The HST Large Programme on NGC 6752. III. Detection of the Peak of the White Dwarf Luminosity Function

TL;DR

This study uses Hubble Space Telescope imaging to identify the peak of the white dwarf luminosity function in NGC 6752, providing an age estimate consistent with other methods and confirming white dwarf cooling models.

Contribution

First detection of the white dwarf luminosity function peak in NGC 6752 using HST, confirming theoretical age estimates and white dwarf cooling models.

Findings

Peak of the white dwarf luminosity function at m_F606W=29.4

White dwarf cooling age of approximately 14 Gyr

Consistent age with main-sequence turnoff estimates

Abstract

We report on the white dwarf cooling sequence of the old globular cluster NGC 6752, which is chemically complex and hosts a blue horizontal branch. This is one of the last globular cluster white dwarf (WD) cooling sequences accessible to imaging by the Hubble Space Telescope. Our photometry and completeness tests show that we have reached the peak of the luminosity function of the WD cooling sequence, at a magnitude m_F606W=29.4+/-0.1, which is consistent with a formal age of ~14Gyr. This age is also consistent with the age from fits to the main-sequence turnoff (13-14Gyr), reinforcing our conclusion that we observe the expected accumulation of white dwarfs along the cooling sequence.

Tables

Table 1. Table 1: Completeness-corrected white dwarf luminosity function (see text).

$m_{\mathrm{F606W}}$	$N_{c_g}$	${\sigma }_{N_{c_g}}$	$N$	[cg]	$m_{\mathrm{F606W}}$	$N_{c_g}$	${\sigma }_{N_{c_g}}$	$N$	[cg]	$m_{\mathrm{F606W}}$	$N_{c_g}$	${\sigma }_{N_{c_g}}$	$N$	[cg]
24.025	1.0863	1.0863	1	0.9205	26.025	4.2538	2.1269	4	0.9403	28.025	30.3241	5.9470	26	0.8574
24.075	0.0000	0.0000	0	0.9212	26.075	0.0000	0.0000	0	0.9397	28.075	24.5926	5.3665	21	0.8539
24.125	1.0846	1.0846	1	0.9220	26.125	5.3244	2.3811	5	0.9391	28.125	24.6935	5.3886	21	0.8504
24.175	1.0838	1.0838	1	0.9227	26.175	2.1312	1.5070	2	0.9384	28.175	21.2531	5.0094	18	0.8469
24.225	1.0829	1.0829	1	0.9234	26.225	3.1989	1.8469	3	0.9378	28.225	27.2691	5.6860	23	0.8434
24.275	1.0821	1.0821	1	0.9242	26.275	5.3369	2.3867	5	0.9369	28.275	34.6987	6.4434	29	0.8358
24.325	2.1624	1.5291	2	0.9249	26.325	1.0688	1.0688	1	0.9356	28.325	43.6949	7.2825	36	0.8239
24.375	0.0000	0.0000	0	0.9256	26.375	4.2811	2.1405	4	0.9344	28.375	38.1762	6.8566	31	0.8120
24.425	2.1590	1.5266	2	0.9264	26.425	1.0717	1.0717	1	0.9331	28.425	32.4937	6.3725	26	0.8002
24.475	0.0000	0.0000	0	0.9271	26.475	2.1463	1.5177	2	0.9318	28.475	29.1773	6.0839	23	0.7883
24.525	1.0778	1.0778	1	0.9278	26.525	2.1492	1.5197	2	0.9306	28.525	24.4714	5.6141	19	0.7764
24.575	0.0000	0.0000	0	0.9285	26.575	2.1521	1.5218	2	0.9293	28.575	31.3912	6.4077	24	0.7645
24.625	1.0761	1.0761	1	0.9293	26.625	3.2326	1.8663	3	0.9281	28.625	46.5008	7.8601	35	0.7527
24.675	1.0753	1.0753	1	0.9300	26.675	5.3950	2.4127	5	0.9268	28.675	37.7967	7.1429	28	0.7408
24.725	3.2233	1.8609	3	0.9307	26.725	4.3218	2.1609	4	0.9255	28.725	30.1810	6.4346	22	0.7289
24.775	0.0000	0.0000	0	0.9314	26.775	3.2454	1.8737	3	0.9244	28.775	29.3904	6.4135	21	0.7145
24.825	1.0729	1.0729	1	0.9320	26.825	2.1660	1.5316	2	0.9234	28.825	35.8392	7.1678	25	0.6976
24.875	0.0000	0.0000	0	0.9327	26.875	6.5053	2.6558	6	0.9223	28.875	47.0173	8.3116	32	0.6806
24.925	0.0000	0.0000	0	0.9333	26.925	2.1709	1.5350	2	0.9213	28.925	31.6437	6.9052	21	0.6636
24.975	1.0707	1.0707	1	0.9339	26.975	8.6931	3.0735	8	0.9203	28.975	41.7517	8.0351	27	0.6467
25.025	3.2100	1.8533	3	0.9346	27.025	5.4393	2.4325	5	0.9192	29.025	52.4042	9.1224	33	0.6297
25.075	0.0000	0.0000	0	0.9352	27.075	5.4454	2.4353	5	0.9182	29.075	50.5908	9.0864	31	0.6128
25.125	2.1372	1.5112	2	0.9358	27.125	8.7224	3.0838	8	0.9172	29.125	48.6741	9.0385	29	0.5958
25.175	1.0679	1.0679	1	0.9365	27.175	6.5492	2.6737	6	0.9161	29.175	62.1934	10.3656	36	0.5788
25.225	0.0000	0.0000	0	0.9371	27.225	4.3710	2.1855	4	0.9151	29.225	58.7314	10.2238	33	0.5619
25.275	2.1328	1.5081	2	0.9377	27.275	3.2870	1.8977	3	0.9127	29.275	92.1133	13.0268	50	0.5428
25.325	0.0000	0.0000	0	0.9384	27.325	7.7016	2.9109	7	0.9089	29.325	86.2680	12.8601	45	0.5216
25.375	1.0650	1.0650	1	0.9390	27.375	5.5243	2.4705	5	0.9051	29.375	107.9029	14.6837	54	0.5005
25.425	1.0642	1.0642	1	0.9396	27.425	6.6571	2.7177	6	0.9013	29.425	100.1523	14.4557	48	0.4793
25.475	1.0635	1.0635	1	0.9403	27.475	8.9136	3.1515	8	0.8975	29.475	78.5872	13.0979	36	0.4581
25.525	1.0628	1.0628	1	0.9409	27.525	6.7137	2.7408	6	0.8937	29.525	68.6640	12.5363	30	0.4369
25.575	3.1862	1.8396	3	0.9416	27.575	10.1135	3.3712	9	0.8899	29.575	40.8919	9.9177	17	0.4157
25.625	0.0000	0.0000	0	0.9422	27.625	7.8998	2.9858	7	0.8861
25.675	0.0000	0.0000	0	0.9428	27.675	13.6008	3.9262	12	0.8823
25.725	1.0599	1.0599	1	0.9435	27.725	5.6915	2.5453	5	0.8785
25.775	5.2995	2.3700	5	0.9435	27.775	12.5735	3.7911	11	0.8749
25.825	1.0606	1.0606	1	0.9429	27.825	8.0334	3.0363	7	0.8714
25.875	2.1226	1.5009	2	0.9422	27.875	11.5224	3.6437	10	0.8679
25.925	2.1241	1.5019	2	0.9416	27.925	16.1965	4.3287	14	0.8644
25.975	1.0627	1.0627	1	0.9410	27.975	12.7774	3.8525	11	0.8609

Full text

††pagerange: The HST Large Programme on NGC 6752. III. Detection of the Peak of the White Dwarf Luminosity Function††thanks: Based on observations with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS 5-26555. – The HST Large Programme on NGC 6752. III. Detection of the Peak of the White Dwarf Luminosity Function††thanks: Based on observations with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS 5-26555. ††pubyear: 201X

1 Introduction

Over 97% of stars end their lives as white dwarfs (WDs). The WD cooling sequence (CS) of a globular cluster (GC) is shaped by age and star formation history of that cluster, and provides a unique opportunity to conduct a census of its already evolved massive star population. WD CSs in old stellar populations can also provide critical information on how the chemical composition of a stellar remnant influences its thermal evolution.

For the oldest stellar populations, the WD CS lies in the faintest and largely unexplored regions of the colour-magnitude diagram (CMD). Deep imaging with the Hubble Space Telescope (HST) has for the first time reached the peak of the luminosity distribution of the WD CS in three classical GCs, namely: NGC 6397, M 4 and 47 Tucanae [Anderson et al. 2008a, Bedin et al. 2009, Kalirai et al. 2012], and has revealed an unexpectedly complex, and double-peaked, WD CS in the metal rich old open cluster NGC 6791 (Bedin et al. 2005a, 2008a,b). Each of the studied GCs hosts multiple stellar populations (mPOPs) and they show small mean spreads of the initial He abundances (Milone et al. 2018). Their WD CSs are consistent with predictions for single-population systems (Richer et al. 2013, Campos et al. 2016).

One more HST investigation of WD CSs is in progress on the massive GC $\omega$ Centauri ($\omega$ Cen), where at least 15 sub-populations are known to exist (Bellini et al. 2017). The massive $\omega$ Cen has also long been known to host mPOPs, with a large spread in both [Fe/H] and (important for that investigation) initial Helium abundance, as deduced from its divided main sequence (MS, Bedin et al. 2004, King et al. 2012). The upper part of the WD CS in $\omega$ Cen bifurcates into two sequences (Bellini et al. 2013): a blue CS consisting of standard CO-core WDs, and a red CS consisting of low-mass WDs with both CO and He-cores. The current hypothesis is that the blue WD CS is populated by the end products of the He-normal stellar population of $\omega$ Cen, while the red WD CS is populated by the end products of the He-rich population. Observing the WD CS down to the peak of its luminosity distribution provides a critical test of this hypothesis, and a resolution of the origin of the multiple WD CSs observed. That object can help to answer to key questions about He-dependence in the evolution of WDs. Fortunately, $\omega$ Cen is close enough that its entire WD CS is within the reach of HST, and it is the subject of an investigation in progress (GO-14118+14662 PI: Bedin; Milone et al. 2017, Bellini et al. 2018, Libralato et al. 2018).

While almost every globular cluster is known to host multiple populations, every single cluster is unique. The GC NGC 6752 represents a transition between the relatively simple globular clusters, and $\omega$ Cen, the most complex globular cluster known. NGC 6752 has an extended blue horizontal branch, a collapsed core and 3 chemically distinct populations (Milone et al. 2010, 2013). It is one of our final opportunities with HST to increase the diversity in our very limited sample of WD CSs, thus far containing only 3 globular clusters, one old open cluster, and the complex $\omega$ Cen system. While the imminent James Webb Space Telescope will also be able to observe the faintest WDs in closest GCs in the IR, currently, there is no foreseeable opportunity in the post-HST era to observe WD CSs in the homogeneous optical photometric system of HST.

2 Observations

All images for this study were collected with the Wide Field Channel (WFC) of the Advanced Camera for Surveys (ACS) of the HST in program GO-15096 (PI: Bedin). Unfortunately, five out of the planned 40 orbits failed because of poor guide star acquisition and will be re-observed at a later time (currently scheduled for August 2019). Usable data were collected between September 7 and 18, 2018, and consist of deep exposures of 1270 s each, 19 in the F814W filter, and 56 in F606W. At the beginning of each orbit shallow images, of $\sim$45 s each, were also collected, in total 27 in F606W and only 10 in F814W. Note that this is an astrometric multi-cycle programme, and a second epoch (also of 40 orbits) has already been approved (GO-15491) and is currently scheduled for late 2020. Proper motions will eventually provide a near perfect decontamination of NGC 6752 members from field objects which are both in the foreground and background of the cluster.

Paper I of this series (Bedin et al. 2019) presented the discovery of a serendipitous dwarf galaxy in the main studied field, while Paper II (Milone et al. 2019) focused on multiple stellar populations detected on the low main sequence of NGC 6752 in our parallel observations with the Infra-Red channel (IR) of the Wide Field Camera 3 (WFC3).

3 Data Reduction and Analysis

All images were pre-processed with the pixel-based correction for imperfections in the charge transfer efficiency (CTE) with methods similar to those described in Anderson & Bedin (2010). Photometry and relative positions were obtained with the software tools described by Anderson et al. (2008b). In addition to solving for positions and fluxes, important diagnostic parameters were also computed, such as the image-shape parameter (RADXS, introduced in Bedin et al. 2008a), which quantifies the fraction of light that a source has outside the predicted point-spread function (PSF), the local sky noise (rmsSKY, Bedin et al. 2009), and the level of crowding (o, Anderson et al. 2008b). The RADXS parameter is useful for eliminating most of the faint unresolved galaxies that tend to plague studies of faint point sources, while the “o” parameter tells how much of the flux within the PSF fitting radius comes from detected and modeled neighbors with respect to the target source. The parameter rmsSKY has a minor role in selection of sources, however it contains precious information on how suitable the surroundings of each source are for accurate recovery and photometry.

The astrometry was registered to International Celestial Reference System (ICRS) using sources in common with Gaia DR2 (Gaia collaboration 2018) with tabulated proper motions transformed to the epoch 2018.689826 of HST data, following the procedures in Bedin & Fontanive (2018).

The photometry from shallow exposures was linked to that from deep images. The photometry was calibrated on the ACS/WFC Vega-mag system following the procedures given in Bedin et al. (2005b) using encircled energy and zero points available at STScI.111http://www.stsci.edu/hst/acs/analysis/zeropoints Finally, we applied shifts of the order of $\sim$0.02 mag to link our photometry to the state of the art HST photometric catalogue by Nardiello et al. (2018) in F606W and F814W, which is obtained for a different field centred on the core of the cluster (Sarajedini et al. 2007). This registration, is important, as it enables us to have in the same system a better populated upper part of the CMD. For these calibrated magnitudes we will use the symbols $m_{\rm F606W}$ and $m_{\rm F814W}$.

Artificial star tests (ASTs) have a fundamental role in this investigation, as they are used to estimate the completeness level, the errors, to optimize selection criteria, and to detect the presence of systematic errors. We performed ASTs by adding artificial stars to the individual images as described in Anderson et al. (2008b), with a random flat spatial distribution, a flat luminosity distribution in $m_{\rm F814W}$ between magnitude 24 and 30, and with colours chosen to place them along the observed WD CS (a fiducial line defined by hand). Indeed, following the prescriptions in Bedin et al. (2009, Sect.  2.3), we used ASTs to perform an input-output correction, i.e., to correct the difference between the inserted and recovered values of magnitudes of artificial stars, a well known systematic error (often referred to as stellar migration) that tends to increasingly overestimate the fluxes of stars towards fainter magnitudes. Basically, stars landing on positive peaks of noise are preferentially detected but also recovered systematically brighter (as much as $\sim$0.2 mag). Hereafter, our magnitudes for both real and artificial stars are intended as corrected for such effects.

The software presented in Anderson et al. (2008b) also corrects for distortion in all the images and transforms their coordinates to a common reference frame after removal of cosmic rays and most of the artifacts. It then combines them to produce stacked images that, at any location, give the sigma-clipped mean of the individual values of pixels of all images at that location.

In Fig. 1 we show a pseudo-trichromatic stack of the ACS/WFC field of view (FoV). [Note that we adopted the F814W images for the red channel, the F606W images for the blue channel, and computed a wavelength weighted (blue/red$\sim$3) average of the two for the green channel.] As part of this work, we made this image publicly available (with WCS header linked to Gaia DR2) as supplementary on-line electronic material.

4 The Colour-Magnitude Diagram

In this section we demonstrate that we have confidently detected a sharp peak in the observed luminosity function (LF) of the WD CS, which we will show in the next sections to be consistent with the peak of the CO WD luminosity distribution in NGC 6752.

In order to detect the faintest sources we allow our algorithms to find any local maximum with a non-null flux in both filters above the local sky, as a potential real object. In principle, one every 9 pixels could generate such a peak, we ended up with ’just’ $\sim$162 000 local maxima, or 1 every $\sim$10$\times$10 pixel2. The CMD for these detected local peaks –without any selection– is shown in panel (a) of Fig 2.

Paper I presented the discovery of a dwarf spheroidal galaxy (dSG) –designated Bedin I– in the background of NGC 6752, which is close to the upper-left corner of the FoV in Fig 1. The RGB stars of this stellar system happen to contaminate the region of the CMD where WDs are located. For this reason we have chosen to completely mask out and not use any sources in the region of the FoV occupied by this resolved background dwarf galaxy. The mask is circular, centred on Bedin I and has a radius of 800 pixels ($\sim$40 arcsec). We also require that any of the detected sources generates a peak in at least 9 out of the 19 F814W images and in at least 26 out of the 56 F606W images. With these first selections the number of suitable peaks drops to $\sim$116 000, and their CMD is shown in panel (b). Still many of these peaks are artifacts and spurious detections.

Our next selection is illustrated in panel (c) and removes objects falling in the regions of the field that are too noisy to detect such objects at all. As we will see, this mild selection in rmsSKY has more significant implications when used to estimate the completeness limited to portions of the FoV suitable for actual detection of faint sources.

The most effective parameter for the selection of real point sources is RADXS (Bedin et al. 2008a, 2009, 2010, 2015). This parameter is a measure of how much flux there is in the pixels just outside of the core, in excess of the prediction from the PSF: RADXS is positive if the object is broader than the PSF, and negative if it is sharper. Even mild selection criteria imposed on this parameter is sufficient to reject 2/3 of the initially detected peaks. The result of this selection of peaks with $-0.35<\texttt{RADXS}<+0.25$ is shown in panel (d).

Our last selection criterion is applied through the parameter o (Anderson et al. 2008b), the fraction of light due to neighbours. We require that the flux of light within the PSF normalized-aperture due to neighbours does not exceed 4 times the flux of the target source (o$<$4). This criterion further reduces the sample by $\sim$10%, and the $\sim$25 000 surviving sources are shown in panel (e).

It is clear, however, that there is only so much we can do to optimize selection criteria parameters at the faintest end, as going fainter, it becomes increasingly difficult to estimate the real shape of sources, or to get any meaningful intrinsic parameters at all. This according to the general principle that *“all cats are grey in the dark”.*222 The phrase is attributed to Benjamin Franklin. ;) At this point it becomes more important to merely assess whether, and at what level, a given local peak is significant with respect to the noise of the local sky. Therefore we proceed as follows: Given that in filter F814W our PSFs contains $\sim$16% of the light in its central pixel, and about $\sim$18% in F606W, we can use the standard deviation ($\sigma$) of pixel values in empty regions of the sky to estimate the magnitude level that corresponds to 3-$\sigma$ and 5-$\sigma$ levels in the sky noise. To transform these noise levels into total flux, we divide these 3-5 $\sigma$ by the fraction of total flux (i.e., 0.16 for F814W and 0.18 for F606W), then convert these values into instrumental magnitudes $-2.5\log{\rm(total\,flux)}$, and finally calibrate them with ACS/WFC Vega-mag zero points. As expected, the bulk of these peaks have white noise, and as such an instrumental colour centred at instrumental magnitudes 0.00 (or once calibrated, $\sim$1 in $(m_{\rm F606W}-m_{\rm F814W}$).

The grey shaded areas in all panels of Fig. 2 show the regions for peaks with a significance less than 5-$\sigma$, as estimated in the least noisy portion of the field (the rectangular region in red in Fig. 1). The green dotted lines instead, mark the 3-$\sigma$ significance level. This essentially is the estimated noise floor-level in the darkest portion of the FoV. Hereafter, we choose to consider all the detections below these green lines as not significant.

Panels (a′), (b′), (c′), (d′), and (e′) repeat these selections for artificial stars. However, there are two additional conditions for ASTs: We know where we added artificial stars, and at which magnitudes. Therefore we impose generous requirements for the recovered artificial stars. An artificial star is successfully recovered if it lies within 1 pixel from its positions in both coordinates, and within 0.75 magnitude from its added flux in both filters.

These criteria are sufficient to remove all the potential mismatches of sources with local maxima in the noise, and that is the reason why the bulk of the floor-level noise is not seen in CMDs of ASTs. These CMDs show that added stars are successfully recovered down to the estimated 3 $\sigma$ level.

The concept of effective (or local) completeness was first introduced in Bedin et al. (2008a), and successively used in other independent studies (Bedin et al. 2009, 2010, 2015). The idea is simple: we cannot hope to recover faint stars around the bright halos of saturated stars, where the shot-noise is already higher than the signal from the faint sources we would hope to recover/detect. Therefore we should limit our search for faint sources to only those regions of the FoV where the sky noise is sufficiently faint to allow the faint sources to be detected.

We use ASTs to map these regions and, in practice, we limit our search for faint stars to only those regions. Naturally, if the completeness is limited to only suitable search areas that exclude the bright halos of saturated stars (see Fig. 1), this approach improves with respect to the completeness computed for the overall field. Panel (e′) also shows both the traditional (overall) completeness and the effective (local) completeness, where the top-axis (in blue) refers to completeness values. The traditional completeness is indicated with a black line, and this seems to never exceed $\sim$70% for the faint magnitude range considered here. Instead, the effective completeness (blue line) can be as high as 95%. Note that the ratio of the two tells us what is the usable fraction of the area to search for stars of a given magnitude.

The conventional rule of trusting only completeness down to 50% would sets the limit of validity of our WD CS LF study down to $m_{\rm F606W}\simeq 29.4$. Interestingly enough, however, this is brighter than the estimated 5-$\sigma$ regions highlighted in grey in Fig. 2.

4.1 The Observed WD CS LF of NGC 6752

Lacking proper motions until the acquisition of our second epoch, our next best option is to obtain as pure as possible a sample of NGC 6752 WD members based on their position in the CMD.

We do expect several field objects to fall along the WD CS locus of NGC 6752, and to affect the exact shape of the WD LF we derive here. However, we cannot foresee a scenario in which the field objects would introduce a well-defined peak mimicking that of the WD LF. In the following we will review three possible sources of contaminants (background/foreground field stars; resolved galaxies, and unresolved galaxies) and address why these sources are very unlikely to introduce the observed feature.

We tested the best available models of star distribution for Galactic field stars in the background and foreground of NGC 6752 (such as the Besançon models333https://model.obs-besancon.fr/modele_descrip.php by Robin et al. 2003) and obtained a rather flat distribution with no peaks. Only $\sim$200 field stars are expected within the WD CS region between $m_{F606W}=22$ and 29.6, compared to a total of over 1200 observed, with between 5 and 9 stars per 0.05-magnitude bin in the region of the WD LF peak.

Empirically, these numbers and color-magnitude distributions of field stars are also supported by similar studies on WDs of star clusters at any height above the Galactic plane (Bedin et al. 2008, 2009, 2010, 2015, Anderson et al. 2008a, Kalirai et al. 2012).

Similarly, we do not expect contamination by background galaxies to mimick the observed feature. All background galaxies sufficiently large and bright to have well established shapes (i.e., above the 5$\sigma$ lines) were easily removed by the selection in the RADXS parameter (even sharp quasar-like object, at HST resolution, reveal departures from PSFs, e.g., Bedin et al. 2003).

A possible concern could be the contamination by faint, compact, blue and unresolved-galaxies in the background of NGC 6752.444Those not already removed by the RADXS photometric selections described in previous section. However, thanks to decades of deep HST observations their numbers and loci in the CMD are well known. For this purpose, a region in the CMD was carefully defined by Bedin et al. (2008a, 2009) and demonstrated to not overlap with the clusters WD CSs.

In Fig. 3 we show the CMDs for the selected sources defined in panels (e) and (e′) of Fig. 2. Again, ASTs turn out to be very useful to define the location of reliably measured WDs along the fiducial CS of NGC 6752 for the entire magnitude range of interest. Sources between the two red lines defined by-hand in the CMD for ASTs (right panels) are assumed to be good WD members of NGC 6752 including photometric errors. Identical lines are also used to define the sample of WDs among real stars (left panels). The blue triangle shows the location of blue-compact unresolved background galaxies adopted by Bedin et al. (2009) using Hubble Ultra Deep Field (HUDF) data, and transformed into the F814W filter as was done in Bedin et al. (2008a). Note that the adopted WD-region is not contaminated by this blue-compact galaxies triangular region; this is a conservative statement, as the lower reddening of NGC 6752 with respect to NGC 6791 would make this triangular region about 0.1 mag bluer (and so even more distant from the WD-region). Note also, that these selections are large enough to take into account the photometric error distributions of unresolved point sources.

On the CMD for real sources of Fig. 3, there are many sources that lie outside the red boundary that defines the white dwarf sequence. These are the expected mixture of stars in the Galactic field, many of which are WDs. These field stars are expected to contaminate both the triangular region of the blue-compact galaxies and the WD region, therefore, they affect the exact shape of the WD LF.

We then count the observed objects ($N$) at the various $m_{\rm F606W}$ magnitudes, and correct their values for the effective-completeness ($c_{g}$), obtaining the completeness-corrected WD LF values ($N_{c_{g}}$). In Table 1 we report the values of the entire WD LF and relative uncertainties. We must treat these numbers with care, however. Not only because energy equipartition is expected to cause the more massive WDs to migrate toward the centre of the cluster, but also because of unaccounted for residual contamination due to field objects or to artifacts. Nevertheless, these uncertainties can only bias the relative numbers, while the WD LF peaks at a magnitude where our completeness is still reliable, and where contaminants cannot affect its true positions.

We define the peak of the WD LF as the bin with the highest value, which is at m_{\rm F606W}$$\simeq29.4. As formal uncertainty we assume $\sigma$=FWHM/2.354$\simeq$0.1 mag, where FWHM is the full-width half maximum of the peak, which is 0.2 mag. At the magnitude of the peak, we observe a rather flat distribution of field objects on the blue-side of the WD CS region, at maximum $\sim$20 objects/bin (completeness corrected). Assuming a similar field contamination with a similar distribution within the WD CS region (the best we can do), we do not expect the position of the peak to change.

5 Theoretical assessment of the LF and conclusions

We now present the first preliminary comparison of the WD LF with models.

Future epochs will enable us to carry out an exhaustive characterization of the observed WD LF of NGC 6752 and of its exact shape, thanks to improved S/N (through the doubling of the exposure time) and proper motion cleaning for artifacts and for field objects (in both the background and foreground). Indeed, the increased S/N planned for the future, will enable us to study in detail not only the exact shape of the WD LF but also the exact shape of the WD CS in the CMD (Anderson et al. 2008a). Here, instead, we only aim to just verify that the rather sharp peak in the observed LF, as supported by the artificial star tests, is roughly consistent with the magnitude theoretically expected for the peak in the LF of the cluster WD CS.

As a first step we have determined the reddening and cluster distance modulus from our data, by fitting theoretical isochrones to the colour of the cluster unevolved main sequence between $m_{\rm F606W}$=18 and 21, and the red giant branch (see middle panel of Fig. 4). To this purpose we have employed our own photometry, complemented by the Nardiello et al. (2018) catalogue, which is consistent with our photometry for the common evolutionary phases. This latter catalogue includes a well populated red giant branch, as well as the horizontal branch. Stars in this cluster display a small mean range of initial helium mass fraction ($\Delta Y$=0.015$\pm$0.005, Milone et al. 2018; Nardiello et al. 2015; Paper II), so that we can proceed by employing isochrones with a single initial He abundance without introducing any major bias in our analysis.

We employed the $\alpha$-enhanced BaSTI (Pietrinferni et al. 2006) isochrones with [Fe/H]=$-$1.6, $Y$=0.246, and employed the extinction law in the ACS filters determined by Bedin et al. (2005b). The isochrone metallicity is close to [Fe/H]=$-$1.48$\pm$0.06 which was determined spectroscopically by Gratton et al. (2005) for NGC 6752. The fit provides $E(B-V)$=0.05 (consistent with $E(B-V)$=0.046$\pm$0.005 as determined by Gratton et al. 2005) and $(m-M)_{0}$=13.10. This distance modulus compares well with $(m-M)_{0}=$12.92$\pm$0.24 obtained from the Gaia DR2 cluster parallax suggested by the Gaia collaboration (2018) after the offset correction discussed by Lindegren et al. (2018). The error on this Gaia distance modulus is, however, still sizable, due to the ‘calibration noise’ described by the Gaia collaboration (2018). The top section (1) of the right panel of Fig. 4 shows that, with this distance modulus and reddening, the theoretical zero age horizontal branch sequence is also a good match to the observed counterpart (the redder part of the observed HB is populated by stars evolving to the asymptotic giant branch phase). Ages of 13 and 14 Gyr bracket the position of the age-sensitive subgiant branch in the CMD (see the middle section of the same panel). The isochrones do not include the effect of atomic diffusion. Its inclusion would decrease these ages by $\sim 1$ Gyr (see, e.g., Vandenberg et al. 2002).

The bottom section (3) of the left panel of Fig. 4 compares the observed CS with BaSTI WD isochrones (Salaris et al. 2010), shifted employing the reddening and distance modulus derived above from the MS-TO isochrones fit. These WD isochrones include CO separation upon cristallization by employing the Segretain & Chabrier (1993) phase diagram for the CO mixture. We have determined with appropriate calculations that the more modern Horowitz et al. (2010) CO phase diagram would decrease the WD cluster ages by $\sim$0.9 Gyr, compared to the results reported below. We show two DA isochrones for an age of 14 Gyr, calculated with progenitor lifetimes (and CO profiles) for the same chemical composition as the MS-TO isochrones. These WD isochrones cover the WD mass range from 0.54 to 1.0 $M_{\odot}$. More massive WDs between 1.0 Mo and $\sim$1.4 $M_{\odot}$ (if they form in metal poor clusters) would be fainter than the faintest magnitude of our isochrones. However, for any reasonable mass function, they would constitute only a fraction of a percent of the total cluster WD population. The first isochrone (displayed in red) has been calculated with the linear initial-final mass relationship (IFMR) from Salaris et al. (2009). The IFMR was extrapolated linearly such that initial masses typical of stars populating Galactic globular cluster main sequence turn off ($\sim$0.8$M_{\odot}$) produce WDs with mass 0.54$M_{\odot}$ (this IFMR was determined down to values of initial masses $\sim$1.8$M_{\odot}$). This mass is consistent with the observational analysis of Kalirai et al.  (2009), who show that bright WDs in the Galactic globular cluster M 4 are generally of DA type, with a typical mass equal to 0.53$\pm$0.01$M_{\odot}$ (we denote this isochrone as isochrone A).

The second DA isochrone (displayed as blue solid line in the figure) is calculated by extrapolating Salaris et al. (2009) IFMR to reach a WD mass equal to 0.49$M_{\odot}$ for initial masses below 1.0$M_{\odot}$ (we denote this isochrone as isochrone B). The reason for this choice is that, as derived from the detailed horizontal branch modelling by Cassisi et al. (2014), a fraction of the stars populating this cluster’s blue HB, have a mass close to the core mass at the He-flash for this metallicity (equal to $\sim$0.49$M_{\odot}$). They will evolve straight to the WD phase skipping the asymptotic giant branch.

Isochrones A and B differ only at magnitudes brighter than $m_{F606W}\sim$29.0, where the CS is populated by fast evolving WDs produced by stars that have just left either the HB, or the post-asymptotic giant branch phase. Overall, the observed CS is closely matched by the theoretical isochrones at magnitudes brighter than $m_{F606W}\sim$28.2 , which however progressively diverge towards bluer colours with increasing magnitudes.

Figure 5 compares the empirical (completeness corrected) WD LF with two theoretical LFs determined from the two 14 Gyr old DA WD isochrones of Fig. 4. Random photometric errors were added to stars, following the results of the artificial star tests, and shifted by the derived extinction and distance modulus.

The total number of stars in the theoretical LFs have been chosen to approximately match the value of the peak of the observed LF at $m_{\rm F606W}\sim$29.4, employing a WD progenitor mass function as a power law with exponent $-$1.9. This exponent was selected to match approximately the observed star counts at least at another magnitude level ($m_{\rm F606W}\sim$28.8), in addition to the peak at $m_{\rm F606W}\sim$29.4. Varying the exponent of the progenitor mass function alters the shape of the theoretical LF, but not the brightness of its peak.

The two different IFMRs employed for isochrone A and B produce identical LFs at the faint parts of the CS. This is not surprising given that the main difference is the value of the initial mass only for low-mass progenitors.

An exhaustive comparison of observed versus theoretical LFs must wait for future epochs that will enable to clean the cluster WD sample more reliably. As things stand, irrespective of the choice of the progenitor mass function exponent, the theoretical LFs cannot match the observed LF along the entire CS. The purpose of this comparison is therefore just qualitative, but it serves to highlight the fact that the magnitude level of the predicted peak of the WD LF does match the magnitude of the observed LF peak, when an age consistent with the cluster TO age is employed.

This strengthens the case for our detection of the peak in the WD LF of NGC 6752.

Acknowledgments

This work is based on observations with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS 5-26555. J.A., R.M.R., D.A., Ad.B, An.B, M.L, and acknowledge support from HST-GO-15096. A.P.M. acknowledges funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research innovation programme (Grant Agreement ERC-StG 2016, No 716082 ‘GALFOR’. A.P.M.  acknowledges support from MIUR through the FARE project R164RM93XW ‘SEMPLICE’. A.F.M.  has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. [797100]. D.N. acknowledges partial support by the University of Padova, Progetto di Ateneo, Grant BIRD178590. This work is supported in part by the NSERC Canada (P.B.). L.R.B. acknowledges support by MIUR under PRIN program #2017Z2HSMF, …and reminds in this astro-ph version that July 16th is also L.R.B. B-day!!! ;)