PHL 5038AB: Is the brown dwarf causing pollution of its white dwarf host star?

We present new results on PHL 5038AB, a widely separated binary system composed of a white dwarf and a brown dwarf, refining the white and brown dwarf parameters and determining the binary separation to be $66^{+12}_{-24}$~AU. New spectra of the white dwarf show calcium absorption lines suggesting the hydrogen-rich atmosphere is weakly polluted, inferring the presence of planetesimals in the system, which we determine are in an S-type orbit around the white dwarf in orbits closer than 17-32 AU. We do not detect any infrared excess that would indicate the presence of a disc, suggesting all dust present has either been totally accreted or is optically thin. In this system, we suggest the metal pollution in the white dwarf atmosphere can be directly attributed to the presence of the brown dwarf companion disrupting the orbits of planetesimals within the system.


INTRODUCTION
It is estimated that 25 to 50 per cent of white dwarfs have atmospheres polluted by metals (Zuckerman et al. 2003(Zuckerman et al. , 2010;;Koester et al. 2014;Hollands et al. 2017).Due to the rapid sinking times of metals in these dense objects, in order for them to be detected, there must be a continual replenishing of the pollutants (Koester 2009;Barstow et al. 2014).
The favoured explanation for the origin of these pollutants is that planetesimals (similar to asteroids) orbiting the white dwarf on eccentric orbits are scattered inwards where they sublimate and eventually accrete onto the white dwarf (e.g.Debes & Sigurdsson 2002;Jura 2003).The composition of the disrupted objects is then inferred from the species present in the white dwarf atmosphere (e.g.Klein et al. 2011).Further evidence for this scenario includes the discovery of planetesimals that transit the polluted white dwarf WD1145+017 (Vanderburg et al. 2015), and further discoveries around other white dwarfs (e.g Vanderbosch et al. 2021).
A range of dynamical mechanisms has been suggested to transport these planetesimals to the inner regions of white dwarf planetary systems, including stellar mass loss via winds, planet(s), and wide-★ E-mail: slc25@leicester.ac.uk binary companions (e.g.Bonsor et al. 2011;Debes et al. 2012;Veras et al. 2014;Bonsor & Veras 2015;Portegies Zwart & Jílková 2015;Petrovich & Muñoz 2017;Hamers & Portegies Zwart 2016;Mustill et al. 2018).However, the most likely explanation is that they are scattered inwards by planets in a surviving outer planetary system, rather like sun-grazing comets in the Solar System.Despite this suggestion, no system has yet been discovered where a definitive mechanism is identified as responsible.
While polluted white dwarfs are relatively common, white dwarfs with substellar companions are relatively rare.Indeed only 0.5 per cent of white dwarfs are thought to have an unresolved substellar companion (Steele et al. 2011;Rebassa-Mansergas et al. 2019).Despite extensive searches (Debes et al. 2011;Dennihy et al. 2017;Hogg et al. 2020;Lai et al. 2021;Owens et al. 2023), there are fewer than 15 white dwarfs known to have brown dwarf companions that have survived common envelope evolution, and now orbit with periods of ∼hrs (e.g.Littlefair et al. 2014;Parsons et al. 2017;Casewell et al. 2020).A similar number of wide, often wide enough to be considered common proper motion companions are also known (e.g.French et al. 2023;Meisner et al. 2020;Day-Jones et al. 2011;Deacon et al. 2014).Recent results from  (Rogers et al. 2024) and   (Mullally et al. 2024) indicate that many more companions may be found in the near future.
PHL 5038AB is a resolvable binary comprising a cool white dwarf with a L-dwarf companion separated on the sky by 1".The white dwarf was first identified by Eisenstein et al. (2006) with Steele et al. (2009) discovering the brown dwarf companion, which at the time, was only the fourth known to orbit a white dwarf after GD 165AB (Becklin & Zuckerman 1988), WD0137-349 (Maxted et al. 2006) and GD 1400AB (Farihi & Christopher 2004).We re-observed the PHL5038AB system 12 years after Steele et al. (2009) in order to begin orbital monitoring of the binary.We present here a new analysis of the system including refined parameters for both the white dwarf and the brown dwarf making use of the parallax from  (Lindegren et al. 2021), and new spectroscopy from XSHOOTER on the Very Large Telescope.Our analysis shows calcium pollution within the white dwarf atmosphere suggesting the brown dwarf companion is disrupting the orbits of planetesimals that are disintegrating into the white dwarf atmosphere, suggesting that this mechanism indeed is possible (e.g.Bonsor & Veras 2015).

Gemini/NIRI Imaging
We imaged PHL 5038 on 2021-05-18 UT with the Near Infrared Imager (NIRI; Hodapp et al. 2003) in the   band as part of programme GN-2021A-FT-207 at the Frederick C. Gillett Gemini Telescope (Gemini North).We obtained 14 60 s exposures at an airmass of 1.09 with the f/6 camera providing a pixel scale of 0.117 ′′ pixel −1 .We reduced the data using the dragons software (Labrie et al. 2019) provided by Gemini Observatory.Flat field and dark frames were obtained for our observations as part of daytime calibrations, and we used dragons to create a bad pixel mask from the 10 s dark frames.The 14 images were reduced and stacked using stars in the image as references.We also stacked the images using the header coordinate system but there was negligible difference between the methods.
We re-reduced the NIRI acquisition images presented in Steele et al. (2009).These images were taken in 2008 in the  band and were reduced using dragons as for the newer NIRI data.

XSHOOTER
There are two sets of observations of PHL 5038AB taken with XSHOOTER (Vernet et al. 2011) on the European Southern Observatory's Very Large Telescope as part of programmes 384.D-0494 (PI: Steele) and 106.213V (PI: Rebassa-Mansergas).The data from the Steele programme were taken on 19-10-2009 with 750 s exposures in the UVB and VIS arms in seeing of 0.93 ′′ and airmass of 1. Data for the Rebassa-Mansergas programme were taken on 14-10-2020 with exposure times of 450 s in seeing of 0.68 ′′ and airmass of 1.1.All observations used the 0.9 ′′ slit and the nodding mode.
We used version 2.11.5 of the ESOReflex (Freudling et al. 2013) 1 data reduction workflow, implementing the XSHOOTER pipeline version 3.5.3, to reduce, combine and calibrate the spectra.The 2010 spectra were combined to create a single 1500 s exposure with S/N of ∼50, and the 2020 spectra were combined to create a 900 s exposure with S/N of ∼20.
1 http://www.eso.org/sci/software/esoreflex/Steele et al. (2009) determined T eff =8000±100 K and log =8.2±0.2 from the Sloan Digital Sky Survey (SDSS) spectrum (Adelman-McCarthy et al. 2006) implying a mass of 0.72± 0.15M ⊙ and distance of 64±10 pc from the flux scaling factor for the white dwarf.However, as the effective temperature is below 12,000 K, these values are now known to be an overestimate due to the lack of 3D corrections to white dwarf models available at the time (Tremblay et al. 2011).Anguiano et al. (2017) used the SDSS DR12 spectra to determine T eff =7575±57 K, log  = 7.59±0.13with a mass of 0.41±0.07M ⊙ and determined the photometric distance to PHL5038AB to be 96±10 pc.The  DR3 distance is measured to be 73.5±0.6 pc, notably different to both the Steele et al. (2009) and the Anguiano et al. (2017) distance.The parameters from Gentile Fusillo et al. (2021) based on the spectral energy distribution and the  eDR3 distance are T eff =8021±186 K, log g=8.03±0.07 and =0.61±0.04M ⊙ .A recent photometric analysis was also performed by Raddi et al. (2022) who used  broad band photometry combined with the  eDR3 parallax to obtain T eff =7762±100 K, log  = 7.95±0.04with a mass of 0.56±0.02M ⊙ .A spectral fit was performed by Kilic et al. (2020) and gave T eff =7525±25 K, log  = 7.89±0.02with a mass of 0.53± 0.02M ⊙ , values consistent with the work of Tremblay et al. (2011).More recently, using the  DR3 spectrum, Jiménez-Esteban et al. ( 2023) determined an effective temperature of 7735±150 K, a log =7.94±0.05 and =0.56±0.03M ⊙ , in agreement with the values of Gentile Fusillo et al. (2021) and Kilic et al. (2020).

White dwarf parameters
We fit the optical spectrum from XSHOOTER following the method in Rebassa-Mansergas et al. ( 2007) and determine T eff = 7800±50 K, log  = 7.97±0.03dex,  = 0.574±0.018M ⊙ and =0.0129±0.0002R ⊙ using the models of Koester (2010) and the 3D correction of Tremblay et al. (2011).Our results are consistent with those of Kilic et al. (2020) and Jiménez-Esteban et al. (2023).We adopt the Kilic et al. (2020) parameters in this work as our abundance models use the same model grid, and our fit to the XSHOOTER spectrum with the Koester (2010) models gives a consistent result.
The white dwarf parameters and the distance measurement have changed since the work of Steele et al. (2009), due to higher resolution spectra with a better signal-to-noise ratio, and increasingly detailed atmospheric models including 3D corrections.Most notably the measurement of the surface gravity of the white dwarf has been revised to a lower value, now that 3D corrections have been applied, and the derived mass has decreased due to the lower  distance, meaning the age estimate for the system and hence the progenitor mass have changed considerably.
We used the wdwarfdate software (Kiman et al. 2022 2 ) to determine the cooling age and likely mass of the white dwarf progenitor as well as the total system age.We used our white dwarf parameters with a DA white dwarf model and the Cummings et al. (2018) initial mass-final mass relation derived using the MIST isochrones (Choi et al. 2016) as input to the code.The resultant fits gave a cooling age of 1.14 +0.05 −0.04 Gyr, and a total system age of 10.26 +3.09 −3.61 Gyr with a white dwarf progenitor mass of 1.07 +0.17 −0.08 M ⊙ .The errors on the progenitor mass and lifetime are dominated by the fact that semiempirical initial mass-final mass relations for white dwarfs heavily rely on open cluster white dwarfs, which tend to have masses that are larger than average, due to their relative youth compared to field Using the bolometric luminosity derived from the  magnitude for the brown dwarf in Steele et al. (2009) and the relations in Dupuy & Liu (2017) we compared to the Sonora Bobcat models (Marley et al. 2021) determining T eff =1577 K and a mass of 0.071 M ⊙ for an age of ∼8 Gyr.An older age of ∼10.5 Gyr gives an effective temperature of 1425 K and a mass of 0.070 M ⊙ which is consistent with the effective temperature suggested in Dupuy & Liu (2017) for the spectral type.

Astrometry and orbital parameters
We measured relative astrometry for PHL 5038AB using the stacked images for each epoch.We fitted an analytic point spread function model to each component, with the model using three concentric 2D Gaussians with different amplitudes, standard deviations, ellipticities, and angles for the ellipticities.This approach is based on work with adaptive optics imaging of low-mass binaries (e.g., Liu et al. 2006).We then converted the positions in pixels of the two binary components into the sky coordinates using the WCS information provided by the telescope in the FITS headers.The system is resolved in both epochs, and the errors in the relative astrometry are dominated by the astrometric calibration of NIRI.We adopt the same uncertainties as in Mann et al. (2019) for their large sample of NIRI astrometry: a fractional uncertainty of 0.23 per cent in pixel scale and 0.1 • in the detector orientation.
We determined the contrast between the two sources is 0.63 ±  2).Using the white dwarf magnitudes predicted by Holberg & Bergeron (2006) and Bergeron et al. (1995) for a 7500 K white dwarf with log  = 8.00 dex (cgs), we obtain a   absolute magnitude of 12.48.While no errors are given on this synthetic photometry, the difference in magnitude between a 7000 K and a 7500 K white dwarf is 0.08.The brown dwarf absolute magnitude for an L8 dwarf from fitting photometry for known brown dwarfs with accurate parallaxes Dupuy & Liu (2012) is   = 13.06 ± 0.03 mag.While our measured difference in photometry is only ∼0.6 mags, the combined errors on the synthetic photometry are less than 0.1 mags, meaning the white dwarf is the brighter component to the southeast in Figure 1.We used the Markov-Chain Monte Carlo (MCMC) orbital analysis tool orvara (v1.0.4;Brandt et al. 2021) to fit orbits to our relative astrometry for PHL 5038AB.We only used one informative prior on the total mass of the system, based on the white dwarf mass estimate of 0.53±0.03M ⊙ and companion mass estimate of 0.070±0.002M ⊙ .This version of orvara does not allow total mass priors, but when fitting relative astrometry that only constrains the total mass, setting a prior on the primary mass of 0.60 ± 0.03 M ⊙ , and a zero mass prior on the companion, is functionally equivalent to employing a total mass prior.For the remaining orbital elements, we used their default priors: linear-flat in the eccentricity () and viewing angles (except inclination, () ∝ sin ), and log-flat in semimajor axis .Our results are based on a run with 100 walkers, 10 6 steps for the MCMC, and 5 temperatures for parallel tempering.We thinned our chains, retaining every 50th step, and discarded the first 75 per cent as burn-in, yielding 5 × 10 5 final samples in our posterior.
The posterior distributions of orbital parameters correspond to semimajor axes of 66 +12 −24 AU, and inclinations of 132 ± 11 • .The eccentricity is essentially unconstrained due to the large errors on the inclination, except that the posterior drops off steeply around the 2- upper limit of  < 0.615 (Figure 2).

PHL 5038A: A polluted white dwarf
Using the white dwarf parameters from Kilic et al. (2020), we fit both epochs of spectra using the 3934 Å Ca ii K line and the models of Dufour et al. (2007) and Coutu et al. (2019) to determine the [Ca/H] abundance (Figure 3).The abundances are -9.44 for the 2008 spectrum and -9.35 for the 2020 spectrum.We also used molly3 to measure the equivalent width of the line, measuring a value of 0.246±0.009and 0.231±0.024Å for the two epochs, determining there is no significant variability in abundance between the two epochs.
We also determined the radial velocity of each epoch using the Ca ii K absorption line and obtained measurements of 70.46 and 69.07 km s −1 , which results in radial velocities of 23.46 and 23.69 km s −1 once the heliocentric corrections (-22.39 and -20.82 km s −1 ) and gravitational redshift (24.6±1 km s −1 ) are applied.These radial velocities are consistent with the 17.59±14.21km s −1 determined by Raddi et al. (2022), and the 44.33±5.45km s −1 from Anguiano et al. (2017) if it is corrected by our gravitational redshift value.These results are also shown in Table 3.This abundance is consistent with other cool DAZ white dwarfs in Zuckerman et al. (2003) -LHS3007 (T eff =7366 K; log g = 7.58) has a similar [Ca/H]=-9.312,as does the more massive GD96 (T eff =7373 K, log g = 8.00), [Ca/H]=-9.409.The abundance is also consistent with the effective temperature [Ca/H] relation in Blouin & Xu (2022), indicating that while this abundance is lower than that of many of the well-known polluted white dwarfs hosting discs, it is not particularly unusual for a white dwarf with PHL 5038A's parameters.We then follow the method in Koester (2009).
For PHL 5038A, the accretion rate of calcium is 1.32 ×10 13 g yr −1 , which if we assume is due to accretion of bodies with chondritic abundance (calcium fraction of 0.057), equates to a total accretion rate of 2.32×10 14 g yr −1 , or 7.4×10 6 g s −1 (Table 1).If we compare this accretion rate to the total amount of Ca expected within the convection zone of the white dwarf, we determine a value of ∼10 18 g.Similarly, this accretion rate is 2.6 × 10 7 g s −1 if we assume the abundance of bulk Earth with 1.6 per cent calcium abundance.
This accretion rate is low, but comparable to other DZ white dwarfs with similar cooling times (Figure 4 in Blouin & Xu 2022): grey dots in Figure 4), but there are very few cool DAZ white dwarfs with similar accretion rates (∼30 triangles on Figure 4), with PHL 5038A having a lower accretion rate than the majority.

A debris disc?
To determine whether there was an infrared excess indicative of a debris disc, we performed an SED fit by using the SDSS photometry apart from the  ′ band, which can suffer from both reddening and atmospheric issues, and the resolved near-infrared photometry from Steele et al. (2009).To determine the white dwarf parameters we   Blouin & Xu (2022).DZ (helium atmosphere) white dwarfs are plotted as grey points, DAZ (hydrogen atmosphere) white dwarfs as outlined triangles.PHL 5038A is plotted as the black square.performed a chi-square minimisation on an interpolated grid of white dwarf cooling models 5 from Bédard et al. (2020) and Blouin et al. (2018).
We determined that PHL 5038A should have T eff =7751 K with a 95 per cent confidence interval between 7460 and 8045 K, and log g of 7.94 (95 per cent confidence interval 7.84 to 8.04), and a mass of 0.56 M ⊙ (95 per cent confidence interval: 0.51 to 0.62 M ⊙ ), which is consistent with the Kilic et al. (2020) spectroscopic fit.We also used the synthetic photometry for DA white dwarfs in the NIR/WISE bands from Holberg & Bergeron (2006) to determine the SED of the white dwarf.To determine the likely magnitudes of the brown dwarf companion we used the observed absolute magnitudes of brown dwarfs from Dupuy & Liu (2012) scaled to the observed  band flux of the brown dwarf to take into account the large rms scatter in the relationship for L8-L9 dwarfs (L8: 0.20 mag, L9: 0.43 mag).Figure 5 shows that the detected excess in the  and  magnitudes is due to the brown dwarf companion alone.
5 https://www.astro.umontreal.ca/~bergeron/CoolingModels/In order to put limits on whether an undetected disc could be present in the system we used the estimated uncertainty in the brown dwarf fluxes from the SED calculation added in quadrature to the measured flux uncertainty from the photometry to adopt a 3- excess criterion above the W1 and W2 observed fluxes.We then estimated the limits on any disc present.We initially considered optically thick discs using the models from Jura (2003).The Roche limit for PHL 5038A is ∼48 R WD , and the sublimation temperature of ∼1600 K is reached at ∼5 R WD .Assuming these radii are the inner and outer radii for an optically thick disc, all such discs are ruled out to nearly edge-on, <89.9 degrees inclination.If we instead consider very narrow rings that are near face-on, a 1 R WD WD thick ring at <33 R WD is ruled out.Narrow rings at smaller radii will be allowed for increasing inclination limits, so are not completely ruled out but are likely to be rare.We therefore conclude that an optically thick disc is unlikely to be present.
We then considered optically thin discs.We used black body discs with temperatures between 600 and 1200 K, spanning the most typical white dwarf disc temperatures in the temperature regime where the W1 and W2 wavelengths are most sensitive.We then calculated the effective limiting area for such discs to be detectable at our 3- excess criterion.We find that for 600 K the limiting emitting area is 0.70 R 2 ⊙ and for 1200 K, it is 0.17 R 2 ⊙ .If we assume a single grain radius for the emitting dust and a rocky composition such that  dust =3 g cm −3 , then these areas correspond to 9.4 × 10 17 (R dust ) g at 600 K and 5.9 × 10 16 (R dust ) g at 1200 K where R dust is the radius of the dust particles.

DISCUSSION
The lack of detection of a disc around PHL 5038A is perhaps not unexpected.For instance, while 25-40 per cent of white dwarfs show signs of metal pollution, only 1.5-4 per cent of white dwarfs show an excess due to a debris disc.It is unlikely the pollution is caused by wind accretion from the brown dwarf.None of the white dwarf primaries in the close white dwarf-brown dwarf binaries with orbital periods of ∼1 hrs shows any metal pollution in their atmospheres, meaning this mechanism is unlikely to have a significant effect on a binary with separation of ∼70 AU.
Figure 9 of Bonsor et al. (2017) suggests our accretion rate for Ca, combined with the low effective temperature of the white dwarf, means PHL 5038A falls close to the boundary between their region A, where the dust has been totally accreted, and region B, where the dust is optically thin and is dominated by Poynting-Robertson drag.2009) is shown as filled triangles for the white dwarf (dark blue) and the combined white dwarf-brown dwarf (purple).The observed WISE magnitudes for the combined white dwarf-brown dwarf are shown as filled red circles.Our predicted photometry for the white dwarf is shown as filled light blue circles.The predicted photometry for the combined system is shown as purple boxes.There is no infrared excess that could be due to a disc detected out to 4.5 microns.Bonsor et al. (2017) suggest that for white dwarfs with a sinking timescale of longer than ∼500 yrs (of which PHL 5038A is one) then a finite dust lifetime can provide an explanation for a lack of detectable disc.Hollands et al. (2018) suggest for DZ white dwarfs where no disc is detected the accretion phase may have indeed finished, and the detected metals are slow sinking tracers of a previous accretion event.However, while PHL 5038A is relatively cool, it is not a DZ.The calcium diffusion timescale is ∼7000 yrs, and the average lifetime of a disc around a white dwarf is predicted to be between 3×10 4 and 5×10 6 years assuming accretion is at a constant rate (Girven et al. 2012).These values would suggest that the accretion phase has not yet ended.Indeed, the level of accretion in this cool, polluted white dwarf is sufficiently low that it could also be supplied by an optically thin dust disc accreting via Poynting-Robertson drag without the dust disc producing a detectable infrared excess, or via a mechanism such as that suggested by (Metzger et al. 2012).Brouwers et al. (2022) however, showed that accretion via Poynting-Robertson drag is unable to produce the necessary accretion on short enough timescales to replicate the abundances detected for many white dwarfs.They place the limit at 10 6 g s −1 , stating that in order to achieve an accretion rate higher than this, the larger asteroids must be ground down into an eccentric tidal dust disc, perhaps via perturbations caused by a giant planet.Our accretion rate is larger, at 7.4×10 6 g s −1 , but still low for a white dwarf.Brouwers et al. (2022) also predict that lower accretion rates are more likely to occur from smaller asteroids, that take longer timescales to grind down and accrete onto the white dwarf.Assuming this is the case, Kenyon & Bromley (2017) found that if the material lost by accretion onto the white dwarf is continually replenished, an equilibrium mass might be achieved for the disc.Assuming a collisional cascade at the tidal disruption radius of the white dwarf, the output of gas accretion onto the white dwarf is then equivalent to the influx of mass into the disk in a steady state.If we use the default values in equation 11 of Kenyon & Bromley (2017), with our calculated accretion rate for PHL 5038A, we find the equilibrium mass is = 3.6×10 17 g assuming particles of radius 1 km.Interestingly, this value also corresponds to just above the accretion rate predicted to show an infrared excess assuming late-stage dust accretion from highly eccentric asteroids (Brouwers et al. 2022).As we have no detectable disc, we cannot put limits on the grain sizes present, but we can combine the estimate from the collisional cascade, with the maximum emitting area we derived in Section 3.6, where we assumed a single grain radius.This combination gives a lower limit of 9.4 ×10 11 g at 600 K, assuming the particle size is 1 micron.The upper limit is found by multiplying this mass by √︁ 1 km/1 micron, to 3×10 16 g.The arguments presented in Burleigh et al. (2002) suggest that PHL 5038B has not always orbited the white dwarf at 66 +12 −24 AU, and has likely increased its orbital separation as the white dwarf progenitor moved off the main sequence 1.14 +0.05 −0.04 Gyr ago.Jeans (1924) calculates that the orbits of planets that do not interact directly with the white dwarf progenitor as it becomes a giant (e.g.those that escape a phase of common envelope evolution) will simply expand their orbits adiabatically by a maximum factor of M MS /M WD .For PHL 5038AB this suggests that the orbit expanded by a factor of two, placing the initial orbit at ∼33 AU, not dissimilar to the location of Neptune in our own solar system.The models of Ventura & Marigo (2009) suggest that the maximum radius of an AGB star of mass 2.5 M ⊙ (the lowest mass they present) should be ∼2.5 AU, considerably smaller than the predicted initial orbit of the brown dwarf confirming this system has had no common envelope phase.
Of the 11 post-common envelope binaries comprising a white dwarf and a brown dwarf, and the ∼10 wider systems (e.g., Meisner et al. 2020 and references therein), only one other white dwarf shows any sort of metal pollution, SDSS J155720.77+091624.6A(Farihi et al. 2017).This white dwarf has a brown dwarf companion on an orbital period of 2.73 hr and the distinctive low mass of the white dwarf is indicative of a post-common envelope system.This binary has a mid-infrared excess seen at 3.6 and 4.5 microns that cannot be attributed to a reflection effect or the white dwarf (Swan et al. 2020;Farihi et al. 2017) and is determined to be emission from a debris disc.However, due to the short period of the system, stable circumstellar material is only permitted at a radius very close to the white dwarf, where the dust would be inconsistent with the observed thermal emission, thus indicating the disc is circumbinary, and is located at ∼3.3 R ⊙ or 0.015 AU.
Unlike SDSS J1557, PHL 5038AB is widely separated, meaning it is possible for debris to be present on a stable S-type orbit between the white dwarf and the brown dwarf.Using Equation 1 of Holman & Wiegert (1999) for our orbital parameters, we took the zero eccentricity case, determining any debris would be stable at a distance <17-32 AU.If we take the maximum eccentricity of 0.6 suggested by the orbital fit, the debris is stable much closer to the white dwarf, closer than 5-8 AU, a significantly narrower range.These values for possible locations of any debris are all outside the typical radius of an AGB star, suggesting the debris may have been a remnant from the binary's formation.
It is therefore possible that the presence of the brown dwarf is responsible for the pollution seen in the white dwarf atmosphere.If the debris belt is currently at the "edge" of the stable zone, with the outer edge of the belt being slowly eroded by interactions with the brown dwarf, the belt could have initially been larger.In this scenario, when the white dwarf was first formed, there would have been an intense period of scattering as the outer edge of the belt was cleared.This scattering decreased with time, leaving us with the low calcium abundance detected in the white dwarf atmosphere.

CONCLUSIONS
We determine that the white dwarf PHL 5038A is polluted by calcium, possibly by rocky material that is being perturbed by the wide brown dwarf companion PHL 5038B which orbits at 66 +12 −24 AU.The brown dwarf likely orbited the ∼1 M ⊙ white dwarf progenitor at ∼33 AU, and migrated outwards as the star evolved off the main sequence, without a common envelope phase.PHL 5038AB is perhaps the first system with a wide substellar companion that could be responsible for the pollution seen at the white dwarf.

Figure 1 .Figure 2 .
Figure 1.NIRI images from 2008 (left) and 2020 (right) showing the positions of the white dwarf and the brown dwarf at each epoch.

3. 5
Accretion ratesWe used the Montreal White Dwarf Database 4(Dufour et al. 2017) and theKilic et al. (2020) values of effective temperature and surface gravity and our average measured [Ca/H] of -9.36 to determine the convection zone mass ratio (log CVZM=-8.298) and the diffusion timescale of calcium in the white dwarf atmosphere (log Ca settle =3.848).

Figure 3 .
Figure3.The Ca K line fit with a DA white dwarf model using the parameters ofKilic et al. (2020) following the methods ofDufour et al. (2007) andCoutu et al. (2019).The [Ca/H] abundances are given for each epoch.

Figure 4 .
Figure 4. Effective temperature vs accretion rate for polluted white dwarfs fromBlouin & Xu (2022).DZ (helium atmosphere) white dwarfs are plotted as grey points, DAZ (hydrogen atmosphere) white dwarfs as outlined triangles.PHL 5038A is plotted as the black square.

Figure 5 .
Figure 5. Observed and predicted photometry for PHL 5038AB.The observed SDSS photometry for the white dwarf is shown as dark blue pluses, and the observed near-IR photometry from Steele et al. (2009) is shown as filled triangles for the white dwarf (dark blue) and the combined white dwarf-brown dwarf (purple).The observed WISE magnitudes for the combined white dwarf-brown dwarf are shown as filled red circles.Our predicted photometry for the white dwarf is shown as filled light blue circles.The predicted photometry for the combined system is shown as purple boxes.There is no infrared excess that could be due to a disc detected out to 4.5 microns.

Table 1 .
Parameters for PHL 5038AB used in this paper.
Marley et al. (2021))as et al. (2002)models for Asymptotic Giant Branch (AGB) stars at first thermal pulse give an initial mass of 1.0 M ⊙ for a core mass of 0.53 M ⊙ which is consistent with the wdwarfdate results for all metallicities.3.2 Brown dwarf parametersOur new, older age estimate for the PHL 5038AB system means that the 60 M Jup mass determination for the brown dwarf fromSteele et al. (2009)is also likely an underestimate.Steele et al. (2009)determined a spectral type of L8-L9 using a  spectrum from NIRI on Gemini North.Using the effective temperature vs spectral type relations inDupuy & Liu (2017)we determine the effective temperature of spectral type L8-9 to be 1300-1450 K.TheMarley et al. (2021)models at an age of 8 Gyr predict masses of 0.068-0.070M ⊙ for PHL 5038B.