Numerical Simulations of Collisional Cascades at the Roche Limits of White Dwarf Stars

To follow the evolution of rocky solid particles orbiting a white dwarf, we rely on Orchestra, a parallel C++/MPI hybrid coagulation + n-body code that tracks the accretion, fragmentation, and orbital evolution of solid particles ranging in size from a few microns to thousands of kilometers (Kenyon 2002; Kenyon & Bromley 2008; Bromley & Kenyon 2011; Kenyon & Bromley 2016a; Kenyon et al. 2016). The ensemble of codes within Orchestra includes a multi-annulus coagulation code for small particles, an n-body code for large particles, and a radial diffusion code to follow the evolution of a gaseous circumstellar disk. Other algorithms link the codes together, enabling each component to react to the evolution of other components.

In this study, we assume that particles lie within a single annulus of width ${\rm{\Delta }}a$ at a distance a from the central star (${\rm{\Delta }}a=0.2a$). Within the annulus, there are M mass batches with characteristic mass m_i and logarithmic spacing $\delta ={m}_{i+1}/{m}_{i};$ adopting δ = 1.4 provides a reasonably accurate solution for the cascade (e.g., Kenyon & Bromley 2015a, 2015b, 2016a, and references therein). Batches contain N_i particles with total mass M_i, average mass ${\bar{m}}_{i}={M}_{i}/{N}_{i}$, horizontal velocity h_i (${e}_{i}=\sqrt{1.6}{h}_{i}/{v}_{K}$), and vertical velocity v_i ($\sin \,{\imath }=\sqrt{2}{v}_{i}/{v}_{K}$). The number of particles, total mass, and orbital velocity of each batch evolve through physical collisions and gravitational interactions with all other mass batches in the ring.

To specify collision rates, we adopt the particle-in-a-box algorithm. In this approach, the collision rate of all particles i with particles j is ${\dot{N}}_{i}={N}_{i}\,{N}_{j}\,\sigma \,v\,{f}_{g}\,\epsilon /V$, where σ is the geometric cross-section, v is the relative velocity, f_g is the gravitational focusing factor, V is the volume occupied by the particles, and is a factor to avoid double counting when i = j (Kenyon & Luu 1998; Kenyon & Bromley 2001, 2002). The relative velocity depends on h_i and v_i. When relative velocities are large (small), f_g is derived in the dispersion (shear) regime with tidal effects included (Kenyon & Luu 1998; Kenyon & Bromley 2004a, 2012).

Collision outcomes depend on the ratio of the center-of-mass collision energy Q_c to the collision energy required to eject half of the mass to infinity ${Q}_{D}^{\star }$. When two particles collide, the instantaneous mass of the merged particle is

$$\begin{eqnarray}&&m={m}_{i}+{m}_{j}-{m}_{\mathrm{esc}}\,,\end{eqnarray} \tag{ 1 }$$

where the mass of debris ejected in a collision is

$$\begin{eqnarray}&&{m}_{\mathrm{esc}}=0.5\,({m}_{i}+{m}_{j}){\left(\displaystyle \frac{{Q}_{c}}{{Q}_{D}^{* }}\right)}^{{b}_{d}}\,,\end{eqnarray} \tag{ 2 }$$

the center-of-mass collision energy is

$$\begin{eqnarray}&&{Q}_{c}=\displaystyle \frac{{m}_{i}\,{m}_{j}\,{v}^{2}}{2\,{({m}_{i}+{m}_{j})}^{2}}\,,\end{eqnarray} \tag{ 3 }$$

the binding energy of a merged pair of particles is (Benz & Asphaug 1999; Leinhardt & Stewart 2012)

$$\begin{eqnarray}&&{Q}_{D}^{\star }={Q}_{b}{r}^{{\beta }_{b}}+{Q}_{g}{\rho }_{p}{r}^{{\beta }_{g}}\,,\end{eqnarray} \tag{ 4 }$$

and b_d is a constant of order unity. In the expression for ${Q}_{D}^{\star }$, the first (second) term corresponds to the bulk strength (gravity) component of the binding energy. We adopt b_d = 1 and set fragmentation parameters in the ${Q}_{D}^{\star }$ relation to those appropriate for rocky material: ${Q}_{b}\approx 3\times {10}^{7}\,\mathrm{erg}\,{{\rm{g}}}^{-1}\,{\mathrm{cm}}^{-{\beta }_{b}}$, ${\beta }_{b}\approx -0.40$, ${Q}_{g}\approx 0.3\,\mathrm{erg}\,{{\rm{g}}}^{-2}\,{\mathrm{cm}}^{3-{\beta }_{g}}$, and ${\beta }_{g}\approx 1.35$ for particles with mass density ${\rho }_{p}=3\,$ ${\rm{g}}\,{\mathrm{cm}}^{-3}$ (see also Davis et al. 1985; Holsapple 1994; Love & Ahrens 1996; Benz & Asphaug 1999; Housen & Holsapple 1999; Ryan et al. 1999; Arakawa et al. 2002; Giblin et al. 2004; Burchell et al. 2005). Particles in the debris have a power-law differential size distribution, $N(r)\propto {r}^{-3.5}$. The mass of the largest particle in the debris is

$$\begin{eqnarray}&&{m}_{\max ,d}={m}_{L,0}\,{\left(\displaystyle \frac{{Q}_{c}}{{Q}_{D}^{* }}\right)}^{-{b}_{L}}\,{m}_{\mathrm{esc}}\,,\end{eqnarray} \tag{ 5 }$$

${m}_{L,0}\approx 0.01\mbox{--}0.5$, and ${b}_{L}\approx 0\mbox{--}1.25$ (Dohnanyi 1969; Wetherill & Stewart 1993; Williams & Wetherill 1994; O'Brien & Greenberg 2003; Kenyon & Bromley 2008; Kobayashi & Tanaka 2010; Weidenschilling 2010; Kenyon & Bromley 2016a). We adopt ${m}_{L,0}=0.2$ and ${b}_{L}\approx 1$.

Near the Roche limit, the ability of colliding particles to merge into a larger particle depends on the tidal field of the white dwarf (e.g., Weidenschilling et al. 1984; Ohtsuki 1993; Canup & Esposito 1995; Karjalainen 2007; Porco et al. 2007; Tiscareno et al. 2013; Hyodo & Ohtsuki 2014; Yasui et al. 2014). We define the Hill radius

$$\begin{eqnarray}&&{r}_{H}={\left(\displaystyle \frac{{m}_{i}+{m}_{j}}{{M}_{\mathrm{wd}}}\right)}^{1/3}a\,,\end{eqnarray} \tag{ 6 }$$

which separates the volume where material is bound to the particles, $r\lesssim {r}_{H}$, from the volume where material is bound to the central star, $r\gtrsim {r}_{H}$. When the radius of the merged particle ${r}_{s}={r}_{i}+{r}_{j}$ is smaller than ${r}_{H}$, the particles merge into a larger object. Otherwise, the collision results in two particles with total mass m from Equation (1). For convenience, we scale the mass lost from each particle by the initial mass, ${m}_{i,\mathrm{new}}\,={m}_{i}(1-{m}_{\mathrm{esc}}/({m}_{i}+{m}_{j}))$.

The orbital elements e_i and ı_i evolve due to collisional damping from inelastic collisions, gravitational interactions, and interactions with the radiation field of the central star. For inelastic and elastic collisions, we follow the statistical, Fokker–Planck approaches of Ohtsuki (1992) and Ohtsuki et al. (2002), which treat pairwise interactions (e.g., dynamical friction and viscous stirring) between all objects (see also Kenyon & Luu 1998; Kenyon & Bromley 2001, 2002, 2004a, 2004b, 2008, 2015a, and references therein). For short-range interactions within 5–10 Hill radii, the Fokker–Planck formalism matches results from detailed n-body simulations. To treat interactions between solids with larger separations, we also calculate long-range stirring (Weidenschilling 1989). When the central star is a low-luminosity white dwarf (${L}_{\mathrm{wd}}\lesssim {10}^{-2}\,{L}_{\odot }$), radiation pressure has negligible impact on the orbital elements. However, we include radial drift and eccentricity damping from PR drag (Burns et al. 1979). Our approach includes tidal terms appropriate for material within the Roche limit. Several test calculations with our algorithms reproduce results from previous studies of velocity evolution near the Roche limit (e.g., Canup & Esposito 1995; Ohtsuki 2000).

Throughout the evolution, collisional disruption generates particles with radii smaller than ${r}_{\min }$, the smallest size included in the grid. We assume that these particles are removed by vaporization and do not interact with larger particles in the grid. Vaporized solids add material to a gaseous disk; over time, the gas accretes onto the white dwarf. We consider the likely impact of a gaseous disk on the solids in Section 4.2.

Aside from treating tidal effects in collision outcomes and dynamical evolution, we must consider tidal disruption of solids near the Roche limit. Among various options for analyzing tidal stability (e.g., Aggarwal & Oberbeck 1974; Dobrovolskis 1990; Davidsson 1999, 2001), we infer constraints using failure criteria developed for terrestrial soils and applied to satellites of Mars and the giant planets in the solar system (e.g., Holsapple & Michel 2006, 2008; Sharma 2009, 2014). As in Holsapple & Michel (2006, 2008), we divide solids into small objects with finite cohesiveness and negligible self-gravity and large objects dominated by gravity. Approaches outlined in Sharma (2009, 2014) yield similar results.

For small objects, the upper limit of the spin periods of asteroids as a function of their measured diameter implies a cohesiveness $k=2.25\times {10}^{7}{r}^{-1/2}$ dyne cm⁻². Larger objects have smaller cohesiveness (more faults in their structure). Applying the Drucker–Prager model outlined in Holsapple & Michel (2008), we calculate ${({a}_{d}/{R}_{\mathrm{wd}})(\rho /{\rho }_{\mathrm{wd}})}^{1/3}$, where a_d is the minimum stable distance for a solid with a given shape, spin, and mean density ρ orbiting a more massive central object with radius ${R}_{\mathrm{wd}}$ and mean density ${\rho }_{\mathrm{wd}}$. For simplicity, we assume a spin axis parallel to the orbital axis.

Figure 1 shows the results for solids with r = 1 cm to 10 km. The dashed gray lines denote the classical Roche limit, ${a}_{R}/{R}_{\mathrm{wd}}$, for fluids with ρ = 3 ${\rm{g}}\,{\mathrm{cm}}^{-3}$ (upper curve) and ρ = 6 ${\rm{g}}\,{\mathrm{cm}}^{-3}$ (lower curve) orbiting a 0.6 ${M}_{\odot }$ white dwarf with a mean density³ ${\rho }_{\mathrm{wd}}=4.45\times {10}^{5}$ ${\rm{g}}\,{\mathrm{cm}}^{-3}$, where

$$\begin{eqnarray}&&\displaystyle \frac{{a}_{R}}{{R}_{\mathrm{wd}}}\approx {C}_{R}{\left(\displaystyle \frac{{\rho }_{\mathrm{wd}}}{{10}^{6}{\rm{g}}{\mathrm{cm}}^{-3}}\right)}^{1/3}{\left(\displaystyle \frac{3{\rm{g}}{\mathrm{cm}}^{-3}}{\rho }\right)}^{1/3}\,\end{eqnarray} \tag{ 7 }$$

and ${C}_{R}\approx 170$. Adopting the standard cohesiveness (solid curves), solids with $r\leqslant 3\mbox{--}10\,\mathrm{km}$ are stable inside the classical Roche limit. Formally, particles with radii $r\leqslant 10$ m are stable even when they orbit at the surface of the white dwarf. Larger objects are progressively less stable until $r\approx 1\mbox{--}10\,\mathrm{km}$, when the stability limit approaches the Roche limit. Prolate solids are more stable than spherical particles. A factor of 10 smaller cohesiveness (dashed lines) has little impact on the results.

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Other approaches to deriving the Roche limit for small solids yield fairly similar results. In his landmark paper, Jura (2003) adopted an expression for the Roche limit from Davidsson (1999) and set⁴ the disruption radius ${a}_{d}/{R}_{\mathrm{wd}}$ as roughly 40% of ${a}_{R}/{R}_{\mathrm{wd}}$. In more recent studies, ${a}_{d}\approx 80 \%$ (Bear & Soker 2013) to 120% (Veras et al. 2014b) of a_R (see also Farihi 2016). All of these investigations assume the solids are rigid spheres, which tends to overestimate the disruption radius (e.g., Davidsson 1999, 2001; Holsapple & Michel 2008; Sharma 2014).

In the gravity limit, 10–100 km objects are probably also stable at ${a}_{d}\approx 0.5\mbox{--}0.7{a}_{R}$. In the solar system, the small satellites of Mars and the gas giants are tidally stable at approximately two-thirds of the standard fluid limit. To explain observations of eclipses in WD 1145+017, Veras et al. (2017) simulated asteroid disruption with the N-body code PKDGRAV (e.g., Leinhardt et al. 2000; Richardson et al. 2000, 2005, and references therein). With $a\approx 0.7{a}_{R}$, spherical objects with $r\approx$ 1–100 km and $\rho \gtrsim 3\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ are stable against tidal disruption on three-month to two-year timescales. Less dense solids disrupt within 1–30 days. For a fixed density, more massive asteroids are more stable.

Based on this discussion, we focus on the evolution of tidally stable objects with $r\leqslant 100\,\mathrm{km}$. Formally, tidally stable solids orbiting the white dwarf are prolate ellipsoids with aspect ratios of roughly 2:1:1. Tidal distortion has no impact on stirring rates, which depend on particle masses. Collision rates depend on geometric cross-sections; however, we ignore the slight changes in rates for ellipsoidal particles in this initial study.

In the next section, we consider a suite of numerical simulations designed to infer whether collisional evolution in systems of solids with e₀ = 0.01 and large vertical scale height leads to states with circular orbits and negligible vertical scale height. First, we demonstrate in Section 3.1 that swarms of indestructible mono-disperse particles with the cross-sectional area required to explain the IR-excess emission of white dwarf debris disks damp on very short timescales.

We then consider a standard cascade calculation of a swarm of solids with initial mass M₀ and no input mass from external sources (Section 3.2). Although these systems also evolve rapidly, they never reach a state with small e and negligible ı: collisional damping is negligible.

This failure motivates the model described in Section 3.3 where the input rate of solids is sufficient to balance the loss rate of particles with $r\lt {r}_{\min }$. When the input particles are small, ${r}_{\max }\lesssim 10\mbox{--}30$ km, collisional evolution yields an equilibrium mass proportional to the mass input rate ${\dot{M}}_{0}$. Swarms with ${r}_{\max }$ ≳ 10–30 km cycle through periods of high mass and near-zero mass. The high mass limit of these swarms scales with ${\dot{M}}_{0}$. Once again, though, the swarms never attain a state with small e and negligible vertical scale height.

To demonstrate how a swarm of solids might damp, we consider material⁵ with ${A}_{d}$ = 10²¹ cm² orbiting at $a\,=0.71{a}_{R}=1.15\,{R}_{\odot }$ (P = 4.5 hr) around a 0.6 ${M}_{\odot }$ white dwarf. In these test simulations, we follow a mono-disperse set of indestructible solids which evolve by collisional damping and viscous stirring.

Figure 2 shows our results. With no change in particle properties, the evolution proceeds until (i) collisional damping and viscous stirring balance or (ii) the vertical scale height is equal to 1–2 particle radii. Although damping times for swarms with identical ${A}_{d}$ are identical, larger particles stir the swarm faster than smaller particles. Thus, swarms of 1000 km particles find a balance with larger vertical scale height H than swarms of 1 cm particles. In this example, the equilibrium has $H\approx r$ and $e\approx 2\times {10}^{-11}r$.

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As the vertical scale height of these rings approaches equilibrium, they are subject to gravitational instability (e.g., Goldreich & Ward 1973; Weidenschilling 1995; Youdin & Shu 2002; Chiang & Youdin 2010, and references therein). For a mono-disperse set of particles orbiting a 0.6 ${M}_{\odot }$ white dwarf, the critical scale height for instability is

$$\begin{eqnarray}{H}_{\mathrm{crit}} & \approx & 1\,\mathrm{cm}\,\left(\displaystyle \frac{{A}_{d}}{{10}^{21}\,{\mathrm{cm}}^{2}}\right)\left(\displaystyle \frac{r}{1\,\mathrm{cm}}\right)\\ & & \times \left(\displaystyle \frac{\rho }{3\,{\rm{g}}\,{\mathrm{cm}}^{-3}}\right)\left(\displaystyle \frac{0.2a}{{\rm{\Delta }}a}\right){\left(\displaystyle \frac{1{R}_{\odot }}{a}\right)}^{1/2}\,.\end{eqnarray} \tag{ 10 }$$

When $H\lesssim r\,a\approx 1\,{R}_{\odot }$, systems with ${A}_{d}\,\approx$ 10²¹ cm² tend to be unstable. Closer to the white dwarf, swarms with somewhat smaller ${A}_{d}$ are also unstable.

At the Roche limit of a white dwarf, massive swarms of indestructible solids find an equilibrium with a vertical scale height comparable to the particle radius. Lower mass swarms have larger H. For reasonable total masses, the maximum H is comparable to r for $r\lesssim 1\mbox{--}10$ cm and ≳5–10 r for $r\gtrsim 10$ cm. Swarms of 1 cm or smaller particles with a total cross-sectional area large enough to produce an observable IR excess are probably gravitationally unstable. A system with enough mass in somewhat larger particles avoids the instability.

Although these results are encouraging, solid particles on high e orbits around a white dwarf are likely to suffer catastrophic collisions. We now consider whether systems with destructible particles can also find equilibria with small H.

Particles orbiting near the Roche limit of a white dwarf are fairly easy to break. We define an orbital eccentricity e_c required for catastrophic disruption, where the collision ejects half the mass of the combined mass of two colliding particles. For two equal mass objects with collision velocity $v\approx {{ev}}_{K}$, the center-of-mass collision energy is ${Q}_{c}={v}^{2}/8\approx {e}^{2}{v}_{K}^{2}/8$. Setting ${Q}_{c}={Q}_{D}^{\star }$ (with ${Q}_{D}^{\star }$ from Equation (4)) yields e_c. For our ${Q}_{D}^{\star }$ parameters, Figure 3 shows e_c for orbits with P = 4.5 hr. Starting with e₀ = 0.01 guarantees the catastrophic disruption of all solids with radii between 1 μm and 300 km.

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To quantify the impact of destructive collisions on collisional damping, we follow the evolution of swarms with initial surface density ${{\rm{\Sigma }}}_{0}=100\,{\rm{g}}\,{\mathrm{cm}}^{-2}$ and total mass ${M}_{0}\approx 8\times {10}^{23}$ g in an annulus centered at a = 1.15 ${R}_{\odot }$ from a 0.6 ${M}_{\odot }$ white dwarf. Material in this annulus has orbital period P = 4.5 hr, initial eccentricity e₀ = 0.01, and initial inclination ${{\imath }}_{0}={e}_{0}/2$. Calculations begin with a mono-disperse set of solids with initial radius ${r}_{0}$.

Figure 4 illustrates the time evolution of the total mass in solids. In each calculation, it takes one to two collision times to generate copious amounts of small objects that systematically remove mass from larger objects. Because the collision time is proportional to the cross-sectional area of the swarm, systems composed of objects 1 km and smaller evolve faster than systems of 100 km and smaller objects.

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Initially, the mass drops monotonically with time. As the calculation proceeds, all systems end up with one or two objects that dominate the mass of the swarm. If the debris can slowly grind down these objects, the mass continues to drop with time. Sometimes, however, the debris evolves more rapidly than the largest solids. The mass then remains roughly constant in time until the remaining large particles collide with each other (if there are at least two of them, as in the tracks for calculations with ${r}_{0}=10$ and 30 km) or forever (if there is only one large object, as in the calculation with ${r}_{0}=100$ km).

While the mass in each system declines, collisional damping among small particles with $r\approx 1$ μm to 10 cm overcomes stirring by the largest solids. However, the reduction in orbital e and ı is rather small: e (ı) drops from 0.01 (0.005) to 0.004–0.006 (0.002–0.003). At the end of these calculations, the vertical scale height $H\approx 1250\mbox{--}1500\,\mathrm{km}$ is roughly 10% of the radius of the central white dwarf.

For a short period of time, there is enough debris in any of these systems to produce an observable IR excess (Figure 5). When ${r}_{0}=1\mbox{--}3\,\mathrm{km}$, the IR luminosity maintains ${L}_{d}/{L}_{\mathrm{wd}}=0.01$ for 1–10 yr. After 100 yr, ${L}_{d}/{L}_{\mathrm{wd}}$ falls well below observed levels. For larger ${r}_{0}$, the IR luminosity matches observed levels for 10–100 yr and then drops dramatically.

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Occasional collisions among the largest objects produce sporadic spikes in ${L}_{d}/{L}_{\mathrm{wd}}$. In systems with larger ${r}_{0}$, collisions among the largest objects are less frequent and produce more debris. Thus, the spikes in ${L}_{d}/{L}_{\mathrm{wd}}$ are more pronounced when ${r}_{0}$ is larger.

Throughout the period when the IR luminosity is ${10}^{-2}\,-{10}^{-3}$ ${L}_{\mathrm{wd}}$, the production rate of particles 1 μm and smaller is ${10}^{12}\mbox{--}{10}^{14}$ ${\rm{g}}\,{{\rm{s}}}^{-1}$. This rate is somewhat larger than the inferred accretion rates of solids onto metallic line white dwarfs. If the 1 μm and smaller particles sublimate, they will generate a gaseous ring which then expands into a disk. The rate of accretion onto the white dwarf depends on the viscous timescale and the underlying structure of the accretion disk. Deriving the rate of accretion onto the central white dwarf requires the solution of the radial diffusion equation for the gas (e.g., Metzger et al. 2012), which is beyond the scope of the present effort.

These calculations demonstrate that collisional damping is ineffective in reducing the vertical scale height of ensembles of solid particles with ${r}_{0}=1\mbox{--}100\,\mathrm{km}$ and finite ${Q}_{D}^{\star }$. Tests with ${r}_{0}=0.1\mbox{--}0.3\,\mathrm{km}$ or ${r}_{0}=300\mbox{--}1000\,\mathrm{km}$ yield similar results. Factor of three changes in ${Q}_{D}^{\star }$ also have a modest impact on the evolution. Adopting smaller (larger) values for ${Q}_{D}^{\star }$ slows down (speeds up) the decline in ${M}_{d}$; however, the overall character of the evolution is unchanged. In all cases, destructive collisions reduce the mass in solids to nearly zero before collisional damping can reduce the vertical scale height dramatically.

In these systems, the production rate of small particles and the dust luminosity are only briefly comparable with those required by observations. During these short periods, however, the evolution time is much shorter than typical timescales observed in metallic line white dwarfs.

If the material lost to 1 μm particles is continuously re-supplied by an external source, it might be possible to maintain an equilibrium mass and luminosity for a ring of solid particles. In this equilibrium, the rate of mass input at the upper mass end of the cascade balances the rate the cascade generates solids with $r\leqslant {r}_{\min }$. Defining $\dot{M}$ as the mass loss rate through the cascade, $\dot{M}={M}_{d}/{t}_{c}$. Using ${t}_{c}=\alpha {t}_{0}$, ${M}_{d,\mathrm{eq}}={(\alpha {r}_{0}\rho {Pa}{\rm{\Delta }}a\dot{M}/6)}^{1/2}$. To derive a simple closed form for the equilibrium mass in the gravity regime for ${Q}_{D}^{\star }$ with ${r}_{0}\gtrsim 1\,\mathrm{km}$, we adopt representative values for other variables and set e = 0.01:

$$\begin{eqnarray}{M}_{d,\mathrm{eq}} & \approx & 7\times {10}^{18}\,{\rm{g}}\,{\left(\displaystyle \frac{\dot{M}}{{10}^{10}{\rm{g}}{{\rm{s}}}^{-1}}\right)}^{1/2}{\left(\displaystyle \frac{0.6{M}_{\odot }}{{M}_{\mathrm{wd}}}\right)}^{9/20}\\ & & \times {\left(\displaystyle \frac{{r}_{0}}{1\mathrm{km}}\right)}^{1.04}{\left(\displaystyle \frac{\rho }{3{\rm{g}}{\mathrm{cm}}^{-3}}\right)}^{9/10}\\ & & \times {\left(\displaystyle \frac{0.01}{e}\right)}^{4/5}{\left(\displaystyle \frac{{\rm{\Delta }}a}{0.2a}\right)}^{1/2}{\left(\displaystyle \frac{a}{{R}_{\odot }}\right)}^{43/20}\,{r}_{0}\gtrsim 1\,\mathrm{km}\,.\end{eqnarray} \tag{ 11 }$$

The equilibrium mass is sensitive to the location of the ring. At fixed a, ${M}_{d,\mathrm{eq}}$ varies roughly linearly with ${r}_{0}$ and as the square root of $\dot{M}$.

Maintaining this equilibrium requires that the timescale to replenish the ring, ${t}_{r}={M}_{d,\mathrm{eq}}/\dot{M}$, is not much longer than the collision time. For large objects and small $\dot{M}$, the mass in solids required to begin the cascade is large; t_r is also large. In these situations, we expect the ring to grow slowly in mass until the cascade begins; collisions then rapidly deplete the ring. The total mass oscillates.

As the evolution proceeds, the vertical scale height depends on the timescale for mass to flow from ${r}_{\max }$ to ${r}_{\min }$. In most cascades, material flows from ${r}_{\max }$ to ${r}_{\min }$ in a few collision times. In some systems, however, there is enough mass in small particles for collisional damping to reduce the vertical scale height substantially (Kenyon & Bromley 2015a, 2016a, 2016b). For rings of solids around white dwarfs, the most massive equilibrium rings require high input rates of massive particles. Thus, these systems have the best chance of developing the physical conditions that promote collisional damping.

To test these ideas, we consider the evolution of a ring of solids with initial mass ${M}_{0}$ = 0. Particles with radius ${r}_{0}$, e₀ = 0.01, and ${{\imath }}_{0}={e}_{0}/2$ are added to the ring at a rate ${\dot{M}}_{0}$. In each time step of length ${\rm{\Delta }}t$, the number of particles with mass ${m}_{0}$ added to the grid is ${\rm{\Delta }}N={\dot{M}}_{0}\,{\rm{\Delta }}t/{m}_{0}$. Our algorithm uses a random number generator to round ${\rm{\Delta }}N$ up or down to the nearest integer. For systems with large ${r}_{0}$, this procedure introduces some shot noise into the input rate.

Each calculation follows the same pattern. Large solids are added to the swarm until they reach a critical cross-sectional area and begin to collide. Debris produced from the first collision then interacts with all other particles in the grid. As this swarm evolves, large objects are continually added to the grid at the nominal rate ${\dot{M}}_{0}$. These new objects continue to power the cascade.

Systems with small ${r}_{0}$ and large ${\dot{M}}_{0}$ easily attain an equilibrium where the mass and cross-sectional area of the swarm are roughly constant. In these calculations, there is little shot noise in the collision rate among the particles in the swarm or in the input rate of large objects. The equilibrium (${M}_{d}$, ${A}_{d}$) depends on (${r}_{0}$, ${\dot{M}}_{0}$).

Figure 6 shows the evolution in mass for systems with ${r}_{0}$ = 1 km (${m}_{0}\ \approx {10}^{16}$ g). Swarms always find an equilibrium where the mass is nearly constant in time. Oscillations about this equilibrium mass are negligible (modest) for large (small) input rates. Shot noise produces these oscillations. At the lowest (highest) rates, a new object is added every 30 yr (20 min). At the lowest rates, a small degree of shot noise disrupts the smooth transport of mass from the largest to the smallest objects.

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Systems with large ${r}_{0}$ and small ${\dot{M}}_{0}$ cannot find an equilibrium. Shot noise dominates the evolution. The swarm repeats a standard sequence of events, where (i) material is added until the cascade begins, (ii) the cross-sectional area rises dramatically, (iii) a robust cascade depletes the small particles in the swarm faster than collisions of large objects can replenish them, and (iv) ${A}_{d}$ and to a lesser extent ${M}_{d}$ decline dramatically. The duty cycle of this process depends on ${r}_{0}$ and the input rate. In these systems, the maximum mass is within a factor of 10 of the equilibrium mass in Equation (11).

Figure 7 repeats Figure 6 for ${r}_{0}$ = 100 km (${m}_{0}\ \approx {10}^{22}$ g). At the highest (lowest) input rates, an object is added every 30 yr (every 30 Myr). It takes 30–40 large objects to commence the cascade. The mass in solids is then sensitive to the input rate. At the largest rates, shot noise produces variations in the rate mass flows down the cascade. Thus, the equilibrium mass varies. At the lowest rates, the system gradually grows in mass until it has the requisite number of large objects to begin the cascade. Collisions then deplete the system at a rate much faster than the rate of adding large objects to the swarm. The mass drops and remains at some minimum level until the input of large objects raises the mass to the level required for a cascade.

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When ${\dot{M}}_{0}$ is larger, the systems come closer to reaching an equilibrium. Large drops in the total mass become more and more infrequent. Once the input rate reaches 10¹² ${\rm{g}}\,{{\rm{s}}}^{-1}$, the solids find a rough equilibrium with a mass close to the expected mass from Equation (11). Shot noise in the input rate generates fluctuations about the equilibrium.

For systems with large ${r}_{0}$, all input ${\dot{M}}_{0}$ lead to roughly the same maximum mass in solids. This mass is close to the equilibrium mass for high input ${\dot{M}}_{0}$. In these rings, the mass in solids grows until the collision rate among the high mass objects reaches the rate required to power the cascade. This rate only depends on the cross-sectional area of the largest objects. While the rate is sensitive to ${r}_{0}$, it is independent of ${\dot{M}}_{0}$.

When ${r}_{0}$ = 1–100 km, the equilibrium disk mass agrees amazingly well with the analytical prediction (Equation (11)). To make this comparison, we examine the mass ratio $\xi ={M}_{\mathrm{eq},n}/{M}_{d,\mathrm{eq}}$, where ${M}_{\mathrm{eq},n}$ is derived from simulations and ${M}_{d,\mathrm{eq}}$ is the analytical prediction (Equation (11)). For large ${r}_{0}$, we consider only those calculations with large ${\dot{M}}_{0}$ where the fluctuations in the mass are small. Among 46 simulations with ${\dot{M}}_{0}={10}^{7}\mbox{--}{10}^{13}\,{\rm{g}}\,{{\rm{s}}}^{-1}$, ξ = 2.0–2.5 (Figure 8); the average ratio is $\bar{\xi }=2.19\pm 0.15$. Despite the factor of two offset, the numerical simulations match the predicted variation of the equilibrium mass with ${\dot{M}}_{0}$ and ${r}_{0}$. Considering the approximations made in deriving the analytical prediction, the good agreement is more than satisfactory.

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Within a large suite of simulations with ${r}_{0}$ = 0.1–300 km and ${\dot{M}}_{0}={10}^{7}\mbox{--}{10}^{13}$ ${\rm{g}}\,{{\rm{s}}}^{-1}$, collisional damping has negligible impact on the evolution. Figure 9 shows an example of the evolution for ${r}_{0}$ = 10 km and ${\dot{M}}_{0}$ = 10¹³ ${\rm{g}}\,{{\rm{s}}}^{-1}$. When the cascade begins, the solids have a vertical scale height H = 2800 km. As the simulation proceeds, large objects with $r\gtrsim 0.1\mbox{--}1\,\mathrm{km}$ maintain this scale height. Collisional damping is ineffective. Although collisional damping often reduces the vertical scale height of small objects with $r\lesssim 1\mbox{--}10$ m, the reduction is modest. Once systems reach the equilibrium mass, small particles have $H\approx 1250\mbox{--}1750\,\mathrm{km}$. When systems cannot reach an equilibrium, collisional damping is more sporadic, $H\approx 2500\mbox{--}2800\,\mathrm{km}$.

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In all simulations, the reprocessed luminosity of the particles ${L}_{d}$ closely follows the total mass ${M}_{d}$. In Figure 10, the predicted ${L}_{d}/{L}_{\mathrm{wd}}$ for systems with ${r}_{0}$ = 1 km rises when the solid mass reaches ${M}_{d}\approx {10}^{18}$ g. Shortly thereafter, the luminosity finds an equilibrium with ${L}_{d}/{L}_{\mathrm{wd}}\approx {10}^{-3}{({\dot{M}}_{0}/{10}^{12}{\rm{g}}{{\rm{s}}}^{-1})}^{1/2}$. Systems with smaller ${\dot{M}}_{0}$ display larger fluctuations about this equilibrium luminosity.

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Once ${r}_{0}$ ≳ 100 km, rings of solids with input ${\dot{M}}_{0}$ ≲ 10¹⁰ ${\rm{g}}\,{{\rm{s}}}^{-1}$ spend most of their time in a very low-luminosity state with ${L}_{d}/{L}_{\mathrm{wd}}\approx {10}^{-8}\mbox{--}{10}^{-7}$ (Figure 11). During this period, the mass in large objects slowly grows. Eventually, the mass reaches a critical level of roughly 10²³ g. Collisions then rapidly generate a luminous system, which quickly fades back to the faint minimum.

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Rings with ${r}_{0}$ = 100 km and larger ${\dot{M}}_{0}$ attain a stable state where the luminosity fluctuates around a rough equilibrium. For any ${\dot{M}}_{0}$, the typical luminosity is identical to the ${L}_{d}/{L}_{\mathrm{wd}}$ achieved by systems with smaller ${r}_{0}$. However, the large flares in ${L}_{d}/{L}_{\mathrm{wd}}$ are much larger than those in rings with smaller ${r}_{0}$.

To quantify how often these calculations generate detectable IR excesses, we define the detection probability as the fraction of time where the reprocessed luminosity ${L}_{d}/{L}_{\mathrm{wd}}\gtrsim {10}^{-3}$. Figure 12 summarizes our results. At large ${\dot{M}}_{0}$, systems with ${r}_{0}$ = 1–100 km spend nearly all of their time above the nominal detection limit. When ${r}_{0}$ = 200–300 km, the time required to accumulate enough mass for the cascade is a significant fraction of the evolution time. Thus, these systems are rarely detectable even when ${\dot{M}}_{0}$ is large.

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For intermediate accretion rates, ${\dot{M}}_{0}\approx {10}^{11}\mbox{--}{10}^{12}\,{\rm{g}}\,{{\rm{s}}}^{-1}$, the detection probability is a few percent. These systems spend more than 90% of their time in low states, where the cascade is fairly dormant. Once the cascade has enough mass, it briefly produces a detectable debris disk.

When the input ${\dot{M}}_{0}$ is small, the IR excess is rarely detectable. Cascades with ${r}_{0}$ ≲ 10 km never generate enough mass in small particles to reach ${L}_{d}/{L}_{\mathrm{wd}}$ = 10⁻³. Although systems with larger ${r}_{0}$ sometimes achieve large ${L}_{d}/{L}_{\mathrm{wd}}$, the fraction of time spent in the bright state is small.

These results are remarkably independent of other input parameters in our calculations. The exponents b_d and b_l in our algorithms for debris production have a limited impact on ${M}_{d}$ and ${L}_{d}$. Adopted values for ${Q}_{D}^{\star }$ are more important: increasing (reducing) ${Q}_{D}^{\star }$ slows down (speeds up) the conversion of particles with $r\lesssim 1\,\mathrm{km}$ into smaller particles (Kobayashi & Tanaka 2010; Wyatt et al. 2011; Kenyon & Bromley 2016a, 2017). Thus, systems with smaller (larger) ${Q}_{D}^{\star }$ have smaller (larger) equilibrium values for ${M}_{d}$ and ${L}_{d}$. For factor of three changes in ${Q}_{D}^{\star }$, however, collisional damping still has a negligible impact on the evolution. Cascades of destructive collisions of objects with ${r}_{0}$ = 0.1–300 km always prevent small particles from attaining a small vertical scale height.

At first glance, the differences between Figures 2 and 9 might seem remarkable. In systems with indestructible particles, damping reduces the vertical scale height by 50% in roughly 10 collision times, ${t}_{50}\approx 10{t}_{0}\approx {10}^{4}\,{\rm{s}}$. Once collisions become destructive, damping is negligible.

However, destructible particles with ${e}_{0}\approx 0.01$ do not survive long enough to damp. When two 1 cm particles with $e\approx 0.01$ collide, their center-of-mass collision energy is ${Q}_{c}\approx {v}^{2}/8\approx {10}^{10}$ $\mathrm{erg}\,{{\rm{g}}}^{-1}$. With ${Q}_{D}^{\star }=3\times {10}^{7}\,\mathrm{erg}\,{{\rm{g}}}^{-1}$, ${Q}_{c}/{Q}_{D}^{\star }\approx 375$; the particles are completely destroyed. From Equation (5), the radius of the largest particle in the debris is roughly 1 mm. The collision of two 1 mm particles yields material with sizes of 0.13 mm and smaller. It then takes another three to four collisions to reduce the debris to submicron sizes. Overall, the five to six collisions required to convert a pair of centimeter-sized particles into copious amounts of micron-sized particles is a factor of two smaller than the number of collisions required to damp their velocities by 50%.

Collisions can reduce the vertical scale height of large objects with mass m₁ after interactions with an equivalent mass in smaller objects (e.g., Goldreich et al. 2004). When these large objects orbit with $e\approx 0.01$ at the Roche limit of a white dwarf, however, they lose mass faster than they damp. For example, a 10 km (100 km) object colliding with a single smaller particle of mass m₂ loses a mass ${\rm{\Delta }}{m}_{1}\approx {10}^{3}{m}_{2}$ (${\rm{\Delta }}{m}_{1}\approx 50{m}_{2}$). Defining $N={m}_{1}/{m}_{2}$ as the number of collisions required for the larger object to interact with an equivalent mass in smaller objects, it is clear that the total mass lost ($N{\rm{\Delta }}{m}_{1}$) is much larger than m₁.

Overall, the level of damping in Figure 9 agrees with expectations based on the survival times. Over five to six collisions among small particles, we anticipate a 25%–30% reduction in the scale height, which is reasonably close to the 33% reduction derived in the calculations. Because collisions with small particles rapidly destroy larger particles, damping of 1 m and larger particles should be smaller than that of smaller particles, as shown in Figure 9. Once the mass in the annulus reaches a steady state, the vertical scale height also reaches a steady state which is set by the amount of damping achieved during the fairly short residence times of the small particles in the grid.

Compared to other situations where damping has been effective during a cascade (e.g., Kenyon & Bromley 2009, 2015a, 2016a, 2016b), the Roche limit of a white dwarf is very harsh. At 1–100 au, $e\approx 0.01$ produces modest ratios ${Q}_{c}/{Q}_{D}^{\star }\approx 1\mbox{--}3$ instead of the ${Q}_{c}/{Q}_{D}^{\star }\approx 100\mbox{--}1000$ discussed here. When ${Q}_{c}/{Q}_{D}^{\star }\approx 1$, it takes ≳25–50 collisions to reduce a pair of 1 cm particles into submicron debris. If it takes 10 collisions to reduce the vertical scale height by a factor of two, there is a reasonable chance that damping can reduce the vertical scale height before particles are ground to dust. At the Roche limit, however, large and small particles are ground to submicron sizes much more rapidly than at 1–100 au. Thus, damping is ineffective.

Since collisions and gravitational dynamics are unable to reduce H significantly, it is important to consider other physical processes capable of circularizing particle orbits. Within a protoplanetary disk, gas drag is a vital component of the growth of planetesimals into protoplanets (e.g., Youdin & Kenyon 2013). The Yarkovsky and YORP effects modify the orbits of asteroids and satellites in the solar system (e.g., Bottke et al. 2006). In the next two subsections, we examine whether any of these processes can circularize the orbits of particles involved in a cascade.

To estimate the impact of gas, we consider a simple model for a steady gaseous disk fed at a constant rate by the vaporization of 1 μm or smaller particles. The disk surface density ${{\rm{\Sigma }}}_{g}={\dot{M}}_{0}/3\pi \nu$, where $\nu =\alpha {c}_{s}{H}_{g}$ is the viscosity, α is the viscosity parameter, c_s is the sound speed, and H_g is the vertical scale height of the gas. Adopting $\alpha \approx {10}^{-3}$ (Metzger et al. 2012) and a gas temperature T_g = 4000 K at $a\approx 1\,{R}_{\odot }$ (e.g., Melis et al. 2010, 2012; Metzger et al. 2012), ${c}_{s}\approx 1\,\mathrm{km}\,{{\rm{s}}}^{-1}$, ${H}_{g}\approx 2000\,\mathrm{km}$, and ${{\rm{\Sigma }}}_{g}\approx 0.05\,{\rm{g}}\,{\mathrm{cm}}^{-2}$ for ${\dot{M}}_{0}={10}^{10}\,{\rm{g}}\,{{\rm{s}}}^{-1}$. This estimate is similar to the ${{\rm{\Sigma }}}_{g}\,\lesssim 0.01\mbox{--}0.1\,{\rm{g}}\,{\mathrm{cm}}^{-2}$ derived in numerical simulations (e.g., Rafikov 2011b; Metzger et al. 2012).

In this example, the sound speed is comparable to the vertical velocity of solid particles. With ${H}_{g}\approx {H}_{s}$, solids spend most of their time interacting with the gas. For simplicity, we assume the solids "see" a typical gas density, ${\rho }_{g}\approx {{\rm{\Sigma }}}_{g}/H\approx 3\times {10}^{-10}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$. Factor of 2–3 changes in ${\rho }_{g}$ have little impact on our discussion.

Within the gas, drag forces circularize the orbits of small particles (Adachi et al. 1976; Weidenschilling 1977; Rafikov 2004; Chiang & Youdin 2010; Youdin & Kenyon 2013). Defining F_d as the drag force, the "stopping time" is ${t}_{s}={{mv}}_{g}/{F}_{d}$, where v_g is the velocity of a particle relative to the gas. For small particles, we consider drag in the Epstein and Stokes regimes, which balance when the particle size $r\approx 9\lambda /4$, where $\lambda =\mu {m}_{H}{c}_{s}P/2\pi {{\rm{\Sigma }}}_{g}{\sigma }_{c}$ is the mean free path, $\mu =28$ is the mean molecular weight, and ${\sigma }_{c}=5\,\times {10}^{-15}$ cm² is the collision cross-section for two Si atoms (Weidenschilling 1977; Rafikov 2004; Metzger et al. 2012). In our model disk, $\lambda \approx 4$ cm; drag is in the Epstein (Stokes) regime for $r\lesssim 4$ cm ($r\gtrsim 4$ cm). At 10 cm, the relevant stopping time is ${t}_{s}\approx {10}^{6}\,{\rm{s}}$. The stopping time is comparable to the orbital period when $r\approx 1$ mm.

Knowledge of the stopping time allows us to assess the response of particles to the gas. When ${\dot{M}}_{0}\approx {10}^{10}\,{\rm{g}}\,{{\rm{s}}}^{-1}$, particles with $r\approx 1$ mm encounter a mass in gas roughly equal to their own mass every orbital period. These and smaller "coupled" particles become entrained in the gas and maintain a large vertical scale height. With velocities comparable to the sound speed, collisions between these particles are destructive and generate debris which remains coupled to the gas. Thus, the gas does not help to halt the cascade and reduce the vertical scale height of the smallest particles.

Although entrained particles drift radially, the drift time is longer than the collision time. For $\alpha ={10}^{-3}$, the viscous timescale is roughly 10³ yr. In our calculations, ${t}_{c}\lesssim 1\mbox{--}10$ yr. Thus, small particles undergo destructive collisions before they drift out of the ring.

Loosely coupled particles larger than 1 mm (i) drift radially inward and (ii) damp in e and i. The gas has a finite pressure and orbits the central star at a lower velocity than solids on Keplerian orbits (Adachi et al. 1976; Weidenschilling 1977). Solids feel a "headwind" which drags them inward and circularizes their orbits. Large particles with $r\gtrsim 10$ m feel little headwind and experience little radial drift or damping. The maximum radial drift rate for centimeter-sized to meter-sized particles is the difference in orbital velocity between the gas and the solids. For metal-rich gas orbiting a white dwarf, this velocity is roughly 1 ${\rm{m}}\,{{\rm{s}}}^{-1}$ (Metzger et al. 2012). The timescale to drift out of an annulus with width ${\rm{\Delta }}a\approx 0.2a$ is several years. Orbits circularize on a similar timescale.

To estimate the relevance of radial drift and circularization, we compare the drift timescale to the collision time (Equation (8)) and the timescale for the cascade to process small particles from 1–100 cm to 1 μm (${t}_{p}\approx {M}_{s}/{\dot{M}}_{0}$, where M_s is the mass in small particles). For cascades with ${r}_{0}$ = 1 km and ${\dot{M}}_{0}$ = 10¹⁰ ${\rm{g}}\,{{\rm{s}}}^{-1}$, ${t}_{0}\approx {10}^{5}$ s, and ${t}_{p}\lesssim {10}^{6}\,{\rm{s}}$. The drift timescale is much longer than the collision time and somewhat longer than the processing time. Thus, it seems unlikely that gas drag has much impact on the cascade: collisions process small particles faster than the gas drags them inward.

For swarms with the equilibrium mass, ${M}_{d,\mathrm{eq}}$, changing ${\dot{M}}_{0}$ is unlikely to modify these conclusions. Although the radial drift time is insensitive to the gas density, gaseous disks with larger ${\dot{M}}_{0}$ damp small particles more rapidly, ${t}_{\mathrm{damp}}\propto {\dot{M}}_{0}^{-1}$. The collision and processing times scale as ${\dot{M}}_{0}^{-1/2}$. Systems with ${\dot{M}}_{0}$ $\lesssim \,{10}^{10}$ ${\rm{g}}\,{{\rm{s}}}^{-1}$ are less susceptible to gas drag than those with larger ${\dot{M}}_{0}$. When ${\dot{M}}_{0}\approx {10}^{14}\,{\rm{g}}\,{{\rm{s}}}^{-1}$, ${t}_{\mathrm{damp}}\approx {t}_{c}$. Although the gas might then circularize the orbits of small particles before collisions can destroy them, this ${\dot{M}}_{0}$ is much larger than the upper end of the range of observed $\dot{M}$ in metallic line white dwarfs.

When ${r}_{0}$ ≳ 30 km, the maximum mass is comparable to the equilibrium mass for a large ${\dot{M}}_{0}$ (Figure 7). The ability of gas to reduce the vertical scale height of small particles then depends on how the gas and the solids interact during the short period of time when the cascade operates. Although evaluating the outcome requires a time-dependent calculation of the gas and the solids, our expectation is that the gas cannot reduce the vertical scale height of 1–100 cm particles before collisions turn them into millimeter and smaller particles which will become entrained in the gas and maintain a large vertical scale height.

Although improving our assessment requires more rigorous calculations of the gas, these results suggest that the gas has a modest impact on the cascade. Combined with our earlier results, collisional dynamics, gas dynamics, and PR drag are unable to reduce the vertical scale height of solids from several thousand kilometers to several kilometers or several meters.

Aside from radiation pressure and PR drag, other interactions between stellar radiation and small solids generate changes in a and e. In the Yarkovsky effect, differences in the loss of radiation from the hotter parts of a rotating solid relative to the cooler parts change a and to a lesser extent e (see the discussions in Burns et al. 1979; Bottke et al. 2006, and references therein). Differential radiation loss from the irregular surface of a solid (the YORP effect; Rubincam 2000 and references therein) modifies the spin rates of small solids and hence alters a and e. Although these effects change a and e on long timescales (Burns et al. 1979; Bottke et al. 2006; Veras et al. 2014a), they have negligible impact over the typical collision or processing time.

By analogy with the Yarkovsky and YORP effects, differential mass loss across the surface of a solid from sublimation might also induce $\dot{a}$ and $\dot{e}$ (Veras et al. 2015a). Rapid sublimation of ices from solids modify a and e on hundred-year timescales. For systems dominated by collisions, however, all ices are probably liberated from the solids in a few collisions. Subsequent sublimation of the rocky material left behind occurs over much longer timescales.

Overall, radiation dynamics seems unlikely to reduce the vertical scale height of small solids orbiting at the Roche limit.

If these processes are unable to modify e and i significantly, it is necessary to identify other mechanisms. As one possibility, Veras et al. (2014b) consider tidal disruption of a spherical non-rotating asteroid passing within the Roche limit of the white dwarf (see also Richardson et al. 1998; Hahn & Rettig 1998; Scheeres et al. 2000; Sharma et al. 2006; Movshovitz et al. 2012). The asteroid has a rubble-pile structure with modest or negligible internal cohesion (e.g., Richardson et al. 2005; Sharma 2009). With these assumptions, asteroid disruption produces a set of distinct particles distributed along the original orbit of the asteroid. The system is then an eccentric, collisionless ring of small solid particles.

Once the ring forms, radiation from the central white dwarf shapes the dimensions of the ring (Veras et al. 2015b). For 1 cm particles with e₀ = 0.99 at a₀ = 10 au, the timescale to drift inside 1 ${R}_{\odot }$ is roughly 600 ${(r/1\mathrm{cm})({L}_{\mathrm{wd}}/{10}^{-2}{L}_{\odot })}^{-1}$ Myr (Burns et al. 1979; Veras et al. 2015b). As particles pass inside the Roche limit, they have ${e}_{R}\approx 0.01$. These outcomes are very sensitive to e₀. When e₀ = 0.999 (0.98), the drift time drops (rises) to 20 Myr (1600 Myr) with ${e}_{R}\approx 0.16$ (0.004). For a typical white dwarf luminosity, 1 cm particles that reach the Roche limit on a reasonable timescale have fairly eccentric orbits ($e\gtrsim 0.01$).

In these circumstances, the collisional evolution of ring particles seems inevitable. At large a, particles on very eccentric orbits have shorter timescales for PR drag than for collisions. As a and e drop, t₀ gradually becomes shorter than t_PR (see Equations (8)–(9)). Although it is possible that ring particles maintain a collisionless structure inside the Roche limit, small differences in the properties of ring particles probably generate a modest dispersion in a, e, and ı as the orbit contracts from $a\approx 10$ au to $a\approx 1\,{R}_{\odot }$. This dispersion leads to crossing orbits and collisional evolution.

As an example, tidally disrupting a single 1 km asteroid yields roughly 10¹⁶ 1 cm particles with a total cross-sectional area ${A}_{d}\approx {10}^{16}$ cm². For orbits with $e\approx 0.01$ (0.1), $a\approx 1\,{R}_{\odot }$, and ${\rm{\Delta }}a\approx {ea}$, the collision time is a few months (a few years; Equation (8)), which is much shorter than the many centuries required for radiation to pull the solids into the sublimation radius. Because the collision time and the drag time both scale with particle radius, collisions always dominate.

From our calculations, rings of small particles with $e\gtrsim {10}^{-3}$ produce a cascade that grinds solids into smaller and smaller objects. Although this evolution cannot modify e significantly, collisions try to puff up the inclinations of ring particles until $i\approx e/2$. The timescale for this change is 5–10 collision times, roughly the time to grind centimeter-sized to meter-sized objects to dust.

To develop a better understanding of the relationship between solid particles and gas in white dwarf debris disks, Rafikov and collaborators investigated models where solids interact with the radiation field of the white dwarf and a gaseous circumstellar disk (Bochkarev & Rafikov 2011; Rafikov 2011a, 2011b; Metzger et al. 2012; Rafikov & Garmilla 2012). In their picture, ensembles of small solids with $r\sim 1$ cm produced by tidal disruption of a single, much larger object follow circular orbits within a disk or ring inside the Roche limit. PR drag pulls these particles closer to the white dwarf, where the radiation field eventually vaporizes them. Vaporization generates a gaseous disk, which then interacts with the small particles through gas drag. On timescales somewhat longer than the viscous time of 10³ yr, the system may develop phases of runaway accretion, where the rate gas falls onto the white dwarf grows dramatically with time. The runaway depletes the gaseous disk. If enough solids remain, this process can repeat; otherwise, a new runaway requires disruption of another large object.

In this approach, interactions among the small solids are minimal (see also Farihi et al. 2008). For centimeter-sized objects on circular orbits, collisional damping maintains low e and low ı. Any low velocity collisions probably produce rebounds with little or no debris; mergers are rare. Although repeated rebounds gradually spread the ring, this process is slow. Under these conditions, there is little likelihood of a cascade or other rapid evolution of the solids.

From our calculations, attaining the initial configuration of this model seems challenging. Small solid particles generated from a disrupted asteroid or comet probably have modest e inside the Roche limit (e.g., Veras et al. 2015b). Once the orbits of these particles cross, a cascade is inevitable. The solids are then rapidly ground to dust and vaporized into a gas. From our estimates, millimeter-sized to centimeter-sized solids interacting with the gas do not drift very far from the cascade. Thus, it is hard to produce a swarm of small solids on nearly circular orbits.

Understanding the orbital geometry of solids inside the Roche limit requires more comprehensive calculations of the evolution of solids produced during the disruption of a comet or asteroid. If these solids can damp onto circular orbits before a cascade begins, then small particles can feed the structures envisioned in Rafikov & Garmilla (2012 and references therein) and generate gas which eventually accretes onto the central star. Otherwise, collisions are a more likely source of a gaseous disk.

Despite the general failure in providing a clear path to disks with a small vertical scale height, cascade calculations of solids at the Roche limit enrich our understanding of metallic line white dwarfs. Although a disintegrating asteroid is a popular mechanism (e.g., Jura 2003, 2008; Bear & Soker 2013; Farihi et al. 2013; Vanderburg et al. 2015; Farihi 2016; Gurri et al. 2017, and references therein), cascades initiated by collisions of large asteroids (as investigated in Section 3) or by collisions of debris from a tidal disruption are a plausible alternative. Rather than attempt to explain observations, here we consider points of contact between our predictions and existing data. As we include more physical processes in the simulation, we plan more comprehensive comparisons with real systems.

• 1.  
  Dusty material with a large vertical extent is necessary to produce the broad range of eclipses in WD 1145+017 (Vanderburg et al. 2015; Alonso et al. 2016; Gänsicke et al. 2016; Rappaport et al. 2016; Zhou et al. 2016; Croll et al. 2017; Gary et al. 2017; Hallakoun et al. 2017). In a cascade interpretation, eclipses result from small particles in the debris from high velocity collisions. Stochastic collisions lead to debris with variable optical depth, accounting for short timescale variations in the system. Sublimation of particles with $r\lesssim 1$ μm explains the presence of strong absorption features from a circumstellar gaseous disk. Entrainment of small particles within sublimating gas might account for plumes of material surrounding large objects. We plan additional calculations to consider this picture in more detail.
• 2.  
  In other systems, dust with a large vertical scale height can account for IR-excess emission with reprocessed luminosity ${L}_{d}/{L}_{\mathrm{wd}}$ ≈ 10⁻³ to $(3\mbox{--}4)\times {10}^{-2}$ (e.g., Farihi 2016 and references therein). Cascades with high ${\dot{M}}_{0}$ or large ${r}_{0}$ account for these systems. Because these systems must wait to collect enough material to begin the cascade, models with large ${r}_{0}$ and a broad range of ${\dot{M}}_{0}$ naturally explain the low frequency of DBZ/DAZ/DZ white dwarfs with detectable IR excesses or gaseous disks. Our calculations predict stochastic changes in the IR excess with a broad range of amplitudes and timescales. Although we considered geometries where solids are confined to a narrow annulus, solids with a larger range in a probably follow a similar evolution (e.g., Kenyon & Bromley 2008, 2012; Kenyon et al. 2016).
• 3.  
  In the cascade picture, metallic line white dwarfs without IR excesses are systems with ${\dot{M}}_{0}\lesssim {10}^{11}$ ${\rm{g}}\,{{\rm{s}}}^{-1}$ or systems with ${r}_{0}$ ≳ 30 km in a "low" state between periods of intense collisions. From our perspective, placing limits on the frequency of white dwarfs with circumstellar gas but no IR excess constrains models for sublimation of solids and viscous transport of the resulting gas.

From our calculations, it seems possible to incorporate collisional cascades into the current framework for the formation and evolution of debris orbiting metallic line white dwarfs. As high eccentricity debris from disrupting asteroids or comets settles into lower e orbits, destructive collisions may play a useful role in generating a detectable IR excess. Gas from the vaporization of micron-sized particles within the cascade provides a path to produce the gaseous disks observed in many systems.

Developing a more robust model for cascades within the Roche limit requires addressing several issues. Solids with a large vertical scale height are hotter and more susceptible to rapid sublimation than solids in a disk with negligible vertical scale height (e.g., Jura 2003; Farihi 2016, and references therein). As discussed earlier, the magnitude of this effect depends on the vaporization time t_v relative to the collision time t_c and the timescale to replenish the cascade ${t}_{r}={M}_{d}/{\dot{M}}_{0}$. In most of our calculations, ${t}_{v}\gg {t}_{c}$ and ${t}_{v}\ll {t}_{r}$. A proper treatment of vaporization requires (i) identifying mechanisms to feed the cascade and (ii) a robust calculation of the evolution of the gas along with the evolution of the solids.

It is also important to consider outcomes for cascades outside the Roche limit. In our experience, collisional damping is effective at 0.1–2 au (e.g., Kenyon & Bromley 2016a; Bromley & Kenyon 2017). Coupled with the larger equilibrium mass at larger a (${M}_{d,\mathrm{eq}}\propto {a}^{43/20}$, Equation (11)) and the less restrictive conditions for growth by merger (Weidenschilling et al. 1984; Ohtsuki 1993; Canup & Esposito 1995; Porco et al. 2007), the cascade is probably less efficient and much less luminous at 2–3 a_R than at the Roche limit. Investigating outcomes for collisional evolution at 1–10 a_R would enable better comparisons between observations and numerical calculations.

Including the evolution of the gas in cascade calculations is another goal. Although our estimates suggest the gas has a modest impact on the cascade, the inward drift of small particles entrained in the gas might increase the predicted magnitude of the IR excess relative to calculations without the gas.

We also need a better understanding of the orbits, shapes, sizes, and spin characteristics of solids scattered into orbits with periastra inside a_R. In many applications, collisional and dynamical evolution erases the initial state of the system; outcomes often have little relation to the initial conditions (e.g., Kenyon & Luu 1998; Kenyon & Bromley 2004a, 2008, 2010). In the calculations described here, the largest objects do not grow and dynamical evolution is negligible. Thus, outcomes are more sensitive to the initial e and ı of solid material. More robust predictions for the distribution of orbital parameters for solids would allow better tests of cascade models for white dwarf debris disks.