# Rapid destruction of planetary debris around white dwarfs through aeolian
erosion

Mor Rozner,1 Dimitri Veras,2,3 Hagai B. Perets1
1 Physics department, Technion - Israel Institute of Technology, Haifa,
320004, Israel
2Centre for Exoplanets and Habitability, University of Warwick, Coventry CV4
7AL, UK
3Department of Physics, University of Warwick, Coventry CV4 7AL, UK E-mail:
<EMAIL_ADDRESS>Ernest Rutherford Fellow

(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 role in 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
in shaping the properties and size-distribution of planetesimals, boulders and
pebbles in gaseous protoplanetary disks. Our analysis of aeolian erosion in WD
debris disks shows it can also play an important role in these environments.
We study the effects of aeolian erosion under different conditions of the
disk, and its erosive effect on planetesimals and boulders of different sizes.
We find that solid bodies smaller than $\sim 5\rm{km}$ will be eroded within
the short disk lifetime. We compare the role of aeolian erosion in respect to
other destructive processes such as collisional fragmentation and thermal
ablation. We find that aeolian erosion is the dominant destructive process for
objects with radius $\lesssim 10^{3}\rm{cm}$ and at distances $\lesssim
0.6R_{\odot}$ from the WD. Thereby, aeolian erosion constitutes the main
destructive pathway linking fragmentational collisions operating on large
objects with sublimation of the smallest objects and Poynting-Robertson drag,
which leads to the accretion of the smallest particles onto the photosphere of
WDs, and the production of polluted WDs.

###### keywords:

stars: white dwarfs – planets and satellites: formation –planets and
satellites: dynamical evolution and stability – minor planets, asteroids:
general – planets and satellites: physical evolution

††pubyear: 2020††pagerange: Rapid destruction of planetary debris around white
dwarfs through aeolian erosion–A

##  1 Introduction

White dwarfs (WDs) are the final evolutionary stage of the vast majority of
all stars. Metal-polluted WDs, which represent $25-50\%$ of the whole WD
population (Zuckerman et al., 2010; Koester et al., 2014), constitute an
observational signature of the accretion of heavy elements. A few per cent of
the WDs are surrounded by planetary debris disks (Manser et al., 2020), and
over sixty WD disks have already been discovered (Zuckerman et al., 2010;
Farihi, 2016). Most of the observed disks are dusty, although about twenty
disks with gaseous components were discovered as well (Dennihy et al., 2020;
Melis et al., 2020; Gentile Fusillo et al., 2020).

The quickly growing population of these disks motivates theoretical
explorations of their structure and evolution, which can reveal both how they
are formed (Veras et al., 2014, 2015; Malamud & Perets, 2020a, b) and how they
accrete onto WDs photospheres, creating observable metal pollution (Jura &
Young, 2014; Hollands et al., 2018; Harrison et al., 2018; Doyle et al., 2019;
Swan et al., 2019; Bonsor et al., 2020). An interesting observed feature of WD
disks is that nearly all the disks are now thought to showcase flux
variability on timescales of weeks to decades (Farihi et al., 2018; Xu et al.,
2018; Swan et al., 2020).

The abundance of objects with different sizes in WD disks – from grain size to
minor planets, reveals rich and interesting dynamics (see a detailed review of
post-main sequence evolution in Veras, 2016), which includes many physical
processes, among them are replenishment and accretion. Frequent collisions in
WD disks (Jura, 2008; Metzger et al., 2012; Kenyon & Bromley, 2017a, 2016, c)
lead to replenishment of grains in the disk – where large objects break into
smaller ones, and gradually are removed by radiation pressure and accretion.
The debris disks might give rise to dynamical excitations and perturbations of
mass that will eventually drive matter onto the WD within the disk lifetime
(Girven et al., 2012; Veras & Heng, 2020). Poynting-Robertson drag results
from the radiation force and causes loss of angular-momentum for small
pebbles, carrying them to the WD (Burns et al., 1979; Rafikov, 2011a, b).

The dynamics and architecture of WD disks are similar to protoplanetary disks
in some aspects, and different in others. While there are some processes that
take place in both of them, such as fragmentation, their parameters might
differ significantly. The scales of WD disks are much smaller, the density and
temperature profiles are different, 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 discussed in the context of sand
dunes (Bagnold, 1941) . Recently, we showed that aeolian erosion can play an
important role in planet formation by setting a new growth-barrier for
pebbles/boulders in protoplanetary disks, and affecting pebble accretion and
streaming instability (Grishin et al., 2020; Rozner et al., 2020). Aeolian
erosion in protoplanetary disks is rapid and efficient, as also verified in
lab experiments and numerical simulations of the conditions of protoplanetary
disks (Paraskov et al., 2006; Demirci et al., 2020b; Demirci & Wurm, 2020;
Demirci et al., 2020a; Schaffer et al., 2020). Aeolian erosion leaves
signatures on the dynamics of objects in the disk, e.g. it fuels pebble-
accretion and induces a redistribution of sizes in the disk, and might
potentially lead to reshaping or complete destruction of objects (Grishin et
al., 2020; Rozner et al., 2020). In contrast with protoplanetary disks, the
temperatures in WD disks are high enough to maintain thermal ablation (Podolak
et al., 1988; Pollack et 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 in WD disks
and interact symbiotically with other dynamical processes in the disk. In
fact, we show that for certain regimes, for objects with radius $\lesssim
10^{3}\rm{cm}$ and at distances $\lesssim 0.6R_{\odot}$ from the WD, in a disk
with $M_{disk}=10^{24}\rm{g}$ for aeolian erosion is the dominant destruction
process. The dominance regime changes with the disk and object’s parameters.
As we will discuss later, the mass of the disk should be $\gtrsim
10^{21}\rm{g}$ 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 small pebbles, Poynting-Robertson drag
becomes efficient and carries the pebbles onto the WD. Hence, aeolian erosion
might set a lower limit on the accretion flux onto the WD and enhance the disk
gas replenishment, by assuring repeatedly regenerating the large abundance of
small objects. Moreover, as we showed in Grishin et al. (2020), aeolian
erosion induces redistribution of the particle sizes and their abundance in
the disk. The size-distribution of objects in WD disks as well as the sizes of
objects which accrete on WDs are not well constrained observationally,
although some theoretical constraints have been established (Kenyon & Bromley,
2017b, c). Aeolian erosion might shed light on in this direction as well.

The paper is organized as follows: in section 2 we briefly review the
parameters space and various models of WD disks. In section 3 we review the
models of aeolian erosion and thermal ablation in WDs disks. In section 4 we
present our results and include eccentric orbits, 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
including collisional cascade, external seeding and further disk generations.
In section 6, we discuss the caveats and limitations of our study. In section
7 we summarize the paper and suggest future directions.

##  2 White Dwarf Disks

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 parameters that will be used in the rest of the
paper.

The total mass of a WD disk ranges between $10^{12}-10^{25}\rm{g}$ (see a
detailed discussion in Metzger et al., 2012; Veras & Heng, 2020 and references
therein). The mass of the gas in the disk, parametrized by the gas-to-dust
ratio, is highly unconstrained and ranges between $10^{-5}$ and unity (Veras,
2016). Observations set the lower limit of the inner-radius to be $\lesssim
0.2R_{\odot}$ (e.g. Rafikov, 2011a) and the outer-radius to be $\gtrsim
1.2R_{\odot}$ (e.g. Gänsicke et al., 2006).

The surface density profile is given as (Metzger et al., 2012),

$\displaystyle\Sigma_{g}(a)=\Sigma_{g,0}\left(\frac{4.72a}{0.6R_{\odot}}\right)^{-\beta}\left(\frac{M_{{\rm
disk}}}{10^{24}\rm{g}}\right),$ (1)

where $a$ is the distance from the center of the disk, $\Sigma_{g,0}$ is the
fiducial surface density and $\beta$ is an arbitrary exponent which is
parametrized as $\beta=n+1/2$, where $n$ describes the viscosity power law
$\nu(a)\propto a^{n}$. The gas density $\rho_{g}$ is determined by the surface
density,

$\displaystyle\rho_{g}=\frac{\Sigma_{g}}{2h(a)},$ (2)

where $h$ is the height of the disk.

This scaling of $\beta$ also induces a temperature scaling (Metzger et al.,
2012),

$\displaystyle T(a)\propto\begin{cases}{\rm constant},\ n=3/2;\\\ a^{-1/2},\
n=1\end{cases}$ (3)

where $n=3/2$ corresponds to an optically thick disk, and $n=1$ corresponds to
an optically thin disk. The aspect ratio of the disk is loosely constrained
and strays over some orders of magnitude. We will use an aspect ratio of
$10^{-2}$ unless stated otherwise. The aforementioned choice of parameters
yields typical values of $\Sigma_{g}=5.1\times 10^{3}g\ cm^{3},\
\rho_{g}=2.55\times 10^{-5}\rm{g\ cm^{-3}},$ and $T=10^{3}\rm{K}$ at
$a=0.6R_{\odot}$.

Unless stated otherwise, we will use $n=3/2$ for all our profiles.

The lifetime of WD disks (at least the ones within a couple of Solar radii
from the WD) ranges between $\sim 10^{4}-10^{6}\ \rm{yr}$ (Girven et al.,
2012; Veras & Heng, 2020). Girven et al. (2012) estimated the age of WD disks
assuming a constant accretion rate to the WD. Recently, Veras & Heng (2020)
introduced a different estimation method, arising from the dynamical processes
that the disk should go through and their typical timescales.

##  3 Aeolian erosion & Thermal Destructive Processes in Disks

###  3.1 Aeolian erosion

Consider a spherical object with radius $R$ that resides at a constant
distance $a$ from the WD and is built from grains with a typical size $d$,
that move in a gaseous medium with a density $\rho_{g}$. The pressure support
present in gaseous disks leads to a difference between the Keplerian velocity
$\Omega_{K}$ around the WD and the actual angular velocity $\Omega_{g}$,
$\Omega_{g}-\Omega_{k}\approx(2\Omega_{K}a\rho_{g})^{-1}\partial P/\partial r$
where $\partial P/\partial r$ is the pressure gradient. Furthermore, the
pressure support leads to the radial drift, which presents one of the
fundamental problems in planet formation in protoplanetary disks – the meter-
size barrier (Weidenschilling, 1977).

The radial velocity in protoplanetary disks obtains a maximum for $\sim
1\rm{m}$ size objects, and these objects inspiral to the inner parts of the
disk on timescales shorter than the expected growth timescale. The drift
velocity is lower for smaller objects – they are better coupled to the gas and
hence have slower relative velocities; larger objects are loosely coupled to
the gas and experience slower drift velocities. For objects in WD disks, the
maximal drift velocity is obtained for smaller objects, in size $\lesssim
1\rm{cm}$ (Kenyon & Bromley, 2017c).

$\displaystyle v_{\rm rel,r}$ $\displaystyle=-\frac{2\eta v_{k}\rm
St}{1+\rm{St}^{2}},$ (4) $\displaystyle v_{\rm rel,\phi}$ $\displaystyle=-\eta
v_{k}\left(\frac{1}{1+\rm{St}^{2}}-1\right)$ (5)

where the Stokes number is defined by

$\displaystyle{\rm St}=\Omega_{K}t_{\rm stop},$ (6) $\displaystyle t_{\rm
stop}=\frac{mv_{\rm rel}}{F_{D}}$ (7)

where $m$ is the mass of the object, $F_{D}$ is the drag force, which is given
by

$\displaystyle\textbf{F}_{D}=\frac{1}{2}C_{D}({\rm Re})\pi R^{2}\rho_{g}v_{\rm
rel}^{2}\hat{\textbf{v}}_{\rm rel}$ (8)

For the drag coefficient, we adopted an emperical fitted formula, based on
experminal data in the regime $10^{-3}\leq Re\leq 10^{5}$ (Perets & Murray-
Clay, 2011 and references therein)

$\displaystyle C_{D}({\rm Re})=\frac{24}{{\rm Re}}(1+0.27{\rm
Re})^{0.43}+0.47\left[1-\exp\left(-0.04{\rm Re}^{0.38}\right)\right]$ (9)

where $Re$ is the Reynolds number, defined by

$\displaystyle{\rm Re}=\frac{4Rv_{{\rm rel}}}{v_{{\rm th}}\lambda},$ (10)

where $v_{\rm{th}}=(8/\pi)^{1/2}c_{s}$ is the mean thermal velocity,
$c_{s}=\sqrt{k_{B}T/\mu}$ is the speed of sound, $k_{B}$ is Boltzmann
constant, $T$ is the temperature of the disk, $\mu$ is the mean molecular
weight, taken to be $2.3m_{H}$ (following Perets & Murray-Clay, 2011) with
$m_{H}$ the mass of a hydrogen atom and $\lambda=\mu/(\rho_{g}\sigma)$ is the
mean free path of the gas.

Figure 1: The relative velocity between objects and the gas, in different
constant distances from the WD. The solid lines are for $v_{\rm rel}$ and the
dashed ones are for the radial component, $v_{\rm r}$.

In fig. 1, we present the relative velocities between the gas and objects for
different sizes. The radial component of the velocity peaks for $\sim
1\rm{cm}$ size objects, and is about $\sim 10^{5}\rm{cm/sec}$. Much smaller
objects are well-coupled to the gas, which leads to smaller velocities, while
much larger objects are weakly-coupled to the gas and hence are not affected
significantly by it. The relative velocity could change for different
parameters of the disk, and as we have mentioned, the parameter space is
currently wide.

Objects in gaseous disks experience gas drag, which depends strongly on their
velocity relative to the gas, $v_{\rm{rel}}$. The gas-drag can play 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 the threshold conditions.

The threshold velocity is dictated by the balance between cohesion forces,
self-gravity and gas-drag, i.e. the headwind should be stronger than the
attraction force between grains in order to initiate aeolian erosion.

The threshold velocity is given by

$\displaystyle
v_{\star}=\sqrt{\frac{A_{N}}{\rho_{g}}\left(\rho_{p}gd+\frac{\gamma}{d}\right)},$
(11)

where $A_{N}=1.23\times 10^{-2}$, and $\gamma=0.165\ \rm{g\ s^{-2}}$ are
determined empirically from Shao & Lu (2000). Both of them are intrinsic
characteristic of the materials that rise from the cohesion forces that hold
the particles together – mostly electrostatic forces and van der Waals forces.
The gravitational acceleration is $g=Gm/R^{2}$. Above the threshold velocity,
i.e. $v_{{\rm rel}}>v_{\star}$, the aeolian erosion rate is given by (Rozner
et al., 2020),

$\displaystyle\frac{dR}{dt}=-\frac{\rho_{g}v_{\rm rel}^{3}}{4\pi
R\rho_{p}a_{\rm coh}}$ (12)

where $a_{{\rm coh}}$ is the cohesion acceleration. The derivation rises from
the work done on the eroded object by the shear pressure; see a detailed
derivation from equations 8 and 9 of Rozner et al. (2020). The derivation is
based on estimation of the typical sweeping rate of grains from the outer
layer and calculating the work done by them.

###  3.2 Thermal Destructive Processes

Along with aeolian erosion, the high temperatures that are usually found in WD
disks might give rise to destructive thermal processes such 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 might lead to phase transitions and then
mass loss. There are two regimes, separated by a critical temperature
$T_{cr}$, in which the latent heat required 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 vaporization removes mass is
dictated by Hertz-Knudsen-Langmuir equation; see a discussion in D’Angelo &
Podolak (2015). In this regime, aeolian erosion significantly dominates for
our choice of parameters, and the timescale for thermal ablation below the
critical temperature is $\gtrsim 95\ \rm{yrs}$ – much longer than typical
aeolian erosion timescale, which enables us to neglect the thermal effect and
focus on the mechanical processes.

Above the critical temperature, the contribution from thermal processes might
add a significant contribution to aeolian erosion and should be added to eq.
12; assuming blackbody emission, the thermal ablation term is given by

$\displaystyle\frac{dR}{dt}\bigg{|}_{{\rm
ablation}}=\frac{1}{L_{s}\rho_{p}}\epsilon_{s}\sigma_{SB}(T_{cr}^{4}-T_{g}^{4}),$
(13)

where $\epsilon_{s}$ is the thermal emissivity of the object ($\epsilon_{s}=1$
for a perfect blackbody), $L_{s}$ is the particle specific vaporization
energy, $T_{cr}$ is the critical temperature – which depends on composition of
the material – and $T_{g}$ is the gas temperature. At high enough
temperatures, close to the WD, small grains sublimate to gas (e.g Metzger et
al., 2012; Shestakova et al., 2019).

##  4 Results

In this section we present the evolution of objects in WD disks due to aeolian
erosion and thermal ablation, starting from a fiducial set of parameters and
then vary them.

Figure 2: The time evolution of objects embedded in a WD disk due to aeolian
erosion. Left: Evolution at a constant distance $a_{0}=0.005\rm{AU}\approx
1R_{\odot}$ from the WD. Right: Different distances from the WD, with a
constant initial size of embedded objects $R_{0}=10^{3}\rm{cm}$.

Aeolian erosion in WD disks is quite efficient, as manifested in Fig. 2. The
timescales for aeolian erosion are extremely short, and for a constant
distance $a=0.005\rm{AU}\approx 1R_{\odot}$ from the WD disk, objects with
radii as large as $\sim 5\times 10^{5}\rm{cm}$ are eroded within the expected
disk lifetime. The chosen fixed distance from the WD, along with the rest of
the parameters that we don’t vary currently, dictates the final size to which
objects are ground down – from $\sim 0.01\rm{cm}$ for $0.001\rm{AU}$ to $\sim
0.18\rm{cm}$ for $0.005\rm{AU}$.

The final size is determined by the initial conditions of the eroded object,
and induces a size re-distribution according to distance from the WD. The
outer parts of the disk are more dilute then the inner ones, and since the
rate of aeolian erosion is proportional to the gas density, as can be seen in
Eq. 12, aeolian erosion is more efficient in the inner part of the disks,
which leads to shorter timescales and smaller final sizes. Aeolian erosion has
a Goldilocks region of sizes in which it attains its maximal efficiency, since
the relative velocity, which plays a significant role in this process, varies
with the coupling of objects to the gas.

###  4.1 Eccentric Orbits

Aeolian erosion depends strongly on the relative velocity (to the third
power), as can be seen from Eq. 12. Hence, its effects on objects on eccentric
orbits may differ substantially from the effects in the circular case, and
lead to stronger more significant erosion due to the higher velocities
involved, that might even be supersonic and under some conditions lead to the
prompt disruption of objects (Demirci et al., 2020a).

The velocity of a planetesimal in an eccentric orbit is given by

$\displaystyle\textbf{v}_{p}=v_{k}\sqrt{2-\frac{r}{a}}\hat{\textbf{v}}_{p}=\sqrt{GM\left(\frac{2}{r}-\frac{1}{a}\right)}\hat{\textbf{v}}_{p}$
(14)

where $M$ is the mass of the WD, $v_{k}$ is the Keplerian velocity, $r$ is the
distance and $a$ is the semimajor axis.

Assuming a circular disk, the gas moves with a velocity

$\displaystyle\textbf{v}_{g}=v_{k}\sqrt{1-\eta}\hat{\textbf{v}}_{g}$ (15)

The magnitude of the relative velocity between the gas and the planetesimal is
given by

$\displaystyle v_{{\rm
rel}}=|\hat{\textbf{v}}_{p}-\hat{\textbf{v}}_{g}|=\sqrt{v_{p}^{2}+v_{g}^{2}-2v_{p}v_{g}{\hat{\textbf{v}}}_{g}\cdot{\hat{\textbf{v}}}_{p}}.$
(16)

For simplicity, we will assume that the evolution of objects experiencing
aeolian erosion is dominated by the maximal relative velocity in the orbit,
when the phase between the gas and the objects is maximal – $\pi/2$, and also
assume that the object is at the pericenter, i.e. $r=r_{p}=a(1-e)$. See Mai et
al. (2020) for more detailed results.

Figure 3: The effect of aeolian erosion on an $10^{3}\rm{cm}$ object embedded
in a WD disk at a distance of $1R_{\odot}$ from the center of the WD. The
relative velocity of the object is considered as the maximal velocity in the
orbit.

Motion in eccentric orbits shortens the timescales of aeolian erosion and the
final size of eroded objects is smaller and could attain $\sim 0.08\,\rm{cm}$
for $e=0.9$.

###  4.2 Multilayer Objects

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 distribution. In the following
we relax homogeneous composition assumption, and consider the effects of
aeolian erosion on inhomogeneous objects. To the best of our knowledge no
previous study considered the internal structure of grains within WD disks,
and we adopt internal size distribution models usually considered for
asteroids.

The Brazil nut effect in asteroids (Matsumura et al., 2014) suggests that in a
mixture of particles, the larger ones tend to end up on the surface of
objects. Therefore, when this process acts, the inner structure of an asteroid
is such that the larger grains are in the outer layers.

Figure 4: The evolution of two-layered differentiated objects (see inset
legend), with an outer layer comprising $20\%$ of the radius and an inner core
of the remaining $80\%$ of the radius, under the effects of aeolian erosion;
all of the layers have the same density of $3.45\rm{g\ cm^{-3}}$. The object
is embedded in a WD disk and resides at a distance of $0.005\rm{AU}$ from the
host WD.  Figure 5: The evolution of triple-layered objects, with three equal
layers, with each layer comprising $1/3$ of the radius, of the same density
$3.45\rm{g\ cm^{-3}}$, but with different size of inner grains that build each
layer (see inset legend). The object is embedded in a WD disk and resides at a
distance of $0.005\rm{AU}$ from the WD.

In Figs. 4, 5, we present how differentiated objects react to aeolian erosion.
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 different sizes of grains impose different
rates of aeolian erosion, the total aeolian erosion timescales change such
that considering larger grains shortens 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.

###  4.3 Thermal Effects

At high temperatures, thermal processes become more significant and might
strengthen the effect of aeolian erosion and give rise to further destruction
and shorten the timescales.

As can be seen in Fig. 6, thermal ablation shortens the timescales in which
objects are ground down to their final scales – here manifested for icy
objects. Rocky objects will require higher disk temperatures for thermal
ablation to be significant, since the critical temperature for rock is
$4000\rm{K}$; due to the large range of possibilities for WD disks in general
– and for temperatures in particular – the temperature in the disk might even
exceed the critical temperature for rocks and other materials, at least for
the hottest youngest WDs. For coated objects, there could be a combined
process of ablating the outer icy layer and then mechanically eroding inner
layers.

Figure 6: The relationship between aeolian erosion and ablation for an icy
object with initial size of $10^{3}\rm{cm}$, embedded at a constant distance
of $0.005\rm{AU}$ from the WD. Solid lines describes aeolian erosion only, and
dashed lines describe the combined effect of aeolian erosion and thermal
ablation (above the threshold temperature for ice).

##  5 Discussion & Implications

###  5.1 Parameters dependence

The possible parameter range for WD disks is wide and enables us to study a
multitude of combinations of parameters. Here we will present a parameter
space exploration for solid bodies embedded in WD disks and subjected to the
effects of aeolian erosion.

Figure 7: The dependence on disk mass for an object with initial size of
$10^{3}\rm{cm}$, and constant distance from the center of $0.005\rm{AU}$.

In Fig. 7 we examine the dependence of aeolian erosion on the disk 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. Since the the aeolian erosion rate
is proportional to the gas density, the rate becomes stronger for larger disk
masses.

Figure 8: The dependence on aeolian erosion on the aspect ratio of the disk,
$h/r=c_{s}^{2}/{r\Omega}$ where $\Omega$ is the angular velocity, for objects
of initial size $10^{3}\rm{cm}$, embedded at the disk in a distance of
$0.005\rm{AU}$.

In Fig. 8, we present the dependence of aeolian erosion on the aspect ratio.
Higher aspect ratios lead to stronger aeolian erosion. And in our default
choice of parameters, the aspect ratio should be $\gtrsim 10^{-3}$ in order to
maintain significant aeolian erosion.

###  5.2 Symbiotic Relations with Other Processes in The Disk

####  5.2.1 Collisional Cascade

Fragmentational collisional erosion is another destructive process that
gradually diminishes the mass of objects in WD disks. This destructive process
generates a collisional cascade which grinds down $10^{5}-10^{7}\rm{cm}$
objects to $10^{-6}\rm{cm}$ objects within $10^{2}-10^{6}\rm{years}$ (e.g.
Kenyon & Bromley, 2017b, c). Collisional cascades and aeolian erosion both
cause objects to lose mass, such that in the presence of gas, these two
processes are symbiotic.

The aeolian erosion timescale is given by

$\displaystyle t_{\rm erosion}=\frac{R}{|\dot{R}|}=\frac{4\pi
R^{2}\rho_{p}a_{\rm coh}}{\rho_{g}v_{\rm rel}^{3}}.$ (17)

The fragmentation timescale is estimated from the collisional timescale. The
collision timescale for a mono-disperse swarm is given by (Kenyon & Bromley,
2017a, b, c and references therein)

$\displaystyle t_{0}=\frac{r_{0}\rho P}{12\pi\Sigma_{g}},$ (18)

where $r_{0}$ is the radius of all the objects in the swarm, $\rho_{p}$ is
their density, $P$ is their orbital period and $\Sigma_{0}$ is the initial
surface density. When a multi-disperse swarm is considered, the modification
of the timescale is parametrized by a collision parameter
$\alpha_{c}\propto(v^{2}/Q_{D}^{*})^{-1}$, where $v$ is the collision velocity
and $Q_{D}^{*}$ is the binding energy of the object (see Leinhardt & Stewart,
2012 and references therein), such that the timescale from multi-species is
given by $t_{c}=\alpha t_{0}$, which can be shorter compared with the mono-
disperse collision timescales. When the replenishment is efficient enough, the
background distribution of objects remains roughly constant. However,
generally, the background density changes with time as well, adding
complications to the analysis.

The ratio between the timescales of aeolian erosion and fragmentational
collisions is given by

$\displaystyle\frac{t_{{\rm erosion}}}{t_{{\rm
fragment}}}=\frac{96\pi^{2}ha_{{\rm coh}}}{\alpha_{c}Pv_{{\rm rel}}^{3}}R$
(19)

such that the timescales are comparable for a ratio around unity. From
equating the ratio to unity, one can set the transition radius between the
aeolian erosion dominated regime and the collisional fragmantation dominaneted
regime, which depends on the rest of the parameters.

Figure 9: The typical timescales of aeolian erosion (solid lines) and
fragmentational collision (dashed lines) as a function of the destructed
object’s radius, at different constant distances from the WD. $R_{{\rm max}}$
is the radius of the largest object in the swarm.

In Fig. 9, we compare the typical timescales of aeolian erosion and
fragmentational collisions at which objects are destroyed. For the
fragmentation collision timescales, we consider a swarm with a maximal size
given by $R_{{\rm max}}$, and assume for simplicity that the fragmentation of
intermediate objects (i.e., not the largest or the smallest object in the
swarm) starts just when they become the largest of the swarm, which is
justified by the hierarchical character of the fragmentational cascade. For
the background distribution, we assume a power law distribution of $N(r)\sim
r^{-3.5}$ (Kenyon & Bromley, 2017a), and a value of $v^{2}/Q_{D}^{*}$ that
yields $\alpha_{c}\sim 10^{2}$. Aeolian erosion dominates at small sizes, and
fragmentational collisions at larger sizes. The transition point between the
regimes varies with the distance from the WD, such that the regime of aeolian
erosion grows with smaller distances. The aeolian erosion timescales are very
short in these regimes, but might be comparable to or longer than the orbital
period timescales of the objects.

We would like to stress that although the destructive character of both
aeolian erosion and collisional cascades leads to shrinkage of objects in the
disks, they differ by their intrinsic physical mechanisms. Aeolian erosion
originates from shear pressure that is induced by gas drag. Hence, the
presence of gas in the disk is a necessary condition to initiate this process.
The collisional cascade arises from collisions between particles in the disk.

####  5.2.2 External Seeding of Objects into The Disk

The abundance of large objects in WD disks might be replenished as a result of
seeding (Grishin et al., 2019; Grishin & Veras, 2019). Exo-planetesimals from
external sources could enrich the abundance of objects in the disk that will
eventually be eroded. External capture especially contributes to the number of
large objects, although the total captured mass is small compared to the dust
initial abundance. Furthermore, seeding could bring into the disk new
materials that might eventually end as pollutants on the WD. The capture rate
is dictated by the supply rate and the capture probability; both vary with the
origin and the size of the captured objects.

The captured objects change the size distribution in the disk, and might
contribute to the steady-state distribution. The injection of external objects
might be at a sufficient rate to cancel out the destructive processes and
maintain a distribution with larger objects than expected. The revised size
distribution should be derived from a full swarm simulation which includes
collisional cascade, aeolian erosion, growth and seeding.

####  5.2.3 Further Disk Generations

The parameters and characteristics of the disk might vary from one generation
to another. Wide-binary evolution could lead to an evolved donor that
transfers mass to its companion, such that the captured material forms a disk
(Perets, 2010; Perets & Kenyon, 2013; Schleicher & Dreizler, 2014; van
Lieshout et al., 2018). A binary main-sequence system evolves such that the
more massive star sheds material which is accreted on the secondary and forms
a protoplanetary disk; the mass that is lost from the system, which is much
greater than the mass transferred from one star to the other, leads to
expansion of the binary orbit (Veras et al., 2011; Kratter & Perets, 2012;
Veras & Tout, 2012). Afterwards, a second generation of debris and planets
form in the pre-evolved system, such that the secondary evolves off the main-
sequence and sheds material to its WD companion and a protoplanetary 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 composition, since
metallicity varies from one generation to another, such that the formed
accretion disks are metal and dust rich. Higher metallicity environments are
better for planet formation (Fischer & Valenti, 2005), such that the formed
disks are likely to have planetary and planetesimal structures.

###  5.3 Possibilities for Wind Erosion of SDSS 1288+1040

One notable potential application of aeolian erosion theory is the disk
orbiting the WD SDSS J1228+1040 (Manser et al., 2019). This disk contains a
planetesimal on an orbit with $a=0.73R_{\odot}$ and $e\approx 0.54$. This
orbit is fully embedded in a dusty and gaseous planetary debris disc, which
extends beyond $1R_{\odot}$.

Because of this compact planetesimal orbit, and because the planetesimal has
been observed over at least 4,000 orbits, it cannot represent a typical
rubble-pile solar system asteroid. Instead, it likely represents an iron-rich
remnant core of a planet, and harbours nonzero internal tensile strength. The
internal structure and size are poorly constrained: the estimated radius range
is $R=1.2-120\ \rm{km}$.

Hence, for only the lowest end of this size range could aeolian erosion could
be effectual. Figure 2 illustrates that km-sized objects may be eroded in
timescales under $\sim 10^{6}$ yr. However, the structures of the objects in
that figure are likely to be very different than the strong and dense
planetesimal around SDSS 1228+1040. Furthermore, a potentially competing
effect are bodily gravitational tides. Veras et al. (2019) show that these
tides could instigate a WD to engulf objects containing just
$10^{-3}M_{\oplus}$, but only for objects with sufficiently low internal
viscosities.

##  6 Caveats

Aeolian erosion is a very effective process in WD disks, as long as the amount
of gas in the disk is non-negligible. The current observations of WD disks
with a gaseous component cannot yet well constrain several 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 understood, with suggested
origins include sublimation (e.g. Metzger et al., 2012), grain-grain
collisional vaporization and sputtering. Recently, Malamud, Grishin, &
Brouwers (2020) presented another channel of gas production in white dwarf
disks, via interactions between an eccentric tidal stream and a pre-existing
dusty compact disk.

Along with the destructive processes, there could be mass influx that we did
not take into consideration in this paper. As manifested in Kenyon & Bromley
(2016, 2017c) in the context of collisional cascade, the rate of mass input
might equalize the mass loss such that objects that are a-priori expected to
be pulverized and will maintain their mass for long timescales. However, here
the combined effect of fragmantational erosion and aeolian erosion might play
a role, and these two processes together will lead to mass depletion of
objects.

We have neglected any asphericity in the object, which is assured due to the
lack of perfect packing efficiency of its constituent grains. Furthermore, we
have neglected any asphericity that develops as a result of an aeolian
erosion, which acts in the direction of the headwind, and might reshape the
eroded objects.

We focus on aggregates, with a weak outer layer. Some of the objects that
could be potentially affected from aeolian erosion are destroyed already in
tidal shredding or collisions. The cohesion forces hold for loosly bound
objects, i.e. they describe aggregates and other forms of cohesion laws should
be taken into account in case of different internal physics. Once the cohesion
law is dictated, the suggested prescription of aeolian erosion will be very
similar to the one we sketched in this paper.

For aeolian erosion to be effective, the dominant stripping force on the weak
outer layers of the aggregates would need to be erosive rather than tidal. The
strength of the tidal force can vary significantly depending on physical
properties; for large homogeneous rubble-piles, this value can vary by a
factor of about $2$ (around $1R_{\odot}$) depending on spin and fluidity, and
can cause stripping on an intermittent, yearly timescale (Veras et al., 2017).
Such intermittency perhaps suggests that 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 the asteroid spin
barrier, which refers to the minimum spin rate at which an asteroid breaks
itself apart. This barrier is well-established at about $2.2\rm{hrs}$ in the
gravity-dominated regime, for spherical rubble piles larger than about
$300\rm{m}$ (Pravec, Harris, & Michalowski, 2002; Hu et al., 2020). However,
because this spin limit is a general function of both the asphericity of the
object and its internal cohesion (see the Appendix of Veras, McDonald, &
Makarov, 2020), incorporating this limit into our modeling would not be
trivial. Our aggregates change shape through time, and spin variations are not
necessarily monotonic.

##  7 Summary

The growing number of WD disks (both gaseous and non-gaseous) that have been
observationally detected and characterized leave open the possibility for
constraining theoretical models for the origins and evolution of such disks.
However, even at the theoretical level, there are important gaps in our
physical understanding of the dynamics and 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 made use of an analytical
model for aeolian erosion in WD disks, based on our studies of such processes
in protoplanetary disks as presented in Rozner et al. (2020). We find that the
typical timescales of aeolian erosion in WD disks are extremely short, with
aeolian erosion grinding down even km-size objects within the disk lifetime.
Consequently, such processes are likely to play an important role in the
evolution of small solid bodies in the disk. We also studied the relationship
between aeolian erosion and other physical processes in WD disks and its
amplification due to the combined effect (see subsections 3.2,5.2). Along with
collisional cascade and thermal ablation, aeolian erosion grinds down
efficiently large objects into small ones with a characteristic final size.
The eroded objects experience dynamical processes that finally grind down
planetesimals/rocks/pebbles/boulders into sufficiently small particles such
that these could drift towards the WD via Poynting-Robertson drag and
contribute to its pollution.

Aeolian erosion is the most efficient and becomes the dominant destruction
process for small objects, and the critical radius for its dominance is
determined by the parameters of the disk, the physical characteristics of the
eroded objects, the distance from the WD and the parameters of the collisional
cascade (see eq. 19).

Similarly to protoplanetary disks, aeolian erosion in WD disks induces a re-
distribution of particles size, according to the distance of the particles
from the WD. Hence, aeolian erosion sets constraints on the parameters of WD
disks that might narrow down the current parameter space. Due to the extremely
short timescales of aeolian erosion, it is not likely that observable
variations in WD disks would be explained by replenishment from aeolian
erosion.

## Acknowledgments

We thank the anonymous reviewer for useful comments which have improved the
manuscript. MR and HBP acknowledge support from the Minerva center for life
under extreme planetary conditions, the Lower Saxony-Israel Niedersachsisches
Vorab reseach cooperation fund, and the European Union’s Horizon 2020 research
and innovation program under grant agree- ment No 865932-ERC-SNeX. DV
gratefully acknowledges the support of the STFC via an Ernest Rutherford
Fellowship (grant ST/P003850/1).

## Data Availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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##  Appendix A Table of commonly used parameters

In this appendix, we present the default values for the used parameters,
unless stated otherwise.

Symbol | Definition | Fiducial Value | Reference
---|---|---|---
$\gamma$ |  | $0.165\rm{g/sec^{2}}$ | Kruss et al. (2019)
$A_{N}$ |  | $1.23\times 10^{-2}$ | Shao & Lu (2000)
$\beta$ |  | $10^{2}\rm{g\ s^{-1}}$ | scaled from Paraskov et al. (2006) and refs. therein
$\rho_{p}$ | planetesimals’ density | rock $3.45\rm{g}\ \rm{cm}^{3}$, ice $1.4\rm{g}\ \rm{cm}^{3}$ | Pollack et al. (1996)
$\mu$ | mean molecular weight | $3.85\times 10^{-24}g$ | Perets & Murray-Clay (2011)
$\sigma$ | neutral collision cross-section | $10^{-15}\rm{cm}$ | Perets & Murray-Clay (2011)
$T$ | temperature (optically thick) | $1000\rm{K}$ |
$M_{disk}$ | disk mass | $10^{24}\rm{g}$ |
$\Sigma_{g}$ | surface density profile (optically thick) | $5.1\times 10^{3}\rm{g\ cm^{2}}$ | Grishin & Veras (2019)
$h/r$ | aspect ratio | $10^{-2}$ |
$d$ | typical ’building-block’ grain size | $0.1\rm{cm}$ |
$T_{cr}$ | critical temperature | ice $648\rm{K}$, rock $4000\rm{K}$ | Podolak et al. (1988)
$L_{s}$ | particle specific vaporization energy (solid) | ice $2.83\times 10^{10}\rm{erg\ g^{-1}}$, rock $8.08\times 10^{10}\rm{erg\ g^{-1}}$ | D’Angelo & Podolak (2015)
| Size dist. in the disk (fragmentation only) | $N(r)\propto r^{-3.5}$ | Kenyon & Bromley (2017b)