The frequency of metal-enrichment of cool helium-atmosphere white dwarfs using the DESI Early Data Release

There is overwhelming evidence that white dwarfs host planetary systems; revealed by the presence, disruption, and accretion of planetary bodies. A lower limit on the frequency of white dwarfs that host planetary material has been estimated to be roughly 25-50 per cent; inferred from the ongoing or recent accretion of metals onto both hydrogen-atmosphere and warm helium-atmosphere white dwarfs. Now with the unbiased sample of white dwarfs observed by the Dark Energy Spectroscopic Instrument (DESI) survey in their Early Data Release (EDR), we have determined the frequency of metal-enrichment around cool-helium atmosphere white dwarfs as 21 $\pm$ 3 per cent using a sample of 234 systems. This value is in good agreement with values determined from previous studies. With the current samples we cannot distinguish whether the frequency of planetary accretion varies with system age or host-star mass, but the DESI data release 1 will contain roughly an order of magnitude more white dwarfs than DESI EDR and will allow these parameters to be investigated.


INTRODUCTION
Planetary systems around main-sequence stars are ubiquitous.Multiple occurrence-rate studies of planetary-mass bodies show that on average there is at least one planet for every star in the Milky Way (Cassan et al. 2012;Poleski et al. 2021;Zhu & Dong 2021).The fraction of stars that host planetary systems is harder to constrain, with estimates that ≃ 30 per cent of Sun-like stars host planets with radii greater than one Earth radius, and orbital periods less than 400 d (Zhu et al. 2018).As the host stars of planetary systems evolve into the giant-phase, it is expected that many of their planets orbiting at distances greater than several au will survive and continue to orbit their stars after they exhaust their fuel and evolve off the main sequence (Villaver & Livio 2009;Mustill et al. 2014;Veras & Gänsicke 2015;Lagos et al. 2021;Veras & Hinkley 2021).
White dwarfs are the dense, degenerate stellar remnants of mainsequence stars with initial masses ≤ 8 M ⊙ (Iben et al. 1997;Dobbie et al. 2006).Their high surface gravities lead to the chemical stratification of their atmospheres, resulting in the majority of white dwarfs having either hydrogen-or helium-dominated atmospheres.Heavier elements settle out of the observable atmosphere on the diffusion timescale, which varies from a few days in hot (≃ 20 000 K) hydrogen-atmosphere white dwarfs, to several million years in cool (< 12 000 K) helium-atmosphere white dwarfs (Koester 2009).Thus the presence of photospheric metal-line absorption features in the spectra of white dwarfs usually 1 reveals the recent or ongoing accretion of metals.
There is a large wealth of evidence that these metals arrive at the white dwarf predominantly from a remnant planetary system that has survived the evolution of its host star to the white dwarf phase.This is identified by (i) direct observations of disintegrating and transiting planetary-sized bodies (Vanderburg et al. 2015(Vanderburg et al. , 2020;;Gänsicke et al. 2019;Vanderbosch et al. 2020Vanderbosch et al. , 2021;;Guidry et al. 2021;Farihi et al. 2022b), (ii) compact (≃ 1 R ⊙ ) debris discs made of dust (Zuckerman & Becklin 1987;Jura 2003;Rocchetto et al. 2015;Wilson et al. 2019), and/or gas (Gänsicke et al. 2006;Manser et al. 2020;Dennihy et al. 2020;Melis et al. 2020;Gentile Fusillo et al. 2021) produced from the complete or partial disruption of a planetary body like an asteroid or comet, (iii) the emission of x-rays from the direct bombardment of the white dwarf surface from the accretion of such a disc (Cunningham et al. 2022), and (iv) the resultant metal-enrichment of white dwarf atmospheres with material consistent in composition with primordial chondritic material (Trierweiler et al. 2023), bulk-Earth (Zuckerman et al. 2007;Gänsicke et al. 2012;Hollands et al. 2017Hollands et al. , 2018a)), and more diverse bodies showing volatiles (Farihi et al. 2013;Xu et al. 2017;Johnson et al. 2022).
For white dwarfs, the frequency of metal-enrichment of their atmospheres can be used as a proxy for a lower limit on the occurrence rate of planetary systems that orbit them.This has been performed in several studies using hydrogen-atmosphere white dwarfs (Zuckerman et al. 2003;Koester et al. 2014;Wilson et al. 2019), as they are the most abundant type of white dwarf.Zuckerman et al. (2003) looked at a sample of approximately 100 hydrogenatmosphere white dwarfs without close stellar companions, with the majority of systems having relatively cool effective temperatures ( eff < 10 000 K), and identified that roughly 25 per cent of them had absorption features consistent with metal-enrichment from planetary 1 Radiative levitation can sustain the presence of heavier elements in the atmosphere of hot (≳ 25 000 K) white dwarfs (Chayer et al. 1995), and convection can "dredge-up" substantial amounts of carbon up from the core in cool (≲ 10 000 K), helium-atmosphere white dwarfs (Pelletier et al. 1986).
material.This result provided the first lower limit on the occurrence rate of planetary systems around white dwarfs.Koester et al. (2014) utilised the Hubble Space Telescope (HST) to look at a sample of 85 hydrogen-atmosphere white dwarfs with effective temperatures in the range 27 000 K >  eff > 17 000 K. This sample of warm hydrogendominated white dwarfs provided a frequency of white dwarfs that have accreted planetary material ranging from ≃ 25 -50 per cent.A more recent study using HST by Wilson et al. (2019) using 143 hydrogen-atmosphere (DA) white dwarfs (which includes the sample of Koester et al. 2014) corroborates these findings with an occurrence rate of the accretion of planetary material of 45 ± 4 per cent.
The frequency of planetary systems around warm heliumatmosphere white dwarfs has been investigated by Zuckerman et al. (2010), who determined an occurrence rate of white dwarfs accreting planetary material of ≃ 30 per cent in the temperature range 19 500 K >  eff > 13 500 K.This is again consistent with the occurrence rates determined from hydrogen-atmosphere white dwarfs, and the frequency of planets around main-sequence stars.This would suggest that the frequency of planetary systems around white dwarfs with hydrogen-and helium-dominated atmospheres do not differ.
The Dark Energy Spectroscopic Instrument (DESI, DESI Collaboration et al. 2016a,b) observes white dwarfs as part of its 5-year survey (Allende Prieto et al. 2020), and the Early Data Release (EDR, DESI Collaboration et al. 2023) contains 2706 spectroscopically confirmed white dwarfs (Manser et al., submitted to MNRAS).These white dwarfs were selected using Gaia photometry and astrometry with the selections described in Cooper et al. (2023), and based on those of Gentile Fusillo et al. (2019Fusillo et al. ( , 2021)).The DESI EDR white dwarf sample closely follows the  < 20 magnitude limited sample of high-confidence white dwarfs of Gentile Fusillo et al. (2021), making it far less biased than the sample of white dwarfs obtained by the Sloan Digital Sky Survey (SDSS, Gunn et al. 2006;Abdurro'uf et al. 2022) which were obtained through multiple differing targeting strategies (Kleinman et al. 2004), and therefore applicable for statistical analyses.
In this paper, we use a subset of cool (11 500 K >  eff > 5 000 K) helium-atmosphere white dwarfs with and without signs of metalenrichment to determine a lower limit on the occurrence rate on planetary systems around white dwarfs with cooling ages longer than ≃ 400 Myr.We give details on the white dwarfs we use for this calculation obtained from the DESI EDR white dwarf sample in Section 2. We then calculate the frequency of white dwarf planetary systems and test the robustness of our calculations in Section 3, and discuss our results and conclude in Section 4.

SAMPLE SELECTION
In this analysis we use the set of spectroscopically confirmed (i.e.excluding uncertain classifications, noted by ":") systems in the DESI EDR white dwarf sample (Manser et al., submitted to MNRAS).DESI on the Mayall 4 m telescope at Kitt Peak National Observatory (KPNO) is a multi-object spectroscopic instrument capable of collecting fibre spectroscopy on up to ≃ 5000 targets per pointing (DESI Collaboration et al. 2022).The fibres are positioned by robot actuators and are grouped into ten petals which feed ten identical three-arm spectrographs, each spanning 3600 -9824 Å at a FWHM resolution of ≃ 1.8 Å.A full description of the DESI reduction pipeline is given by Guy et al. (2023).
To determine the frequency of metal-enriched white dwarfs, we need to identify a subsample of white dwarfs with similar properties where the presence or absence of extrinsic metal can be deduced.
Helium-atmosphere white dwarfs that display only the signatures of metals (excluding carbon), the DZ spectral class, are relatively easy to identify, with calcium abundances being detectable several orders of magnitude lower than their hotter counterparts that show He i features (DBZs) and optical absorption features that dramatically alter the emergent spectral energy distribution (Dufour et al. 2007;Koester & Kepler 2015;Hollands et al. 2017).Additionally, the convection zones of DZs are deep, leading to long diffusion timescales (> 1 Myr) for metals to sink out of their atmospheres and thus making metal-enrichment detectable for significantly longer than their warmer counterparts (Koester 2009).These attributes make DZs good proxies for determining limits on the occurrence rate of planetary systems around white dwarfs.
A DZ white dwarf can arise from the accretion of planetary material onto a featureless DC white dwarf, or a DQ showing carbon dredge-up features.The presence of metals in a white dwarf atmosphere can eradicate the spectral features of dredged-up carbon at optical wavelengths (Pelletier et al. 1986;Blouin 2022;Hollands et al. 2022), but at sufficiently low metal abundances, carbon dredgeup features and metal-enrichment can be observed simultaneously, such as in the DZQ2 , WD J101453.59+411416.94, and the DQZ, WD J142018.91+324921.12,identified within the DESI EDR sample.This observational bias is thought to be the reason metal-enriched DQs appear to be so rare.
As the Balmer lines remain present in the optical spectrum of white dwarfs down to cooler temperatures than He i lines (Bergeron et al. 1997), cool helium-atmosphere white dwarfs with traces of hydrogen can have spectral types DA, DAZ, and DZA (e.g.GD362; Zuckerman et al. 2007).These are excluded from the following analysis, as without full modelling of their spectra it is difficult to distinguish them from the more common hydrogen-atmosphere white dwarfs that share those spectral types.
Based on the above arguments, we select the metal-enriched helium-atmosphere DZs, DQZs, and DZQs, along with their nonenriched counterparts: DCs and DQs, from the DESI EDR sample (Manser et al., submitted to MNRAS).This results in a subsample of 298 white dwarfs containing 197 DCs, 50 DQs, 49 DZs, 1 DQZ, and 1 DZQ (Fig. 1).We further restrict our sample to exclude likely contaminants using a Gaia colour cut 0 < ( BP −  RP ) < 1.The majority of DQs bluer than ( BP −  RP ) = 0 sit significantly below the white dwarf cooling track due to them having high masses.These systems are thought to be white dwarfs produced in binary mergers (Dunlap & Clemens 2015;Kawka et al. 2023), and therefore do not have the same origin as the more abundant cool DQs which are likely to be the non-enriched equivalents to DZ, DZQ and DQZ white dwarfs.DCs redder than ( BP −  RP ) = 1 are thought to be dominated by hydrogen-rich atmosphere white dwarfs as the Balmer absorption lines stop being present in the emergent spectrum (Caron et al. 2023), and we therefore exclude systems above this colour limit.For helium-dominated atmospheric models with no carbon or metals, the 0 < ( BP −  RP ) < 1 colour range spans the broad temperature range of ≃ 11 500 K >  eff > 5000 K (Bédard et al. 2020).

RESULTS
We calculate the fraction,  , of cool metal-enriched heliumatmosphere white dwarfs as the number of DZ, DQZ, and DZQ white dwarfs divided by both the number of these metal-enriched systems and the number of the types they would appear as without the accretion of planetary bodies -DCs and DQs,  = ( DZ +  DQZ +  DZQ )/( DC +  DQ +  DZ +  DQZ +  DZQ ) arriving at a value of  = 0.21 ± 0.03.The uncertainty on  has been determined by sampling from a binomial distribution, and this method is used throughout this study.The value of  obtained here is in reasonable agreement with values determined using cool hydrogen-atmosphere white dwarfs (Zuckerman et al. 2003), but significantly lower than those determined for warmer hydrogen-and helium-atmosphere white dwarfs (Zuckerman et al. 2010;Koester et al. 2014;Wilson et al. 2019).
We investigate the dependence of  on ( BP −  RP ) in Fig. 1, where we calculate  in six equal-sized bins of 39 white dwarfs across the 0 < ( BP −  RP ) < 1.  varies between ≃0.15 -0.36 throughout the selected colour range of 0 < ( BP −  RP ) < 1, with five out of six bins in excellent agreement with the constant value of  = 0.21 ± 0.03, which would suggest there is no detectable trend between  and ( BP −  RP ).
Our calculation also relies on the assumption that the colour cut 0 < ( BP −  RP ) < 1 we make cleanly selects a sample of coolhelium dominated white dwarfs and rejects contaminant systems.Relaxing this cut and using all 298 white dwarfs with spectral types of DC, DQ, DZ, DZQ and DQZ results in a value of  = 0.17 ± 0.03, and can be taken as a conservative lower-limit on the fraction of metal-enriched cool helium-atmosphere white dwarfs.This value is also consistent with our previous estimate and a larger sample size is needed to identify a significant difference between the two values of  obtained so far.We also compare our values of  to those obtainable using the volume limited 20 pc sample of white dwarfs from Hollands et al. (2018b), which contains 29 DC and DCP white dwarfs, 13 DQ, DQPec and DQPecP white dwarfs, and 9 DZ white dwarfs, where 'P' and 'Pec' identify the detection of polarisation due to magnetism and peculiar/unknown features respectively.The resulting value of  = 0.18 ± 0.06 obtained from the entire 20 pc sample is in excellent agreement with the value of  we obtain from the DESI EDR sample.Finally, there are some arguments that all DQ white dwarfs are the product of binary merger events (Farihi et al. 2022a), and as such should not be considered to be in the same population as the standard, isolated cool DC white dwarfs.If this is the case, we also calculate  values excluding DQs, DQZs and DZQs, which are given in Table 1, and are still in good agreement with  values including carbon-line DQ white dwarfs.
These tests demonstrate the robustness of our assumptions with the available data sets and that the frequency we determine here is representative for planetary accretion at cool helium-dominated white dwarfs.We present all the occurrence rates of planetary accretion at white dwarfs we discuss in Table 1.

DISCUSSION AND CONCLUSIONS
Helium-atmosphere white dwarfs in the temperature range ≃ 11 500 K >  eff > 5000 K have cooling ages, , ranging from 460 Myr <  < 6.4 Gyr for a 0.6 M ⊙ white dwarf (Bédard et al. 2020).Our sample therefore probes a significantly older range of observed white dwarfs compared with the relatively young samples of Koester et al. ( 2014 2020) for pure helium-atmosphere white dwarfs with masses from top to bottom of 0.4 M ⊙ , 0.6 M ⊙ , 0.8 M ⊙ and 1.0 M ⊙ are plotted as dashed lines on the Gaia HRD, with vertical tabs highlighting  eff values in steps of 1000 K.These cooling tracks are only strictly applicable to helium-atmosphere white dwarfs with no traces of metals or carbon.The red shaded regions shows the ranges excluded from the 0 < ( BP −  RP ) < 1 selection used in our  calculations.One DC, WD J095106.36+645400.46,sits far below the cooling track with  Abs > 16 and is thought to have a spectrum dominated by collisionally-induced absorption (Kilic et al. 2020;Bergeron et al. 2022).We still include this system in our analyses.Bottom panel: The fraction of metal-enriched helium-dominated white dwarfs with no trace hydrogen detectable as a function of  BP - RP is shown in six, equal sized bins of 39 white dwarfs.Vertical lines on the gray points represent 1  errors on the value of  in the bin, while the horizontal lines represent the span of the ( BP −  RP ) values for each bin.The horizontal dashed line and shaded region is set to the average value and uncertainty of  = 0.21 ± 0.03 calculated using all 234 white dwarfs in the range 0 < ( BP −  RP ) < 1.
Table 1.The fraction of white dwarfs that have accreted planetary material,  from various studies.The sample size column gives both the total sample size, with the number of metal-enriched white dwarf in brackets.Cooling ages, , are estimated based on the temperature range of the sample and white dwarf evolutionary models of Bédard et al. (2020) which span ages in the range 9 Myr <  < 300 Myr.The sample of hydrogen-dominated white dwarfs investigated by Zuckerman et al. (2003) extends over a temperature and age range consistent with our sample.
The occurrence rates of planetary systems around white dwarfs obtained by Wilson et al. (2019) and Koester et al. (2014) are at least as high as 0.45 ± 0.04 and 0.56 ± 0.10 respectively, significantly higher than the value of 0.21 ± 0.03 we derive here.Conversely the occurrence of planetary systems in the samples of Zuckerman et al. (2003) and Zuckerman et al. (2010) reduces to ≃ 0.25 -0.30, slightly higher than but still in agreement with the value we obtain here.While this could be suggestive of the depletion of planetary material around white dwarfs as a function of cooling age, Koester et al. (2014) note that this discrepancy is likely due to the high resolution, high signal-to-noise ratio, and ultraviolet wavelengthrange obtained by the HST Cosmic Origins Spectrograph compared with the ground-based observations of Zuckerman et al. (2003Zuckerman et al. ( , 2010) ) and the DESI EDR sample.Higher-resolution and deeper follow-up of the full DESI EDR sample of cool helium-dominated white dwarfs could reveal additional metal-enriched systems undetected by DESI or identify contaminant systems (e.g.white dwarfs with weak or magnetically-split hydrogen lines), both of which would increase the lower limit of  determined here.Hollands et al. (2018a) found a depletion in the accretion rates on an -folding timescale of ≃ 1 Gyr around cool DZ white dwarfs in the cooling-age range 1 Gyr <  < 8 Gyr.However, this result has been disputed by (Blouin & Xu 2022) who used updated white dwarf models along with Gaia parallaxes and identified at best only a factor ten drop off in accretion rates over the range of 1 Gyr <  < 8 Gyr.
While these studies suggest a potential depletion or constant rate of planetary material accreted by white dwarfs over several billion years, they cannot be used to directly probe changes to the occurrence rate of planetary systems around white dwarfs over the same time-frame.The detection sensitivity of planetary material in the atmospheres of white dwarfs is dependent on both the amount of material that is being delivered to and accreted by white dwarfs in addition to the frequency of planetary systems around them.Forward-modelling of the samples of white dwarfs in Table 1 would allow the evolution of the accretion and frequency of planetary systems to be constrained based on the current understanding of model atmospheres (Bauer & Bildsten 2018;Cunningham et al. 2019), and inform future observing campaigns.
While we ignore the presence of hydrogen in our analysis, there is a significant link between the presence of hydrogen in a heliumatmosphere white dwarf, and the presence of metals (Gentile Fusillo et al. 2017).This correlation in the presence of hydrogen and metals has been explained as either: (i) the accretion of planetary debris being linked to the accretion of hydrogen in the form of water (Gentile Fusillo et al. 2017), or the accretion of giant planet atmospheres (Schreiber et al. 2019), or (ii) that helium-atmosphere white dwarfs born without detectable trace hydrogen have a significantly reduced planetary system occurrence rate (Bédard et al. 2023).Investigating the link between the presence of hydrogen and metals is beyond the scope of this paper, but the frequencies we calculate here for cool helium-atmosphere white dwarfs along with those in Table .1 will allow the degeneracy between the two hypotheses listed to be lifted.
The DESI survey is well into its five year observing program, and the first data release (DR1, currently internal) contains spectroscopy of over 47 000 white dwarf candidates (Manser et al. 2023).Based on the DESI EDR sample of spectroscopically confirmed white dwarfs, it is expected that the DR1 sample with contain over 2 000 helium-dominated white dwarfs, and roughly 500 metal-enriched white dwarfs in the colour range 0 < ( BP −  RP ) < 1.Such a sample would allow the frequency of planetary systems around white dwarfs to be probed in much greater detail, such as whether there is a drop-off in the occurrence of planetary systems at late cooling ages.Additionally, this sample would allow other studies to be performed, such as an independent test of observed accretion rates with white dwarf cooling ages, or differences in accreted material as a function of white dwarf mass and therefore host-star progenitor mass, studies of which are currently limited (Johnson et al. 2010;Zhu & Dong 2021).

Figure 1 .
Figure 1.Top panel:The Gaia Hertzsprung-Russell Diagram (HRD) showing the metal-enriched (orange and magenta triangles) and non-enriched (blue stars cyan squares) systems used in the calculation of the frequency of planetary systems around cool, helium-atmosphere white dwarfs,  (see Section 3).Cooling tracks fromBédard et al. (2020) for pure helium-atmosphere white dwarfs with masses from top to bottom of 0.4 M ⊙ , 0.6 M ⊙ , 0.8 M ⊙ and 1.0 M ⊙ are plotted as dashed lines on the Gaia HRD, with vertical tabs highlighting  eff values in steps of 1000 K.These cooling tracks are only strictly applicable to helium-atmosphere white dwarfs with no traces of metals or carbon.The red shaded regions shows the ranges excluded from the 0 < ( BP −  RP ) < 1 selection used in our  calculations.One DC, WD J095106.36+645400.46,sits far below the cooling track with  Abs > 16 and is thought to have a spectrum dominated by collisionally-induced absorption(Kilic et al. 2020;Bergeron et al. 2022).We still include this system in our analyses.Bottom panel: The fraction of metal-enriched helium-dominated white dwarfs with no trace hydrogen detectable as a function of  BP - RP is shown in six, equal sized bins of 39 white dwarfs.Vertical lines on the gray points represent 1  errors on the value of  in the bin, while the horizontal lines represent the span of the ( BP −  RP ) values for each bin.The horizontal dashed line and shaded region is set to the average value and uncertainty of  = 0.21 ± 0.03 calculated using all 234 white dwarfs in the range 0 < ( BP −  RP ) < 1.
. No colour cuts have been placed and it is likely that the sample includes hydrogen-atmosphere DCs and high-mass binary-merger DQ white dwarfs.
a b Sample observed exclusively with HST rather than ground-based observatories.