arXiv subjects

A. Bonsor

Publications and source records attributed to A. Bonsor.

18 recordsLinked to original sources

arXiv2024

Seven white dwarfs with circumstellar gas discs II: Tracing the composition of exoplanetary building blocks

This second paper presents an in-depth analysis of the composition of the planetary material that has been accreted onto seven white dwarfs with circumstellar dust and gas emission discs with abundances reported in Paper I. The white dwarfs are accreting planetary bodies with a wide range of oxygen, carbon, and sulfur volatile contents, including one white dwarf that shows the most enhanced sulfur abundance seen to date. Three white dwarfs show tentative evidence (2-3$\sigma$) of accreting oxygen-rich material, potentially from water-rich bodies, whilst two others are accreting dry, rocky material. One white dwarf is accreting a mantle-rich fragment of a larger differentiated body, whilst two white dwarfs show an enhancement in their iron abundance and could be accreting core-rich fragments. Whilst most planetary material accreted by white dwarfs display chondritic or bulk Earth-like compositions, these observations demonstrate that core-mantle differentiation, disruptive collisions, and the accretion of core-mantle differentiated material are important. Less than one percent of polluted white dwarfs host both observable circumstellar gas and dust. It is unknown whether these systems are experiencing an early phase in the disruption and accretion of planetary bodies, or alternatively if they are accreting larger planetary bodies. From this work there is no substantial evidence for significant differences in the accreted refractory abundance ratios for those white dwarfs with or without circumstellar gas, but there is tentative evidence for those with circumstellar gas discs to be accreting more water rich material which may suggest that volatiles accrete earlier in a gas-rich phase.

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arXiv2024

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.

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arXiv2023

Seven white dwarfs with circumstellar gas discs I: white dwarf parameters and accreted planetary abundances

Observations of planetary material polluting the atmospheres of white dwarfs are an important probe of the bulk composition of exoplanetary material. Medium- and high-resolution optical and ultraviolet spectroscopy of seven white dwarfs with known circumstellar dust and gas emission are presented. Detections or meaningful upper limits for photospheric absorption lines are measured for: C, O, Na, S, P, Mg, Al, Si, Ca, Ti, Cr, Fe, and Ni. For 16 white dwarfs with known observable gaseous emission discs (and measured photospheric abundances), there is no evidence that their accretion rates differ, on average, from those without detectable gaseous emission. This suggests that, typically, accretion is not enhanced by gas drag. At the effective temperature range of the white dwarfs in this sample (16,000-25,000K) the abundance ratios of elements are more consistent than absolute abundances when comparing abundances derived from spectroscopic white dwarf parameters versus photometric white dwarf parameters. Crucially, this highlights that the uncertainties on white dwarf parameters do not prevent white dwarfs from being utilised to study planetary composition. The abundances of oxygen and silicon for the three hydrogen-dominated white dwarfs in the sample with both optical and ultraviolet spectra differ by 0.62 dex depending on if they are derived from the optical or ultraviolet spectra. This optical/ultraviolet discrepancy may be related to differences in the atmospheric depth of line formation; further investigations into the white dwarf atmospheric modelling are needed to understand this discrepancy.

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arXiv2022

Rapid Formation of Exoplanetesimals Revealed by White Dwarfs

The timing of formation for the first planetesimals determines the mode of planetary accretion and their geophysical and compositional evolution. Astronomical observations of circumstellar discs and Solar System geochronology provide evidence for planetesimal formation during molecular cloud collapse, much earlier than previously estimated. Here, we present distinct observational evidence from white dwarf planetary systems for planetesimal formation occurring during the first few hundred thousand years after cloud collapse in exoplanetary systems. A significant fraction of white dwarfs have accreted planetary material rich in iron core or mantle material. In order for the exo-asteroids accreted by white dwarfs to form iron cores, substantial heating is required. By simulating planetesimal evolution and collisional evolution we show that the most likely heat source is short-lived radioactive nuclides such as Al-2 (half life of approximately 0.7 Myr). Core-rich materials in the atmospheres of white dwarfs, therefore, provide independent evidence for rapid planetesimal formation, concurrent with star formation.

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arXiv2022

High resolution ALMA and HST imaging of $\kappa$CrB: a broad debris disc around a post-main sequence star with low-mass companions

$\kappa$CrB is a ${\sim}2.5\,$Gyr old K1 sub-giant star, with an eccentric exo-Jupiter at ${\sim}2.8\,$au and a debris disc at tens of au. We present ALMA Band 6 ($1.3\,$mm) and HST scattered light ($0.6\,\mu$m) images, demonstrating $\kappa$CrB's broad debris disc, covering an extent $50{-}180\,$au in the millimetre (peaking at $110\,$au), and $51{-}280\,$au in scattered light (peaking at $73\,$au). By modelling the millimetre emission, we estimate the dust mass as ${\sim}0.016\,M{\oplus}$, and constrain lower-limit planetesimal sizes as $D_{\rm{max}}{>}1\,$km and the planetesimal belt mass as $M_{\rm{disc}}{>}1\,M_{\oplus}$. We constrain the properties of an outer body causing a linear trend in 17 years of radial velocity data to have a semi-major axis $8{-}66\,$au and a mass $(0.4{-}120)\,M_{\rm{Jup}}$. There is a large inner cavity seen in the millimetre emission, which we show is consistent with carving by such an outer massive companion with a string of lower mass planets. Our scattered light modelling shows that the dust must have a high anisotropic scattering factor ($g{\sim}0.8{-}0.9$) but an inclination ($i{\sim}30{-}40\,$degrees) that is inferred to be significantly lower than the $i{\sim}61\,$degrees millimetre inclination. The origin of such a discrepancy is unclear, but could be caused by a misalignment in the micron and millimetre sized dust. We place an upper limit on the CO gas mass of $M_{\rm{CO}}{<}(4.2{-}13){\times}10^{-7}\,M_{\oplus}$, and show this to be consistent with levels expected from planetesimal collisions, or from CO-ice sublimation as $\kappa$CrB begins its giant branch ascent.

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arXiv2021

How planets grow by pebble accretion IV: Envelope opacity trends from sedimenting dust and pebbles

The amount of nebular gas that a planet can bind is limited by its cooling rate, which is set by the opacity of its envelope. Accreting dust and pebbles contribute to the envelope opacity and, thus, influence the outcome of planet formation. Our aim is to model the size evolution and opacity contribution of solids inside planetary envelopes. We then use the resultant opacity relations to study emergent trends in planet formation. We design a model for the opacity of solids in planetary envelopes that accounts for the growth, fragmentation and erosion of pebbles during their sedimentation. We formulate analytical expressions for the opacity of pebbles and dust and map out their trends as a function of depth, planet mass, distance and accretion rate. We find that the accretion of pebbles rather than planetesimals can produce fully convective envelopes, but only in lower-mass planets that reside in the outer disk or in those that are accreting pebbles at a high rate. In these conditions, pebble sizes are limited by fragmentation and erosion, allowing them to pile up in the envelope. At higher planetary masses or reduced accretion rates, a different regime applies where the sizes of sedimenting pebbles are only limited by their rate of growth. The opacity in this growth-limited regime is much lower, steeply declines with depth and planet mass but is invariant with the pebble mass flux. Our results imply that the opacity of a forming planetary envelope can not be approximated by a value that is constant with either depth or planet mass. When applied to the Solar System, we argue that Uranus and Neptune could not have maintained a sufficiently high opacity to avoid runaway gas accretion unless they both experienced sufficiently rapid accretion of solids and formed late.

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arXiv2021

A near-infrared interferometric survey of debris-disk stars. VII. The hot/warm dust connection

(abridged) Context. The origin of hot exozodiacal dust and its connection with outer dust reservoirs remains unclear. Aims. We aim to explore the possible connection between hot exozodiacal dust and warm dust reservoirs (> 100 K) in asteroid belts. Methods. We use precision near-infrared interferometry with VLTI/PIONIER to search for resolved emission at H band around a selected sample of nearby stars. Results. Our observations reveal the presence of resolved near-infrared emission around 17 out of 52 stars, four of which are shown to be due to a previously unknown stellar companion. The 13 other H-band excesses are thought to originate from the thermal emission of hot dust grains. Taking into account earlier PIONIER observations, and after reevaluating the warm dust content of all our PIONIER targets through spectral energy distribution modeling, we find a detection rate of 17.1(+8.1)(-4.6)% for H-band excess around main sequence stars hosting warm dust belts, which is statistically compatible with the occurrence rate of 14.6(+4.3)(-2.8)% found around stars showing no signs of warm dust. After correcting for the sensitivity loss due to partly unresolved hot disks, under the assumption that they are arranged in a thin ring around their sublimation radius, we however find tentative evidence at the 3σ level that H-band excesses around stars with outer dust reservoirs (warm or cold) could be statistically larger than H-band excesses around stars with no detectable outer dust. Conclusions. Our observations do not suggest a direct connection between warm and hot dust populations, at the sensitivity level of the considered instruments, although they bring to light a possible correlation between the level of H-band excesses and the presence of outer dust reservoirs in general.

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arXiv2018

Dust Production and Depletion in Evolved Planetary Systems

The infrared dust emission from the white dwarf GD 56 is found to rise and fall by 20% peak-to-peak over 11.2 yr, and is consistent with ongoing dust production and depletion. It is hypothesized that the dust is produced via collisions associated with an evolving dust disk, temporarily increasing the emitting surface of warm debris, and is subsequently destroyed or assimilated within a few years. The variations are consistent with debris that does not change temperature, indicating that dust is produced and depleted within a fixed range of orbital radii. Gas produced in collisions may rapidly re-condense onto grains, or may accrete onto the white dwarf surface on viscous timescales that are considerably longer than Poynting-Robertson drag for micron-sized dust. This potential delay in mass accretion rate change is consistent with multi-epoch spectra of the unchanging Ca II and Mg II absorption features in GD 56 over 15 yr, although the sampling is sparse. Overall these results indicate that collisions are likely to be the source of dust and gas, either inferred or observed, orbiting most or all polluted white dwarfs.

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arXiv2016

ALMA observations of the $η$ Corvi debris disc: inward scattering of CO-rich exocomets by a chain of 3-30 M$_\oplus$ planets?

While most of the known debris discs present cold dust at tens of AU, a few young systems exhibit hot dust analogous to the Zodiacal dust. $η$ Corvi is particularly interesting as it is old and it has both, with its hot dust significantly exceeding the maximum luminosity of an in-situ collisional cascade. Previous work suggested that this system could be undergoing an event similar to the Late Heavy Bombardment (LHB) soon after or during a dynamical instability. Here we present ALMA observations of $η$ Corvi with a resolution of 1."2 (~22au) to study its outer belt. The continuum emission is consistent with an axisymmetric belt, with a mean radius of 152au and radial FWHM of 46au, which is too narrow compared to models of inward scattering of an LHB-like scenario. Instead, the hot dust could be explained as material passed inwards in a rather stable planetary configuration. We also report a 4sigma detection of CO at ~ 20au. CO could be released in situ from icy planetesimals being passed in when crossing the H$_2$O or CO$_2$ ice lines. Finally, we place constraints on hidden planets in the disc. If a planet is sculpting the disc's inner edge, this should be orbiting at 75-100au, with a mass of 3-30 M$_\oplus$ and an eccentricity < 0.08. Such a planet would be able to clear its chaotic zone on a timescale shorter than the age of the system and scatter material inwards from the outer belt to the inner regions, thus feeding the hot dust.

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arXiv2016

How to design a planetary system for different scattering outcomes: giant impact sweet spot, maximising exocomets, scattered disks

This paper considers the dynamics of scattering of planetesimals or planetary embryos by a planet on a circumstellar orbit. We classify six regions in the planet's mass versus semimajor axis parameter space according to the dominant outcome for scattered objects: ejected, accreted, remaining, escaping, Oort Cloud, depleted Oort Cloud. We use these outcomes to consider which planetary system architectures maximise the observability of specific signatures, given that signatures should be detected first around systems with optimal architectures (if such systems exist in nature). Giant impact debris is most readily detectable for 0.1-10Mearth planets at 1-5au, depending on detection method and spectral type. While A stars have putative giant impact debris at 4-6au consistent with this sweet spot, that of FGK stars is typically <<1au contrary to expectations; an absence of 1-3au giant impact debris could indicate a low frequency of terrestrial planets there. Three principles maximise cometary influx from exo-Kuiper belts: a chain of closely separated planets interior to the belt, none of which is a Jupiter-like ejector; planet masses not increasing strongly with distance (for a net inward torque on comets); ongoing replenishment of comets, possibly by embedded low-mass planets. A high Oort Cloud comet influx requires no ejector and architectures that maximise the Oort Cloud population. Cold debris disks are usually considered classical Kuiper belt analogues. Here we consider the possibility of detecting scattered disk analogues, which could be betrayed by a broad radial profile and lack of small grains, as well as spherical 100-1000au mini-Oort Clouds. Some implications for escaping planets around young stars, detached planets akin to Sedna, and the formation of super-Earths, are also discussed.

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arXiv2016

A near-infrared interferometric survey of debris-disc stars. V. PIONIER search for variability

Context: Extended circumstellar emission has been detected within a few 100 milli-arcsec around > 10% of nearby main sequence stars using near-infrared interferometry. Follow-up observations using other techniques, should they yield similar results or non-detections, can provide strong constraints on the origin of the emission. They can also reveal the variability of the phenomenon. Aims: We aim to demonstrate the persistence of the phenomenon over time scales of a few years and to search for variability of our previously detected excesses. Methods: Using VLTI/PIONIER in H band we have carried out multi-epoch observations of the stars for which a near-infrared excess was previously detected with the same observing technique and instrument. The detection rates and distribution of the excesses from our original survey and the follow-up observations are compared statistically. A search for variability of the excesses in our time series is carried out based on the level of the broadband excesses. Results: In 12 of 16 follow-up observations, an excess is re-detected with a significance of > 2 sigma, and in 7 of 16 follow-up observations significant excess (> 3 sigma) has been re-detected. We statistically demonstrate with very high confidence that the phenomenon persists for the majority of the systems. We also present the first detection of potential variability in two sources. Conclusions: We conclude that the phenomenon responsible for the excesses persists over time scales of a few years for the majority of the systems. However, we also find that variability intrinsic to a target can cause it to have no significant excess at the time of a specific observation.

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arXiv2014

ALMA and Herschel Observations of the Prototype Dusty and Polluted White Dwarf G29-38

ALMA Cycle 0 and Herschel PACS observations are reported for the prototype, nearest, and brightest example of a dusty and polluted white dwarf, G29-38. These long wavelength programs attempted to detect an outlying, parent population of bodies at 1-100 AU, from which originates the disrupted planetesimal debris that is observed within 0.01 AU and which exhibits L_IR/L = 0.039. No associated emission sources were detected in any of the data down to L_IR/L ~ 1e-4, generally ruling out cold dust masses greater than 1e24 - 1e25 g for reasonable grain sizes and properties in orbital regions corresponding to evolved versions of both asteroid and Kuiper belt analogs. Overall, these null detections are consistent with models of long-term collisional evolution in planetesimal disks, and the source regions for the disrupted parent bodies at stars like G29-38 may only be salient in exceptional circumstances, such as a recent instability. A larger sample of polluted white dwarfs, targeted with the full ALMA array, has the potential to unambiguously identify the parent source(s) of their planetary debris.

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arXiv2014

Stochastic accretion of planetesimals onto white dwarfs: constraints on the mass distribution of accreted material from atmospheric pollution

This paper explores how the stochastic accretion of planetesimals onto white dwarfs would be manifested in observations of their atmospheric pollution. Archival observations of pollution levels for unbiased samples of DA and non-DA white dwarfs are used to derive the distribution of accretion rates, confirming that rates become systematically lower as sinking time is decreased, with no discernable dependence on cooling age. The accretion rates expected from planetesimals that are all the same mass (ie, a mono-mass distribution) are explored both analytically and using a Monte Carlo model, quantifying how measured accretion rates inevitably depend on sinking time, since different sinking times probe different times since the last accretion event. However, that dependence is so dramatic that a mono-mass distribution can be excluded. Consideration of accretion from a broad distribution of planetesimal masses uncovers an important conceptual difference: accretion is continuous (rather than stochastic) for planetesimals below a certain mass, and the accretion of such planetesimals determines the rate typically inferred from observations; smaller planetesimals dominate the rates for shorter sinking times. A reasonable fit to the observationally inferred accretion rate distributions is found with model parameters consistent with a collisionally evolved mass distribution up to Pluto-mass, and an underlying accretion rate distribution consistent with that expected from descendants of debris discs of main sequence A stars. With these parameters, while both DA and non-DA white dwarfs accrete from the same broad planetesimal distribution, this model predicts that the pollution seen in DAs is dominated by the continuous accretion of <35km objects, and that in non-DAs by >35km objects (though the dominant size varies between stars by around an order of magnitude from this value)...

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arXiv2013

Can eccentric debris disks be long-lived? A first numerical investigation and application to $ζ^2$ Reticuli

Imaging of debris disks has found evidence for both eccentric and offset disks. One hypothesis is that these provide evidence for massive perturbers that sculpt the observed structures. One such disk was recently observed in the far-IR by the Herschel Space Observatory around $ζ^2$ Ret. In contrast with previously reported systems, the disk is significantly eccentric, and the system is Gyr-old. We aim to investigate the long-term evolution of eccentric structures in debris disks caused by a perturber on an eccentric orbit. Both analytical predictions and numerical N-body simulations are used to investigate the observable structures that could be produced by eccentric perturbers. The long-term evolution of the disk geometry is examined, with particular application to the $ζ^2$ Ret system. In addition, synthetic images of the disk are produced for comparison with Herschel observations. We show that an eccentric companion can produce both the observed offsets and eccentric disks. Such effects are not immediate and we characterise the timescale required for the disk to develop to an eccentric state. For the case of $ζ^2$ Ret, we place limits on the mass and orbit of the companion required to produce the observations. Synthetic images show that the pattern observed around $ζ^2$ Ret can be produced by an eccentric disk seen close to edge-on, and allow us to bring additional constraints on the disk parameters of our model (disk flux, extent). We determine that eccentric planets or stellar companions can induce long-lived eccentric structures in debris disks. Observations of such eccentric structures provide potential evidence of the presence of such a companion in a planetary system. We consider the example of $ζ^2$ Ret, whose observed eccentric disk can be explained by a distant companion at tens of AU, on an eccentric orbit ($e_p\gtrsim 0.3$).

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arXiv2013

An interferometric study of the Fomalhaut inner debris disk. III. Detailed models of the exozodiacal disk and its origin

[Abridged] Debris disks are extrasolar analogs to the solar system planetesimal belts. The star Fomalhaut harbors a cold debris belt at 140 AU as well as evidence of a warm dust component, which is suspected of being a bright analog to the solar system's zodiacal dust. Interferometric observations obtained with the VLTI and the KIN have identified near- and mid-infrared excesses attributed to hot and warm exozodiacal dust in the inner few AU of the star. We performed parametric modeling of the exozodiacal disk using the GRaTeR radiative transfer code to reproduce the interferometric data, complemented by mid- to far-infrared measurements. A detailed treatment of sublimation temperatures was introduced to explore the hot population at the sublimation rim. We then used an analytical approach to successively testing several source mechanisms. A good fit to the data is found by two distinct dust populations: (1) very small, hence unbound, hot dust grains confined in a narrow region at the sublimation rim of carbonaceous material; (2) bound grains at 2 AU that are protected from sublimation and have a higher mass despite their fainter flux level. We propose that the hot dust is produced by the release of small carbon grains following the disruption of aggregates that originate from the warm component. A mechanism, such as gas braking, is required to further confine the small grains for a long enough time. In situ dust production could hardly be ensured for the age of the star, so the observed amount of dust must be triggered by intense dynamical activity. Fomalhaut may be representative of exozodis that are currently being surveyed worldwide. We propose a framework for reconciling the hot exozodi phenomenon with theoretical constraints: the hot component of Fomalhaut is likely the tip of the iceberg since it could originate from a warm counterpart residing near the ice line.

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arXiv2013

Asteroid Belts in Debris Disk Twins: VEGA and FOMALHAUT

Vega and Fomalhaut, are similar in terms of mass, ages, and global debris disk properties; therefore, they are often referred as "debris disk twins". We present Spitzer 10-35 um spectroscopic data centered at both stars, and identify warm, unresolved excess emission in the close vicinity of Vega for the first time. The properties of the warm excess in Vega are further characterized with ancillary photometry in the mid infrared and resolved images in the far-infrared and submillimeter wavelengths. The Vega warm excess shares many similar properties with the one found around Fomalhaut. The emission shortward of ~30 um from both warm components is well described as a blackbody emission of ~170 K. Interestingly, two other systems, eps Eri and HR 8799, also show such an unresolved warm dust using the same approach. These warm components may be analogous to the solar system's zodiacal dust cloud, but of far greater. The dust temperature and tentative detections in the submillimeter suggest the warm excess arises from dust associated with a planetesimal ring located near the water-frost line and presumably created by processes occurring at similar locations in other debris systems as well. We also review the properties of the 2 um hot excess around Vega and Fomalhaut, showing that the dust responsible for the hot excess is not spatially associated with the dust we detected in the warm belt. We suggest it may arise from hot nano grains trapped in the magnetic field of the star. Finally, the separation between the warm and cold belt is rather large with an orbital ratio >~10 in all four systems. In light of the current upper limits on the masses of planetary objects and the large gap, we discuss the possible implications for their underlying planetary architecture, and suggest that multiple, low-mass planets likely reside between the two belts in Vega and Fomalhaut.

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arXiv2011

The scattering of small bodies in planetary systems: constraints on the possible orbits of cometary material

The scattering of small bodies by planets is an important dynamical process in planetary systems. We present an analytical model to describe this process using the simplifying assumption that each particle's dynamics is dominated by a single planet at a time. As such the scattering process can be considered as a series of three body problems during each of which the Tisserand parameter with respect to the relevant planet is conserved. This constrains the orbital parameter space into which a particle can be scattered. Such arguments have previously been applied to the process by which comets are scattered to the inner Solar System from the Kuiper belt. Our analysis generalises this for an arbitrary planetary system. For particles scattered from an outer belt directly along a chain of planets, based on the initial value of the Tisserand parameter, we find that it is possible to (i) determine which planets can eject the particles from the system, (ii) define a minimum stellar distance to which particles can be scattered, and (iii) constrain range of particle inclinations (and hence the disc height) at different distances. Applying this to the Solar System, we determine that the planets are close to optimally separated for scattering particles between them. Concerning warm dust found around stars that also have Kuiper belt analogues, we show that, if there is to be a dynamical link between the outer and inner regions, then certain architectures for the intervening planetary system are incapable of producing the observations. Furthermore we show that for certain planetary systems, comets can be scattered from an outer belt, or with fewer constraints, from an Oort cloud analogue, onto star-grazing orbits, in support of a planetary origin to the metal pollution and dustiness of some nearby white dwarfs.

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arXiv2010

Post-main sequence evolution of A star debris discs

While the population of main sequence debris discs is well constrained, little is known about debris discs around evolved stars. This paper provides a theoretical framework considering the effects of stellar evolution on debris discs, particularly the production and loss of dust within them. Here we repeat a steady state model fit to disc evolution statistics for main sequence A stars, this time using realistic grain optical properties, then evolve that population to consider its detectability at later epochs. Our model predicts that debris discs around giant stars are harder to detect than on the main sequence because radiation pressure is more effective at removing small dust around higher luminosity stars. Just 12% of first ascent giants within 100pc are predicted to have discs detectable with Herschel at 160um. However this is subject to the uncertain effect of sublimation on the disc, which we propose can thus be constrained with such observations. Our model also finds that the rapid decline in stellar luminosity results in only very young white dwarfs having luminous discs. As such systems are on average at larger distances they are hard to detect, but we predict that the stellar parameters most likely to yield a disc detection are a white dwarf at 200pc with cooling age of 0.1Myr, in line with observations of the Helix Nebula. Our model does not predict close-in (<0.01AU) dust, as observed for some white dwarfs, however we find that stellar wind drag leaves significant mass (~10^{-2}Msolar), in bodies up to ~10m in diameter, inside the disc at the end of the AGB phase which may replenish these discs.

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