# Circularization of tidal debris around white dwarfs: implications for gas
production and dust variability

Uri Malamud,1 Evgeni Grishin2 and Marc Brouwers3
1School of the Environment and Earth Sciences, Tel Aviv University, Ramat
Aviv, 6997801 Tel Aviv, Israel
2Department of Physics, Technion - Israel Institute of Technology, Technion
City, 3200003 Haifa, Israel
3Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge
CB3 0HA
E-mail<EMAIL_ADDRESS>
(Accepted for publication in MNRAS, 14/12/2020.)

###### Abstract

White dwarf (WD) pollution is thought to arise from the tidal disruption of
planetary bodies. The initial fragment stream is extremely eccentric, while
observational evidence suggest that discs are circular or nearly so. Here we
propose a novel mechanism to bridge this gap and show that the fragments can
rapidly circularise through dust or gas drag when they interact with a pre-
existing compact disc. We assume that the tidal stream mainly consists of
small cohesive fragments in the size range 10-1000 m, capable of resisting the
WD tidal forces, whereas the compact discs span a wide mass range. We provide
an analytical model, accompanied by N-body simulations, and find a large
parameter space in fragment sizes and orbital separation that leads to full
circularization. Partial circularization is possible for compact discs that
are several orders of magnitudes less massive. We show that dust-induced
circularization inherently produces gas as tidal fragments collisionally
vaporize the pre-existing dust along their path. We show that ongoing gas
production has a higher probability to occur during the early stages of tidal
disruption events, resulting from the fact that smaller fragments are the
first to circularize. Intermittent gas production however becomes more likely
as the tidal stream matures. This could explain why only a small subset of
systems with dusty compact discs also have an observed gaseous component.
Additionally, the interaction yields fragment erosion by collisional
shattering, sputtering, sublimation and possibly ram-pressure. Material
scattered by the collisions might form a thin dusty halo that evolves through
PR drag, in compatibility with observed infrared variability.

###### keywords:

white dwarfs – planet–disc interactions

††pubyear: 2020††pagerange: Circularization of tidal debris around white
dwarfs: implications for gas production and dust variability–D

##  1 Introduction

Between 25-50% of white dwarf (WD) atmospheres are polluted with heavy
elements (Zuckerman et al., 2003, 2010; Koester et al., 2014), originating
from accretion of planetary material (Debes & Sigurdsson, 2002; Jura, 2003;
Kilic et al., 2006; Jura, 2008). The details and especially the sequence of
events leading to such an acceretion are however still not fully understood.

Tens of WD systems are known to host debris, of either dust or gas (or both),
which are in close vicinity to the WD, extending to $10^{1}-10^{2}$ WD radii
(Farihi, 2016). In terms of the WD tidal radius, observed debris are thought
to be near or within the WD Roche limit, which is approximately equal to the
progenitor star’s main-sequence physical radius (Bear & Soker, 2015). The
fiducial Roche value is $R_{\odot}$, the Sun’s present-day radius. The
presence of orbiting dust is deduced from measurements of infrared excess
(Rocchetto et al., 2015), while gas is inferred from metal emission lines or
absorption features (see recent review by Manser et al., 2020). The origin of
material at such close proximity to the WD is clearly not primordial (Graham
et al., 1990). During the post main-sequence evolution of the progenitor star
it greatly expands and could swallow any pre-exiting planetary material up to
a distance of a few AU (Mustill & Villaver, 2012). Hence, the region that has
been cleared prior to the WD stellar phase is many orders of magnitude larger
than the domain of typically observed debris.

The debris are instead thought to originate from planetary bodies which are
perturbed by various potential mechanisms (Debes & Sigurdsson, 2002; Bonsor et
al., 2011; Debes et al., 2012; Kratter & Perets, 2012; Perets & Kratter, 2012;
Shappee & Thompson, 2013; Michaely & Perets, 2014; Veras & Gänsicke, 2015;
Stone et al., 2015; Hamers & Portegies Zwart, 2016; Veras, 2016; Payne et al.,
2016; Caiazzo & Heyl, 2017; Payne et al., 2017; Petrovich & Muñoz, 2017;
Stephan et al., 2017; Smallwood et al., 2018) to highly eccentric (and tidal-
crossing) orbits, and are therefore subsequently tidally disrupted to form an
initial eccentric circumstellar tidal stream of planetary fragments.

During this initial formation phase, the morphology of the resulting tidal
stream depends on the properties of the object being tidally disrupted. An
asteroid would tidally disrupt to form a narrow ring of material, with very
little spread in the orbital energy of its constituent fragments (Veras et
al., 2014; Malamud & Perets, 2020a; Nixon et al., 2020), unless it originates
very far from the WD (as in Kuiper belt objects or Oort cloud objects) in
which case the relative energy spread compared to the original orbital energy
could be much greater. Larger objects that are the size of dwarf planets,
moons or even terrestrial planets, would typically form a much more dispersed
debris disc; i.e, the constituent fragments would have a large spread in
orbital energy. The disc would be comprised of tightly bound interlaced
elliptic eccentric annuli with semi-major axes extending from as little as
$\sim$0.05 AU (eccentricities ranging from at least 0.9) to well beyond the
original planet orbit (Malamud & Perets, 2020a, b). While these scenarios
result in completely different debris discs, the outcomes nevertheless share a
common morphological characteristic – high eccentricity (eccentricities
ranging from at least 0.9 and typically $\sim$ 1).

Taking a leap forward in time, the studies of Kenyon & Bromley (2017a, b)
focus on the final formation sequence of the disc, when it is assumed that the
disc attains a much more compact state (with maximum eccentricity of order
$\sim 0.01$). Here, collisional grinding of large particles rapidly pulverize
them to mere dust and (sublimated) gas. Such discs are circular or nearly so.
They are relatively quiescent and are typically regarded as the canonical
discs one often discusses in the context of observed WD discs with infrared
excess (Jura, 2003). In order to avoid confusion with evolving eccentric
discs, throughout the paper we shall refer to this compact canonical disc
configuration simply as ’compact discs’.

A recent study explores the possibility that highly eccentric, ring-like tidal
debris, are also compatible with infrared excess evidence instead of compact
discs (Nixon et al., 2020). Earlier work by Dennihy et al. (2016) also shows
that it is possible to relax the typical assumption of a completely circular
dusty compact disc, and that elliptical discs present an alternative
interpretation. However, they acknowledge that the absence of ample long
wavelength data makes it quantitatively difficult to draw important
distinctions which discern between various high eccentricity models.
Furthermore, both studies lack concrete quantitative modeling (while Nixon et
al. (2020) provide a qualitative argument) accounting for the moderate
eccentricites observed in a variety of discs (see Section 2.1 for details). We
find it more likely that the extremely eccentric configurations proposed in
Malamud & Perets (2020b) (hereafter MP20) or (Nixon et al., 2020) simply
reflect an early or intermediate stage in the formation process of the compact
discs that are normally associated with infrared excess.

We are thus _missing_ an important link in between the initial formation stage
and the compact disc stage. How do discs evolve from a state of such extreme
initial eccentricity? Veras et al. (2015) consider disc shrinkage through the
drifting of small micron-to-cm sized particles by Poynting-Robertson (PR)
drag, however it is unclear that this particle size range constitutes a
significant mass fraction of the initial disc. Most studies actually advocate
tidal fragments in the approximate size range of 10-1000m (Brown et al., 2017;
Kenyon & Bromley, 2017a; Rafikov, 2018; Malamud & Perets, 2020a). For this
size range, various arguments in Malamud & Perets (2020a) and particularly in
their Appendix A emphasize the potential importance of the Yarkovsky effect,
which could be an important agent for circularizing the disc, however it
remains to be properly explored in this context (Veras et al., 2015).
Alternatively, collisional cascade might potentially break the fragments on
long evolutionary timescales, such that a more significant fraction of the
disc could then evolve to circularize through PR drag (Wyatt et al., 2011).

In this manuscript we now suggest a third alternative, in which orbital
dissipation and circularization of tidally formed fragments is facilitated
through interaction with a pre-existing dusty or gaseous compact disc (or
having both components). Our goal is to stipulate the conditions for this
mechanism plays a significant role in the process of shrinking the disc
(eccentricity) and in changing its fundamental properties. Few previous
studies have began to explore the role of gas as a possible circularizing
agent (Grishin & Veras, 2019; O’Connor & Lai, 2020), however in this paper we
greatly extend the discussion and focus primarily on pre-existing dust, since
it is observationally more prevalent than gas. It is estimated that between 1%
and 3% of all single WDs with cooling ages less than around 0.5 Gyr possess
circumstellar dust (Farihi et al., 2009; Rebassa-Mansergas et al., 2019),
while the occurrence rate for dust discs to have observable gaseous component
(in emission) is only a few percent (Manser et al., 2020). However, several
newly-available pre-prints (Dennihy et al., 2020; Gentile Fusillo et al.,
2020; Melis et al., 2020) may have increased this fraction to about a third or
more. Gaseous components are nevertheless rarer. The relative rarity of
detectable gas may simply reflect an earlier stage in the circularization
process.

In what follows, we will not only show that dust-assisted circularization is
likely, but also that gas production is an inescapable by-product of the
interaction between an eccentric tidal stream and a pre-existing dusty compact
disc. In turn, we will show that detectable gas components are rarer since
they correlate with (more probable to be observed in) the initial stages of
disc circularization and evolution, when the tidal stream is still abundant
with small fragments and gas production is continuous rather than
intermittent. Finally we will show that if material is deposited far enough
from the WD it can form an outer dust halo, compatible with observations of
infrared variability seen in dusty WD discs.

Recently, Grishin & Veras (2019) (hereafter GV19) proposed an analytical
formalism suitable for calculating gas-assisted circularization, based on an
earlier work by Grishin et al. (2019). They suggest that small exo-Kuiper or
exo-Oort like objects in the 0.1-10 km size range may be captured by a gaseous
disc in the vicinity of a WD. Here we use a similar formalism but now
investigate the evolution of realistic debris disc configurations, by using
accurate initial conditions that emerge in the theoretical study of MP20, in
addition to more realistic fragment sizes than in the GV19 study. More
importantly, we present an extension of the GV19 formalism to include dust-
assisted circularization in addition to gas-assisted circularization, as
observational constraints dictate that the former should be the more central
circularization agent. Our model assumptions and initial conditions are
detailed in Section 2. Our analytical model and additional numerical
simulations are described in Section 3. The results are presented in Section 4
and discussed in Section 5.

Our model of course implies that a pre-existing compact disc is present,
generated in the past by different mechanisms. The discussion elaborates on
various formation possibilities, as well as the mass ratio between the tidal
stream and the compact disc, which determines the outcomes of partial versus
full circularization. A summary of the paper is provided in Section 6.

##  2 Assumptions and initial conditions

We consider four vital elements of the model: (i) the mass and spatial domain
of the dust/gas components in a pre-existing compact disc ; (ii) the size of
interacting eccentric fragments; (iii) the orbit of eccentric fragments; and
(iv) the point where fragments intersect the compact disc. They are
respectively discussed in the following subsections.

###  2.1 Disc masses and spatial extent

Dust discs are commonly characterized by a typical infrared excess that can be
explained via circumstellar dust at a temperature of around 1000 K. In the
standard technique for modelling infrared excess (Jura, 2003) it is assumed
that the WD illuminates a passive, opaque, flat circumstellar annulus where
the incident optical energy is re-radiated in the infrared. This yields a
typical declining disc temperature as a function of radial distance. In turn
this constrains the flux, and when compared with the observations, the inner
and outer radii of the annulus can be extracted. There is a considerable
degeneracy between the disc inclination and the disc radial extent. For
example in Dennihy et al. (2016) the disc width of WD EC 05365 can be modelled
with varying inner and outer radii in the range of approximately $\sim
0.4-0.9R_{\odot}$ (see their table 3). In some cases the inclination can be
constrained. E.g. for a transiting system as in WD 1145+017 (Vanderburg et
al., 2015), the inclination must be high (most observational studies define
$i=90^{\circ}$ for when the disc is roughly edge-on to account for the
transit, whereas we consider $i=0$ in this case) and Zhou et al. (2016)
constrain the dust position from infrared excess to $\sim 0.9R_{\odot}$. The
inclination may also be constrained in discs that have both dust and emitting
gas (Gänsicke et al., 2006; Manser et al., 2016b). For a given inclination of
70∘, Brinkworth et al. (2009) constrain the dust disc of SDSS J1228+1040 to
range between 0.2-1.2 $R_{\odot}$.

While up to half of all WDs have atmospheres polluted with heavy elements
(Koester et al., 2014), only a small fraction of order few % are observed to
possess circumstellar dust (Farihi et al., 2009; Rebassa-Mansergas et al.,
2019). Unless most WDs accrete material directly, which is unlikely (Debes &
Sigurdsson, 2002), accretion has to be mediated by a disc, suggesting that
most discs simply evade detection. Indeed Rocchetto et al. (2015) study the
distribution of infrared fractional luminosities as a function of the WD
cooling age and show that they are significantly less than the maximum value
allowed for flat discs. In other words, these discs seem to be less extended
compared with their allowed theoretical range, and for WDs with cooling age
under $\sim$ 0.5 Gyr there is no increase in the fractional luminosities as
the WDs mature, despite the expected inward migration of the inner sublimation
zone. Evading infrared detection therefore must involve insufficient surface
area of the emitting dust grains. Rocchetto et al. (2015) suggest three
potential compact disc configurations that could escape detection: (1) high
fraction of gas / completely gaseous discs (Jura, 2008) ; (2) narrow (in
radial extent) opaque dust rings (Farihi et al., 2010); (3) optically thin
dust discs (Bochkarev & Rafikov, 2011). Similar ideas were proposed by Bonsor
et al. (2017). We will return to these points later on in Section 5.6.

This last paragraph importantly indicates that the classical view of a pre-
existing dust disc as a Saturn-like, geometrically thin and optically thick
disc is far from certainty. Rather, at least in WDs with cooling age less than
0.5 Gyr, the majority of discs _do not_ conform with the canonical compact
disc configuration, as evident by the small fraction of IR excess detections,
despite the favourable observational bias in this class of WDs. Nevertheless,
in the context of our paper we will assume that pre-existing dust discs
initially adhere to the classical view, but we will also discuss how their
structure and thus detectability can change in the aftermath of a tidal
disruption event. Our assumed inner and outer disc radii for pre-existing dust
are 0.2 and 1 $R_{\odot}$ respectively, in accordance with the inferred
observational constraints previously discussed.

The first clue for uncovering disc masses (both gas and dust) comes from the
average WD accretion rates. The latter can be estimated via the WD observed
photospheric metal abundances. Using theoretical estimates for the size of the
outer convective zone of the WD, the amount of polluting mass may be derived
from the aforementioned metal abundances, and divided by the theoretical
estimates for the heavy elements sinking times to yield the average accretion
rate (see Koester (2009) and references therein for both the former and the
latter). The ensuing average accretion rates fall within a relatively wide
range of $10^{5}-10^{11}$ g $\times$ s-1 (Farihi et al., 2009; Farihi et al.,
2010; Koester et al., 2014). As a heuristic approach, let us assume a constant
accretion rate throughout the lifetime of the disc, which is empirically
estimated by Girven et al. (2012) as $\sim 10^{4}-10^{6}$ yr. The total disc
mass $M$ then falls within an even wider range of $10^{16}-10^{24}$ g. The
upper limit estimate of $10^{24}$ g is agreeable with the maximal mass
observed in He-dominated WDs (Farihi et al., 2010; Xu & Jura, 2012; Girven et
al., 2012). We consider this mass range appropriate for dust discs. In the
following paragraphs we will also discuss gas, its radial extent and possible
mass range.

Gas discs are typically found in combination with observed infrared excess,
i.e in dust discs that have a gaseous component. They are extremely rare with
only 12 known WD discs that contain a gaseous component (Manser et al., 2020),
hence pre-existing gas discs are not central to the ideas presented in this
paper, but we nevertheless thoroughly discuss them for completion and since
they help us constrain the spatial extent of compact discs. In 7 of the 12
aforementioned discs the presence of gas has been detected via the Doppler-
broadened, double-peaked line emission of the Ca II 8600 Å triplet, which
results from the Keplerian rotation of a flat disc that is photo-ionised by
the WD. In the other five discs the gas was detected via absorption features.
For the moment, the underlying differences between gas emission and absorption
features in these systems is not well understood (Manser et al., 2020). Only
one WD (SDSS 1228+1040) stands out in having both emission and absorption
(Gänsicke et al., 2012). To contrast, WD 1145+017 has a similarly highly
inclined (i.e edge-on) disc and has been heavily monitored (see Cauley et al.
(2018); Rappaport et al. (2018); Izquierdo et al. (2018); Karjalainen et al.
(2019); Xu et al. (2019); Fortin-Archambault et al. (2020) and numerous older
references therein), yet it does not appear to have emitting gas. The
situation is similar with SDSS J1043+0855 (Manser et al., 2016b; Melis &
Dufour, 2017). The low statistics involved currently make it difficult to
speculate on the root cause that differentiates between these systems. We also
note that in the time interval between the submission and revision of this
manuscript, several pre-prints (Dennihy et al., 2020; Gentile Fusillo et al.,
2020; Melis et al., 2020) reported (in total) 15 new WD discs with gaseous
emission, tripling the number of previously known discs with gas. We mention
these studies for completion.

Some spatial constraints on the shape and radial extent of the gas may be
obtained by modelling the gas emission. Significant asymmetry in the line
profiles suggests that the gas component has non-negligible eccentricity,
ranging between $\sim 0.2-0.4$ (Gänsicke et al., 2006; Gänsicke et al., 2008;
Melis et al., 2010). If modelled via a series of co-aligned elliptical orbits
of identical eccentricity, as aforementioned, the radial distribution of the
gas may be constrained. Correcting the velocities by accounting for the dust
disc best-fit inclination angles, previous estimates have placed the gas at a
distance of around $\sim$0.15-1.2 $R_{\odot}$ (Gänsicke et al., 2006; Gänsicke
et al., 2008; Melis et al., 2010).

Constraints on the spatial extent of the circumstellar gas from absorption
features have so far only been modelled for one system, that of WD 1145+017
(Fortin-Archambault et al., 2020). Despite several approximations used in the
construction of this model, it places the gas at a compatible distance of
$\sim$0.2-0.5 $R_{\odot}$.

Finally, over long periods, looking at the time evolution of the gas emission
and line morphology shows smooth progression from a red-dominated asymmetry to
a blue-dominated one. The morphology of the peaks also changes, with the blue-
shifted peak becoming stronger and sharper, while the redshifted peak becomes
shallower and weaker and extends to higher velocities. Overall this evolution
is commensurate with relativistic apsidal precession of an eccentric gas
component in the gravitational potential of the WD (Manser et al., 2016a;
Cauley et al., 2018; Fortin-Archambault et al., 2020). This places interesting
constraints for the precession periods, ranging from 4.6 yr for WD 1145+017
(Cauley et al., 2018; Fortin-Archambault et al., 2020) to 27 yr for SDSS
J1228+1040 (Manser et al., 2016a). In the latter study, Manser et al. (2016a)
find that precession periods of 1.54, 27.8 and 134 yr correspond to orbits
with semi-major axes of 0.2, 0.64 and 1.2 $R_{\odot}$, respectively (adopting
a WD mass of 0.7 $M_{\odot}$, and in the limit of small eccentricity). These
estimates are in approximate agreement with previously mentioned gas regions
inferred through other techniques.

Taken at face value, the aforementioned constraints imply that when gas is
observed, it inhabits the inner portions of WD compact discs, and may extend
approximately up to $R_{\odot}$. As already mentioned, this line of evidence
is also very compatible with the radial extent evident in dust discs. In both
cases $R_{\odot}$ is suggested as an approximate upper limit, in excellent
agreement with the typical WD Roche limit. We will take the most conservative
approach, allowing the gas, if it exists, to extend only to $\sim 1R_{\odot}$,
which is similar to the approach in the study of van Lieshout et al. (2018).
For the gas inner edge we choose 0.2 $R_{\odot}$, based on the aforementioned
observational constraints.

The mass of the gas in the debris is even more uncertain from observations.
The mass ratio of gas-to-dust in the disc is essentially unconstrained.
According to Veras (2016) it ranges from $10^{-5}$ to unity. Given the
aforementioned dust disc mass range of $10^{16}-10^{24}$ g, and adopting
$10^{-5}$ as the lowest possible (most limiting) gas-to-dust mass ratio, the
gass mass ranges from $10^{11}-10^{24}$ g. We nevertheless note in the
interest of accuracy, that purely gaseous discs of masses $10^{23}-10^{24}$,
in the upper range suggested above, are not likely to apply to polluted WDs.
This mass is rather typical of cataclysmic variable discs, and requires an
input mass accretion rate of at least $10^{13}$ g$\times$s-1 (Figure 3 in
Hameury (2020)), which is both incompatible with WD disc accretion rates, as
noted previously, and should also appear as a bright X-ray source, contrasting
with observations.

In combination, what we know about both dust and gas discs seems to indicate
that material is roughly delineated between 0.2-1 $R_{\odot}$, which is what
we will assume in our baseline model. In our forthcoming analysis we will
further assume for simplicity that the material in the disc initially spreads
throughout the entire radial extent of the disc rather than confined to a
narrow region. We treat the mass range as a free parameter in the range
suggested for dust mass, i.e covering 8 orders of magnitude between
$10^{16}-10^{24}$ g. Masses under $10^{16}$ g are ignored as they are too
small to yield effective circularization as we shall see in Section 4. Masses
close to $10^{24}$ are probably rare or inapplicable, yet we consider them for
completion.

###  2.2 Fragment sizes

The GV19 study considers gas interactions with objects in the 0.1-10 km size
range, appropriate to their Oort-cloud origin. We however consider tidally
disrupted fragments, and therefore regardless of the size of their progenitor,
the correct size range should ensue from the intrinsic properties of the
progenitor as well as its precise origin and how deeply it penetrates the
tidal sphere. Overall we suspect that the fragment size range is reduced
compared to the GV19 study. We refer to the Brown et al. (2017) study, in
which tidal fragments emerging from tidal disruptions of larger parent objects
and are considered as monolithic objects with some internal strength, when
their size is estimated at tens to hundreds of m. Similar estimates of
fragment sizes, in the range 10-1000 m, are given in the Kenyon & Bromley
(2017a) study (see e.g. their Figure 1).

In a different study, Rafikov (2018) considers the vertical tidal collapse of
a large planetesimal as it deeply penetrates the tidal sphere of a WD. In this
context, he envisions the planetesimal as a collection of fragments which
collide with each other as they collapse inwards towards the orbital mid-
plane. These collisions start off as catastrophic, reducing fragments sizes in
a collisional cascade. However, below a minimum size of roughly $\sim$100-300
m, depending on the exact material (Leinhardt & Stewart, 2009), the collision
velocity for catastrophic fragmentation rather increases as material strength
becomes more important. Rafikov (2018) thus hypothesizes that the final size
distribution of fragments resulting in this process should peak around the
size at which the collision velocity leading to catastrophic disruption is
minimized. This size is estimated to be 0.1-1 km in accordance with the
Leinhardt & Stewart (2009) findings.

The study by MP20 also performed detailed tidal disruption simulations,
finding that as much as 70-80% of the tidal fragments have relatively short
rotation periods less than twice the known 2.2 h cohesion-less asteroid spin-
barrier (only sufficiently small and cohesive/monolithic asteroids are able to
spin faster than this limit). Hence, the cohesion-less asteroid spin-barrier
might also serve as clue to the potential size of the tidal fragments.
Asteroids spinning faster than the spin-barrier fall in the size range of
150-300 m according to Pravec et al. (2002). Given all these independent
estimates, we treat the fragment sizes as a free parameter ranging some 2
orders of magnitude, between 10-1000 m. Smaller or larger fragments are
ignored since we consider their initial mass fraction in the stream to be
negligible.

Throughout this paper we consider only fully formed tidal stream. The MP20 and
Veras et al. (2014) studies show that several orbits (of the progenitor) are
required for the stream to fully evolve. Before this, the fragments could be
more clumped up and not spread uniformly throughout the stream, and if the
pericentre is far from the WD, partial disruptions result in large fragments
that require many flybys before they finish dissecting to their smallest
possible constituents. However since the stream formation timescale is
relatively short, we ignore this phase entirely.

###  2.3 Initial fragment orbits

Simple analytical arguments given in Section 1 of Malamud & Perets (2020a)
provide compelling evidence that the tidal fragments following a disruption
event should occupy diverse initial semi-major axes, based on the precise
origin and size of the progenitor object. For example, a small km-sized
asteroid originating from a minimal distance of at least several AU (required
in order to survive the host star’s post main-sequence evolution, see Mustill
& Villaver (2012) and Mustill et al. (2014)) would disrupt to form a ring of
debris on the original asteroid semi-major axis. Such a ring would have a very
tiny spread in the orbital energies of the fragments (Veras et al., 2014;
Malamud & Perets, 2020a; Nixon et al., 2020).

A common misunderstanding is to assume that the same outcome applies to all
small asteorids. For example, consider an analogue Kuiper belt object akin to
recently visited Arrokoth (Grishin et al., 2020a; McKinnon et al., 2020). It
orbits its main-sequence host star at a distance of several tens of AU, and
orbital expansion primarily during the AGB phase of the host star would likely
drive its semi-major axis to $\sim$150 AU when the star reaches the WD phase
(depending on the host star’s mass, see e.g. Malamud & Perets (2017a) and
(Malamud & Perets, 2017b)). Since its size is of the order $\sim 10^{1}$ km,
it is large enough and distant enough to form a moderately dispersed debris
disc rather than a ring, when it tidally disrupts upon reaching a distance of
$R_{\odot}$ from the WD (its outer tidal sphere limit). It may even marginally
produce unbound debris (as evident in figure 2 of Malamud & Perets (2020a)),
while the innermost semi-major axis $a$ in the tidal stream (see their
equation 3) would be halved. If its close approach distance is smaller than
$R_{\odot}$, then the fragment dispersion increases considerably. Thus, a non
ring-like outcome is clearly plausible and even likely in small asteroids that
originate in exo-Kuiper belts or beyond.

As we consider tidal disruptions of larger objects, in the size range
$10^{2}-10^{3}$ km or more (or else the pericentre distance is much less than
$R_{\odot}$), the tidal fragments transition from forming a ring
configuration, or a disc with moderate orbital dispersion as previously shown,
to a bi-modal semi-major axis distribution (Malamud & Perets, 2020a). In the
latter configuration, half of the progenitor material becomes unbound and the
other half becomes tightly bound to the WD, with $a$ as little as 0.05 AU. The
fragment semi-major axis distribution is approximately that given by Figure 2
of MP20, the peak in the distribution ranging between 0.1 to 1.5 AU for
progenitors of mass 1$M_{\oplus}$ to 10${}^{-4}M_{\oplus}$ respectively.

We conclude that different tidal disruption scenarios lead to very different
semi-major axes of the tidal fragments interacting with the pre-existing
compact disc. To quantify the range of $a$, let us recap the aforementioned
cases as discussed above, assuming the pericentre $q$ at $R_{\odot}$ for our
analysis: (i) disruption of large progenitors result in a bi-modal fragment
distribution with $a$ between $10^{-1}-10^{0}$ AU; (ii) asteroids originating
at several AU form rings. Hence fragments would orbit on the original
progenitor semi-major axis of few $10^{0}$ AU; (iii) typical Kuiper belt
analogues form moderately dispersed debris if their size is $\sim$10 km. If
smaller, then their debris remain close to the original progenitor orbit.
Overall the likely fragment semi-major axis distribution is of order few $\sim
10^{1}-10^{2}$ AU; (iv) typical Oort cloud analogues with original orbits in
the range $10^{3}-10^{5}$ AU form anything from moderately dispersed debris
($\sim$1 km objects at $10^{3}$ AU) to fully bi-modal debris ($\sim$10 km at
$10^{3}$ AU, or $\sim$1 km objects at $10^{4}$ AU and beyond). In a bi-modal
disruption regime the fragment semi-major axes converge onto a value of
$q^{2}/2L$ (Malamud & Perets, 2020a), where $L$ is the progenitor radius.
Hence we expect in all of these cases that most of the fragments will have $a$
around $\sim 10^{3}$ AU (while very few fragments may indeed extend to the
original orbit or more, however their numbers would be entirely negligible).

We note that the cases discussed in the previous paragraph were analysed
assuming a pericentre $q$ of $R_{\odot}$. If a lower $q$ is used in the same
analysis the fragment semi-major axes are further reduced since deeper
disruptions are necessarily more dispersive. In conclusion, we treat the semi-
major axis of bound fragments as a free parameter, ranging between
$10^{-1}-{10^{3}}$ AU. Tidal fragments would rarely have $a$ larger than about
$10^{3}$ AU hence this range is strongly constrained.

###  2.4 Intersection between tidal fragments and compact disc

The intersection point or points between the tidal fragments and the pre-
existing compact disc will occur at roughly the pericentre of the tidal stream
(i.e. the pericentre of the progenitor object). Our approach for selecting the
preicentre distance is similar to that of the MP20 study. They focus primarily
on tidal disruptions in which the progenitor’s pericentre is Roche-grazing and
corresponds to 1$R_{\odot}$, an approximate outer-limit. However, their
complete investigation includes three choices for the pericentre which cover
the full range of distances relative to $R_{\odot}$: deep, intermediate and
grazing.

Here we also consider three cases for the pericentre distance: an inner
intersection of 0.2$R_{\odot}$ (i.e intersecting the compact disc along its
inner-edge), an intermediate intersection point of 0.6$R_{\odot}$ and an
outer, grazing intersection of 1$R_{\odot}$. We make the simplification that
the argument of pericentre is zero, so that the fragments have one
intersection point with the compact disc rather than. Later in Section 3.1 we
show that if it were not the case, our results would only change
quantitatively within a factor unity, which is why such an assumption is
judicious. We refer the reader to Figure 4, where a schematic diagram shows
the intersection point between a crossing fragment with the compact disc. Most
of the details in this diagram, however, will only be elaborated on in Section
5.

##  3 Model

###  3.1 Analytical formalism

We consider a generic profile for a pre-existing compact disc, which is
typically composed of dust, and in some cases could have a gaseous component
as well. We then describe the circularization mechanism due to the drag forces
between fragments in the tidal stream and the pre-existing compact disc, with
the goal of calculating the circularization timescale. The model was initially
developed by GV19 strictly for gas discs, however here we extend the model
discussing how dust-induced drag may be equivalent, and therefore applicable
in the limit of very high velocities.

Regardless of disc material type, we assume in accordance with the
observational constraints Section 2.1 that the radial extent of the compact
disc ranges from $r_{\rm in}=0.2R_{\odot}$ to $r_{{\rm out}}=R_{\odot}$. Our
surface density profile is given by

$\Sigma(r)=\Sigma_{0}\left(\frac{r}{R_{\odot}}\right)^{-\beta},$ (1)

where $\beta$ is an arbitrary exponent. For both dust and gas we shall take
$\beta=3/2$, as follows. When considering gaseous WD discs, Metzger et al.
(2012) find that the gaseous surface density $\Sigma_{g}(r)\propto
r^{-n-1/2}$, where $n$ describes the viscosity power law $\nu(r)\propto
r^{n}$. For an optically thin gas disc with $T(r)\propto r^{1/2}$, they take
$n=1$ and $\beta=3/2$. When considering dusty WD discs, the radial profile is
essentially unknown. However, in young proto-planetary discs the canonical
minimal mass Solar nebula predicts a power law of $\beta=3/2$
(Weidenschilling, 1977). This scaling is physically justified if the solids
distribute the specific angular momentum $\ell$ in the disc as
$\ell\propto\sqrt{a}$, similarly to the Keplerian motion of the planets.
Without any other knowledge of the structure of compact discs around the WD,
we also follow this prescription. Thus, $\beta=3/2$ is an acceptable choice
for both material types. Additionally, we also note that the model is not very
sensitive to different choices of $\beta$ in the range 1-2.

The total mass of the disc is then

$\displaystyle M\approx\intop_{r_{\rm in}}^{r_{\rm out}}\Sigma_{(}r)2\pi
rdr\approx\frac{4\pi}{3}\Sigma_{0}R_{\odot}^{2},$ (2)

with the surface density constant

$\displaystyle\Sigma_{0}\approx 5\cdot 10^{-4}\rm{g\
cm^{-2}}\left(\frac{M}{10^{19}{\rm g}}\right).$ (3)

Consider a tidal fragment of size $R$, density $\rho$ and mass $m$, orbiting a
WD of mass $M_{\rm WD}$ with a semi-major axis $a$ and a grazing pericentre
$q=a(1-e)\ll a$. Grishin et al. (2019) calculate the energy loss due to drag
during a single passage (see justification in Appendix A) through the disc at
pericentre if the coincident material in the disc is gaseous. They further
assume, in order to minimize the number of free parameters, that the fragment
crosses the disc face-on, neglecting the inclination $i$ which contributes an
order unity pre-factor. This means that our forthcoming analytical model
circularization timescale may be considered as a lower limit result. The drag
induced is simply the familiar aerodynamic gas drag. The energy loss is
therefore

$\Delta E=-\frac{\pi}{2}CR^{2}\Sigma(q)v_{\rm rel}^{2},$ (4)

where $C$ is the drag coefficient, $v_{\rm rel}^{2}$ is the relative velocity
between the fragment and the compact disc and $\Sigma(q)$ is replaced with
$\Sigma_{\rm g}(q)$, the gas surface density. For a highly eccentric orbit,
the velocity is approximated by the escape velocity $v_{\rm rel}^{2}=2GM_{\rm
WD}/q$, where $M_{\rm WD}$ is the mass of the WD.

The drag coefficient measures the strength of the pressure difference in a
gaseous medium. For low velocities (or low Reynolds numbers) the drag force is
in the linear (Stokes) regime, which means that $C\propto v_{\rm rel}^{-1}$.
Conversely, passage at extremely high velocity is very fast compared to the
response of the gas, and the drag force is quadratic and proportional to
$v_{\rm rel}^{2}$ (ram-pressure regime), which means that $C$ is of order
unity.

What happens when the coincident material is dusty rather than gaseous? Here
the analogy of a Stokes drag law is unclear and it is not certain that dust
drag is even active for low velocities. However, for very high velocities the
details of the ambient medium are unimportant and the drag law behaves as in
the ram-pressure regime with $C\approx 1$. Indeed, analogies between
collisionless and gaseous medium have been proposed in the context of
dynamical friction, or gravitational drag, either in 3D geometry (Rephaeli &
Salpeter, 1980; Ostriker, 1999), or in 2D slab geometry (Muto et al., 2011;
Grishin & Perets, 2015). While significant differences were present for the
subsonic case, the hypersonic case of very high velocity essentially behaves
universally, regardless of the property of the medium. We thus conclude that
in the limit of high velocities, Equation 4 is (to order unity) indifferent to
the type of compact disc material coincident with the passing fragments.
Summarizing the previous paragraphs, both $\beta$ and $C$ are approximately
independent of the coincident material type. From this point onwards, this
section will extend the GV19 study while treating both dust and gas via a
generic, identical formalism.

In order to complete their circularization, the tidal fragments must interact
with a sufficient amount of material to move from their eccentric orbital
energy $E=-GM_{\rm WD}m/2a$ to a contracted and circular energy state $E_{\rm
circ}=-GM_{\rm WD}m/2q$, where $m$ is the fragment mass.

The fractional energy change per orbit is given by

$\mathcal{Q}^{-1}\equiv\left|\frac{\Delta
E}{E}\right|=\frac{3C\Sigma(q)}{2\rho R}\frac{a}{q},$ (5)

where the quantity $\mathcal{Q}$ is the "quality factor" that measures how
many (equivalent to the original) orbits the object will do before it will
lose a substantial amount of energy and will be circularized. The definition
is analogous to the tidal factor $\mathcal{Q}$ that is frequently used to
describe tidal dissipation in two-body systems (Goldreich & Soter, 1966; Hut,
1981), only that here the source of the dissipation is the induced drag. Note
that $E$ is given by the initial orbital energy while in reality $a$ changes
as a function of time. $\mathcal{Q}$ is therefore only a zeroth-order
approximate to the number of orbits. In Section 4 we however compare this
simple expression with numerical computations and show that it is indeed
accurate to order of magnitude.

For full circularization within one to a few orbits, or $\mathcal{Q}\sim 1$,
the fragment should be sufficiently small, i.e. $R\lesssim
1.5C(a/q)\Sigma/\rho$. Larger fragments with $\mathcal{Q}\gg 1$ will take
longer to fully circularize. The typical circulariztion time is $\tau_{\rm
circ}=\langle|E/\dot{E}|\rangle$, where the brackets indicate averaging over
the orbit. Since the loss of energy is impulsive and occurs at the pericentre,
the energy loss can be estimated as $\dot{E}=\Delta E/\Delta t$, where $\Delta
E$ is given in Eq. 4, and $\Delta t=P=2\pi\sqrt{a^{3}/GM_{\rm WD}}$ is the
orbital period, which leads to the circularization time simply being

$\tau_{\rm circ}=\mathcal{Q}P=\frac{4\pi\rho
Rq}{3C\Sigma}\sqrt{\frac{a}{GM_{\rm WD}}}.$ (6)

Note that the timescale increases with the separation as $a^{1/2}$, meaning
that distant fragments will require more time to circularize, however it will
require less iterations since $\mathcal{Q}\propto a^{-1}$.

###  3.2 Caveats and restrictions

There are three restrictions entailed in our simple model. The first
restriction relates to time. Drag-induced dissipation is active as long as the
ratio between the orbital period and the disc lifetime is small. The most
distant tidal fragments, with $a$ around $10^{3}$ AU (Section 2.3) have
orbital periods of the order of several $10^{4}$ yr, which roughly correspond
to the lower limit empirical disc lifetime of $10^{4}$ yr (Girven et al.,
2012). Hence, if the pre-existing compact disc lifetime is not affected by the
interaction itself (a point which is further discussed in Section 5.6), we may
conclude that most fragments are capable of significant drag-induced
circularization, however they require $\mathcal{Q}\lesssim 1$. Luckily
$\mathcal{Q}$ is inversely proportional to $a$ as we have shown, and indeed
yields low values. Conversely, tightly orbiting fragments with $a$ of the
order of 1 AU have an orbital period of merely 1 yr. In line with the previous
argument, they can have $\mathcal{Q}$ as high as $10^{4}$ and still
circularize.

The second restriction relates to conservation of energy. When we say that the
drag coefficient $C$ is of order unity, it is the equivalent of saying that
the pre-existing dust or gas attains the same velocity as the fragment. In
other words, the energy loss in the fragment’s orbit (Equation 4) must be
compensated by energy gain of the interacting compact disc material. One other
possibility is that some of the energy goes to collisional vaporization of
pre-existing dust, however it is easy to show that in the limit of the very
high relative velocities in our scenario, $E_{vap}\sim 1.5*10^{11}$ erg
$\times$ g-1 (for SiO2, see Podolak et al. (1988)) is not very energy-
expensive which means that energy conservation implies gaining roughly
identical velocity (although not necessarily in the same direction). This in
turn implies that the passage of a single fragment both scatters material and
carves a gap in the compact disc. The gap has to be filled prior to
interacting with another fragment, otherwise the above model is incorrect. The
proper filling time is quantitatively discussed in Section 5.1.

The third restriction relates to conservation of momentum. Since the
collisional interactions conserve angular momentum, keeping the compact disc
angular momentum pointing in the same direction cannot work unless $M\gg
M_{\rm progenitor}$, or else compact disc and progenitor object are co-orbital
from the beginning. Even more importantly, if instead $M\lessapprox M_{\rm
progenitor}$, this would entail a catastrophic fate to the part of the compact
disc which interacts with the tidal stream, and diffusion would not be able to
continue filling the gaps as previously discussed. Therefore, full
circularization is necessarily truncated and only partial circularization can
occur. We nevertheless note that even in the case of partial circularization
our model is still very useful, as we discuss in the next section.

Finally, we note that our model does not directly consider any effects from
shocks, although the compact disc may indeed have a gaseous component, and the
interaction is obviously extremely hypersonic. This aspect is beyond the scope
of this paper, having already introduced many novel aspects, however it should
be explored further in future dedicated studies.

###  3.3 Partial circularization

The calculation in Section 3.1 assumes that the mass of the compact disc far
exceeds that of the tidal stream, such that circularization can proceed fully.
However, since the compact disc forms from bodies that are likely ejected to
tidal-crossing orbits from the same reservoir of planetesimals as the tidal
stream itself, it is probable that the fragments often encounter a disc whose
mass is insufficient for complete circularization. In this case, the limit in
dust mass to collide with leaves the fragments on shrunken orbits
($a_{\mathrm{new}}$) that lie within their initial semi-major axis $a$ and the
pericentre $q$. Let us assume for simplicity that the fragments all have the
same size and mass, and therefore we may equate the total orbital energy of
the stream plus the energy loss from Equation 4 to the new total energy of the
stream:

$\frac{-GM_{\rm WD}M_{\rm progenitor}}{2a}-\frac{\pi}{2}Cv_{\rm rel}^{2}\Delta
M=-\frac{-GM_{\rm WD}M_{\rm progenitor}}{2a_{\rm new}},$ (7)

where $\Delta M$ is some fraction of the compact disc mass $M$ that the stream
has collided with, $M_{\rm WD}$ is the WD mass and $M_{\mathrm{progenitor}}$
is the mass of the tidal stream progenitor. Equation 7 then simplifies to

$\frac{a}{a_{\rm new}}=1+2\pi C\frac{a}{q}\frac{\Delta M}{M_{\rm
progenitor}},$ (8)

Equation 8 is valid up to the last stages of circularization, when the
differential velocity shifts away from $v_{\rm rel}^{2}=2GM_{\rm WD}/q$ to a
lower value (as further discussed in Section 4). Interestingly, it indicates
that while complete circularization of the stream requires a mass that is at
least comparable to that of the progenitor, significant partial
circularization already occurs for very small disc masses. As an example,
shrinking the initial orbit from $a$ to $a/2$ only requires a disc-to-stream
mass ratio of $\Delta M/M_{\mathrm{progenitor}}=q/(2\pi Ca)$. This amounts to
tiny mass fractions of merely $10^{-2}-10^{-7}$ for the stream semi-major axis
being 0.1 to 1000 AU respectively (and $q$ in the range $0.2-1~{}R_{\odot}$).
Hence, while complete circularization is only expected to occur in a limited
fraction of cases, nearly all tidal streams that encounter a pre-existing
compact disc will experience significant drag-induced orbital shrinkage.

The likely prevalence of partial circularization begs the question of what
happens after this process completes. We note that since the tidal stream is
composed of different-sized fragments, partial circularization of the
fragments will result in orbital differences based on fragment size. This in
turn will lead to different apsidal precession rates and can ultimately
facilitate catastrophic collisions among remaining fragments in the stream
until constituent particles are sufficiently small to be transported by PR
drag. We do not work out the detailed consequences of partial circularization
here but will dedicate a future work to this process.

Finally, full circularization occurs when we set $a_{\rm new}=q$ in Equation
8, which works out to give

$\Delta M=\frac{1}{2\pi C}\left(M_{\rm
progenitor}-\frac{q}{a}\right)\approx\frac{M_{\rm progenitor}}{2\pi C}.$ (9)

Equation 9 is tantamount to the simple statement that the fragments in the
tidal stream should encounter their own mass (within a factor of unity) inside
the compact disc for them to achieve full circularization.

###  3.4 Numerical simulations

In order to test and better quantify the analytical results, we employ
numerical N-body simulations of debris on initially eccentric orbits passing
through a pre-existing disc. For the N-body integration, we use the publicly
available code REBOUND (Rein & Liu, 2012). We use IAS15, a fast, adaptive,
high-order integrator for gravitational dynamics, accurate to machine
precision over a billion orbits (Rein & Spiegel, 2015). The drag force is
modelled with a constant drag coefficient $C$ equals unity, as discussed in
detail in Section 3.1.

The surface density is prescribed according to Equations 1 and 3. For the mid-
plane volume density, it is given by $\rho(z=0,r)=\Sigma(r)/h/\sqrt{2\pi}$,
where $h$ is the scale height of the disc and is set to the fiducial value of
one percent of the radial distance, $10^{-2}r$. The local density in the $z$
plane is set by a Gaussian profile,
$\rho(z,r)=\rho(z=0,r)\exp(-z^{2}/2h^{2})$. While this treatment is typical of
gas discs (taking after the GV19 study), we apply it here in a broader sense
to both dust and gas.

The disc is assumed to rotate with a Keplerian velocity of a circular orbit
and is assumed to far exceed the mass of the fragment stream, such that its
circularization can fully proceed as discussed in Section 3.2. Numerical
results would be presented and compared to analytical ones in the following
section, where we explore the full parameter space.

##  4 Results

The analytical model described in Section 3 allows us to estimate the typical
time for circularization as $\tau_{\rm circ}$. Here we explore the dependence
of $\mathcal{Q}$ and $\tau_{\rm circ}$ on the size and orbit of the fragments,
in addition to the disc mass $M$. We consider the canonical fragment density
of $\rho=3\ \rm g\ cm^{-3}$ and WD mass of $M_{\rm WD}=0.6~{}M_{\odot}$.

(a) $q=1R_{\odot}$

(b) $q=0.6R_{\odot}$

(c) $q=0.2R_{\odot}$

Figure 1: Contour plot of lines of equal $\log_{10}\mathcal{Q}$ in separation
$a$ as a function of disc mass $M$, given a fragment pericentre of
$q/R_{\odot}=1,0.6,0.2$ for panels (a), (b) and (c) respectively. The contours
are normalized for fragment size $R=10\ \rm{m}$. The contour line labelled 0
means that a 10 m fragment is instantaneously captured, and the same outcome
applies for the overlying dashed lines with negative numbers. Different
fragment sizes should be changed according to $\mathcal{Q}\to\mathcal{Q}\times
R/10\rm m$. Recall that in our model $M$ also provides an upper limit for the
progenitor’s mass.

Figure 1 shows contours of equal levels of $\log_{10}\mathcal{Q}$ as a
function of the disc mass and fragment semi-major axes. The pericentre $q$ are
indicated on top of each panel. The contours are normalized to $R=10\ \rm m$,
while for other fragment sizes the scaling is simply
$\mathcal{Q}\to\mathcal{Q}\times R/10\rm m$. We see that the contours have
identical slopes since $\mathcal{Q}\propto(Ma)^{-1}$. Larger separations and
elevated disc masses both lower $\mathcal{Q}$, since the mass increases the
energy extraction rate, while large separations decrease the orbital energy. A
smaller pericentre distance $q$ results in a higher surface density at the
intersection point and thus reduces $\mathcal{Q}$. Large fragments lose energy
more slowly, since $\mathcal{Q}\propto R$.

(a) $M=10^{19}$ g

(b) $a=10$ AU

Figure 2: Typical circularization time, $\tau_{\rm circ}=\mathcal{Q}P$ as a
function of (a) the initial fragment semi-major axis (fixing $M$ at $10^{19}$
g, however for different $M$ change $\tau_{\rm circ}$ according to $\tau_{\rm
circ}\to\tau_{\rm circ}\times 10^{19}{\rm g}/M$); and (b) the disc mass $M$
(fixing $a$ at 10 AU, however for different $a$ change $\tau_{\rm circ}$
according to $\tau_{\rm circ}\to\tau_{\rm circ}\times\sqrt{a/10{\rm AU}}$).
Fragment size is indicated by line style, such that $R=$10/100/1000 m
corresponds to solid/dashed/dotted lines respectively. Line colours correspond
to fragment pericentre distances, lighter tones denoting shorter distances.
The shaded area between $10^{4}-10^{6}$ yr is the empirically-based lifetime
which might apply to the compact disc.

Figure 2 shows the circularization timescale. In Panel 1(a) it is shown as a
function of the separation when the compact disc mass is fixed at $M=10^{19}$
g. This timescale must be shorter or comparable to the compact disc lifetime
(shaded area, based on Girven et al. (2012)). Note that these disc lifetimes
are based on empirical evidence, rather than grounded by direct measurement or
any physically motivated values. As we discuss later in Section 3.3, the
compact disc fate may sometimes be closely affected by its interaction with
the tidal stream, so its lifetime can in fact be significantly shorter than
that suggested by this empirical range.

The circularization timescale increases with the separation since
$\mathcal{Q}P\propto a^{1/2}$. We see that small 10 m fragments (solid lines)
are able to circularize within $\lesssim 10^{6}$ yr for any separation and any
value of $q$. Intermediate 100 m sized fragments (dashed lines) also
circularize within that time frame however they must intersect the disc at
$q=0.6R_{\odot}$ or less, unless their separation is less than $10^{2}$ AU, in
which case any value of $q$ is appropriate. Lastly, large 1000 m fragments
(dotted lines) are considerably harder to circularize. They must be either of
small separation or penetrate very deeply into the tidal sphere. Panel 1(b)
shows the circularization timescale as a function of the disc mass $M$ when
$a$ is fixed at 10 AU. It is indicative of a wide range of fragment sizes and
disc masses that circularize in propitious time. The minimum $M$ required at
least for the smallest fragments is about $10^{16}$ g.

(a) $q=1R_{\odot}$

(b) $q=0.6R_{\odot}$

Figure 3: Time evolution of the orbital elements of a fragment on initially
eccentric orbit with $a=10\rm au$ and initial inclination of
$i_{0}=90^{\circ}$. The disc mass is $M=10^{22}$ g. The pericentre is
$q=1R_{\odot}$ in (a) and $q=0.6R_{\odot}$ in (b). The initial mean anomaly is
$\mathcal{M}=3\pi/2$.

Finally, Figure 3 shows the time evolution by the numerical simulations
described in Section 3.4, of 10 m fragments initially separated by $a=10$ AU.
The disc mass is fixed in the initial conditions of our code to be $10^{22}$
g. This choice is motivated by obtaining sufficiently short simulation
runtime, of the order of $\sim$1 day. Panels 2(a) and 2(b) are similar except
in having different choices of $q$. Each contain two subplots of the change in
the semi-major axis $a$ and the corresponding change in the eccentricity $a$
and inclination $i$. Of course $e$ is initially $\sim$1 and $i$ is initially
set to the maximum possible value of $90^{\circ}$ (face-on orientation).

We first discuss the damping timescale of $a$ versus $e$ and $i$. Since
$\tau_{\rm circ}$ is defined above as the energy dissipation timescale, and
since $E$ and $a$ are anti-correlated, $\tau_{\rm circ}$ is strictly speaking
the migration timescale, or the rate of change of the semi-major axis. If we
start with a very eccentric orbit of $e=1$ (or $a\gg q$), we eventually end up
with a circular orbit of $e=0$ (or $a=q$). It is worth noting that the times
for complete circularization and total elimination of the eccentricity are the
same, however the eccentricity damping timescale, i.e the rate of change of
the eccentricity with respect to its own magnitude, is different from
$\tau_{\rm circ}$. This comes from the fact that the eccentricity often has
initial values close to 1, and even after significant decrease of $a$ (say, of
the order of itself) the eccentricity is still close to 1. Only when $a$
approaches $q$ does $e$ start to change significantly. As can be seen, $i$ and
$e$ behave very similarly.

The numerical simulations show that during the last phase of circularization
fragments indeed become co-orbital with the compact disc and therefore are
effectively embedded in it. Most importantly, Figure 3 and other simulation
runs have all shown that there is a good agreement between the full numerical
calculation and the analytical model. One expects the numerical simulations to
reach full circularization faster than the analytical calculation since it is
only an approximation and the exact drag changes as $a$ evolves. By comparing
Figure 3 to Figure 21(b) using the same parameters, the timescales are clearly
in agreement to order of magnitude, and the numerical circularization is
indeed faster, as expected. Furthermore, in Panel 2(b) it takes around $\sim
10$ orbits for $a$ to change in the order of itself, corresponding to the
expected value of $\mathcal{Q}\approx 10$ from the analytical calculation.
Conversely, in panel 2(b), $a$ changes significantly in one orbit,
corresponding to the expected value of $\mathcal{Q}\approx 1$. The numerical
simulations and analytical formalism are consistent.

##  5 Discussion

###  5.1 Gap formation and filling

In this section we quantitatively expand the discussion in Section 3.2, where
it was suggested that each fragment effectively carves a column of dimension
$R$ through the compact disc. In Appendix B we derive from first principles a
heuristic timescale for the diffusion filling timescale that will act to close
the carved gap by local dust (ignoring the possibility of fully gaseous
discs). We obtain the following equation:

$\tau_{{\rm fill}}=\frac{3MR^{2}q^{1/2}}{8h^{2}\rho_{\rm d}r_{\rm out}r_{\rm
d}(GM_{\rm WD})^{1/2}},$ (10)

where $M$ is the compact disc mass with an outer boundary $r_{\rm out}$, $q$
the pericentre distance, $h$ the compact disc scale height, $M_{\rm WD}$ the
WD mass and $\rho_{\rm d}$ and $r_{\rm d}$ denote the density and radius of
compact disc dust grains.

When a gap forms by a fragment crossing the compact disc, our conceptual
circularization model is valid as long as it fills up before another fragment
overlaps the same region. All fragment crossings must encounter coincident
dust rather than gaps, or else our model would be inconsistent with the
calculation in Section 3.1. We also recall from Sections 3.2 and 3.3 that our
only other condition is $M\gg M_{\rm progenitor}$.

Our goal is now to consider the typical time between fragment crossings, and
in particular the typical time between overlapping crossings. For simplicity,
let us consider only same-size tidal fragments of dimension $R$, originating
from a tidally disrupted progenitor of size $L$, such that all fragments
occupy a similar semi-major axis $a$, as in the case of typical asteroid tidal
disruptions (see Section 2.3). The intersection between the compact disc and
each fragment occurs at the progenitor’s pericentre distance $q$. Based on the
MP20 study, all fragments intersect the compact disc within a relatively
narrow cross-section approximately equal to the original progenitor’s cross-
section, i.e an area proportional to $L^{2}$. Clearly $L$ is many orders of
magnitudes smaller than the dimension of the entire compact disc which is of
order $R_{\odot}$, so the entire cross section is point-like in relation to
the compact disc area. While this effective intersection point remains fixed
in 3D space, the compact disc itself rotates around the WD.

Additionally, since $R\ll L$, the probability of successive fragments in the
tidal stream to overlap exactly at the same intersection point is of the order
of $R^{2}/L^{2}$, assuming a uniform spatial distribution inside the stream.
The total number of fragments in the stream $N$ is given by the mass in the
stream divided by the mass of each fragment, or $N=L^{3}/R^{3}$ (constant
$\rho$ assumed). Thus the typical time between crossings in this simplified
setup is the fragment orbital period $\sqrt{a^{3}/GM_{\rm WD}}$ divided by N:

$\tau_{\rm cross}=\sqrt{\frac{a^{3}}{GM_{\rm WD}}}\frac{R^{3}}{L^{3}}.$ (11)

However considering the probability to cross in the same exact point, the
typical time between overlapping crossings is $\sqrt{a^{3}/GM_{\rm WD}}$
divided by $L/R$:

$\tau_{\rm overlap}=\sqrt{\frac{a^{3}}{GM_{\rm WD}}}\frac{R}{L}.$ (12)

If the gap is unable to rotate in time to avoid an overlapping crossing, the
model is invalid. The compact disc rotation period at a radial distance of $q$
from the WD is (using Kepler’s law) $P=2\pi q^{3/2}(GM_{\rm WD})^{-1/2}$. The
disc must rotate a small distance of approximately $R$ in order to avoid a
second hit by an overlapping crossing in the same point. The fractional
rotation time is $R/2\pi q\times P$. We equate this with $\tau_{\rm overlap}$
to constrain the semi-major $a$. The resulting expression is
$a<q^{1/3}L^{2/3}$. Since the progenitor size $L$ obviously cannot be larger
than the size of the disc, $a$ cannot exceed the pericentre. The conclusion is
that $\tau_{\rm overlap}$ must be much larger than the fractional rotation
time. Overlapping crossings are sufficiently temporally separated that the
disc always rotates away.

The gap thus has to complete _at least_ one full orbit around the WD before it
has some probability to intercept an additional fragment. Therefore, one
rotation period of the compact disc is a minimal requirement for the time
interval between gap fillings. Based on this, $\tau_{\rm fill}$ from Equation
10 must be smaller than $P$, and we can rearrange this equation to obtain a
strong limit on the maximum disc mass allowed for the model to work:

$\begin{split}M\lessapprox 2\times
10^{29}\>\mathrm{g}\>\left(\frac{R}{10~{}\rm{m}}\right)^{-2}\left(\frac{q}{R_{\odot}}\right)\left(\frac{h}{1000~{}\rm{km}}\right)^{2}\\\
\left(\frac{\rho_{\rm
d}}{3~{}\rm{g}\times\rm{cm}^{-3}}\right)\left(\frac{r_{\rm
d}}{1~{}\mu\rm{m}}\right).\\\ \end{split}$ (13)

We take the fiducial values $\rho_{\rm d}=3~{}\rm{g}\times\rm{cm}^{-3}$ and
$r_{\rm d}=1~{}\mu\rm{m}$. We then consider extremum values $R=1000$ m and
$q=0.2R_{\odot}$ to obtain $M\lessapprox 4\times
10^{24}~{}\rm{g}\times(h/1000~{}\rm{km})^{2}$. Based on our compact disc mass
range from Section 2.1 ($10^{16}~{}\rm{g}<M<10^{24}~{}\rm{g}$), Equation 13
implies that our model is robust even for 1000 m gaps, unless $h$ is
considerably smaller than 1000 km. Since we consider fragments that follow a
power law size distribution, most gaps will in fact be of dimension $R=10$ m,
in which case the model is robust for any $M$ and $h$ down to 1-10 km.

The scale height $h$ is observationally an unconstrained parameter.
Nevertheless, the Kenyon & Bromley (2017a, b) theoretical studies investigate
the evolution of compact dusty discs around WDs. It is shown that if the
debris inside the compact disc are of infinite strength, collisional evolution
leads to a scale height as small as just one or two times the size of the
largest debris. However, in the more plausible case where debris suffer
catastrophic disruptions, the scale height might be reduced by a factor of at
most 2. The authors then show that the scale height will generally remain in
the order of $\sim 10^{3}\ \rm km$, corresponding to $f=0.002-0.004$. Given
these arguments we conclude that $h$ is very likely compatible with our model
being robust, and the gaps fill quickly prior to completing one orbital period
(of order $\sim$ few hours). Empirically, in WD 1145+017 (Vanderburg et al.,
2015) $h$ must be large, otherwise eclipses of the WD by orbiting solids would
not be observed. However this is merely one example. We also show in Appendix
C that the $h$ range considered in this work is safely larger than the limit
case that leads to gravitational instability in the compact disc.

###  5.2 A ram-pressure catastrophe?

Could the response of the fragment to passing through the compact disc be
catastrophic? As discussed in Section 3, gas or dust collectively lead to a
large pressure difference between the front and the back of the fragment. This
so called ram pressure (or dynamical pressure) may be seen as equivalent to
the energy density, which can be worked out from Equation 4.

$P_{\rm ram}=\rho_{\rm disc}v_{\rm rel}^{2}=\frac{(C\sim 1)\Sigma(q)v_{\rm
rel}^{2}}{h}.$ (14)

As previously mentioned the scale height $h$ is a quantity that is poorly
constrained by physical models, ranging from one or two dust radii to a few
thousand km (Kenyon & Bromley, 2017a). Because the relative velocities at
pericentre are so large (few $\sim 10^{2}$ km/s), we check if the resulting
ram pressure can exceed the threshold for compressive fracture. Measurements
of meteoroids in the Earth’s atmosphere yield compressive strengths in the
broad range of 1-500 Mpa (Petrovic, 2002; Popova et al., 2011; D’Angelo &
Podolak, 2015). Even in the limit of the smallest compressive strength in this
range (1 MPa), the scale height must be considerably less than $\sim 10^{3}$
km and the disc mass must exceed $\sim 10^{23}$ g for $P_{\rm ram}$ to be
important. In reality, the actual compressive strength of large cohesive
fragments inside the Roche limit could be in the neighbourhood of 500 Mpa,
otherwise they are not likely to withstand the tidal forces and remain
cohesive monolithic objects (for example, the assumed strength in Brown et al.
(2017) is 1 GPa). Likewise, disc masses approaching $\sim 10^{23}-10^{24}$ g
should be extremely rare, as already discussed in detail in Section 2.1.
Therefore, plugging in a more plausible disc mass of $M=10^{21}$ g, we
conclude that the compact disc scale height $h$ has to be on the order of
meters, same as our smallest fragments, in order for the ram pressure to be
important. Ram-pressure is thus deemed as an unlikely mechanism to bring about
catastrophic outcomes in strong cohesive or monolithic fragments.

However, for weak comet-like material, it is an open question to what extent
fragments are susceptible to catastrophically disintegrate. As an example
consider the compressive strength recently measured for the surface of comet
67P/Churyumov–Gerasimenko, an astonishingly low compressive strength of merely
12 Pa (O’Rourke et al., 2020). Such weak material will certainly not survive.
Nevertheless, the surface strength mentioned above probably does not reflect
on the bulk strength of 67P/Churyumov–Gerasimenko. Other estimates have
yielded compressive strengths ranging from 1 kPa (Biele et al., 2015) to 2-4
MPa (Spohn et al., 2015), so the effective strength is probably related to
depth variations.

Given the above discussion, it remains to be determined if comet-like
fragments are resistant to ram-pressure breakup, yet cohesive rocky fragments
are in all likelihood strong enough and will not be affected. Independent of
ram-pressure, fragment erosion is likely, and its effect is discussed in the
following section.

###  5.3 Gas production and fragment erosion

How gas is actually being produced in WD discs is one of the fundamental open
questions in the field, although as previously discussed, gas is rarely
observed and is much more infrequent than dust (Manser et al., 2020), although
the fraction of gaseous discs has recently increased considerably. We discuss
several possible mechanisms. First, gas may be produced by sublimation of dust
at a sufficiently close distance from the WD (see for example Metzger et al.
(2012) and references therein). For this mechanism, the presence and radial
extent of the gas should clearly correlate to some degree with the cooling age
(hence luminosity) of the WD. If collisions among grains at velocities of at
least few $10^{-1}-10^{0}$ km $\times$ s-1 occur, they would lead to
shattering (Tielens et al., 1994), thereby reduce the grain sizes and enhance
the effectiveness of sublimation. The classical dust sublimation ’radius’ is
in fact not a radius but an annulus of different grain-size-dependent
sublimation rates. Additionally, grains of volatile materials are more prone
to sublimation and if they are introduced for any reason, even within an
optically thick compact disc, they would certainly sublimate at greater
distances than the dust.

A second possibility is gas being produced via grain-grain collisional
vaporization. In this process the collisions between projectile grains and
similar-sized or else much larger target grains result in compression shocks
which lead to increase in pressure and temperature, and subsequently phase
transformations in the material (Ahrens & O’Keefe, 1972). Driven essentially
by the kinetic energy of the impacting bodies, studies have shown that partial
vaporization can be incepted when the internal energy behind the shock is
about 1-2 times the material surface binding energy, which is the energy
necessary in order to remove an atom from the top surface layer in vacuum
(Kudriavtsev et al., 2005). Full vaporization is possible at about 5 times the
material surface binding energy (see Tielens et al. (1994) and discussion
therein). The corresponding collision velocities required of similar-sized
silicate grains is of the order of a few $10^{1}$ km $\times$ s-1 for partial
vaporization, while full vaporization requires $\sim 10^{2}$ km $\times$ s-1.
If not similar-sized, but instead having a lower projectile:target mass ratio
(e.g, 0.1 in Figure 9 of Tielens et al. (1994)), the smaller projectile is
more likely to fully vaporize while the vaporized fraction in the target is
reduced compared to same-size collisions. Full vaporization in the target then
requires typical collision velocities a few times larger than those mentioned
in the case of similar-sized grains.

Tidally disrupted fragments are however larger than mere dust grains (say,
tens or hundreds of meters versus mere micron-sized projectile particles).
While collisional vaporization also occurs when the target is infinitely
larger than the projectile, in such cases the vaporized mass in the target is
negligible compared to the mass lost to shattered dust particles, excavated
from the underlying (i.e deeper) shocked surface layers (Tielens et al.,
1994). The term shattering here is equivalent to cratering, and it is very
important for fragment erosion. An impacting dust particle hits the fragment,
producing a crater on the surface. As a rule of thumb craters have about 10-20
the linear dimension of the impactor, hence the excavated volume (or mass) is
some $\sim 10^{3}$ times larger than that of the impacting grain. However, as
we show in Appendix D, the dusty ejecta from the fragment’s surface collides
with dust grains in the compact disc at high velocities, causing grain-grain
vaporization. This quickly forms a blanket of vaporizing material that
effectively prevents almost all subsequent compact disc projectile dust from
reaching the fragment’s surface. Instead of a high yield of the order of
$10^{3}$ we have an effective yield which is shown to be $Y_{\rm
shat}\approx$1.

An influx of tidally disrupted fragments, infinitely larger than mere dust
grains, could therefore interact with a pre-existing compact disc. Given the
previously mentioned tidal radius of typical WDs, the parabolic pericentre
Keplerian velocity of an extremely eccentric ring of fragments would be
several times $\sim 10^{2}$ km $\times$ s-1, while the circular (or near-
circular) Keplerian velocity of a pre-existing compact disc is a factor
$\sqrt{2}$ lower for the same distance. Hence even if the fragments are
exactly co-orbital with the compact disc, the relative collision velocities
are extremely high. Additionally, any inclination relative to the pre-existing
compact disc considerably raises the relative collision velocities. For a more
dispersive tidal stream, as in the MP20 study, fragments of different
eccentricities converge at periecntre. In this case, even the velocity
dispersion within the stream itself is significant (however we do not discuss
mutual fragment collisions in this study, only their interaction with the
compact disc). Therefore, given the previously mentioned shattering and
vaporization threshold velocities, it is easy to speculate in this scenario
that conditions are highly appropriate for collisional gas formation, driven
by vaporization of two components: pre-existing compact disc dust and eroded
dust originating from the shattered fragments.

A third possibility is gas generation via sputtering, a process in which
energetic gas ions collide with solid material. Colliding ions set off
collision cascades among the atoms inside the target material, and some atoms
may recoil back toward the surface of the target. If indeed a collision
cascade reaches the surface of the target, and its remaining energy is greater
than the target’s surface binding energy, an atom is ejected (Behrisch &
Eckstein, 2007). Thus, sputtering requires the presence of pre-existing gas
intersecting solid material. The sputtering yield (i.e, atoms sputtered per
gas ion) depends on the deposited energy per unit length to the surface
binding energy. Hence, not merely the target material is important, but also
the collision velocity and mass of the gas ions. For typical silicate material
(of both the gas and solid targets) and given the likely velocities as argued
in the previous paragraph, the sputtering yield $Y_{sput}$ is of the order of
0.1-1, and approximately an order of magnitude larger if the material is
$H_{2}O$ (see figure 10 in Tielens et al. (1994)). A similar sputtering yield
of 0.1 was adopted in the study of Jura (2008) for silicates. The only pre-
requisite is coincident gas to trigger the process.

A forth and final possibility might be related to a recent discovery of a
planetesimal held together by its own internal strength, embedded inside the
well studied debris disc of SDSS J1228+1040 (Manser et al., 2019). These
authors have used short-cadence spectroscopy in order to search for signs of
random variations in the gaseous emission of the WD, which they hypothesized
would be produced by random gas-generating collisions among dust particles.
Instead they found periodic variations, and concluded that the variability in
emissions must result from an excited cloud of gas trailing a cohesive object
whose size is unconstrained but may be up to a few hundred km in radius, and
having a two-hour orbital period. Or else, they hypothesized that the
planetesimal itself could be producing the gas if its orbit is close enough to
the WD to trigger surface sublimation. In the first case, however, it is not
yet clear what sort of interaction between the planetesimal and the disc could
generate the gas.

The relative velocity of an eccentric fragment with eccentricity $e$ embedded
within a circular (assumed for simplicity) compact disc is given by
$(\sqrt{1+e}-1)v_{\rm circ}$. Even $e$ as low as 0.1-0.2 should induce
sufficient relative velocities (a few dozen km $\times$ s-1) for partial
collisional vaporization of dust and also erode an embedded fragment through
shattering to replenish dust. Once the relative velocity is small enough, none
of the above is expected. The SDSS J1228+1040 embedded object’s semi-major
axis is however modelled as 0.73 $R_{\odot}$, without any appreciable change
between observations separated by almost a year (Manser et al., 2019).
Therefore it does not seem likely that in this case a small primordial
eccentricity is responsible for producing the gas. I.e, the planetesimal is
fully embedded in the compact disc rather than still in the process of
reducing its eccentricity to match that of the compact disc. The precise
details of how the variable gas emission is generated thus requires further
investigation. Nevertheless, it is clear that at least during the onset of
fragment entrapment in the compact disc, continuous and widespread gas
generation is expected. Also, the case of SDSS J1228+1040 emphasizes that
additional physics might be important.

For completion, we also note that for a gaseous compact disc, aeolian wind
erosion is also capable of eroding the surface of fragments for typical
relative velocities of a few dozen m $\times$ s-1, which corresponds to
extremely low $e$ in the range 0.0001-0.0002. Wind erosion is however a purely
mechanical process, not unlike the erosion of sand dunes. It applies to weakly
consolidated surfaces, and was initially invoked for protoplanetary discs
(Rozner et al., 2020b; Grishin et al., 2020b). Recent study of wind erosion by
Rozner et al. (2020a) assumes a-priori that fragments with the appropriate
mechanical properties reside in WD disc. In our study however the fragments
must first _become_ embedded in the compact disc by circularization, hence
they inescapably first interact with the disc at much higher velocities. In
turn, if they are weakly consolidated, it is unclear that they would survive
those interactions or the ram pressure analysis discussed in Section 5.2.

A conceivable picture finally emerges from the above arguments. At the initial
stages of tidal debris formation, following a tidal disruption event of a
planetary object, an influx of tidally disrupted fragments should interact
with a pre-existing compact disc and generate gas via sputtering (Jura, 2008)
if fragments meet obstructing gas along their path. In addition, any
coincident dust grains in the pre-existing compact disc would be _entirely_
collisionally vaporized, and so would similar-sized or slightly larger grains
in the tidal stream itself. Larger tidal fragments (of sizes ranging between
$10^{1}-10^{3}$ m) will collisionally self-erode mainly through shattering,
however the secondary effect is likewise grain-grain collisions, again leading
to gas production. Hence, large fragment self-erosion would replenish the
compact disc in new dust and gas, however not instantaneously, since the
arguments in Section 3.1 make it clear that material in the compact disc
initially attains the same velocity as the fragments themselves.

Finally, if some fragments bear volatiles or super-volatiles, all of the
above-mentioned mechanisms will be enhanced. Namely, the sputtering yield
would increase by at least an order of magnitude, and any collisionally
shattered grains would have an increased chance of being sublimated even if
shadowed by dust inside the opaque compact disc. Fragments whose orbital
energy has been sufficiently dissipated by the interactions with the compact
disc become trapped in it. Then there is at least a short phase during which
relative velocities are sufficient to continuously rather than impulsively
generate more widespread gas. Arguments presented in the MP20 and Veras et al.
(2014) studies indeed favour such an influx of fragments at the earlier stages
of tidal debris disc formation. Hence, we predict gas formation to be an
inescapable part our model. In the following section we continue discussing
the conditions for having a steady gas component in the compact disc.

###  5.4 Observational consequences

Until now certain studies (e.g., Grishin & Veras (2019) and O’Connor & Lai
(2020)) have assumed that a pre-existing compact disc exists and that it has a
gaseous component, however without specifying why. In the previous section we
have shown how a passing tidal fragment can interact with a compact disc to
generate gas, and we shall now further discuss under what conditions the gas
production can be ongoing. This will be the first observational consequence of
our model, and will depend on the balance between gas production and
condensation. The second potential consequence of our model is the formation
of a dust halo, leading to infrared variability.

Consider again the various gas forming mechanisms, presented in Section 5.3.
Even if we completely neglect gas production by grain-grain collisional
vaporization, sputtering or sublimation of volatiles, collisional vaporization
of pre-existing dust grains in the compact disc (by a crossing fragment) would
convert an entire column of dust with volume $hR^{2}$ and mass $\Sigma R^{2}$
to gas. Additionally, more dust would be produced from self-erosion of the
fragment by shattering. According to the discussion in Section 3.2, all this
material should initially obtain a velocity of similar magnitude to that of
the fragment. It is reasonable to assume that not all of this material will
attain exactly the fragment trajectory. More likely, at least some part of it
might scatter in different directions by the interaction. This material would
attain different orbits than the stream. Neglecting the details of this
chaotic scatter, we can simply assume that the interaction leads to various
concentrations of gas and dust around the WD, forming a sort of halo, in
addition to material accompanying the fragment.

Let us now consider a tidal fragment following the interaction with the
compact disc. We have just outlined that as it emerges from the compact disc
it is surrounded by a cloud of gas and dust of mass $\sim\Sigma R^{2}$,
initially travelling along the original fragment trajectory. However, if
indeed even a thin halo of material exists, gas and small dust grains would
quickly decelerate to match the velocity of the halo whereas the infinitely
larger (relatively speaking) fragment would continue uninterrupted. What will
be the probable fate of such gas and dust? First we note that the temperatures
above and below an opaque compact dust disc are necessarily much higher than
inside the compact disc due to the absence of shadowing (see e.g. Melis et al.
(2010)), and therefore even refractory dust should sublimate on some
timescale, unless sufficiently far from the WD (also depending on its cooling
age). Whether dust or gas, Farihi et al. (2018) estimate that any material
should be quickly re-assimilated into the underlying compact disc, unless
radially extending beyond the disc extremities (inner or outer edge). This is
because if their orbits cross that of the compact disc, the latter would damp
relative dust inclinations and/or act to re-condense gas on orbital
timescales. In what follows we discuss these two options, namely, halo
material internal to the compact disc extremities, which could lead to
detectable ongoing gas, and halo material outside the outer edge of the disc
which could lead to infrared variability.

####  5.4.1 Ongoing gas production

As we have previously described, a constant flux of fragments necessarily
produces gas. However the gas might only be present at quasi steady state if
the rate of production exceeds the rate of gas condensation. Let us then
consider the condensation timescale of the gas. In the study of Metzger et al.
(2012), the authors state that it is not clear how gas can exist
simultaneously in stable phase equilibrium with adjacent dust. The compact
dust disc is believed to be optically thick and the temperature of the solids
is necessarily below the sublimation temperature for all but the innermost
annulus of dust (otherwise the entire compact disc would all have been
gaseous). If hotter gas exists at any time, atoms of gas should stick upon
colliding with the surface of dust particles, resulting in condensation. The
probability of sticking is given by the so-called sticking coefficient
$\alpha$ (Leitch-Devlin & Williams, 1985). The condensation timescale
$\tau_{\rm cond}$ is shown to be $\sim P/\alpha$, where $P$ is the orbital
time of the gaseous component at separation $q$. We thus have:

$\tau_{\rm cond}=\alpha^{-1}\sqrt{\frac{q^{3}}{GM_{\rm WD}}}.$ (15)

In the study of Metzger et al. (2012), Equation 15 constituted as a serious
problem since in the absence of steady gas formation in their model, any
existing gas should condense in typical timescales of a few to a few hundreds
of orbital periods, depending on how small $\alpha$ is (one orbital period is
$\sim$3-4 h at $r=R_{\odot}$). In our model this problem is greatly alleviated
by the fact that we do have a steady mechanism that indeed forms gas
continuously. As shown by MP20 and other studies, during the early phase of
tidal disruption and debris disc formation, an influx of eccentric tidal
fragments ensues. For continuous gas we require $\tau_{\rm cross}<\tau_{\rm
cond}$, otherwise gas is only produced intermittently. We use Equations 11 and
15 to obtain a condition on the semi-major axis $a$ such that

$a\lessapprox
5000~{}\rm{AU}\left(\frac{q}{R_{\odot}}\right)\left(\frac{L}{10~{}\rm{km}}\right)^{2}\left(\frac{R}{10~{}\rm{m}}\right)^{-2},$
(16)

where we assume $\alpha$ is approximately of order unity (Leitch-Devlin &
Williams, 1985). Taking fiducial values of $q=R_{\odot}$ and $L=10$ km (an
asteroid), we see that gas production is continuous for 10 m fragments
virtually irrespective of the separation in the range $a<10^{3}$ AU. On the
other hand, 1000 m fragments require $a<0.5$ AU which almost never happens in
our model.

Equation 16 therefore provides an interesting prediction. During the initial
stages of a tidal disruption event, the tidal stream is still abundant with
small fragments. These however circularize much faster than the large
fragments, as clearly evident in Figures 1 and 2. After the small fragments
are embedded in the compact disc, the number of remaining large fragments is
much lower (given an initial power-law fragment size distribution) and
therefore while gas is still being produced, it happens discontinuously and
the gas quickly condenses. Therefore, gas is probably much easier to detect
during the initial phases following tidal disruptions, and much harder to
detect as tidal streams mature and evolve. This prediction is in agreement
with the observation that fewer discs are observed to have a gas component
compared to the number of discs that have dust (Manser et al., 2020), if the
progenitor is assumed to be asteroid-sized. It is also seemingly compatible
with the case of SDSS J1617+1620, where strong double-peaked CaII emission
lines appeared in 2008 and monotonically faded over a period of a few years
(Wilson et al., 2014). In our model, a sufficiently small asteroid progenitor
might produce this pattern, having only a small mass in the ensuing tidal
stream, which evolves rapidly. We note to contrast, that large progenitors
have large $L$, in addition to tidal fragments that typically have small $a$
and a wide spread in orbital energies. This requires a different derivation
for $\tau_{\rm cross}$. Also, their mass has a larger probability to exceed
that of the pre-existing compact disc, suggesting that at the end of a partial
circularization phase, gas production (if it occurs) must ensue from other
processes and no longer through interacting with a pre-existing compact disc.

We also note that gas production in the early stages following a tidal
disruption event is contributed by self-erosion of small fragments, as will be
discussed in more details in Section 5.7.

Finally we note that another potential consequence could be variability in the
UV. Since most circumstellar absorption lines are concentrated in the UV (Xu
et al., 2019), variable concentration of halo gas might yield UV variability
and this could be checked by performing UV photometry of known gaseous discs.

####  5.4.2 Detection of infrared variability

Figure 4: A schematic diagram (not to scale) depicting the interaction of
tidal fragments with the pre-existing compact disc, with possible
observational consequences. An influx of fragments from a tidal stream is
collisionally interacting with the compact disc. During each crossing a gap
forms in the rotating compact disc, whose size depends on the fragment size.
The fragment scatters material in the collision and continues on its new
trajectory after drag-induced orbital dissipation. As the compact disc rotates
gaps are filled through diffusion, however large gaps fill more slowly.
Fragments are accompanied by a trailing cloud of material (from compact disc
and self-erosion). These clouds experience sublimation/condensation depending
on radial distance from the WD. They decelerate through interacting with an
inner halo of gas and outer halo of dust. High concentrations of gas in the
inner halo can exist if the influx of tidal fragments exceeds the rate of
condensation onto the compact disc. Dust in the outer halo infalls through PR
drag and may produce infrared variability. Eventually migration inwards of the
compact disc’s outer edge results in re-assimilation.

Assuming that some small fraction of material is scattered beyond the compact
disc outer edge (hereafter the ’outer halo’), what is the significance and
possible implications of such material? As previously stated, the temperatures
above and below an opaque compact dust disc are higher than in the shadowed
internal parts (see e.g. Jura (2003) and Melis et al. (2010)). Depending on
the exact WD cooling age, scattered and unshielded refractory dust might
sublimate on some timescale, unless the dust grains are too large. However,
material reaching far enough from the WD would necessarily condense to form
dust. Regardless of WD cooling age, we thus speculate that an outer halo
exists, far enough from the WD that it may be composed primarily of optically
thin dust grains. Some of this dust could initially scatter to these wide
orbits around the WD during the collisions between fragments and the compact
disc. More dust is perhaps accumulated in this outer halo through interacting
with the tidal fragments (and associated clouds of material around them) that
pass through it. A third option is dust produced by large (in our model larger
than about 1000 m), tidally dissecting fragments. Although the dissected sub-
fragments are much larger than mere dust (Rafikov, 2018), a small fraction of
dust may be released, adding to the outer halo dust. In this work we do not
provide a calculation of the exact amount of dust to reach the outer halo,
since this calculation is beyond the scope of our current paper. This requires
a more specific and detailed model which should be left to future dedicated
studies. We however discuss the fate of such material if indeed it exists.

Extending beyond the outer edge of the compact disc, rapid damping by direct
contact with the compact disc would no longer be possible. However, PR drag
would cause the dust grains to drift inward. At some point they might
sublimate, depending on their size and the WD cooling age. When the gas/dust
eventually falls below the outer edge of the compact disc it would be re-
assimilated in it as discussed in Section 5.4.1. The dominant timescale in the
above picture is $\tau_{\rm PR}$, the timescale of radial drift via PR drag.
From Kenyon & Bromley (2017a) we have:

$\tau_{\rm PR}\cong 5{\rm yr}\left(\frac{d}{1\mu
m}\right)\left(\frac{a}{R_{\odot}}\right)^{2}\left(\frac{10^{-2}L_{\odot}}{L_{\rm
WD}}\right),$ (17)

where $r_{\rm d}$ is the grain size, $a$ its semi-major axis (assuming small
eccentricity) and $L_{\rm WD}$ the WD luminosity. The radial drift timescale
is in the order of years, however the grains must only travel a fraction of
the distance to the WD (they cease to be optically thin dust when they fall
below their sublimation radius or reach the outer edge of the compact disc).
Additionally, grains of various sizes drift at different speeds and therefore
collisional damping might also expedite the inward radial drift. Overall the
required timescale might even be weeks or months.

Interestingly, Farihi et al. (2018), Swan et al. (2019), Wang et al. (2019)
and more recently Swan et al. (2020) have shown evidence for widespread
infrared variability which is found across a large population of stars. Not
only that, increased variability correlates with stars which also feature
emitting gas. The timescale of highest variability is months rather than hours
or days, noting however that very few observations in the sample are spaced
less than a week apart. Farihi et al. (2018) and Swan et al. (2020) interpret
these findings as related to collisions, without providing a detailed model.
However, as outlined in Section 5.4.1, if the optically thin dust is internal
to the compact disc extremities, these authors cannot easily accommodate their
proposed collisions with the long variability timescales observed.

If however an outer halo exists, the timescales are more compatible. Any
infrared variability on timescales longer than hours (approximately
corresponding to re-assimilation in the compact disc) should be governed by
how frequently large deposits of dust reach the outer halo. If the time
between such deposits is longer than the aforementioned PR drag timescale then
the PR drag timescale would be subdominant. If the opposite is true, then
variability could be a signature of the fragment size distribution, or perhaps
a sign of clumping of fragments in the tidal stream (which usually correlates
with an early phase in the evolution of the tidal stream). More accurate and
detailed models are required in order to develop this idea further.

For completeness we also note that in a recent 3 yr photometric near-infrared
survey of 34 WDs in the J, H and especially the $\sim 2\mu$m K band which is
more sensitive to dust in comparison, a largely stable flux with no
significant variability has been reported (Rogers et al., 2020). This
contrasts with previous K-band variability reports ranging between 13% (Xu et
al., 2018) and 18.5% (Xu & Jura, 2014) in two respective WDs. Of course the
Spitzer $3.6\mu$m and $4.5\mu$m bands which were mentioned earlier (Farihi et
al., 2018; Swan et al., 2019; Wang et al., 2019; Swan et al., 2020) are more
sensitive to the dust variability and thus we interpret our model based on
these findings.

Finally, we wish to emphasize that the idea raised in this section is
essentially _independent_ of whether or not gas production is ongoing or
intermittent. However, if gas production is ongoing it probably means that
there is a much higher concentration of gas and perhaps dust inside the inner
halo (i.e the part of the halo having a radial distance smaller than $r_{\rm
out}$). In turn fragments travelling through the inner halo have a greater
chance of scattering material from the inner to the outer halo, hence one
might expect increased variability. That is in addition to the fact that the
fragment crossings are more numerous at earlier times implying more material
is scattered. Swan et al. (2020) have indeed shown that discs that also have
gas in emission feature higher variability.

Figure 4 visually captures the ideas proposed in Sections 5.4.1 and 5.4.2.
Interestingly, somewhat similar ideas regarding halo structures have already
been proposed based on the merit of fitting the observational evidence (e.g.,
see section 3.3 in Farihi (2016)).

###  5.5 Forming the previous compact disc

A caveat of the model is obviously the requirement for a pre-existing compact
disc. In other words, in the framework of our model it is impossible to shrink
an eccentric tidal stream generated by a recent tidal disruption event without
first having a compact disc around the WD with which it can interact. Within
the confines of our model we nevertheless hypothesize that the recent
progenitor that generated the tidal stream eventually adds its own mass and
replenishes the exiting compact disc. Under certain conditions, which are
discussed in the next Section (5.6), this could allow for longevity of the
compact disc, ensuring its survival over multiple tidal disruption events.
Alternatively, the compact disc may also be dispersed under other conditions,
which raises the questions - what other processes can initially generate the
pre-existing compact disc for our model to work?

We would argue that this is not a ’chicken and egg’ type of problem, since the
compact disc can probably form without dust-assisted circularization, only on
longer timescales or else assuming a different composition. In Section 1 we
have already mentioned some of the other potential mechanisms that may enable
the formation of a compact disc, and we would repeat them briefly: (1) the
Yarkovsky effect is in theory capable of affecting tidal fragments of the size
we have explored (Veras et al., 2015); (2) if collisional cascade may break
initially large fragments to mere dust (Wyatt et al., 2011) then the disc may
shrink via PR drag (Veras et al., 2014); (3) perhaps our favourite alternative
is a tidal stream consisting of fragments so highly rich in volatile/super-
volatile materials that it deposits a large fraction of its mass in sublimated
gas during its preliminary or secondary flyby around the WD. This gas would
trigger the onset of circularization via our model, initially through gas drag
only, without the need for other models. Nor does it require long timescales
in order to operate.

###  5.6 Implications of partial circularization

We recall from Sections 3.2 and 3.3 that when the tidal stream progenitor mass
is comparable or larger than the compact disc mass, i.e $M\lessapprox M_{\rm
progenitor}$, we have partial circularization. The compact disc is dispersed
and we are left with a partially circularized tidal stream + scattered
optically thin material in close WD orbit. As we shall see in the following
section, fragment erosion in that case is negligible.

If the the tidal stream progenitors are equal-sized, what would be the model
implications? Suppose the time interval between planetesimal injections (which
give rise to tidal disruptions and subsequent tidal streams) is shorter than
time it takes the compact disc to form without a pre-existing disc (i.e,
through a combination of Yarkovsky, catastrophic collisions, PR drag or some
other processes). In such a case, we have no mechanism to form the canonical
compact disc in the first place, since if it would form - it would be
dispersed. Accretion onto the WD would be mainly controlled by PR drag. If
instead some WD systems are configured to inject planetesimals at a lower rate
compared to the compact disc formation timescale, then these systems would be
capable of hosting compact discs. Such compact discs would persist until
dispersed by some future tidal disruption event of comparable progenitor mass
or dispersed by intrinsic processes on some characteristic timescale.

Another possibility is that occasionally massive planetesimals are injected to
form unusually massive compact discs compared with the typical planetesimal
masses. Such a compact disc would resist a catastrophic fate until the next
tidal disruption of a similarly massive planetesimal. As long as it encounters
injected planetesimals of significantly less mass, it would be able to fully
circularize the ensuing tidal streams. How long such a compact disc may
persist depends on the balance between its intrinsic lifetime (rate of mass
loss to WD accretion) and the mass contribution from fragment self-erosion
(i.e the tidal streams also contribute their own mass, as we discuss next).

The possibilities above lack the details to support a decisive conclusion
among the two possibilities, however the point we wish to emphasize is that
the scarce observations of dusty compact discs in combination with our model,
generally imply a dependence on either planetesimal injection rate or size
distribution.

We also suggested that in the absence of a compact disc, the accretion rate
onto the WD might be steadily controlled by PR drag. However, in the presence
of a compact disc interacting with a tidal stream, the rate of accretion might
increase via brief and intense accretion episodes from tidal disruption
events. Similar ideas were suggested by Jura (2008); Farihi et al. (2012) in
order to explain why polluted He-dominated WDs exhibit average accretion rates
that exceed that of polluted H-dominated WDs by one or two orders of
magnitudes, while the rate in H-dominated WDs is considered to be in agreement
with the theoretical mass transfer limit via PR drag (Girven et al., 2012). We
note that other ideas were also suggested, such as a runway accretion model
(Rafikov, 2011) and differential volatile accretion in rare water-bearing
planetesimals (Malamud & Perets, 2016).

###  5.7 Erosion vs. circularization timescales

In section 5.3 we identify two fragment erosion channels. In a gaseous compact
disc we have sputtering, producing a yield $Y$ in the range 0.1-1. In a dusty
compact disc we have shattering, producing an effective yield of $\sim$1\. By
simple arguments, we may now estimate if fragment self-erosion can potentially
hinder the circularization process.

For a fragment of mass $m$, radius $R$ and density $\rho$ we have $m=4\pi\rho
R^{3}/3$. The yield is by definition the eroded fragment mass over the
incident compact disc mass, and therefore if it crosses the compact disc face-
on, $\Delta m$ is $\pi R^{2}\Sigma Y$. We now have

$\frac{dR}{dt}=\frac{\Delta m}{\Delta t}\frac{dR}{dm}=\frac{\Sigma
Y}{8\pi\rho}\sqrt{\frac{GM_{\rm WD}}{a^{3}}},$ (18)

where $\Delta t=P=2\pi\sqrt{a^{3}/GM_{\rm WD}}$ is the orbital period. Using
Equation 18 we may obtain the erosion timescale $\tau_{\rm erosion}$ from

$\tau_{\rm erosion}=\frac{R}{dR/dt}=\frac{8\pi\rho R}{\Sigma
Y}\sqrt{\frac{a^{3}}{GM_{\rm WD}}}.$ (19)

Erosion outpaces circularization when, using Equation 6,

$\frac{\tau_{\rm erosion}}{\tau_{\rm circ}}<1.$ (20)

Or equivalently, when

$Y>6\frac{a}{q}.$ (21)

Even if we take the minimum semi-major axis $a=0.1$ AU, and the maximum
pericentre $q=1R_{\odot}$, we require $Y>129$ for the erosion timescale to
replace the circularization timescale. Since we have found that $Y\leq 1$, we
conclude that the circularization time is essentially unaffected by erosion.

We also conclude that for partial circularization, erosion is entirely
negligible. We have shown in Section 3.3 that even when $M_{\rm progenitor}$
is many orders of magnitude larger than the compact disc mass $M$, the tidal
stream undergoes significant circularization. The mass eroded away
(collectively) from the tidal stream is roughly equal to $MY$, hence it would
be negligible compared to the remaining mass in the stream.

Finally, if instead $M\gg M_{\rm progenitor}$ and full circularization ensues,
Equation 9 shows that the tidal stream encounters nearly its own mass in the
compact disc. If the yield $Y$ is equal or close to 1, as we have found, by
definition the tidal stream is greatly eroded by the time it circularizes
since the eroded mass is the mass encountered in the compact disc times
$\sim$1\. Fully circularized fragments embedded in the disc will always turn
out to be a small fraction of their initial mass by the time they circularize.
Additionally, eroded gas and dust grains quickly conjoin the compact disc (as
in Section 5.4).

##  6 Summary

Various theoretical studies have shown that planetary tidal disruptions by WDs
initially form extremely eccentric fragment streams, with $e>0.9$ and often
approaching unity. Around a subset of systems, observational evidence of
infrared excesses indicates that some structures eventually shrink to form
compact, nearly circular discs. Further observational support comes from rare
compact discs that have a gaseous component. Explaining why this change occurs
poses a major theoretical challenge, as the tidal streams need to lose large
amounts of both energy and angular momentum. Previous theoretical works have
suggested the reprocessing of stellar light, which can alter the orbits of the
tidal stream constituents either by PR drag or the Yarkovsky force. However
the former requires that the stream is primarily composed of small dust, which
is (at least initially) incorrect, and the latter remains to be thoroughly
explored. In this paper, we suggest a new mechanism: drag-assisted
circularization as an effective alternative in systems that posses a
circumstellar disc.

Our model is based around the observation that these dust/gas discs around WDs
are typically compact and located within the stellar Roche radius. This means
that fragments from subsequent tidal disruptions will collide with this
material at their pericenter, leading to high velocity collisions that
generate strong drag forces on the fragments, while carving gaps in the
compact disc. We calculate the resulting changes in the fragment’s orbits with
an analytical model, where we account for the diffusive filling of the gaps.
We then verify our expressions with more detailed N-body simulations.

Our results indicate that drag-assisted circularization is a rapid process
that can completely circularize large fragments up to a kilometre in size
within the expected disc lifetimes. This efficiency is primarily due to the
high pericentre velocities at near-unity ecccentricity, that greatly speed up
the process. We find that full circularization of tidal fragments can occur if
the compact disc mass is as low as $10^{16}$ g. Interestingly, drag-assisted
circularization is efficient for fragments on a wide range of semi-major axes,
as the longer orbital timescales at wider orbits are compensated by their
increased perictenter velocity.

We also find that the outcome of drag-based circularization is primarily
determined by the mass ratio of the compact disc and stream progenitor. This
comes from the fact that complete circuarization requires a compact disc that
is ideally much more massive than the tidal stream, in which case the tidal
stream would eventually conjoin the compact disc. Significant partial
circularization is on the other hand possible even when the tidal stream
exceeds the compact disc by up to seven orders of magnitude in mass, but then
the compact disc is dispersed, and continued circularization requires other
mechanisms. We therefore expect partial drag-assisted circularization to be a
ubiquitous feature of the fragment’s orbital evolution, as even small
progenitors that form the pre-existing compact disc can significantly affect
the orbits of major bodies. We speculate that because the scale of partial
circularization is also a function of fragment size, it can trigger a
subsequent phase of collisions between fragments that grind down the remaining
material, facilitating further circularization by PR drag. We will investigate
this further in a future work.

Besides drag, the high collision velocities between fragments and the compact
disc necessarily lead to the production of significant amounts of gas mainly
through vaporization of pre-existing compact disc dust and dust eroded from
the fragments, but also via other channels. Given a size-distribution of tidal
stream fragments, gas production starts off continuously, yielding an
observable signature of ongoing early circularization. But as the smaller
fragments circularize their collision velocities decline, the gas production
then originates from increasingly larger fragments and thus it becomes
increasingly intermittent and harder to observe. Gas production finally halts
completely when either all the tidal fragments are circularized or the compact
disc has dissipated. In this way, our model offers a natural explanation for
the fact that gaseous material is only observed around a small fraction of
polluted WDs.

We also speculate that fragments can scatter material far from the WD, giving
rise to a dusty halo which evolves towards the WD through PR drag until
eventually the material would be re-assimilated inside the compact disc. The
timescale could be approximately compatible with recent observational evidence
for infrared variability.

A necessary requirement of the model is the pre-existence of a compact disc
around the WD. We show that this problem is not insurmountable since a number
of mechanisms may enable its formation in the first place.

## Acknowledgements

We would like to thank Jay Farihi and Na’ama Hallakoun for extremely helpful
discussions relating to the observational constraints associated with debris
discs, and also Christopher Manser for helpful references regarding observed
systems. We would like to thank anonymous referee for highly useful
suggestions that greatly improved the quality of the paper. UM acknowledges
support from the Pazy Foundation grant 01023882.

## Data availability

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

## References

  * Ahrens & O’Keefe (1972) Ahrens T. J., O’Keefe J. D., 1972, Moon, 4, 214
  * Bear & Soker (2015) Bear E., Soker N., 2015, Monthly Notices of the Royal Astronomical Society, 450, 4233
  * Behrisch & Eckstein (2007) Behrisch R., Eckstein W., 2007, Sputtering by Particle Bombardment. Springer
  * Biele et al. (2015) Biele J., et al., 2015, Science, 349, 1.9816
  * Bochkarev & Rafikov (2011) Bochkarev K. V., Rafikov R. R., 2011, The Astrophysical Journal, 741, 36
  * Bonsor et al. (2011) Bonsor A., Mustill A. J., Wyatt M. C., 2011, Monthly Notices of the Royal Astronomical Society, 414, 930
  * Bonsor et al. (2017) Bonsor A., Farihi J., Wyatt M. C., van Lieshout R., 2017, Monthly Notices of the Royal Astronomical Society, 468, 154
  * Brinkworth et al. (2009) Brinkworth C. S., Gänsicke B. T., Marsh T. R., Hoard D. W., Tappert C., 2009, The Astrophysical Journal, 696, 1402
  * Bromley & Kenyon (2013) Bromley B. C., Kenyon S. J., 2013, The Astrophysical Journal, 764, 192
  * Brown et al. (2017) Brown J. C., Veras D., Gänsicke B. T., 2017, Monthly Notices of the Royal Astronomical Society, 468, 1575
  * Caiazzo & Heyl (2017) Caiazzo I., Heyl J. S., 2017, Monthly Notices of the Royal Astronomical Society, 469, 2750
  * Cauley et al. (2018) Cauley P. W., Farihi J., Redfield S., Bachman S., Parsons S. G., Gänsicke B. T., 2018, The Astrophysical Journal Letters, 852, L22
  * D’Angelo & Podolak (2015) D’Angelo G., Podolak M., 2015, The Astrophysical Journal, 806, 203
  * Debes & Sigurdsson (2002) Debes J. H., Sigurdsson S., 2002, The Astrophysical Journal, 572, 556
  * Debes et al. (2012) Debes J. H., Walsh K. J., Stark C., 2012, The Astrophysical Journal, 747, 148
  * Dennihy et al. (2016) Dennihy E., Debes J. H., Dunlap B. H., Dufour P., Teske J. K., Clemens J. C., 2016, The Astrophysical Journal, 831, 31
  * Dennihy et al. (2018) Dennihy E., Clemens J. C., Dunlap B. H., Fanale S. M., Fuchs J. T., Hermes J. J., 2018, The Astrophysical Journal, 854, 40
  * Dennihy et al. (2020) Dennihy E., et al., 2020, arXiv e-prints, p. arXiv:2010.03693
  * Farihi (2016) Farihi J., 2016, New Astronomy Reviews, 71, 9
  * Farihi et al. (2009) Farihi J., Jura M., Zuckerman B., 2009, The Astrophysical Journal, 694, 805
  * Farihi et al. (2010) Farihi J., Jura M., Lee J. E., Zuckerman B., 2010, The Astrophysical Journal, 714, 1386
  * Farihi et al. (2012) Farihi J., Gänsicke B. T., Wyatt M. C., Girven J., Pringle J. E., King A. R., 2012, Monthly Notices of the Royal Astronomical Society, 424, 464
  * Farihi et al. (2018) Farihi J., et al., 2018, Monthly Notices of the Royal Astronomical Society, 481, 2601
  * Fortin-Archambault et al. (2020) Fortin-Archambault M., Dufour P., Xu S., 2020, The Astrophysical Journal, 888, 47
  * Gänsicke et al. (2006) Gänsicke B. T., Marsh T. R., Southworth J., Rebassa-Mansergas A., 2006, Science, 314, 1908
  * Gänsicke et al. (2008) Gänsicke B. T., Koester D., Marsh T. R., Rebassa-Mansergas A., Southworth J., 2008, Monthly Notices of the Royal Astronomical Society, 391, L103
  * Gänsicke et al. (2012) Gänsicke B. T., Koester D., Farihi J., Girven J., Parsons S. G., Breedt E., 2012, Monthly Notices of the Royal Astronomical Society, 424, 333
  * Gentile Fusillo et al. (2020) Gentile Fusillo N. P., et al., 2020, arXiv e-prints, p. arXiv:2010.13807
  * Girven et al. (2012) Girven J., Brinkworth C. S., Farihi J., Gänsicke B. T., Hoard D. W., Marsh T. R., Koester D., 2012, The Astrophysical Journal, 749, 154
  * Goldreich & Soter (1966) Goldreich P., Soter S., 1966, Icarus, 5, 375
  * Goldreich & Tremaine (1978) Goldreich P., Tremaine S. D., 1978, Icarus, 34, 227
  * Graham et al. (1990) Graham J. R., Matthews K., Neugebauer G., Soifer B. T., 1990, The Astrophysical Journal, 357, 216
  * Grishin & Perets (2015) Grishin E., Perets H. B., 2015, The Astrophysical Journal, 811, 54
  * Grishin & Veras (2019) Grishin E., Veras D., 2019, Monthly Notices of the Royal Astronomical Society, 489, 168
  * Grishin et al. (2019) Grishin E., Perets H. B., Avni Y., 2019, Monthly Notices of the Royal Astronomical Society, 487, 3324
  * Grishin et al. (2020a) Grishin E., Malamud U., Perets H. B., Wand el O., Schäfer C. M., 2020a, Nature, 580, 463
  * Grishin et al. (2020b) Grishin E., Rozner M., Perets H. B., 2020b, The Astrophysical Journal Letters, 898, L13
  * Hamers & Portegies Zwart (2016) Hamers A. S., Portegies Zwart S. F., 2016, Monthly Notices of the Royal Astronomical Society, 462, L84
  * Hameury (2020) Hameury J. M., 2020, Advances in Space Research, 66, 1004
  * Hut (1981) Hut P., 1981, Astronomy & Astrophysics, 99, 126
  * Izquierdo et al. (2018) Izquierdo P., et al., 2018, Monthly Notices of the Royal Astronomical Society, 481, 703
  * Jura (2003) Jura M., 2003, The Astrophysical Journal, 584, L91
  * Jura (2008) Jura M., 2008, The Astronomical Journal, 135, 1785
  * Karjalainen et al. (2019) Karjalainen M., de Mooij E. J. W., Karjalainen R., Gibson N. P., 2019, Monthly Notices of the Royal Astronomical Society, 482, 999
  * Kenyon & Bromley (2017a) Kenyon S. J., Bromley B. C., 2017a, The Astrophysical Journal, 844, 116
  * Kenyon & Bromley (2017b) Kenyon S. J., Bromley B. C., 2017b, The Astrophysical Journal, 850, 50
  * Kilic et al. (2006) Kilic M., von Hippel T., Leggett S. K., Winget D. E., 2006, The Astrophysical Journal, 646, 474
  * Koester (2009) Koester D., 2009, Astronomy and Astrophysics, 498, 517
  * Koester et al. (2014) Koester D., Gänsicke B. T., Farihi J., 2014, Astronomy and Astrophysics, 566, A34
  * Koschny & Grün (2001) Koschny D., Grün E., 2001, Icarus, 154, 402
  * Kratter & Perets (2012) Kratter K. M., Perets H. B., 2012, The Astrophysical Journal, 753, 91
  * Kudriavtsev et al. (2005) Kudriavtsev Y., Villegas A., Godines A., Asomoza R., 2005, Applied Surface Science, 239, 273
  * Leinhardt & Stewart (2009) Leinhardt Z. M., Stewart S. T., 2009, Icarus, 199, 542
  * Leitch-Devlin & Williams (1985) Leitch-Devlin M. A., Williams D. A., 1985, Monthly Notices of the Royal Astronomical Society, 213, 295
  * Malamud & Perets (2016) Malamud U., Perets H. B., 2016, The Astrophysical Journal, 832, 160
  * Malamud & Perets (2017a) Malamud U., Perets H. B., 2017a, The Astrophysical Journal, 842, 67
  * Malamud & Perets (2017b) Malamud U., Perets H. B., 2017b, The Astrophysical Journal, 849, 8
  * Malamud & Perets (2020a) Malamud U., Perets H. B., 2020a, Monthly Notices of the Royal Astronomical Society, 492, 5561
  * Malamud & Perets (2020b) Malamud U., Perets H. B., 2020b, Monthly Notices of the Royal Astronomical Society, 493, 698
  * Manser et al. (2016a) Manser C. J., et al., 2016a, Monthly Notices of the Royal Astronomical Society, 455, 4467
  * Manser et al. (2016b) Manser C. J., Gänsicke B. T., Koester D., Marsh T. R., Southworth J., 2016b, Monthly Notices of the Royal Astronomical Society, 462, 1461
  * Manser et al. (2019) Manser C. J., et al., 2019, Science, 364, 66
  * Manser et al. (2020) Manser C. J., Gänsicke B. T., Gentile Fusillo N. P., Ashley R., Breedt E., Hollands M., Izquierdo P., Pelisoli I., 2020, Monthly Notices of the Royal Astronomical Society, 493, 2127
  * McKinnon et al. (2020) McKinnon W. B., et al., 2020, Science, 367, aay6620
  * Melis & Dufour (2017) Melis C., Dufour P., 2017, The Astrophysical Journal, 834, 1
  * Melis et al. (2010) Melis C., Jura M., Albert L., Klein B., Zuckerman B., 2010, The Astrophysical Journal, 722, 1078
  * Melis et al. (2020) Melis C., Klein B., Doyle A. E., Weinberger A. J., Zuckerman B., Dufour P., 2020, arXiv e-prints, p. arXiv:2010.03695
  * Metzger et al. (2012) Metzger B. D., Rafikov R. R., Bochkarev K. V., 2012, Monthly Notices of the Royal Astronomical Society, 423, 505
  * Michaely & Perets (2014) Michaely E., Perets H. B., 2014, The Astrophysical Journal, 794, 122
  * Mustill & Villaver (2012) Mustill A. J., Villaver E., 2012, The Astrophysical Journal, 761, 121
  * Mustill et al. (2014) Mustill A. J., Veras D., Villaver E., 2014, Monthly Notices of the Royal Astronomical Society, 437, 1404
  * Muto et al. (2011) Muto T., Takeuchi T., Ida S., 2011, The Astrophysical Journal, 737, 37
  * Nixon et al. (2020) Nixon C. J., Pringle J. E., Coughlin E. R., Swan A., Farihi J., 2020, arXiv e-prints, p. arXiv:2006.07639
  * O’Connor & Lai (2020) O’Connor C. E., Lai D., 2020, arXiv e-prints, p. arXiv:2005.05977
  * O’Rourke et al. (2020) O’Rourke L., et al., 2020, Nature, 586, 697
  * Ostriker (1999) Ostriker E. C., 1999, The Astrophysical Journal, 513, 252
  * Payne et al. (2016) Payne M. J., Veras D., Holman M. J., Gänsicke B. T., 2016, Monthly Notices of the Royal Astronomical Society, 457, 217
  * Payne et al. (2017) Payne M. J., Veras D., Gänsicke B. T., Holman M. J., 2017, Monthly Notices of the Royal Astronomical Society, 464, 2557
  * Perets & Kratter (2012) Perets H. B., Kratter K. M., 2012, The Astrophysical Journal, 760, 99
  * Petrovic (2002) Petrovic J., 2002, in 34th COSPAR Scientific Assembly.
  * Petrovich & Muñoz (2017) Petrovich C., Muñoz D. J., 2017, Astrophysical Journal, 834, 116
  * Podolak et al. (1988) Podolak M., Pollack J. B., Reynolds R. T., 1988, Icarus, 73, 163
  * Popova et al. (2011) Popova O., Borovička J., Hartmann W. K., Spurný P., Gnos E., Nemtchinov I., Trigo-Rodríguez J. M., 2011, Meteoritics and Planetary Science, 46, 1525
  * Pravec et al. (2002) Pravec P., Harris A. W., Michalowski T., 2002, Asteroid Rotations. University of Arizona Press, Tucson, p.113-122, pp 113–122
  * Rafikov (2011) Rafikov R. R., 2011, Monthly Notices of the Royal Astronomical Society: Letters, 416, L55
  * Rafikov (2018) Rafikov R. R., 2018, The Astrophysical Journal, 861, 35
  * Rappaport et al. (2018) Rappaport S., Gary B. L., Vanderburg A., Xu S., Pooley D., Mukai K., 2018, Monthly Notices of the Royal Astronomical Societ, 474, 933
  * Rebassa-Mansergas et al. (2019) Rebassa-Mansergas A., Solano E., Xu S., Rodrigo C., Jiménez-Esteban F. M., Torres S., 2019, Monthly Notices of the Royal Astronomical Society, 489, 3990
  * Rein & Liu (2012) Rein H., Liu S.-F., 2012, Astronomy & Astrophysics, 537, A128
  * Rein & Spiegel (2015) Rein H., Spiegel D. S., 2015, Monthly Notices of the Royal Astronomical Society, 446, 1424
  * Rephaeli & Salpeter (1980) Rephaeli Y., Salpeter E. E., 1980, The Astrophysical Journal, 240, 20
  * Rocchetto et al. (2015) Rocchetto M., Farihi J., Gänsicke B. T., Bergfors C., 2015, Monthly Notices of the Royal Astronomical Society, 449, 574
  * Rogers et al. (2020) Rogers L. K., Xu S., Bonsor A., Hodgkin S., Su K. Y. L., von Hippel T., Jura M., 2020, Monthly Notices of the Royal Astronomical Society, 494, 2861
  * Rozner et al. (2020a) Rozner M., Veras D., Perets H. B., 2020a, arXiv e-prints, p. arXiv:2011.12299
  * Rozner et al. (2020b) Rozner M., Grishin E., Perets H. B., 2020b, Monthly Notices of the Royal Astronomical Society, 496, 4827
  * Shappee & Thompson (2013) Shappee B. J., Thompson T. A., 2013, The Astrophysical Journal, 766, 64
  * Smallwood et al. (2018) Smallwood J. L., Martin R. G., Livio M., Lubow S. H., 2018, Monthly Notices of the Royal Astronomical Society, 480, 57
  * Smrekar et al. (1985) Smrekar S., Cintala M. J., Horz F., 1985, in Lunar and Planetary Science Conference. Lunar and Planetary Science Conference. pp 793–794
  * Spohn et al. (2015) Spohn T., et al., 2015, Science, 349, 2.464
  * Stephan et al. (2017) Stephan A. P., Naoz S., Zuckerman B., 2017, The Astrophysical Journal Letters, 844, L16
  * Stone et al. (2015) Stone N., Metzger B. D., Loeb A., 2015, Monthly Notices of the Royal Astronomical Society, 448, 188
  * Swan et al. (2019) Swan A., Farihi J., Wilson T. G., 2019, Monthly Notices of the Royal Astronomical Society Letters, 484, L109
  * Swan et al. (2020) Swan A., Farihi J., Wilson T. G., Parsons S. G., 2020, Monthly Notices of the Royal Astronomical Society, 496, 5233
  * Tielens et al. (1994) Tielens A. G. G. M., McKee C. F., Seab C. G., Hollenbach D. J., 1994, The Astrophysical Journal, 431, 321
  * Vanderburg et al. (2015) Vanderburg A., et al., 2015, Nature, 526, 546
  * Veras (2016) Veras D., 2016, Royal Society Open Science, 3, 150571
  * Veras & Gänsicke (2015) Veras D., Gänsicke B. T., 2015, Monthly Notices of the Royal Astronomical Society, 447, 1049
  * Veras et al. (2014) Veras D., Leinhardt Z. M., Bonsor A., Gänsicke B. T., 2014, Monthly Notices of the Royal Astronomical Society, 445, 2244
  * Veras et al. (2015) Veras D., Leinhardt Z. M., Eggl S., Gänsicke B. T., 2015, Monthly Notices of the Royal Astronomical Society, 451, 3453
  * Wang et al. (2019) Wang T.-g., et al., 2019, The Astrophysical Journal Letters, 886, L5
  * Weidenschilling (1977) Weidenschilling S. J., 1977, Astrophysics and Space Science, 51, 153
  * Wilson et al. (2014) Wilson D. J., Gänsicke B. T., Koester D., Raddi R., Breedt E., Southworth J., Parsons S. G., 2014, Monthly Notices of the Royal Astronomical Society, 445, 1878
  * Wyatt et al. (2011) Wyatt M. C., Clarke C. J., Booth M., 2011, Celestial Mechanics and Dynamical Astronomy, 111, 1
  * Xu & Jura (2012) Xu S., Jura M., 2012, The Astrophysical Journal, 745, 88
  * Xu & Jura (2014) Xu S., Jura M., 2014, The Astrophysical Journal Letters, 792, L39
  * Xu et al. (2018) Xu S., et al., 2018, The Astrophysical Journal, 866, 108
  * Xu et al. (2019) Xu S., et al., 2019, The Astronomical Journal, 157, 255
  * Zhou et al. (2016) Zhou G., et al., 2016, Monthly Notices of the Royal Astronomical Society, 463, 4422
  * Zuckerman et al. (2003) Zuckerman B., Koester D., Reid I. N., Hünsch M., 2003, The Astrophysical Journal,, 596, 477
  * Zuckerman et al. (2010) Zuckerman B., Melis C., Klein B., Koester D., Jura M., 2010, The Astrophysical Journal, 722, 725
  * van Lieshout et al. (2018) van Lieshout R., Kral Q., Charnoz S., Wyatt M. C., Shannon A., 2018, Monthly Notices of the Royal Astronomical Society, 480, 2784

##  Appendix A Two vs. one intersection points

In Section 3.1 we point out that energy loss in Equation 4 ensues from a
single intersection between a fragment and the compact disc, which in turn
originates from our simplification that the argument of pericentre is assumed
zero.

As already mentioned in Section 2.4, the general case may result in two
intersection points with the compact disc instead of one. For example, let us
consider the orthogonal case where the argument of pericentre equals $\pi/2$.
Then it is easy to show that the true anomaly at the two crossing points is
$\pi/2$, and the distance between the central star and the two crossing points
is $2q$ instead of $q$. Now having two intersection points, $2\Delta E(2q)$
replaces $\Delta E(q)$ in Equation 4. $v_{rel}^{2}$ goes like $\propto 1/r$
for large eccentricities, so increasing $q$ by a factor of 2 cancels out with
the 2 prefactor. The only difference then comes from replacing $\Sigma(q)$
with $\Sigma(2q)$, which translates into a factor $2^{-\beta}$, or $\sim$0.35.
This difference is merely of order unity, and can therefore be neglected.

##  Appendix B Gap filling timescale

###  B.1 Estimating the diffusion filling time

Consider a disc of total mass $M$ with an outer boundary $r_{\rm out}.$ For
simplicity, the radial profile of the compact disc surface density is now
neglected, and $r_{\rm in}=0$. The scale height is generally given by
$h=fr_{{\rm out}},$ where $f$ is a small number which depends on the disc
properties and will be discussed later. The latter dictates a relationship
between the velocity dispersion in the disc vertical axis and the Keplerian
velocity $v_{\rm z}/v_{\rm K}\sim h/r_{\rm out}\sim f$, noting also that
$v_{\rm h}=2v_{\rm z}$ is the horizontal velocity dispersion.

Consider that a cylindrical gap of radius $R$ (i.e comparable to the radius of
the interacting fragment) has been carved in the compact disc. The dusty
particles around it will scatter to fill it. A particle at location $D$ from
the gap moving horizontally at a random direction with velocity $v_{\rm h}$.
After passing a mean free path $\lambda$ it encounters and scatters again at a
different horizontal velocity and preforms a random walk. From diffusion, the
number of steps for a particle to traverse a distance $D$ and fill the gap is
$X=(D/\lambda)^{2}$. The mean time between collisions is $\tau_{{\rm
coll}}=\lambda/v_{\rm h}$ thus the total diffusion time to fill the gap is

$\tau_{{\rm fill}}=X\tau_{{\rm
coll}}=\frac{D^{2}}{\lambda^{2}}\frac{\lambda}{v_{\rm h}}=\frac{D^{2}}{\lambda
v_{\rm h}}=\frac{D^{2}}{2\lambda fv_{\rm K}}.$ (22)

It is reasonable to assume that in order to fill a gap of dimension $R$ the
particle must also travel a similar distance, so we take $D\approx R$.

###  B.2 Estimation of the mean free path

Generally, the mean free path is $\lambda=1/nA$ where $A=\pi r_{\rm d}^{2}$ is
the surface area given a dust grain size $r_{\rm d}$, and the dust number
density $n$ is given by

$n=\frac{\rho_{\rm disc}}{m_{d}}=\frac{3\rho_{\rm disc}}{4\pi\rho_{\rm
d}r_{\rm d}^{3}},$ (23)

where $\rho_{\rm disc}$ is the density of a the pre-existing compact disc,
$m_{d}$ the mass of a dust grain and $\rho_{d}$ the density of a dust grain.
The mean free path is then

$\lambda=\frac{4\pi\rho_{\rm d}r_{\rm d}^{3}}{3\rho_{{\rm disc}}\pi r_{\rm
d}^{2}}=\frac{4}{3}\frac{\rho_{\rm d}}{\rho_{\rm disc}}r_{\rm
d}=\frac{4}{3}f\frac{\rho_{\rm d}r_{\rm out}^{3}}{M}r_{\rm d}.$ (24)

The diffusion filling time is obtained from Equation 22 as

$\tau_{{\rm fill}}=\frac{3MR^{2}}{8f^{2}\rho_{\rm d}r_{\rm out}^{3}r_{\rm
d}v_{\rm K}}=\frac{3MR^{2}q^{1/2}}{8h^{2}\rho_{\rm d}r_{\rm out}r_{\rm
d}(GM_{\rm WD})^{1/2}}.$ (25)

Note that $\tau_{\rm fill}\propto M$, meaning that as the compact disc mass
decreases, it takes less time to fill the gap. This might at first seem
counter-intuitive, but in fact it is a logical outcome. When $M$ lowers, the
disc density $\rho_{\rm disc}$ lowers. Then the dust number density decreases,
so that the mean free path is larger. Although the time between collisions
$\tau_{\rm coll}$ increases, $X$ decreases more and less steps are required to
traverse the same distance.

###  B.3 Comparison with existing literature

Above we derive the filling timescale from first principles. In order to
verify our result we compare it with previous calculation of gap filling by
Bromley & Kenyon (2013) applied to Saturn’s rings. There the time to fill the
gap is

$\tau_{{\rm fill}}=\frac{R^{2}}{\nu_{\rm rad}},$ (26)

where the radial viscosity $\nu_{\rm rad}$ is given by (Goldreich & Tremaine,
1978):

$\nu_{\rm rad}=\frac{v_{\rm h}^{2}P}{4\pi}\frac{\tau_{\rm d}}{1+\tau_{\rm
d}^{2}},$ (27)

such that $P$ is the orbital period and $\tau_{d}=3\Sigma/4\rho_{\rm d}r_{\rm
d}$ is the "collisional depth" of the dust, analogous to optical depth in the
context of photon diffusion111The optical depth concept is borrowed from
photon diffusion, and estimates the number of collisions a dust particle need
to traverse while passing in the disc. To avoid confusion, we call it
”collisional depth” to better relate to the nature of the interactions..

In the optically thick limit $\tau_{\rm d}\gg 1$, Equation 27 reduces to
$\nu_{\rm rad}=v_{\rm h}^{2}P/4\pi\tau_{\rm d}$. In terms of the disc
parameters, we recast $P=2\pi r_{\rm out}/v_{\rm K}=\pi r_{\rm out}fv_{\rm h}$
and $\Sigma\approx M/r_{\rm out}^{2}$. We then replace $\nu_{\rm rad}$ in
Equation 26 and obtain the exact same filling time as in Equation 25.

Indeed by taking typical values for $\rho_{\rm d}$, $r_{\rm d}$ and $r_{\rm