Rapid destruction of planetary debris around white dwarfs through aeolian erosion
MMNRAS , 1–9 (2020) Preprint 4 February 2021 Compiled using MNRAS L A TEX style file v3.0
Rapid destruction of planetary debris around white dwarfs throughaeolian erosion
Mor Rozner, ★ Dimitri Veras, , † Hagai B. Perets Physics department, Technion - Israel Institute of Technology, Haifa, 320004, Israel Centre for Exoplanets and Habitability, University of Warwick, Coventry CV4 7AL, UK Department of Physics, University of Warwick, Coventry CV4 7AL, UK
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
The discovery of numerous debris disks around white dwarfs (WDs), gave rise to extensive study of such disks and their rolein polluting WDs, but the formation and evolution of these disks is not yet well understood. Here we study the role of aeolian(wind) erosion in the evolution of solids in WD debris disks. Aeolian erosion is a destructive process that plays a key role inshaping the properties and size-distribution of planetesimals, boulders and pebbles in gaseous protoplanetary disks. Our analysisof aeolian erosion in WD debris disks shows it can also play an important role in these environments. We study the effects ofaeolian erosion under different conditions of the disk, and its erosive effect on planetesimals and boulders of different sizes. Wefind that solid bodies smaller than ∼ (cid:46) cm and at distances (cid:46) . 𝑅 (cid:12) from the WD. Thereby, aeolianerosion constitutes the main destructive pathway linking fragmentational collisions operating on large objects with sublimationof the smallest objects and Poynting-Robertson drag, which leads to the accretion of the smallest particles onto the photosphereof WDs, and the production of polluted WDs. Key words: stars: white dwarfs – planets and satellites: formation –planets and satellites: dynamical evolution and stability –minor planets, asteroids: general – planets and satellites: physical evolution
White dwarfs (WDs) are the final evolutionary stage of the vastmajority of all stars. Metal-polluted WDs, which represent 25 − ★ E-mail: [email protected] † STFC Ernest Rutherford Fellow
The abundance of objects with different sizes in WD disks – fromgrain size to minor planets, reveals rich and interesting dynamics(see a detailed review of post-main sequence evolution in Veras2016), which includes many physical processes, among them arereplenishment and accretion. Frequent collisions in WD disks (Jura2008; Metzger et al. 2012; Kenyon & Bromley 2017a, 2016, 2017c)lead to replenishment of grains in the disk – where large objectsbreak into smaller ones, and gradually are removed by radiationpressure and accretion. The debris disks might give rise to dynamicalexcitations and perturbations of mass that will eventually drive matteronto the WD within the disk lifetime (Girven et al. 2012; Veras &Heng 2020). Poynting-Robertson drag results from the radiation forceand causes loss of angular-momentum for small pebbles, carryingthem to the WD (Burns et al. 1979; Rafikov 2011a,b).The dynamics and architecture of WD disks are similar to proto-planetary disks in some aspects, and different in others. While thereare some processes that take place in both of them, such as frag-mentation, their parameters might differ significantly. The scales ofWD disks are much smaller, the density and temperature profiles aredifferent, and the typical velocities could be much higher.One important process in protoplanetary disks is aeolian (wind)erosion. Aeolian erosion is a purely mechanical destructive process,which is very common in many occasions in nature, mainly dis-cussed in the context of sand dunes (Bagnold 1941) . Recently, weshowed that aeolian erosion can play an important role in planet © a r X i v : . [ a s t r o - ph . E P ] F e b Rozner et al. formation by setting a new growth-barrier for pebbles/boulders inprotoplanetary disks, and affecting pebble accretion and streaminginstability (Grishin et al. 2020; Rozner et al. 2020). Aeolian erosionin protoplanetary disks is rapid and efficient, as also verified in labexperiments and numerical simulations of the conditions of proto-planetary disks (Paraskov et al. 2006; Demirci et al. 2020b; Demirci& Wurm 2020; Demirci et al. 2020a; Schaffer et al. 2020). Aeolianerosion leaves signatures on the dynamics of objects in the disk, e.g.it fuels pebble-accretion and induces a redistribution of sizes in thedisk, and might potentially lead to reshaping or complete destruc-tion of objects (Grishin et al. 2020; Rozner et al. 2020). In contrastwith protoplanetary disks, the temperatures in WD disks are highenough to maintain thermal ablation (Podolak et al. 1988; Pollacket al. 1996; D’Angelo & Podolak 2015) along with aeolian erosion,and accelerate the destruction of small objects.In this paper, we suggest that aeolian erosion could take place inWD disks and interact symbiotically with other dynamical processesin the disk. In fact, we show that for certain regimes, for objects withradius (cid:46) cm and at distances (cid:46) . 𝑅 (cid:12) from the WD, in a diskwith 𝑀 𝑑𝑖𝑠𝑘 = g for aeolian erosion is the dominant destructionprocess. The dominance regime changes with the disk and object’sparameters. As we will discuss later, the mass of the disk should be (cid:38) g to enable aeolian erosion.The efficiency of aeolian erosion, as well as of thermal ablation,depends on the composition of the object and the disk temperature,and acts to grind down large objects into small pebbles. For smallpebbles, Poynting-Robertson drag becomes efficient and carries thepebbles onto the WD. Hence, aeolian erosion might set a lower limiton the accretion flux onto the WD and enhance the disk gas replen-ishment, by assuring repeatedly regenerating the large abundanceof small objects. Moreover, as we showed in Grishin et al. (2020),aeolian erosion induces redistribution of the particle sizes and theirabundance in the disk. The size-distribution of objects in WD disksas well as the sizes of objects which accrete on WDs are not wellconstrained observationally, although some theoretical constraintshave been established (Kenyon & Bromley 2017b,c). Aeolian ero-sion might shed light on in this direction as well.The paper is organized as follows: in section 2 we briefly reviewthe parameters space and various models of WD disks. In section3 we review the models of aeolian erosion and thermal ablation inWDs disks. In section 4 we present our results and include eccentricorbits, multilayer objects and the relationship with thermal effects.In section 5 we discuss our results and suggest possible implications:we discuss the dependence of our model on the disk parameters,the symbiotic relations with other processes in the disk includingcollisional cascade, external seeding and further disk generations.In section 6, we discuss the caveats and limitations of our study. Insection 7 we summarize the paper and suggest future directions. In contrast with protoplanetary disks which properties are better-constrained, WD disk parameters are uncertain by orders-of-magnitude. Here we briefly review the ranges of these parametersthat will be used in the rest of the paper.The total mass of a WD disk ranges between 10 − g (see adetailed discussion in Metzger et al. 2012; Veras & Heng 2020 andreferences therein). The mass of the gas in the disk, parametrizedby the gas-to-dust ratio, is highly unconstrained and ranges between10 − and unity (Veras 2016). Observations set the lower limit of the inner-radius to be (cid:46) . 𝑅 (cid:12) (e.g. Rafikov 2011a) and the outer-radiusto be (cid:38) . 𝑅 (cid:12) (e.g. Gänsicke et al. 2006).The surface density profile is given as (Metzger et al. 2012), Σ 𝑔 ( 𝑎 ) = Σ 𝑔, (cid:18) . 𝑎 . 𝑅 (cid:12) (cid:19) − 𝛽 (cid:18) 𝑀 disk g (cid:19) , (1)where 𝑎 is the distance from the center of the disk, Σ 𝑔, is thefiducial surface density and 𝛽 is an arbitrary exponent which isparametrized as 𝛽 = 𝑛 + /
2, where 𝑛 describes the viscosity powerlaw 𝜈 ( 𝑎 ) ∝ 𝑎 𝑛 . The gas density 𝜌 𝑔 is determined by the surfacedensity, 𝜌 𝑔 = Σ 𝑔 ℎ ( 𝑎 ) , (2)where ℎ is the height of the disk.This scaling of 𝛽 also induces a temperature scaling (Metzger et al.2012), 𝑇 ( 𝑎 ) ∝ (cid:40) constant , 𝑛 = / 𝑎 − / , 𝑛 = 𝑛 = / 𝑛 = − unless stated otherwise. Theaforementioned choice of parameters yields typical values of Σ 𝑔 = . × 𝑔 𝑐𝑚 , 𝜌 𝑔 = . × − g cm − , and 𝑇 = K at 𝑎 = . 𝑅 (cid:12) .Unless stated otherwise, we will use 𝑛 = / ∼ − yr (Girvenet al. 2012; Veras & Heng 2020). Girven et al. (2012) estimatedthe age of WD disks assuming a constant accretion rate to the WD.Recently, Veras & Heng (2020) introduced a different estimationmethod, arising from the dynamical processes that the disk shouldgo through and their typical timescales. Consider a spherical object with radius 𝑅 that resides at a constantdistance 𝑎 from the WD and is built from grains with a typical size 𝑑 , that move in a gaseous medium with a density 𝜌 𝑔 . The pressuresupport present in gaseous disks leads to a difference between theKeplerian velocity Ω 𝐾 around the WD and the actual angular velocity Ω 𝑔 , Ω 𝑔 − Ω 𝑘 ≈ ( Ω 𝐾 𝑎𝜌 𝑔 ) − 𝜕𝑃 / 𝜕𝑟 where 𝜕𝑃 / 𝜕𝑟 is the pressuregradient. Furthermore, the pressure support leads to the radial drift,which presents one of the fundamental problems in planet formationin protoplanetary disks – the meter-size barrier (Weidenschilling1977).The radial velocity in protoplanetary disks obtains a maximum for ∼
1m size objects, and these objects inspiral to the inner parts of thedisk on timescales shorter than the expected growth timescale. Thedrift velocity is lower for smaller objects – they are better coupledto the gas and hence have slower relative velocities; larger objectsare loosely coupled to the gas and experience slower drift velocities.For objects in WD disks, the maximal drift velocity is obtained forsmaller objects, in size (cid:46)
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MNRAS000 , 1–9 (2020) eolian Erosion in WD Disks Figure 1.
The relative velocity between objects and the gas, in differentconstant distances from the WD. The solid lines are for 𝑣 rel and the dashedones are for the radial component, 𝑣 r . 𝑣 rel , r = − 𝜂𝑣 𝑘 St1 + St , (4) 𝑣 rel ,𝜙 = − 𝜂𝑣 𝑘 (cid:18) + St − (cid:19) (5)where the Stokes number is defined bySt = Ω 𝐾 𝑡 stop , (6) 𝑡 stop = 𝑚𝑣 rel 𝐹 𝐷 (7)where 𝑚 is the mass of the object, 𝐹 𝐷 is the drag force, which isgiven by F 𝐷 = 𝐶 𝐷 ( Re ) 𝜋𝑅 𝜌 𝑔 𝑣 ˆ v rel (8)For the drag coefficient, we adopted an emperical fitted formula,based on experminal data in the regime 10 − ≤ 𝑅𝑒 ≤ (Perets &Murray-Clay 2011 and references therein) 𝐶 𝐷 ( Re ) = ( + . ) . + . (cid:104) − exp (cid:16) − . . (cid:17)(cid:105) (9)where 𝑅𝑒 is the Reynolds number, defined byRe = 𝑅𝑣 rel 𝑣 th 𝜆 , (10)where 𝑣 th = ( / 𝜋 ) / 𝑐 𝑠 is the mean thermal velocity, 𝑐 𝑠 = √︁ 𝑘 𝐵 𝑇 / 𝜇 is the speed of sound, 𝑘 𝐵 is Boltzmann constant, 𝑇 is the temperatureof the disk, 𝜇 is the mean molecular weight, taken to be 2 . 𝑚 𝐻 (following Perets & Murray-Clay 2011) with 𝑚 𝐻 the mass of ahydrogen atom and 𝜆 = 𝜇 /( 𝜌 𝑔 𝜎 ) is the mean free path of the gas.In fig. 1, we present the relative velocities between the gas andobjects for different sizes. The radial component of the velocity peaksfor ∼ ∼ cm / sec. Much smallerobjects are well-coupled to the gas, which leads to smaller velocities,while much larger objects are weakly-coupled to the gas and henceare not affected significantly by it. The relative velocity could changefor different parameters of the disk, and as we have mentioned, theparameter space is currently wide.Objects in gaseous disks experience gas drag, which dependsstrongly on their velocity relative to the gas, 𝑣 rel . The gas-drag canplay the role that is usually played by the wind in aeolian erosion and trigger loss of the outer layer of objects, as long as they reach thethreshold conditions.The threshold velocity is dictated by the balance between cohesionforces, self-gravity and gas-drag, i.e. the headwind should be strongerthan the attraction force between grains in order to initiate aeolianerosion.The threshold velocity is given by 𝑣 ★ = √︄ 𝐴 𝑁 𝜌 𝑔 (cid:16) 𝜌 𝑝 𝑔𝑑 + 𝛾𝑑 (cid:17) , (11)where 𝐴 𝑁 = . × − , and 𝛾 = .
165 g s − are determinedempirically from Shao & Lu (2000). Both of them are intrinsic char-acteristic of the materials that rise from the cohesion forces thathold the particles together – mostly electrostatic forces and van derWaals forces. The gravitational acceleration is 𝑔 = 𝐺𝑚 / 𝑅 . Abovethe threshold velocity, i.e. 𝑣 rel > 𝑣 ★ , the aeolian erosion rate is givenby (Rozner et al. 2020), 𝑑𝑅𝑑𝑡 = − 𝜌 𝑔 𝑣 𝜋𝑅𝜌 𝑝 𝑎 coh (12)where 𝑎 coh is the cohesion acceleration. The derivation rises fromthe work done on the eroded object by the shear pressure; see adetailed derivation from equations 8 and 9 of Rozner et al. (2020).The derivation is based on estimation of the typical sweeping rate ofgrains from the outer layer and calculating the work done by them. Along with aeolian erosion, the high temperatures that are usuallyfound in WD disks might give rise to destructive thermal processessuch as thermal ablation (Podolak et al. 1988; Pollack et al. 1996;D’Angelo & Podolak 2015) and sublimation (e.g Metzger et al. 2012;Shestakova et al. 2019). The heat that the outer layer absorbs mightlead to phase transitions and then mass loss. There are two regimes,separated by a critical temperature 𝑇 𝑐𝑟 , in which the latent heatrequired for vaporization is zero, and varies according to the material(e.g. Opik 1958; Podolak et al. 1988; D’Angelo & Podolak 2015).See the fiducial parameters for ablation in table A.Below the critical temperature, the rate in which vaporizationremoves mass is dictated by Hertz-Knudsen-Langmuir equation; seea discussion in D’Angelo & Podolak (2015). In this regime, aeolianerosion significantly dominates for our choice of parameters, andthe timescale for thermal ablation below the critical temperature is (cid:38)
95 yrs – much longer than typical aeolian erosion timescale, whichenables us to neglect the thermal effect and focus on the mechanicalprocesses.Above the critical temperature, the contribution from thermal pro-cesses might add a significant contribution to aeolian erosion andshould be added to eq. 12; assuming blackbody emission, the ther-mal ablation term is given by 𝑑𝑅𝑑𝑡 (cid:12)(cid:12)(cid:12)(cid:12) ablation = 𝐿 𝑠 𝜌 𝑝 𝜖 𝑠 𝜎 𝑆𝐵 ( 𝑇 𝑐𝑟 − 𝑇 𝑔 ) , (13)where 𝜖 𝑠 is the thermal emissivity of the object ( 𝜖 𝑠 = 𝐿 𝑠 is the particle specific vaporization energy, 𝑇 𝑐𝑟 is thecritical temperature – which depends on composition of the material– and 𝑇 𝑔 is the gas temperature. At high enough temperatures, closeto the WD, small grains sublimate to gas (e.g Metzger et al. 2012;Shestakova et al. 2019). MNRAS , 1–9 (2020)
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In this section we present the evolution of objects in WD disks dueto aeolian erosion and thermal ablation, starting from a fiducial setof parameters and then vary them.Aeolian erosion in WD disks is quite efficient, as manifested inFig. 2. The timescales for aeolian erosion are extremely short, and fora constant distance 𝑎 = . ≈ (cid:12) from the WD disk, objectswith radii as large as ∼ × cm are eroded within the expecteddisk lifetime. The chosen fixed distance from the WD, along withthe rest of the parameters that we don’t vary currently, dictates thefinal size to which objects are ground down – from ∼ . . ∼ . . Aeolian erosion depends strongly on the relative velocity (to the thirdpower), as can be seen from Eq. 12. Hence, its effects on objects oneccentric orbits may differ substantially from the effects in the circularcase, and lead to stronger more significant erosion due to the highervelocities involved, that might even be supersonic and under someconditions lead to the prompt disruption of objects (Demirci et al.2020a).The velocity of a planetesimal in an eccentric orbit is given by v 𝑝 = 𝑣 𝑘 √︂ − 𝑟𝑎 ˆ v 𝑝 = √︄ 𝐺 𝑀 (cid:18) 𝑟 − 𝑎 (cid:19) ˆ v 𝑝 (14)where 𝑀 is the mass of the WD, 𝑣 𝑘 is the Keplerian velocity, 𝑟 is thedistance and 𝑎 is the semimajor axis.Assuming a circular disk, the gas moves with a velocity v 𝑔 = 𝑣 𝑘 √︁ − 𝜂 ˆ v 𝑔 (15)The magnitude of the relative velocity between the gas and theplanetesimal is given by 𝑣 rel = | ˆ v 𝑝 − ˆ v 𝑔 | = √︃ 𝑣 𝑝 + 𝑣 𝑔 − 𝑣 𝑝 𝑣 𝑔 ˆ v 𝑔 · ˆ v 𝑝 . (16)For simplicity, we will assume that the evolution of objects experi-encing aeolian erosion is dominated by the maximal relative velocityin the orbit, when the phase between the gas and the objects is max-imal – 𝜋 /
2, and also assume that the object is at the pericenter, i.e. 𝑟 = 𝑟 𝑝 = 𝑎 ( − 𝑒 ) . See Mai et al. (2020) for more detailed results.Motion in eccentric orbits shortens the timescales of aeolian ero-sion and the final size of eroded objects is smaller and could attain ∼ .
08 cm for 𝑒 = . Previously, we discussed objects which are composed from a single-size grain distribution. However, the physical reality might be more complicated and could give rise to a non-trivial internal size distribu-tion. In the following we relax homogeneous composition assump-tion, and consider the effects of aeolian erosion on inhomogeneousobjects. To the best of our knowledge no previous study consideredthe internal structure of grains within WD disks, and we adopt inter-nal size distribution models usually considered for asteroids.The Brazil nut effect in asteroids (Matsumura et al. 2014) suggeststhat in a mixture of particles, the larger ones tend to end up onthe surface of objects. Therefore, when this process acts, the innerstructure of an asteroid is such that the larger grains are in the outerlayers.In Figs. 4, 5, we present how differentiated objects react to aeolianerosion. Erosion enables us to decompose the layers of an object –as far as they are in the correct regimes in which erosion is effective– and to reveal the inner layers in short timescales. Since differentsizes of grains impose different rates of aeolian erosion, the totalaeolian erosion timescales change such that considering larger grainsshortens the timescales, and smaller grains lengthen them. Moreover,it can be seen that the final size is determined by the innermost layer,and larger grains lead to smaller final sizes.
At high temperatures, thermal processes become more significantand might strengthen the effect of aeolian erosion and give rise tofurther destruction and shorten the timescales.As can be seen in Fig. 6, thermal ablation shortens the timescales inwhich objects are ground down to their final scales – here manifestedfor icy objects. Rocky objects will require higher disk temperaturesfor thermal ablation to be significant, since the critical temperaturefor rock is 4000K; due to the large range of possibilities for WD disksin general – and for temperatures in particular – the temperature inthe disk might even exceed the critical temperature for rocks andother materials, at least for the hottest youngest WDs. For coatedobjects, there could be a combined process of ablating the outer icylayer and then mechanically eroding inner layers.
The possible parameter range for WD disks is wide and enables usto study a multitude of combinations of parameters. Here we willpresent a parameter space exploration for solid bodies embedded inWD disks and subjected to the effects of aeolian erosion.In Fig. 7 we examine the dependence of aeolian erosion on thedisk mass. As can be seen in eq. 1, the surface density of the disk,and hence the gas density, grow linearly with the disk mass. Sincethe the aeolian erosion rate is proportional to the gas density, the ratebecomes stronger for larger disk masses.In Fig. 8, we present the dependence of aeolian erosion on theaspect ratio. Higher aspect ratios lead to stronger aeolian erosion.And in our default choice of parameters, the aspect ratio should be (cid:38) − in order to maintain significant aeolian erosion. Fragmentational collisional erosion is another destructive processthat gradually diminishes the mass of objects in WD disks. This de-structive process generates a collisional cascade which grinds down
MNRAS , 1–9 (2020) eolian Erosion in WD Disks Figure 2.
The time evolution of objects embedded in a WD disk due to aeolian erosion.
Left : Evolution at a constant distance 𝑎 = . ≈ (cid:12) from theWD. Right : Different distances from the WD, with a constant initial size of embedded objects 𝑅 = cm. Figure 3.
The effect of aeolian erosion on an 10 cm object embedded in aWD disk at a distance of 1 𝑅 (cid:12) from the center of the WD. The relative velocityof the object is considered as the maximal velocity in the orbit. Figure 4.
The evolution of two-layered differentiated objects (see inset leg-end), with an outer layer comprising 20% of the radius and an inner core ofthe remaining 80% of the radius, under the effects of aeolian erosion; all ofthe layers have the same density of 3 .
45g cm − . The object is embedded in aWD disk and resides at a distance of 0 . − cm objects to 10 − cm objects within 10 − years (e.g.Kenyon & Bromley 2017b,c). Collisional cascades and aeolian ero-sion both cause objects to lose mass, such that in the presence of gas,these two processes are symbiotic. Figure 5.
The evolution of triple-layered objects, with three equal layers, witheach layer comprising 1 / .
45g cm − , butwith different size of inner grains that build each layer (see inset legend). Theobject is embedded in a WD disk and resides at a distance of 0 . Figure 6.
The relationship between aeolian erosion and ablation for an icyobject with initial size of 10 cm, embedded at a constant distance of 0 . The aeolian erosion timescale is given by 𝑡 erosion = 𝑅 | (cid:164) 𝑅 | = 𝜋𝑅 𝜌 𝑝 𝑎 coh 𝜌 𝑔 𝑣 . (17)The fragmentation timescale is estimated from the collisional MNRAS , 1–9 (2020)
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Figure 7.
The dependence on disk mass for an object with initial size of10 cm, and constant distance from the center of 0 . Figure 8.
The dependence on aeolian erosion on the aspect ratio of the disk, ℎ / 𝑟 = 𝑐 𝑠 / 𝑟 Ω where Ω is the angular velocity, for objects of initial size10 cm, embedded at the disk in a distance of 0 . timescale. The collision timescale for a mono-disperse swarm isgiven by (Kenyon & Bromley 2017a,b,c and references therein) 𝑡 = 𝑟 𝜌𝑃 𝜋 Σ 𝑔 , (18)where 𝑟 is the radius of all the objects in the swarm, 𝜌 𝑝 is theirdensity, 𝑃 is their orbital period and Σ is the initial surface density.When a multi-disperse swarm is considered, the modification of thetimescale is parametrized by a collision parameter 𝛼 𝑐 ∝ ( 𝑣 / 𝑄 ∗ 𝐷 ) − ,where 𝑣 is the collision velocity and 𝑄 ∗ 𝐷 is the binding energy of theobject (see Leinhardt & Stewart 2012 and references therein), suchthat the timescale from multi-species is given by 𝑡 𝑐 = 𝛼𝑡 , which canbe shorter compared with the mono-disperse collision timescales.When the replenishment is efficient enough, the background distri-bution of objects remains roughly constant. However, generally, thebackground density changes with time as well, adding complicationsto the analysis.The ratio between the timescales of aeolian erosion and fragmen-tational collisions is given by 𝑡 erosion 𝑡 fragment = 𝜋 ℎ𝑎 coh 𝛼 𝑐 𝑃𝑣 𝑅 (19)such that the timescales are comparable for a ratio around unity.From equating the ratio to unity, one can set the transition radiusbetween the aeolian erosion dominated regime and the collisionalfragmantation dominaneted regime, which depends on the rest of theparameters.In Fig. 9, we compare the typical timescales of aeolian erosion Figure 9.
The typical timescales of aeolian erosion (solid lines) and frag-mentational collision (dashed lines) as a function of the destructed object’sradius, at different constant distances from the WD. 𝑅 max is the radius of thelargest object in the swarm. and fragmentational collisions at which objects are destroyed. Forthe fragmentation collision timescales, we consider a swarm with amaximal size given by 𝑅 max , and assume for simplicity that the frag-mentation of intermediate objects (i.e., not the largest or the smallestobject in the swarm) starts just when they become the largest of theswarm, which is justified by the hierarchical character of the frag-mentational cascade. For the background distribution, we assume apower law distribution of 𝑁 ( 𝑟 ) ∼ 𝑟 − . (Kenyon & Bromley 2017a),and a value of 𝑣 / 𝑄 ∗ 𝐷 that yields 𝛼 𝑐 ∼ . Aeolian erosion dom-inates at small sizes, and fragmentational collisions at larger sizes.The transition point between the regimes varies with the distancefrom the WD, such that the regime of aeolian erosion grows withsmaller distances. The aeolian erosion timescales are very short inthese regimes, but might be comparable to or longer than the orbitalperiod timescales of the objects.We would like to stress that although the destructive character ofboth aeolian erosion and collisional cascades leads to shrinkage ofobjects in the disks, they differ by their intrinsic physical mechanisms.Aeolian erosion originates from shear pressure that is induced by gasdrag. Hence, the presence of gas in the disk is a necessary conditionto initiate this process. The collisional cascade arises from collisionsbetween particles in the disk. The abundance of large objects in WD disks might be replenishedas a result of seeding (Grishin et al. 2019; Grishin & Veras 2019).Exo-planetesimals from external sources could enrich the abundanceof objects in the disk that will eventually be eroded. External captureespecially contributes to the number of large objects, although thetotal captured mass is small compared to the dust initial abundance.Furthermore, seeding could bring into the disk new materials thatmight eventually end as pollutants on the WD. The capture rate isdictated by the supply rate and the capture probability; both vary withthe origin and the size of the captured objects.The captured objects change the size distribution in the disk, andmight contribute to the steady-state distribution. The injection ofexternal objects might be at a sufficient rate to cancel out the de-structive processes and maintain a distribution with larger objectsthan expected. The revised size distribution should be derived froma full swarm simulation which includes collisional cascade, aeolianerosion, growth and seeding.
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MNRAS000 , 1–9 (2020) eolian Erosion in WD Disks The parameters and characteristics of the disk might vary from onegeneration to another. Wide-binary evolution could lead to an evolveddonor that transfers mass to its companion, such that the capturedmaterial forms a disk (Perets 2010; Perets & Kenyon 2013; Schleicher& Dreizler 2014; van Lieshout et al. 2018). A binary main-sequencesystem evolves such that the more massive star sheds material whichis accreted on the secondary and forms a protoplanetary disk; themass that is lost from the system, which is much greater than themass transferred from one star to the other, leads to expansion ofthe binary orbit (Veras et al. 2011; Kratter & Perets 2012; Veras &Tout 2012). Afterwards, a second generation of debris and planetsform in the pre-evolved system, such that the secondary evolves offthe main-sequence and sheds material to its WD companion and aprotoplanetary disk formed – similarly to the previous stages. Finally,the binary orbit expands and a third generation debris disk is formed.Higher generation disks might give rise to different disk compo-sition, since metallicity varies from one generation to another, suchthat the formed accretion disks are metal and dust rich. Higher metal-licity environments are better for planet formation (Fischer & Valenti2005), such that the formed disks are likely to have planetary andplanetesimal structures.
One notable potential application of aeolian erosion theory is the diskorbiting the WD SDSS J1228+1040 (Manser et al. 2019). This diskcontains a planetesimal on an orbit with 𝑎 = . 𝑅 (cid:12) and 𝑒 ≈ . 𝑅 (cid:12) .Because of this compact planetesimal orbit, and because the plan-etesimal has been observed over at least 4,000 orbits, it cannot repre-sent a typical rubble-pile solar system asteroid. Instead, it likely rep-resents an iron-rich remnant core of a planet, and harbours nonzerointernal tensile strength. The internal structure and size are poorlyconstrained: the estimated radius range is 𝑅 = . −
120 km.Hence, for only the lowest end of this size range could aeolianerosion could be effectual. Figure 2 illustrates that km-sized objectsmay be eroded in timescales under ∼ yr. However, the structuresof the objects in that figure are likely to be very different than thestrong and dense planetesimal around SDSS 1228+1040. Further-more, a potentially competing effect are bodily gravitational tides.Veras et al. (2019) show that these tides could instigate a WD toengulf objects containing just 10 − 𝑀 ⊕ , but only for objects withsufficiently low internal viscosities. Aeolian erosion is a very effective process in WD disks, as long as theamount of gas in the disk is non-negligible. The current observationsof WD disks with a gaseous component cannot yet well constrain sev-eral of the disk parameters, such as the disc mass and its scale height.Also, the exact origins of the gas in such disk are not well under-stood, with suggested origins include sublimation (e.g. Metzger et al.2012), grain-grain collisional vaporization and sputtering. Recently,Malamud, Grishin, & Brouwers (2020) presented another channelof gas production in white dwarf disks, via interactions between aneccentric tidal stream and a pre-existing dusty compact disk.Along with the destructive processes, there could be mass influxthat we did not take into consideration in this paper. As manifested in Kenyon & Bromley (2016, 2017c) in the context of collisionalcascade, the rate of mass input might equalize the mass loss such thatobjects that are a-priori expected to be pulverized and will maintaintheir mass for long timescales. However, here the combined effect offragmantational erosion and aeolian erosion might play a role, andthese two processes together will lead to mass depletion of objects.We have neglected any asphericity in the object, which is assureddue to the lack of perfect packing efficiency of its constituent grains.Furthermore, we have neglected any asphericity that develops asa result of an aeolian erosion, which acts in the direction of theheadwind, and might reshape the eroded objects.We focus on aggregates, with a weak outer layer. Some of theobjects that could be potentially affected from aeolian erosion aredestroyed already in tidal shredding or collisions. The cohesion forceshold for loosly bound objects, i.e. they describe aggregates and otherforms of cohesion laws should be taken into account in case ofdifferent internal physics. Once the cohesion law is dictated, thesuggested prescription of aeolian erosion will be very similar to theone we sketched in this paper.For aeolian erosion to be effective, the dominant stripping force onthe weak outer layers of the aggregates would need to be erosive ratherthan tidal. The strength of the tidal force can vary significantly de-pending on physical properties; for large homogeneous rubble-piles,this value can vary by a factor of about 2 (around 1 𝑅 (cid:12) ) depending onspin and fluidity, and can cause stripping on an intermittent, yearlytimescale (Veras et al. 2017). Such intermittency perhaps suggeststhat erosive and tidal forces may act in concert in certain cases,particularly as the aggregate changes shape.Another aspect of the physics which we did not model is theasteroid spin barrier, which refers to the minimum spin rate at whichan asteroid breaks itself apart. This barrier is well-established atabout 2 . The growing number of WD disks (both gaseous and non-gaseous)that have been observationally detected and characterized leave openthe possibility for constraining theoretical models for the origins andevolution of such disks. However, even at the theoretical level, thereare important gaps in our physical understanding of the dynamicsand processes that take place in WD disks.In this paper we focused on the processes of aeolian erosion, which,to date, were not considered in the context of WD disks. We madeuse of an analytical model for aeolian erosion in WD disks, based onour studies of such processes in protoplanetary disks as presented inRozner et al. (2020). We find that the typical timescales of aeolianerosion in WD disks are extremely short, with aeolian erosion grind-ing down even km-size objects within the disk lifetime. Consequently,such processes are likely to play an important role in the evolutionof small solid bodies in the disk. We also studied the relationship be-tween aeolian erosion and other physical processes in WD disks andits amplification due to the combined effect (see subsections 3.2,5.2).Along with collisional cascade and thermal ablation, aeolian erosiongrinds down efficiently large objects into small ones with a char-
MNRAS , 1–9 (2020)
Rozner et al. acteristic final size. The eroded objects experience dynamical pro-cesses that finally grind down planetesimals/rocks/pebbles/bouldersinto sufficiently small particles such that these could drift towardsthe WD via Poynting-Robertson drag and contribute to its pollution.Aeolian erosion is the most efficient and becomes the dominantdestruction process for small objects, and the critical radius for itsdominance is determined by the parameters of the disk, the physicalcharacteristics of the eroded objects, the distance from the WD andthe parameters of the collisional cascade (see eq. 19).Similarly to protoplanetary disks, aeolian erosion in WD disksinduces a re-distribution of particles size, according to the distanceof the particles from the WD. Hence, aeolian erosion sets constraintson the parameters of WD disks that might narrow down the currentparameter space. Due to the extremely short timescales of aeolianerosion, it is not likely that observable variations in WD disks wouldbe explained by replenishment from aeolian erosion.
ACKNOWLEDGMENTS
We thank the anonymous reviewer for useful comments which haveimproved the manuscript. MR and HBP acknowledge support fromthe Minerva center for life under extreme planetary conditions, theLower Saxony-Israel Niedersachsisches Vorab reseach cooperationfund, and the European Union’s Horizon 2020 research and inno-vation program under grant agree- ment No 865932-ERC-SNeX.DV gratefully acknowledges the support of the STFC via an ErnestRutherford Fellowship (grant ST/P003850/1).
DATA AVAILABILITY
The data that support the findings of this study are available from thecorresponding author upon reasonable request.
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In this appendix, we present the default values for the used parame-ters, unless stated otherwise.
This paper has been typeset from a TEX/L A TEX file prepared by the author. MNRAS , 1–9 (2020) Rozner et al.
Symbol Definition Fiducial Value Reference 𝛾 . / sec Kruss et al. (2019) 𝐴 𝑁 . × − Shao & Lu (2000) 𝛽 g s − scaled from Paraskov et al. (2006) and refs. therein 𝜌 𝑝 planetesimals’ density rock 3 .
45g cm , ice 1 .
4g cm Pollack et al. (1996) 𝜇 mean molecular weight 3 . × − 𝑔 Perets & Murray-Clay (2011) 𝜎 neutral collision cross-section 10 − cm Perets & Murray-Clay (2011) 𝑇 temperature (optically thick) 1000K 𝑀 𝑑𝑖𝑠𝑘 disk mass 10 g Σ 𝑔 surface density profile (optically thick) 5 . × g cm Grishin & Veras (2019) ℎ / 𝑟 aspect ratio 10 − 𝑑 typical ’building-block’ grain size 0 . 𝑇 𝑐𝑟 critical temperature ice 648K, rock 4000K Podolak et al. (1988) 𝐿 𝑠 particle specific vaporization energy (solid) ice 2 . × erg g − , rock 8 . × erg g − D’Angelo & Podolak (2015)Size dist. in the disk (fragmentation only) 𝑁 ( 𝑟 ) ∝ 𝑟 − . Kenyon & Bromley (2017b)MNRAS000