for a single hot source. We discuss the QSO contamination in section 3.2.

We restricted the catalog to sources with photometric errors mag in both FUV and NUV. Fig. 4, previously discussed, and Table 2 in this work show the effect of the error cuts on the sample’s statistics. We additionally use one optical band (r) to separate the hot sources in “single” star candidates, having color NUV-r0.1, and “binary” candidates having NUV-r0.1. In more detail, the NUV-r boundary between single and binary hot stars has a complex dependence on stellar  and radii of the binary pair, which we will address in a different work. For example, a hot WD with an A-type companion, and pairs with small mass ratios will fall in the color selection of our “single” hot-star sample. The adopted value is a useful cut to eliminate most extragalactic sources contaminating our FUV-NUV-0.13 sample, as they have NUV-r0.1, however it is obviously an oversimplification for a detailed study of binaries. In sum, our so-called (for simplicity) “binary” sample includes stellar pairs with a hot WD and a cooler star, and some QSOs; the “single” sample includes all single stars and some binaries. We impose no error cut on the r magnitudes, in order to not limit the sample of the hottest WDs, which are faint at optical wavelengths. If the restriction of error0.3 mag were imposed to the r-band, the density of objects in the MIS sample would be reduced by about 30% (GALEX is ‘deeper’ than SDSS for very hot stars of low luminosity, see section 3.1); the loss would be much smaller for the brighter AIS sample.

The density of hot-star candidates is shown in Fig. 6 and will be discussed later. The magnitude distribution of the total hot-star candidate sample (FUV-NUV  -0.13) is shown with shadowed histograms in Figures 8 and beyond, and the “single” hot star candidates (FUV-NUV  -0.13 and NUV-r0.1) with solid-color histograms.

3.1 Hot Stars not detected by SDSS

While the SDSS depth of 22.3/23.3/23.1/22.3/20.8 AB mag in u,g,r,i,z provides a fairly complete match to the AIS UV source catalog, the hottest, smallest stars detected in the MIS may fall below the SDSS limit. This can be guessed by comparing in Fig.s 8, 10, 12 the green-dashed histogram (matched GALEX-SDSS sources with FUV-NUV-0.13) with the light-green filled-color histogram (matched sources with FUV-NUV-0.13 and NUV-r0.1, i.e. “single” hot stars). The hottest single WDs will be faintest at optical wavelengths (e.g. a star with log g=9. and  =50,000/100,000 K would have FUV-r=-2.33/-2.61 in AB mag, according to our TLUSTY model grids). Therefore, hot WDs still detectable in UV at the depth of our MIS sample (see Fig. 4) may be below the SDSS detection limit, while hot WDs with an optically-brighter binary companion or QSOs, which have redder UV-optical colors, will be detectable also in the SDSS imaging. This explains why the density of MIS “single” hot matched sources (light-green) drops at faint magnitudes earlier than the dashed-green histogram, the difference between the two being larger than what can be ascribed to QSO contamination (Section 3.2).

In order to estimate the incompletess of faint hot-WD counts in our matched sample, we searched for GALEX MIS sources with FUV-NUV-0.13 that are within the SDSS DR7 footprint, but do not have SDSS counterpart. We found 1,500 such sources, having no optical match in our catalog with a match radius of 3 (section 2.3). We performed a number of tests to verify if these are real sources. We matched them against the SDSS catalog, extending the match radius to 6, and found 399 additional matches, 138 of them are classified as “pointlike” sources at the sdss resolution. These mostly appear to be actual sources in the images, although we note that for faint sources, the SDSS classification of “extended” and “pointlike” is not always reliable, as shown by Bianchi et al. (2009a). It is expected that a small number of sources may have optical coordinates differing by more than 3 from the UV position; sometimes this is due to a nearby source not fully resolved. For the remaining 1,100 UV sources with no SDSS match out to 6, visual inspection of random subsamples revealed some to be part of extended cirrus emission, many seem likely sources, a few cases are parts of a shredded galaxy. The ‘kron radius’ from the GALEX pipeline gives an indication of the spatial extent of the source: 189 sources have kron radius larger than 3.5, hence are probably not stellar sources, but visual inspection suggests about 75% of them to be real sources. Even visual inspection however is not always conclusive, for faint sources or complicated fields, and better resolution or deeper exposures would be needed for a final sample. In sum, a very large fraction of the 1,500 objects unmatched within 3 are actual sources, and an undetermined fraction may be actual hot stars.

In spite of the large uncertainties, we added all these objects to the “single” hot matched sources catalog, and show the total as filled-dark-green histograms in Fig.s 6, 8, 10, 12. They must be considered a very generous upper limit, and only a reminder that the light-green histograms suffer from incompleteness for hot stars with NUV fainter than 22mag.

3.2 Purity and completeness of the sample

In order to estimate the probability that the photometrically-selected hot star candidates actually are hot stars, and the possible contamination of the sample by other types of objects, we examined the subsamples of our catalogs for which SDSS spectra exist. Out of 9,028 MIS matched sources with FUV-NUV-0.13 and FUV,NUV error 0.3mag, 810 have SDSS spectra, 104 of which are sources with NUV-r0.1, i.e. in the “binaries” locus. Of these “binary” candidates with existing spectra, 58 are spectroscopically classified by the SDSS pipeline as stars, 4 as galaxies, 42 as QSOs. Therefore, almost half of the MIS sources with FUV-NUV-0.13 and NUV-r0.1 could be QSOs (as found also by Bianchi et al. (2007a)), and half could be hot stars with a cool companion. In more detail, QSOs numbers increase at fainter magnitudes, and have FUV-NUV colors closer to our limit (-0.13) than the hot WD (Bianchi et al. (2009a)). Because stars and QSO counts vary with magnitude in different ways, and especially because this statistic can be highly biased by the SDSS selection of spectroscopic targets, which is of course unrelated to our present selection, and because the spectroscopic survey does not reach the depth of the MIS photometric survey, we refrain from assuming a correction for the fraction of extragalactic objects in the binaries sample based on current data.

A much higher purity is found for the “single” hot-star candidates (FUV-NUV-0.13 and NUV-r0.1). Out of 4,924 in the MIS sample, 706 have spectra, 703 of which are classified as stars, and two as galaxies. Only one is classified as QSO by the SDSS spectroscopic pipeline, but it is actually a hot WD, as shown by Bianchi et al. (2009a). This implies a purity of almost 100% for the “single” hot-star candidates, down to the magnitude limit of the SDSS spectroscopy at least. Again we stress however that the spectroscopic subset is serendipitous for our purpose, but not necessarily unbiased.

In the AIS sample, out of 28,319 total hot-star candidates (21,606 of which with NUV-r0.1), 4448 (3737) have spectra, classified as 4075 (3721) stars, 309 (9) QSOs, and 59 (7) galaxies; corresponding to 91.6% (99.6% for “single” hot-star candidates) purity. The higher content of stellar sources in the AIS sample, even for the “binaries”, is due to the brighter magnitude limit.

A crude, more direct indication that there is contamination by extragalactic sources in the “binary” hot-star locus, is that the fraction of “binaries” (NUV-r0.1) among the hot sources is about 25% for the AIS and 45% for the MIS. We note that the AIS value of 25% is not reflecting the fraction of actual binaries, because the pairs whose components have similar    are included in the “single” color cut, and on the other hand, AIS “binaries” may also contain QSOs.

In view of the contamination by QSOs in the “binary” hot-star sample, in the following analysis with Milky Way models we will consider the counts of “single” hot-star candidates, and assume a canonical binary fraction of 30% in the models.

3.3 Characteristics of the Hot-Star Candidate Catalog

Figure 6: Density of “single” hot stars at different Galactic latitudes (all magnitudes combined, green for MIS, blue for AIS). The stellar counts are not corrected for Milky Way extinction, which is more severe at low latitudes, and causes the sharp drop seen on the MW plane. The bright-green histogram is the GALEX-SDSS matched MIS single hot-star candidates, the dark-green histogram includes also hot UV sources with no optical match, within the SDSS footprint (see text). The ranges 10-20 N and 70-80 N have only 6 and 3 square degree MIS coverage in the matched footprint, therefore the statistical significance is much less than in other bins.

The number of hot sources selected (pointlike, with error 0.3mag in FUV and NUV, and FUV-NUV-0.13; see sect.3), and of the subset with NUV-r0.1 (“single”), are given in the last two columns of Table 2. Stellar counts as a function of magnitude will be analyzed in the following sections. Figure 6 shows the density of “single” hot-star candidates (all magnitudes combined) as a function of Galactic latitude. The density increases by over a factor of seven, from the poles towards the Galactic plane. Owing to the MW extinction, the counts are a lower limit at all latitudes, but especially near the Galactic disk. Therefore, the variation with latitude shown by this figure is less than the actual one. For the MIS, the density of hot stars in the matched sample (light-green histogram) is incomplete due to the SDSS limit (section 3.1). The dark-green histogram shows the MIS “single” hot-star candidates including the hot GALEX sources without optical match. Sources with NUV-r0.1 are not shown due to the QSO contamination (section 4.3); adding the stellar binaries in this color range would increase the number density but not change the trend.

In the next section we analyze the density of hot stellar candidates as a function of magnitude; the samples are divided in strips of 10 Galactic latitude, in order to examine the structure of this MW stellar component.

4 Analysis. Comparison with Milky Way Models

From the point of view of stellar evolution, the sample presented in this paper includes essentially two kinds of evolved stellar objects: the post-AGB stars that have just expelled their envelopes and are crossing the HR diagram towards higher  ’s at constant luminosity, on their way to become planetary nebule nuclei, and the hot white dwarfs which are fading both in   and luminosity. Post-AGB stars are very elusive because they evolve very fast, on time-scales of the order of 10-10 years (Vassiliadis & Wood (1994)). Hot white dwarfs evolve with longer timescales, but at significantly fainter luminosities. Both are elusive at all wavelengths except the UV. Our hot-star census based on the UV sky surveys provides the first opportunity to examine a comprehensive, unbiased sample of such objects. Our hot post-AGB and WD candidate sample extends over a significant sky coverage, and clearly presents a disk-like distribution, concentrated towards the Galactic disk (Fig. 6). Such sample will be analyzed in the context of a model for the Milky Way geometry below.

4.1 Constraining Milky Way models

In this section we analyze the number counts and sky distribution of our hot star candidates, using the TRILEGAL stellar population synthesis code (Girardi et al. 2005). TRILEGAL creates mock catalogues of stars belonging to the Milky Way, and then distributes them along the line-of-sight. It extracts the simulated stars from extended libraries of evolutionary tracks and synthetic stellar spectra, assuming reasonable prescriptions for the distributions of ages, masses and metallicities of the Milky Way stellar components: thin and thick disks, halo and bulge. We note here that the bulge component does not contribute almost any source to our hot star sample, and is therefore not shown in the plots of our model results.

The thin disk density, , decreases exponentially with the galactocentric radius, , and as a squared hyperbolic secant function in the direction perpendicular to the plane, :

(1)

The scale length is set to  pc, whereas the scale heigth increases with the population age as

(2)

Such an increase is necessary to describe the observed increase of the velocity dispersion W with age (see e.g. Holmberg et al. (2009)).

The thick disk follows the same functional form but with both scale parameters fixed as  pc and  pc. The constants , for the thin and thick disks, are adjusted so that the surface thin disk density is equal to 59 , whereas the local thick disk density is of 0.0015 . The halo is modeled as an oblate spheroid following a deprojected law, with a local density of 0.00015 . Finally, the Sun is located at  pc,  pc.

The default parameters for the geometry of the MW components in TRILEGAL are calibrated to reproduce the star counts in a local sample extracted from the Hipparcos catalogue, and in several multiband catalogues including the shallow, all-sky 2MASS, and a few deep surveys such as EIS-deep and DMS (Girardi et al. (2005)). For regions out of the Galactic plane, i.e. for , errors in the star counts predicted by TRILEGAL are typically of about 10 to 20% down to (Girardi et al., in preparation).

The reddening is taken into account in the Milky Way models as follows. Along a given line-of-sight, the value of at infinity is taken from the Schlegel et al. (1998) maps. The total extinction is then distributed along the line-of-sight as if it were generated by an exponential dust layer with scale height ; its default value is  pc. In this way, the closest simulated stars are unreddened, whereas those at distances of a few hundred parsecs are reddened by the full amount predicted by the Schlegel et al. (1998) maps.

We use a new version of TRILEGAL, which has been modified for the purposes of this analysis in several ways; most of these modifications will be detailed in a subsequent paper (Zabot et al., in preparation). Post-AGB stars, planetary nebulae nuclei and white dwarfs of types DA and DB have been included using the evolutionary tracks from Vassiliadis & Wood (1994) and Althaus et al. (2001, 2005), together with the Koester (2008) synthetic spectra. In the present version we have considered only the DAs, since they are the dominant type among hot WDs (Hansen & Liebert, 2003). The fraction of DBs is known to increase significantly for WDs cooler than 12000 K, thanks to convection in the He layers (Bergeron et al., 1997; Hansen & Liebert, 2003). Such cool WDs, however, are not included in our selection of sources. The TRILEGAL code uses the Koester (2008) grids to assign magnitudes to the theoretical stars. In Figure 7 we show a comparison with magnitudes calculated from TLUSTY models, and in Figure 8 we compare stellar counts obatined by selecting model stars by  and by synthetic colors, in order to illustrate the sometimes neglected effect of the model-atmosphere uncertanties.

Following our preliminary findings of discrepancies between predictions and hot star counts based on earlier data releases (Bianchi et al. 2009b), the TRILEGAL code has been modified to also allow the choice of initial–final mass relation (IMFR) independent from the prescription adopted for the previous evolutionary phases. In this way, the mass distribution of the white dwarfs does not need to follow the constraints imposed by the previous TP-AGB tracks, which come from Marigo & Girardi (2007).

Model predictions for sample Galactic latitudes are shown in Figure 8, computed with default TRILEGAL parameters for the MW geometry and the default initial-final mass relation (IMFR) from Marigo & Girardi (2007, hereafter MG07), as well as the Weidemann (2000) IFMR (hereafter W2000). We show the thin disk, thick disk, and halo stellar components, as well as the total predicted counts. The IFMR from MG07 alrgely overpredicts faint star counts, while the W2000 IFMR produces an overall better match with the observed counts, at all Galactic latitudes (see also Fig. 10), and was therefore adopted in all our calculations that follow. Given the relevance of the IFMR in the context of stellar evolution, we explain in the next section how it affects the hot-star count predictions, which can be tested by our data.

Figure 7: Comparison of model magnitudes (NUV) and color (FUV-NUV) of stars with log(g)=7.0 and 9.0, of varying  . Model magnitudes are constructed from our grid of pure-H TLUSTY NLTE model spectra and from Koester (2008) LTE models. The latter include convection, which TLUSTY (ver198) does not. Our color cut of FUV-NUV-0.13 corresponds to   18,800/17,200K (log(g)=9.0/7.0) in the TLUSTY models, and to  =18,000K in solar Kurucz-model colors with log(g)=5.0 (not shown). While TLUSTY and Koester model colors with log(g)=7.0 agree quite well for  up to 26,000K, the two grids are discrepant by 0.15mag for log(g)=9.0, where the Ly wings are broad enough to enter the GALEX FUV band when convection is taken into account. The situation reverses at hotter  ’s. A color cut at FUV-NUV-0.5 benefits from less scatter among model grids, which we could take as a measure of less uncertainty, however the resulting sample (the very hottest stars) would be drastically reduced, and so the statistical significance.
Figure 8: Modeling WD counts: the effect of the IFMR. The density of hot star candidates at two sample latitudes (10-wide strips) are shown, with model predictions (halo, thin and thick disk components, and total) The models in the top-left panel are computed with the TRILEGAL default IFMR (MG07) and the others with W2000 IFMR. The thick disk and halo counts become significant at magnitudes fainter than 19, and 20, for these two IFMR respectively. Models with W2000 IFMR match better the data down to magnitudes 20-21, below which the AIS becomes incomplete. In the top and bottom-right panels the model counts are obtained selecting stars with  18000 K in the model calculations, and in the bottom-left panel selecting stars by color cut (FUV-NUV-0.13) with TRILEGAL’s transformation from  to magnitudes, from the same calculations. The comparison illustrates the uncertainties introduced by the transformation of the isochrones into magnitudes via model atmospheres. The filled-green histograms are GALEX MIS hot “single” sources with SDSS match (light green), and including sources with no SDSS detection (dark green).
Figure 9: LEFT: Some of the IFMRs implemented in TRILEGAL, and tested in this work. The MG07 IFMR depends on metallicity, so that different age–metallicity relations (AMR) will sample it in different ways; the curve shown here is for the Rocha-Pinto et al. (2000) AMR, which is used to model the MW thin disk in TRILEGAL. Other IFMRs are the semi-empirical one from Weidemann (2000), and the purely empirical ones by Ferrario et al. (2005) and Kalirai et al. (2008). RIGHT: The distribution of initial and final WD masses (top and bottom panels, respectively) derived from TRILEGAL models using the Marigo & Girardi (2007, red lines) and Weidemann (2000, black lines) IFMR. The different lines illustrate the contributions from the different MW components, for one sample latitude, down to NUV.
Figure 10: Density of hot stars, separated in 10 ranges of Galactic latitude. The area coverage for AIS and MIS, indicated on each panel, varies significantly among latitudes, and the statstics is better where the coverage is large. TRILEGAL model counts, computed with W2000 IFMR, are shown with lines for thin disk (dotted), thick disk (dash-dotted) and halo (dashed), as well as total (solid line). The black lines are model stars selected by  18000K (upper line: total, lower line: single model stars); the red lines are stars selected from the same models by color cut (FUV-NUV-0.13) as explained in Fig.8. The match is good overall, at bright magnitudes, but better at Northern high latitudes and Southern low latitudes (except for 20). The dashed green-outlined histograms include both single and binary MIS hot-star candidates (the latter including also QSOs at faint magnitudes). The solid-color histograms are the “single” hot-star candidates (NUV-r0.1). The models were computed covering the centers of the GALEX MIS fields in each latitude strip, except for the lowest latitudes (b20) where the TRILEGAL models are computed following the distribution of the AIS fields, because MIS has little or no coverage. The large difference between North and South 20-30 latitudes, seen both in observations and predictions, is due to the MIS-SDSS overlap including more directions towards the Galactic center in the South, and away from it in the North; the longitude dependence is illustrated in Figure 11.

4.2 The Initial-Final Mass Relation

While the Milky Way geometry has been derived by previous studies of low mass stars, the IMFR is one of the least constrained factors in our understanding of stellar evolution, in spite of its importance for determining the yield of chemical elements, and ultimately the Galaxy chemical evolution. The reasons for this uncertainty include insufficient statistics of post-AGB stars. The final WD mass is confined to a very small range ( 0.5-1.0) compared to the 0.8-8range of their main sequence initial masses, and the post-AGB evolutionary time-scale varies steeply within the small WD mass range. The luminosity remains constant over a large range of  (up to 100,000K), and again depends on the remnant mass, making it much harder to establish their distance and absolute luminosity than for main sequence stars.

Therefore, the mismatch between model predictions computed with TRILEGAL’s default parameters and observed stellar counts (Figure 8, top-left panel), also previously noted by Bianchi et al. (2009b), prompted us to explore different initial-final mass relations in TRILEGAL. Although other factors, such as the WD birthrate, or the assumed MW extinction model, or a different geometry may also affect model predictions, the IFMR is a most critical aspect for modeling hot-WD counts, and was therefore explored in this first analysis. The geometry is better constrained by low-mass stars (e.g. Girardi et al. 2005), which are more numerous, and their counts based on optical-IR bands are less affected by extinction than a UV-based catalog. The right-hand panels of Fig. 8 show that the W2000 IFMR produces a better match of the model calculations with the observed stellar counts than the MG07 IFMR, and this trend is seen at all Galactic latitudes.

These two IFMRs are illustrated in Fig. 9, together with additional IFMRs recently derived by Ferrario et al. (2005) and Kalirai et al. (2008) from extensive high-S/N spectroscopy of white dwarfs in Galactic open clusters. Despite the uncertainties related to the age-dating of the clusters, their field contamination, and the possible dependences on metallicity, they are somewhat similar to the Weidemann (2000) semi-empirical IFMR. The MG07 IFMR, instead, is derived from theoretical evolutionary tracks and, as shown in the figure, yields significantly higher final masses for all .

In order to understand why the model-predicted hot-WD counts depend on the IFMR, let us first consider the distribution of WD masses in present-day surveys of the Solar neighborhood. Empirical determinations have always concluded that the WD mass distribution presents a strong peak at low masses, with a maximum close to 0.55 or 0.65   and a well-defined tail of more massive WDs, as well as a faint low-mass tail believed to be either observational errors or the result of binary evolution (e.g. Bergeron et al. (1992); Bragaglia et al. (1995); Madej et al. (2004); Liebert et al. (2005); Kepler et al. (2007); Hu et al. (2007); Holberg et al. (2008)) .

The origin of this peaked mass distribution can be readily derived from basic population synthesis theory (see Marigo, 2001; Ferrario et al., 2005). Given a volume-limited sample containing stellar populations of all ages between and  Gyr, the distribution of WD initial masses will be given by

(3)

where is the IMF, is the star formation rate at the time of stellar birth , and and are the main sequence and WD lifetimes, respectively. The first 2 terms in the right-hand side of this equation represent the production rate (or, alternatively, the “death rate”) of evolved stars with different masses, in number of stars per unit time. Considering a Salpeter (1955) IMF, , and a reasonably-constant star formation rate over the galaxy history, the equation above indicates a marked peak of the production rate at the minimum initial mass for the formation of a white dwarf, which is about 1  for a galaxy age of  Gyr. For higher initial masses, the production rate should fall as . This behaviour will be shared by any galaxy model with nearly-constant star formation, independently of its IFMR.

Assuming a constant and monotonic IFMR, , the WD mass distribution is given by

(4)

therefore, for almost-linear and linear IFMRs like the Weidemann (2000), Ferrario et al. (2005), and Kalirai et al. (2008) ones, the WD production rate is still expected to behave like a power-law peaked at the smallest masses. This smallest mass corresponds to a WD mass of   (see Fig. 9).

Figure 11: Model-predicted counts at latitudes between 20-30 North, and South, for four symmetric longitude quadrants. Model calculations were performed in 1 fields evenly distributed in longitude, and combined in 90 sections, to illustrate the dependence of the model-predicted hot stars counts on longitude. The strongest differences are seen between center and anti-center directions, as expected, and the North-South asymmetry is most prominent in directions towards the Galactic center. Lines show thin disk (dotted), thick disk (dash-dotted), halo (dashed) and total counts (solid).
Figure 12: Thin disk description. In view of the fact that our present data is most sensitive to the thin disk population, we compare model results computed with three different thin disk geometries: in the top panel the TRILEGAL default geometry, in the middle panel an exponential (option 1), and in the bottom panel a z sech (option 2) description (see text). The top panel, with the default geometry description given in eq. 1 and 2, gives the closest results to the observed counts, confirming previous findings by Girardi et al. (2005). This option was adopted in all our model calculations shown. Colors and symbols as in Fig. 10.

The WD lifetime , defined as the total time after the AGB in which the star will be hotter than 18,000K, is longer for WDs of higher mass, because, although they evolve much faster in the constant-luminosity phase, they reach higher  than lower-mass WDs, and spend a longer time in their WD cooling track before fading below the Teff=18000K limit. Models using different IFMRs simply associate the final masses, hence also these lifetimes, to different initial masses, without affecting the WD production rate. The end result in terms of mass distribution (see equation 4) is that smaller WD masses will weigh the distribution towards shorter , hence decreasing the numbers of observed WDs with  18000K. This effect explains the higher number of hot WDs predicted by the MG07 IFMR respect to the W2000 IFMR (Fig. 8). In fact, as can be inferred from Fig. 9, the W2000 IFMR gives significantly lower WD final masses (hence shorter ) for the interval of initial masses between 0.6 and  . Because of the IFM being skewed towards lower masses, these objects represent numerically the majority within the whole mass range.

Figure 9 (right panel) shows the distribution of initial and final WD masses derived from TRILEGAL for a sample latitude, using either the MG07 or W2000 IFMR. It shows that most of the hot WDs observed by GALEX are expected to derive from low-mass stars, with initial masses below  . Moreover, it indicates that the bulk of observed WDs have low masses, typically   in the case of the W2000 IFMR. There is no direct confirmation of this particular prediction, although it is in line with WD mass determinations derived from spectroscopic surveys of more limited samples (e.g. Bergeron et al., 1992; Bragaglia et al., 1995; Madej et al., 2004; Liebert et al., 2005; Kepler et al., 2007; Hu et al., 2007; Holberg et al., 2008).

In the calculations with MG07 IFMR the distribution of predicted WD masses is quite different from the Weidemann (2000) case, with a broad mass distribution extending from 0.55 to 0.65 , followed by a sort of gap, and a second peak of WD masses located slightly above 0.7 . These features reflect the positive slope of this IFMR up to  , and its plateau at   for   (see Fig. 9 -left). Another particularity of this IFMR is the high masses (about 0.64 ) predicted for the WDs belonging to the halo and thick disk. This prediction conflicts with recent mass determinations of WDs in old star clusters, which indicate values close to 0.53  (Kalirai et al., 2008, 2009). A similar trend is seen at different latitudes.

Overall, an IFMR such as that of W2000 produces predicted counts closer to the observed numbers, at bright magnitudes where our sample is complete, and at the same time mass distributions in agreement with those derived from spectroscopy of nearby WDs. These aspects underscore the difficulties of deriving the IFMR from theoretical evolutionary tracks of AGB stars, even when these tracks are directly calibrated with observations of AGB stars in the Local Group, as is the case of MG07. This work, and future analyses planned on a wider sky area to explore concurrently the effects of the MW dust geometry, suggests that a deeper investigation of AGB evolution is necessary, especially concerning the theoretical prescriptions for mass loss, to reconcile constraints provided by WDs in our Galaxy and those provided by AGB stars in nearby galaxies.

We finally note that even models calculated with W2000 IFMR tend to predict more faint-star counts than our “single” WD sample (Fig. 10). The mismatch is significant if we consider the red model lines (model stars selected by their synthetic FUV-NUV color), which should be the most consistent with the observed sample selection, to the extent that the model atmospheres used by TRILEGAL are correct. Therefore, we explored also the case of a Kalirai et al. (2008) IFMR: the predicted counts are lower at faint magnitudes (especially the halo component fainter than NUV22), as expected from Fig. 9-left, while still generally in line with the bright star counts. Because of the incompleteness of our sample at magnitudes fainter than 21, and other factors that cannot be conclusively constrained with the current sample, we defer more quantitative conclusions which rests on a comparison of faint stars to a future work, and we adopt in this work the W2000 IFMR, that matches well the bright star counts, to explore other effects.

4.3 Milky Way geometry

As explained above, the default parameters, in particular the MW geometry, in the TRILEGAL code were defined from previous applications of this code to analysis of stellar counts from several surveys. Particularly important in this context were the 2MASS star counts, which are sensitive to the low-mass stars (0.8). These low-mass stars do not overlap with the mass range of the hot-WD progenitors (about 0.8-8), however they are very numerous and long-lived, and are little affected by uncertainties in stellar evolutionary models. Therefore we should expect the geometry to be well constrained by previous works, except for the limited extent of the previous surveys.

The agreement between our data and model predictions (with W2000 IFMR) is generally good at bright magnitudes, where our sample is complete. In more detail, we should also explore the dependence of observed and predicted stellar counts with longitude. The effect is more prominent towards the Galactic plane, therefore we examined latitudes between 20-30 North and South, and computed models with evenly spaced coverage along these strips. The models predict, as expected, different counts towards the Galaxy center and anti-center, shown in Fig. 11. The coverage of our present GALEX-SDSS matched catalog at low latitudes is still too sparse for a conclusive comparison, which will be instead attempted by matching the AIS data with GSC2, and with the increased MIS coverage expected in the GALEX “Extended Mission” phase.

Finally, because our UV-optical matched samples are complete to UV magnitudes brighter than 20-21, therefore mostly sensitive to the thin disk population (Fig.s 8 and 10), and because the models with default MW geometry slightly overpredict the stellar counts at low (especially Northern) Galactic latitudes, we explored different geometries of the thin disk with TRILEGAL. In particular, in Fig. 12 we show the effect of adopting more strongly peaked density distributions at the Galactic Plane: either the simple vertical exponential law, (which has been adopted in many MW models, starting from the classical Bahcall & Soneira (1980, 1984)’s), or the simple hyperbolic secant law, . The results confirm that the default geometry adopted by Girardi et al. (2005) and in this work, i.e. the squared hyperbolic secant law, provide the closest prediction to the observed counts. The same distribution does also provide a good match to star counts of “normal stars” in 2MASS and in the local Hipparcos sample (Girardi et al., 2005). The slight overprediction and underprediction of hot star counts at certain latitudes may also be due to extinction by interstellar dust, which may be either patchy (as in fact it is) or have a slightly different distribution than assumed. These remaining discrepancies and the effects of dust distribution may be addressed with a wider MIS coverage in the future.

4.4 The kinematic of Milky Way stars

Figure 13: Left: Distance distributions predicted by our TRILEGAL models for the hot stars in the 20-30N strip of MIS, and for three different cuts in NUV magnitude. This plot indicates that the brightest (and complete) MIS sample of NUV18mag is likely made of WDs located in the thin disk at typical distances between 0.15 and 1 kpc (in this latitude range). The faintest samples could instead contain a significant fraction of stars at distances larger than 2 kpc, comprising also a small fraction of thick disk and halo WDs. Right: distribution of proper motions (amplitudes): observed distribution (green solid histograms) are compared to model predictions (continuous lines), computed for the same area. The numbers in each panel indicate the NUV magnitude cut, the number of predicted hot stars, of observed ones in MIS, and of those which have proper motion information from USNO-B.
Figure 14: Distribution of velocities (left) from the SDSS spectroscopic pipeline for the subsample of MIS hot stars with spectra. The comparison is inconclusive because of the large errors (often larger than the values) in the pipeline velocity measurements, shown in the right panel.

The plots of stellar density versus Galactic latitude presented in the previous Sections clearly indicate that we are looking at a population of objects for which the mean distance is larger than the mean scale height on the thin disk. At typical distances of a few hundred parsecs, there will be no parallax measurements for such stars. The most direct distance information we can find for them is related to their proper motions.

The table “USNO” in the SDSS database contains proper motions for SDSS objects made after recalibrating USNO-B1.0 (Munn et al. (2004)) against SDSS astrometry. The systematic errors are quoted to less than 1 mas/year. The UCAC3 proper motion catalog is strongly incomplete for our sample, owing to its magnitude limit of (which roughly corresponds to ). In fact, we checked and found that just a handful of our hot-WD candidates is present in UCAC3.

Figure 13 presents the histograms of proper motions, , for the hot stars in the 20-30N strip of MIS, and for three different cuts in NUV magnitude. The observed distributions are compared with those predicted from the TRILEGAL models, computed for the same area and for the W2000 IFMR. The model distributions are derived as follows. The observed velocity ellipsoids of each Galaxy component are taken form the literature. Each simulated star is then given a random space velocity that follows the Schwarzschild distribution for this ellipsoid. Space velocities are then corrected for the solar motion (cf. Dehnen & Binney 1998) and projected on the sky using the transformations derived from Hipparcos. The final results are simulated proper motions and radial velocities. We adopt an age-dependent velocity ellipsoid for the thin disk, derived from the Geneva-Copenhagen Survey of Solar neighbourhood (Holmberg et al. 2008). For the thick disk and halo, velocity ellipsoids are taken from Layden et al. (1996). These prescriptions provide a first-order description of the general distributions of proper motions and radial velocities, but are not expected to describe the details of the local velocity streams, nor the effects caused by disk shear in the more distant stars. Moreover, we do not simulate measurement errors in the proper motions. More details will be given elsewhere.

The upper panel of figure 13-right compares the observed and model distributions for a bright WD sample, with a cut at NUV18mag. There is quite good agreement both in the numbers and distributions of proper motions. The agreement in the numbers derives from the fact that the photometry is quite complete, and just 4% of this bright subsample does not have a proper motion listed in USNO-B. This means that both photometry and proper motion information are very complete in this case, and quite similar to the ones expected for a thin disk population.

The middle and bottom panels of Fig. 13-right compare the observed and simulated distribution for fainter samples, with cuts at NUV19.5 and NUV21.5mag, respectively. It shows that these samples become incomplete, first in the proper motions at NUV=19.5, and then in photometry at magnitudes NUV=21.5. Note for instance the increasing fraction of stars without proper motion measurement in USNO-B, which goes from 13 to 54% from NUV19.5 to NUV21.5.

From this comparison we conclude that our hot WDs do follow the kinematics expected for the thin disk, even if the proper motion information is severely incomplete for the faintest WD candidates.

Distance distributions predicted by the same models are shown in Fig. 13-left. This plot indicates that the brigthest (and complete) MIS sample of NUV18mag is likely made of WDs located in the thin disk at typical distances between 0.15 and 1 kpc in the latitude range 20-30 North. The fainter sample could instead contain a significant fraction of stars at distances larger than 2 kpc, comprising also a small fraction of thick disk and halo WDs. Note however that this distribution of distances is inferred from models, and we presently have no data to verify the distances of our sample stars, but for the proper motions already depicted in Fig. 13 (right); future work will address identification of thick disk and halo objects with follow-up spectroscopy. The distance distribution is fairly similar at other latitudes.

Finally, Fig.14 presents the expected distributions of radial velocities for the latitude 20-30N sample, from the models, and the radial velocities derived from SDSS spectra, available for a subsample of our hot star catalog (section 3.2). This is a biased subset because the choice of SDSS spectroscopic targets is obviously unrelated to our selection of hot stellar objects. The worse effect however, making the comparison inconclusive, is the large errors in velocity determinations from the SDSS pipeline (see Fig. 14-right).

5 Conclusions and Summary

From GALEX’s data release GR5 we constructed catalogs of unique UV sources (i.e. eliminating repeated observations), and subcatalogs with matched SDSS optical photometry. We extracted catalogs of hot star candidates (FUV-NUV-0.13), mostly comprised of hot WDs. Over 38,000 such sources with photometric error 0.3mag, have SDSS photometry (74,000 with UV photometric error 0.5mag), excluding UV sources with multiple optical matches, that are between 8 to 30% of the total. The GALEX surveys AIS and MIS cover different latitudes at their respective depths (21 and 23 AB mag), enabling a first quantitative analysis of the hot-WD stellar population with Milky Way models. Current descriptions of MW stellar components (halo, thin and thick disk) are mostly based on stellar counts of low mass (0.8) stars (2MASS, shallow but all-sky, plus a few deep but area-limited surveys). We explored in this work different thin disk geometries and different IFMR. GALEX provides an unprecedented census of the evolved descendants of 0.8-8stars, and the analysis of our current hot-WD sample places some constraints on the initial-final mass relation (IFMR), one of the crucial ingredients to understand the chemical enrichment of the ISM.

The brightest (and complete) sample is likely made of hot WDs located in the thin disk at typical distances between 0.15 and 1 kpc. The faintest samples could instead contain a significant fraction of stars at distances larger than 2 kpc, comprising also hot WDs in the thick disk and halo. This work is based on the analysis of matched GALEX-SDSS sources, and the SDSS magnitude limit prevents analysis of thick-disk and halo components. Hot WDs are however detectable at such larger distances at MIS depth (see dark-green histograms in Fig.s 8 and 10, and section 3.1), and will be investigated in a future work with deeper optical data.

Model predictions of hot star counts at different latitudes, computed with currently accepted MW geometry, and assuming W2000 IFMR, match the data quite well at intermediate latitudes and bright magnitudes, but are somewhat discrepant at low latitudes and faint magnitudes. There is a slight but systematic North-South asymmetry, and MW models with canonical geometry for stars and dust match better the data at Southern low latitudes, and at Northern high latitudes, indicating that the dust distribution on the MW disk may also be better constrained by our data than was previously possible. The dust distribution will be investigated with the future more extended sky coverage from the ongoing GALEX mission. However, the observed asymmetry may also be due to the longitude dependence of the stellar populations, as our present matched-sources sample covers unevenly North and South Galactic latitudes (Fig. 1). Our MIS matched-sources sample is incomplete at magnitudes fainter than about 21.5 due to the SDSS limits for the hottest sources. At all latitudes, and at magnitudes where the samples are complete, MW model predictions of hot WD counts computed with IFMRs favouring lower final masses match the GALEX hot WD counts significanly better than the IFMR currently postulated by Marigo & Girardi (2007) from AGB stars data. This result contributes an important constraint to stellar evolution in the final phases which drive the yield of chemical elements.

5.1 The implications for stellar evolution

What are the implications of constraining the IFMR for the galaxy chemical evolution? The IFMR contains the record of the nucleosynthesis history during the TP-AGB phase, as well as determines the contribution of AGB stars in terms of dark remnants. The IFMR, and the final yield of chemical elements, are significantly affected by the occurrence (or not) of the so-called “third dredge-up”. At each dredge-up episode the core mass is reduced by an amount , where is the classical efficiency parameter, and denotes the mass growth of the core during the preceeding interpulse period.

In the extreme case where , no increment of the C-O core is predicted. As a consequence, for low-mass stars not experiencing the third dredge-up (), the remnant mass scales directly with the duration of the TP-AGB lifetime, while for stars in which the third dredge-up takes place () this simple proportionality is replaced by a more complex interplay between core growth, dredge-up, and mass loss. For such more massive stars (, where , depending on metallicity and model details) the final mass and the chemical yields are related as we will briefly discuss below (see Marigo & Girardi 2001 for more details).

For the generic element , the corresponding AGB yield is defined as:

(5)

where is the initial abundance at the epoch of star formation, is the value at time , and is the current mass-loss rate.

Let us first consider, in the IFMRs in Fig. 9, the part for . The chemical yields produced by these stars (mainly C, N, and He) are determined by the first dredge-up, and the cool-bottom process (Boothroyd & Sackmann 1999) or thermohaline mixing (Charbonnel & Zahn 2007) operating during the RGB, as no subsequent dredge-up events take place on the AGB according to the standard stellar evolution theory. For these low-mass stars the AGB yield, defined by Eq. (5), simplifies as , where is the surface abundance left after the RGB phase, and is the total amount of mass ejected on the AGB. In turn, the expelled mass is related to the final mass through , where denotes the stellar mass at the onset of the AGB phase. The final masses of these stars with can be reduced by simply invoking more efficient mass-loss along the AGB. Their chemical yields would increase correspondingly, since less envelope mass will be locked in the central core following the outward displacement of the H-burning shell, so that will be larger. For instance, for an AGB mass of , a final mass versus would imply an increase of the AGB chemical yield by , as well as a larger total chemical yield, including the RGB wind contribution.

For , in addition to the first dredge-up (and possibly second dredge-up on the Early-AGB), the chemical yields include also the enrichment, mainly in He and primary C, produced by the He-shell flashes and brought up to the surface by the third dredge-up, and the enrichment of N produced by the hot-bottom burning process occurring in the most massive AGB stars (). Again smaller final masses are reached when higher mass-loss rates occur, and/or deeper dredge-up (i.e. larger ). These two factors may affect the chemical yields in opposite ways. On one hand, a high mass loss limits the number of dredge-up episodes, hence lowering the chemical yields; on the other hand, a deeper dredge-up conveys larger amounts of carbon and helium to the surface layer, which are then ejected. In addition, mass loss and dredge-up efficiency are actually inter-related, a larger surface C abundance generally favouring stronger dust-driven mass loss in C stars (e.g. Mattsson et al. 2008), while a significant reduction of the mass of the envelope may decrease its penetration (i.e. lower ) during the third dredge-up (see, e.g. Karakas et al. 2002). In sum, a higher mass expelled does not directly imply that such material is relatively more enriched of processed elements.

An IFMR similar to Weidemann (2000) weighs the final masses towards lower values than e.g. the IFRM of MG07 in the range , while the two relations essentially agree at higher . For , forcing the theoretical IFMR to converge on a W2000-type IFMR by assuming a higher mass loss would produce a modest increase of the chemical yields from these objects (because more mass is ejected), compared to the MG07 IMFR case. As for the initial mass range , a higher mass loss (necessary to reach a smaller final mass, as suggested by our WD counts) may reduce the number of dredge-up episodes on the TP-AGB and this may lead to a lower yield of processed material. In other words, constraining the IFMR translates into constraining the total mass lost in the AGB phase. The chemical composition of the ejecta from these stars, however, is also critically sensitive to the depth of the third dredge-up, which can be calibrated by measurements of abundances in Planetary Nebulae (Marigo et al 2003) and by a better estimate of the duration of the AGB phase at different metallicity (Marigo et al. in prep.).

5.2 Future Work

With the current MIS coverage (1,103 square degrees overlapping with SDSS photometry in GR5), we compared with MW models stellar counts summed in 10 latitude strips, combining all longitudes in order to obtain acceptable statistics. In order to disentangle geometry of the stellar populations, dust extinction, and stellar evolution in more detail, we need to analyze also the longitude dependence of stellar counts, as demonstrated by Fig.s 10 and 11. We will explore dust effects in the MW disk by matching the AIS with GSC2, for a larger sky coverage (see Bianchi et al (2011)). The density of hot stars (the rarest in nature) is very low and large area coverage is needed for good statistics.

The MW halo and thick disk components, however, become significant at UV magnitudes fainter than about 21, therefore require a wider MIS coverage (planned in the GALEX Extended Mission phase) to be usefully constrained (see fig.s 1, 8 and 10), and deeper optical surveys.

Finally, hot-WDs in binaries, again elusive at optical wavelengths, are uniquely and unambiguously revealed and characterized by the GALEX UV sky survey, matched to optical surveys. These binaries sample different types of stellar pairs (different  ranges) than those detected and characterized by optical surveys, thanks to the UV photometry (e.g. Fig.6 of Bianchi et al. (2007b)). Because our current photometric selection includes contamination by extragalactic objects in the “binary” locus, and on the other hand the binaries do not significantly change the distribution of stellar counts, which was the subject of the present analysis, the binary candidates will be investigated in a future work.

5.3 Final remarks. The online catalogs.

The present catalog of “single” hot star candidates has a high purity, as estimated from the serendipitous (but not unbiased) spectroscopic SDSS coverage. While we expect most of the hottest sources to be high gravity objects (CSPN, subdwarfs or WD), we did not apply any cut in gravity, therefore main sequence stars and supergiants hotter than 18,000K are also included in our catalogs. Their number is very small compared to the evolved stars, however their inclusion in the catalog makes it more generally useful.

Finally, we remind future users of our catalogs that GALEX UV sources with more than one optical counterpart were excluded from the analysis of the hot-star sample, (and should be excluded from any sample based on photometric color selection), but their fraction can be corrected statistically using Table 2. They are included (flagged with “rank”=1) in our online catalogs. A few percent of the matched sources may be spurious matches (Fig. 3) but these would mostly have random UV-optical colors. The hot-star candidates with NUV-r0.1, includes also some extragalactic objects, the relative fraction is magnitude and latitude dependent.

The source catalogs (GALEX unique sources, GALEX-SDSS matched sources, hot-stars samples) are available in electronic form only, from the author’s web site (http://dolomiti.pha.jhu.edu/uvsky ) where files description can be found, and will be also posted on MAST (http://galex.stsci.edu and http://archive.stsci.edu/hlsp/). A statistical analysis and discussion of properties, useful to potential users of our (and similar) catalogs for understanding completeness and biases of any sample selection is given by Bianchi et al (2011).

acknowledgments

Data presented in this paper were obtained from the Multimission Archive at the Space Telescope Science Institute (MAST). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. Support for MAST for non-HST data is provided by the NASA Office of Space Science via grant NAG5-7584 and by other grants and contracts.

GALEX (Galaxy Evolution Explorer) is a NASA Small Explorer, launched in April 2003. We gratefully acknowledge NASA’s support for construction, operation, and science analysis of the GALEX mission, developed in cooperation with the Centre National d’Etudes Spatiales of France and the Korean Ministry of Science and Technology. L.B. and J.H. acknowledge partial support from FUSE GI grant H901 (NASA NNX08AG97G). A.Z. acknowledges financial support from CNPq-MCT/Brazil. L.G. acknowledges partial funding by from contract ASI-INAF I/016/07/0. We are very greateful to A. Thakar for discussions of very many issues regarding the SDSS database.

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Latitude ——- area [deg] ——- E(B-V)
range GALEX GR5 GR5+DR7 mean 1
MIS
n.a. n.a.
n.a. n.a.
n.a. n.a.
n.a. n.a.
n.a. n.a.
Total:
AIS
Total:

(a) values are computed from the Schlegel et al. (1998) maps for the centers of the GALEX fields, then averaged. They do not reflect an average Galactic trend, since they follow the specific distribution of fields shown in Fig. 1, which is rather non uniform at low latitudes

Table 1: Sky coverage for GR5 MIS and AIS, and GR5xDR7
Latitude # GALEX # GALEX sources ——————— # matched sources ——————— ———————— # Pointlike matched sources —————————-
range sources FUV-NUV-0.13 err 0.5 err err err 0.5 err err with FUV-NUV-0.13
err err err rank0 rank1 0.5 0.3 0.5 0.3 err NUV-r0.1
0.5 0.5 0.5 0.3 rank0 rank1 rank0 rank1 rank0 rank1 rank0 rank1 rank0 rank1 0.5 0.3 0.3
MIS
-90 -85 66929 16822 2444 622 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
-85 -80 177553 49043 6304 1540 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
-80 -75 112394 32092 4021 971 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
-75 -70 394332 109647 12984 3070 87456 2708 33147 1300 15510 794 17782 638 2584 211 1158 127 321 91 42
-70 -65 798670 223254 25020 6151 156597 3994 63005 2099 34584 1450 33311 1021 5554 345 2759 224 671 201 78
-65 -60 1409943 400165 44106 11391 360403 10991 149854 5826 87609 4079 71801 2433 12151 866 6395 565 1386 426 155
-60 -55 1244707 328909 40804 10773 228705 7106 86721 3448 49148 2388 49844 1659 7401 501 4061 334 931 296 125
-55 -50 632803 143121 21932 6162 164257 5294 49621 2093 25136 1291 41518 1388 4477 308 2374 204 711 257 112
-50 -45 609861 137541 21637 5756 304497 12522 92372 5102 46024 3217 77719 3427 8067 812 4342 506 1230 472 244
-45 -40 428767 89077 13169 3620 207041 10578 57421 3948 29654 2414 61074 3127 5178 651 2869 413 745 311 157
-40 -35 538838 103347 15242 4283 211133 9835 50373 3174 27961 1978 83015 3652 6030 580 3634 349 843 396 228
-35 -30 323258 59584 8370 2715 124660 8736 27139 2794 16141 1929 58244 3735 3658 607 2374 419 558 303 174
-30 -25 214275 31837 5398 1749 133209 8714 23127 2116 11900 1213 75788 4697 4076 529 2404 301 656 365 240
-25 -20 48433 6190 1035 357 34035 2638 4825 494 2441 272 22202 1652 1109 149 646 94 182 93 65
15 20 43887 8626 997 304 26939 1383 6195 433 3488 279 14452 716 1297 120 812 81 132 77 51
20 25 226867 47323 6099 1672 144508 7255 35983 2484 18325 1450 64767 3040 5810 575 3372 358 635 308 218
25 30 403868 91523 10958 2782 281365 12461 79869 4695 39329 2745 101054 4591 10017 974 5594 565 1075 520 346
30 35 711824 152925 21310 5682 474446 18387 132915 6920 66709 4116 155923 6218 14995 1263 8182 727 2079 862 488
35 40 937945 203696 27963 7589 621803 24188 173805 9065 91210 5604 200400 8114 18511 1602 10080 971 2720 1094 601
40 45 641846 152205 21044 5661 430026 15963 133123 6575 68076 4018 119976 4803 12413 1116 6306 654 1892 744 413
45 50 707063 162317 21478 5659 460636 15730 139985 6496 74669 4238 127343 4598 12844 1041 6839 658 1885 639 325
50 55 731762 170967 20334 5143 492531 19798 149059 7947 80742 4944 135058 5360 13351 1257 7468 769 1700 641 371
55 60 690921 170036 20115 4864 461420 18063 146692 7491 76642 4663 119973 4788 12470 1150 6521 704 1657 528 293
60 65 377644 97916 11366 2899 255057 9001 86150 4034 45134 2581 62828 2388 7169 649 3695 415 912 309 142
65 70 99569 25651 3073 716 68009 2407 22683 1058 10337 609 13951 546 1857 151 832 83 238 75 44
70 75 23953 6405 678 136 16778 537 5757 238 2588 137 3126 137 446 46 198 26 62 20 12
-90 90 12597912 3020219 387881 102267 5745511 228289 1749821 89830 923357 56409 1711149 72728 171465 15503 92915 9547 23221 9028 4924
AIS
-85 -90 186959 29800 6088 1187 2118 66 488 22 137 9 686 24 60 2 21 1 5 0 0
-80 -85 639403 107385 20863 4235 15996 448 3458 144 1075 65 4635 152 399 33 177 17 55 28 24
-75 -80 946695 157960 31268 6157 42060 1425 9492 431 2998 179 11306 416 922 79 448 32 134 60 41
-70 -75 1379695 220313 43504 8513 102750 4001 23114 1254 7740 600 28113 1238 2273 223 1058 116 309 138 97
-65 -70 1779025 267234 53565 10329 104406 3670 24373 1188 8059 514 28907 1114 2464 221 1070 102 350 141 99
-60 -65 1823298 273723 54203 10650 151592 6068 34985 1981 11556 944 44240 1714 3466 342 1584 181 489 233 176
-55 -60 2202425 309450 62380 12610 149424 6616 29732 1895 9752 819 50822 2148 3580 355 1700 168 537 294 214
-50 -55 2221574 260072 55813 11764 137391 6620 21591 1640 7251 732 56334 2497 3270 348 1589 154 553 331 234
-45 -50 2526007 271544 59711 12805 195821 10880 29085 2665 10396 1253 81131 4309 4433 562 2221 277 801 473 338
-40 -45 2644017 259939 59142 14759 157644 12248 20715 3038 7274 1400 79249 5390 3708 680 1808 341 758 441 302
-35 -40 2824496 243813 55332 13542 243682 15446 28209 3371 9899 1562 133778 7605 5717 831 2900 417 1208 725 501
-30 -35 3022905 251126 53008 14535 171132 14681 14919 2977 5579 1395 112514 7969 4148 857 2175 454 938 570 394
-25 -30 2725467 171367 39306 12449 219113 20491 15759 3593 6299 1665 159944 12263 5423 1154 2978 571 1330 829 611
-20 -25 2565319 121358 29997 11039 158390 19012 7832 2255 3563 1116 133521 13447 4060 922 2233 504 1037 650 476
-15 -20 2294738 90915 21775 9748 104999 13902 4288 1362 2245 739 94135 10745 2685 665 1536 384 660 406 291
-10 -15 1474126 54091 11682 6026 107441 14988 4086 1142 2582 762 102078 12821 3235 736 2048 511 607 350 231
-5 -10 468735 20256 3196 1719 15893 2690 834 224 547 147 15334 2358 707 158 441 102 77 39 22
0 -5 81792 5544 338 126 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
5 0 112359 9822 673 324 4730 816 364 140 263 120 4577 774 340 133 241 113 37 18 2
10 5 896414 41563 5055 2532 67009 10668 4191 1170 2914 871 64784 9577 3683 921 2520 682 285 169 86
15 10 2209741 93883 17027 7825 133415 20698 6792 1979 4160 1267 124034 17107 5085 1207 3230 806 758 458 301
20 15 2785094 137819 28011 10512 248149 22946 22199 3385 9760 1708 192683 16088 8372 1376 4610 747 1398 892 688
25 20 3078141 180485 38516 11924 482621 34646 50125 5965 19504 2777 328234 21367 14284 2013 7510 1020 2539 1604 1230
30 25 3074765 199400 45162 11949 565885 34039 59037 6033 20866 2622 336893 18763 14374 1854 7217 885 2777 1683 1308
35 30 3068226 221555 50831 11949 770294 40366 91294 7455 30887 3131 420587 20138 18413 1879 9082 840 3615 2135 1714
40 35 2827184 215451 50937 10501 911569 43682 109497 8095 35498 3333 459207 20279 19905 1846 9697 822 3899 2278 1774
45 40 2867019 241576 57082 10742 1149694 53336 144002 10172 45305 4158 490863 21484 21555 2157 10284 980 4171 2345 1858
50 45 2589087 237387 55351 9692 1187271 53682 158334 10737 47012 4333 455468 19957 21918 2243 10131 1013 4017 2139 1679
55 50 2183650 219294 49511 8631 1135384 51155 165181 11031 48979 4336 414632 18158 21966 2136 9994 973 3845 1979 1556
60 55 2031169 227445 49608 8489 1207517 55161 187254 12923 56759 5317 390611 17457 21534 2264 9850 1071 3576 1838 1410
65 60 1617230 193927 41370 7220 1058375 50788 173459 12127 52661 5039 332166 15743 19066 2203 8630 1048 3063 1531 1197
70 65 1419335 168945 36062 6064 927973 42934 151376 10316 44148 4136 278120 12937 16547 1808 7447 822 2578 1325 1022
75 70 1204662 135513 29867 4605 792486 37015 121558 8221 34147 3190 227147 10912 13190 1494 5749 675 2127 1033 792
80 75 860091 88480 19796 2865 563957 27906 78969 5847 21026 2238 155680 8158 8642 1069 3668 475 1370 645 512
85 80 471126 59667 12490 1978 310436 15381 53200 3836 14757 1510 87185 4318 5673 638 2445 290 842 401 307
90 85 164322 20454 4760 795 107786 5417 18269 1330 5094 523 27164 1384 1682 212 754 102 267 138 119
-90 90 65266291 5808556 1253280 290790 13704403 753888 1868061 149944 590692 64510 5926762 340811 286779 35621 139046 17696 51012 28319 21606

Note: The columns “rank0” give the number of UV sources with one SDSS counterpart (within the match radius of 3 ), and “rank1” of those with multiple optical matches. UV sources with multiple matches may have composite UV colors, therefore are excluded from the analysis sample; their exclusion is accounted for by correcting the density of sources by (rank1/(rank0+rank1)). The fraction from columns 16-17 (from which our analysis sample is extracted) is plotted in Fig. 3. Numbers in columns 18, 19 and 20 include only “rank0” sources.

Table 2: Statistical characteristics of the Catalogs and effects of error cuts
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