High-Eccentricity Migration of Planetesimals Around Polluted White Dwarfs

MNRAS 000,1–17 (2020) Preprint 9 October 2020 Compiled using MNRAS LATEX style ﬁle v3.0

High-eccentricity migration of planetesimals around polluted white dwarfs

Christopher E. O’Connor? and Dong Lai Department of Astronomy and Cornell Center for Astrophysics and Planetary Science, Cornell University, Ithaca, NY 14853, U.S.A.

Accepted 2020 August 24. Received 2020 August 6; in original form 2020 May 11.

ABSTRACT Several white dwarfs with atmospheric metal pollution have been found to host small planetary bodies (planetesimals) orbiting near the tidal disruption radius. We study the physical properties and dynamical origin of these bodies under the hypothesis that they underwent high-eccentricity migration from initial distances of several as- tronomical units. We examine two plausible mechanisms for orbital migration and circularization: tidal friction and ram-pressure drag in a compact disc. For each mechanism, we derive general analytic expressions for the evolution of the orbit that can be rescaled for various situations. We identify the physical parameters that determine whether a planetesimal’s orbit can circularize within the appropriate time-scale and constrain these parameters based on the properties of the observed systems. For tidal migration to work, an internal viscosity similar to that of molten rock is required, and this may be naturally produced by tidal heating. For disc migration to operate, a minimal column density of the disc is implied; the inferred total disc mass is consistent with estimates of the total mass of metals accreted by polluted WDs. Key words: minor planets, asteroids – planets and satellites: dynamical evolution and stability – planetary systems – white dwarfs: individual: WD 1145+017, SDSS J1228+1040

1 INTRODUCTION of candidate planetary bodies in a few systems, all of which are apparently in states of ongoing disruption and accretion White dwarfs (WDs) frequently display the signatures of (Vanderburg et al. 2015; Manser et al. 2019; Vanderbosch remnant planetary systems in the forms of atmospheric et al. 2019; Gänsicke et al. 2019). Vanderburg et al.(2015) metal pollution and circumstellar debris (Farihi 2016). The reported that the K2 light curve of WD 1145+017 (here- pollution rate among solitary WDs is between 25 and 50 per after WD1145) contains several deep, asymmetric, transit- cent (Zuckerman et al. 2003, 2010; Koester et al. 2014). A like signals with periods of 4.5 to 5 h; they attributed these small fraction (less than 5 per cent) also exhibits infrared ex- to at least one planetesimal ejecting small, rapidly disin- cess, indicative of circumstellar dust or debris discs in close tegrating fragments (see also Gänsicke et al. 2016; Rap- proximity to the WD (Farihi et al. 2009; Barber et al. 2012). paport et al. 2016, 2018; Xu et al. 2019a). In the case of These phenomena are thought to originate from the tidal SDSS J1228+1040 (hereafter J1228), the presence of a plan- disruption and subsequent accretion of asteroids or minor etesimal was inferred from the variability of Ca emission arXiv:2005.05977v3 [astro-ph.EP] 8 Oct 2020 ii planets that were excited onto highly eccentric orbits by from a known circumstellar gas disc on time-scales ranging companions to the WD (Jura 2003), such as surviving plan- from hours to decades: Manser et al.(2019) identified the ets (e.g., Debes & Sigurdsson 2002; Frewen & Hansen 2014; signal with a 2-h period with the object’s Keplerian orbit. Mustill et al. 2018) or stellar binary partners (e.g., Bonsor They further suggested that, if the orbit is moderately ec- & Veras 2015; Petrovich & Muñoz 2017; Stephan, Naoz & centric, relativistic apsidal precession could account for the Zuckerman 2017). Observations of WD pollution thus allow observed variability of the source over decades (Manser et al. insight into the structure and chemical composition of ex- 2016). trasolar rocky planets and small bodies (Zuckerman et al. 2007; Klein et al. 2011; Xu et al. 2019b; Bonsor et al. 2020). If these observations have been interpreted correctly, The study of polluted WDs and their putative planetary then there arise several questions about the nature and systems has been enriched in recent years by the discovery origin of these planetesimals. Their close proximity to the host WDs renders them vulnerable both to tidal disruption and to sublimation, yet in at least two cases the planetes- ? E-mail: [email protected] imals have survived long enough to be observed. To some

© 2020 The Authors 2 O’Connor & Lai extent, this constrains the properties of the objects them- 2014) selves: Manser et al.(2019) argue that the object around 5 X 2 mN J1228 must have at least the bulk density and rigidity of E˙ = nE0 m[W2mFmN (e)] =(k˜ ), (1a) 4π 2 iron in order to endure the star’s tidal gravity and must be m,N at least several km in diameter in order to withstand UV 5 X 2 mN L˙ = E N[W F (e)] =(k˜ ), (1b) radiation from the WD over several years. 4π 0 2m mN 2 m,N The dynamical history of these objects is likewise puz- zling, especially as it relates to the tidal-disruption theory where the series include terms with m ∈ {0, ±2} and N any 1/2 1/2 of WD pollution. The prevailing view is that rocky bodies integer. Note W2,0 = −(π/5) and W2,±2 = (3π/10) . ˜mN that enter the WD’s tidal radius are completely disrupted, The quantity k2 is the complex-valued Love number, to be ˜mN forming debris streams that circularize near the tidal radius discussed later in this section; =(k2 ) is its imaginary part. 3 1/2 on relatively short time-scales (e.g. Jura 2003; Veras et al. The orbital frequency (or mean motion) is n = (GM∗/a ) 2014, 2015a). A possible observation of an extended debris and the characteristic tidal energy is given by stream transiting a polluted WD helps to corroborate this GM 2s5 idea (Vanderbosch et al. 2019). However, the candidate ob- E = ∗ . (2) 0 a6 jects around WD1145 and J1228 apparently do not conform to this simplistic narrative. They are intact (or partly so), In equations (1), the Hansen coefficient is given by which suggests that they are monolithic bodies with internal 1 Z π cos [mψ(ξ) − Nφ(ξ)] F (e) = dξ, (3) cohesion rather than self-gravitating rubble piles. They also mN π (1 − e cos ξ)2 have quasi-circular orbits, which suggests that they have un- 0 dergone some form of high-eccentricity migration from their where the eccentric anomaly ξ is related to the mean and original orbits beyond 1 au. We will show that the condi- true anomalies φ and ψ according to tions under which this migration can reproduce the orbits φ = ξ − e sin ξ, (4a) of these planetesimals place meaningful constraints on their cos ξ − e properties and those of the WD’s accretion system. cos ψ = . (4b) In this study, we examine two mechanisms that could fa- 1 − e cos ξ cilitate high-eccentricity migration of small bodies into close The evolution of the orbital elements a and e is given orbits around WDs. In Section2, we consider orbital evo- by lution due to static tides raised on a rocky planetesimal by 1 da E˙ the WD’s gravity. In Section3, we explore evolution under = , (5a) drag forces in cases where the WD possesses an accretion a dt |E| disc. In Section4, we discuss the implications of each sce- 1 de 1 − e2 E˙ 2L˙ = − , (5b) nario regarding the properties and origin of the candidate e dt 2e2 |E| L planetesimals. We review our main results in Section5. where the energy and angular momentum of the orbit (assuming Mb M∗) are GM M E = − ∗ b , (6a) 2 TIDAL MIGRATION 2a 21/2 Studies of high-eccentricity migration in planetary systems L = Mb GM∗a 1 − e . (6b) frequently invoke tidal dissipation as a means to shrink and By the conservation of total angular momentum, the body’s circularize planetary orbits. Previous works have focussed rotation rate Ω evolves according to on giant planets and mostly adopted the theory of weak s tidal friction. In this section, we study the migration of 1 dΩs L˙ small bodies into tight orbits under internal tidal dissipa- = − , (7) Ωs dt IbΩs tion (hereafter “tidal migration”). We first present a gen- assuming that the body has zero obliquity. eral formalism to compare different models of dissipation; In this formalism, the physical processes that give rise we then apply that formalism to two dissipation models and to tidal dissipation are encoded by the complex Love number extract constraints on the parameters of these models from k˜ , more specifically its imaginary part =(k˜ ). In general, k˜ the properties of the candidate planetesimals around WDs. 2 2 2 is a function of the forcing frequency ω; because each component (m, N) of the tidal response has a different forcing frequency 2.1 Formalism ωmN ≡ Nn − mΩs, (8) Consider a small body in orbit around a star of mass M∗. ˜mN ˜ Let the orbital semi-major axis be a and eccentricity e. We each has a distinct Love number k2 ≡ k2(ωmN ). These characterize the body by a linear size s a and bulk density quantities constitute the principal source of uncertainty in 3 ρ, from which we estimate its mass Mb = ρs M∗ and studying the tidal evolution of astrophysical systems, as they 5 moment of inertia Ib = ρs . depend both on the internal structure of the dissipating The response of an extended body to tidal forcing can body and the specific microphysical process responsible for be expressed in general as an infinite series of Fourier-like dissipation. In the remainder of this section, we employ two oscillatory components. The resulting expressions for the en- tidal models to illustrate the tidal evolution of a rocky plan- ergy transfer rate and tidal torque are (e.g., Storch & Lai etesimal around a WD.

MNRAS 000,1–17 (2020) Planetesimals around polluted WDs 3

To compare the overall efficiency of tidal dissipation where between different models, we define functions Z1 and Z2 31 255 185 25 p (e) = 1 + e2 + e4 + e6 + e8, (15a) and rewrite equations (1) as: 1 2 8 16 64

nE0 15 2 45 4 5 6 E˙ = − Z (e, Ω /Ω ), (9a) p2(e) = 1 + e + e + e , (15b) (1 − e)6 1 s p 2 8 16 2 3 4 ˙ E0 p5(e) = 1 + 3e + e . (15c) L = − Z2(e, Ωs/Ωp), (9b) 8 (1 − e)9/2 One can see from equation (14b) that weak tidal friction where Ω is the dissipating body’s rotation rate and s drives a body into stable, pseudosynchronous rotation at a 1 + e 1/2 rate given by Ωp = n (10) (1 − e)3 Ωs 1 p2(e) = 2 . (16) is its orbital angular velocity at pericenter. These functions Ωp ps (1 + e) p5(e) are defined in the same spirit as the functions FE and FT used by Vick & Lai(2019) but differ by factors related to the Since L IbΩs, this synchronization occurs rapidly com- Love numbers and eccentricity. The characteristic time-scale pared to the evolution of a and e. Substitution of equation (16) into equation (14a) shows that the tidal evolution of of tidal migration is related to Z1: a weakly dissipating body in pseudosynchronous rotation is 6 1/2 a ρrp a entirely determined by e and k /Q: t ≡ = |Z (e, Ω /Ω )|−1, (11a) 2 a 2 3 1 s p a˙ 2s GM∗ 2 3k2/Q [p2(e)] 6 −3/2 Z (e) = p (e) − . (17) 30 Myr r s −2 M 1 8 1 ≈ p ∗ (1 + e) p5(e) |Z1| R 1 km M For e 1, we find Z ' (21/2)(k /Q)e2; whilst for e → ρ a 1/2 1 2 1, Z1 ' 0.15(k2/Q). Thus the tidal migration time-scale × −3 , (11b) 1 g cm 1 au (equation 11b) for highly eccentric initial orbits is: where rp = a(1 − e) is the separation between the body and 200 Myr r 6 s −2 M −3/2 the star at pericenter. The time-scale of spin–orbit synchro- t ≈ p ∗ a k /Q R 1 km M nization is similarly related to Z2. 2 The dissipation functions Z and Z are sensitive to sev- ρ a 1/2 1 2 × . (18) eral quantities. We have expressed them explicitly as func- 1 g cm−3 1 au tions of the eccentricity of a body’s orbit and of its rotation rate normalized by a characteristic orbital frequency. Implic- itly, but equally importantly, they depend also on several 2.1.2 Viscoelastic Dissipation parameters that characterize the tidal dissipation mecha- Tidal friction in a realistic rocky or icy body may be de- nism under consideration, namely those that appear in the scribed by viscoelastic theory, in which material can behave ˜ function k2(ω). like both an elastic solid and a viscous fluid, depending on the nature of the applied strain (Turcotte & Schubert 2.1.1 Weak Friction 2002). There exist many rheological models for viscoelastic substances. Following Storch & Lai(2014), we adopt the Studies of tidal dissipation in planetary systems frequently Maxwell model for our discussion. adopt the theory of weak tidal friction (e.g. Alexander 1973; A Maxwell material has two rheological parameters: the Hut 1981), in which the tidal bulge raised on a body lags shear modulus (or rigidity) µ and the viscosity η. The transi- behind the orbit by a small, constant time τ. The ansatz for tion between elastic and viscous behaviours is characterized the imaginary part of the complex Love number (for all m, by the Maxwell frequency N) is ωM ≡ µ/η. (19) =(k˜2) = k2ωτ, (12) When forced at frequencies ω ωM, the material behaves where k2 (the real-valued Love number) and τ are indepen- like an elastic solid; at low frequencies ω ωM, it behaves dent of m and N or the forcing frequency ω. We relate the like a viscous fluid. lag-time to the tidal quality factor Storch & Lai(2014) calculated the complex Love num- 1 ber as a function of forcing frequency ω for a homogeneous, Q ≡ . (13) Ωpτ spherical body composed of viscoelastic rock. We adapt their result to the problem at hand simply by replacing the spheri- The summations over the forcing components in equations cal radius with our characteristic size parameter s and by es- (1) yield closed-form expressions for Z1 and Z2: 2 timating the object’s surface gravity as g ∼ GMb/s = Gρs. 3k2/Q 2 Ωs For a Fourier component with frequency ω, the imaginary Z1(e, Ωs/Ωp) = 8 p1(e) − (1 + e) p2(e) , (1 + e) Ωp part of the Love number is (14a) " 2 2#−1 ˜ 57ωη ωη 19µ 3k2/Q 2 Ωs =(k2) = 1 + 1 + , (20) Z2(e, Ωs/Ωp) = 13/2 p2(e) − (1 + e) p5(e) , 4β µ 2β (1 + e) Ωp (14b) where β ≡ G(ρs)2 is a measure of the object’s self-gravity

MNRAS 000,1–17 (2020) 4 O’Connor & Lai with dimensions of stress or pressure. It is convenient to deﬁne dimensionless variables e = 0.30 ω 19µ 19Ω η 0.4 ω¯ ≡ , µ¯ ≡ , η¯ ≡ p (21) Maxwell Z1 Z Ωp 2β 2β Maxwell Z2 n

o Weak Z i 1 such that t 0.2 c Weak Z2 " 2 #−1 n 3 ω¯η¯ u

2 F

˜ =(k2) = ω¯η¯ 1 + (1 +µ ¯) . (22) 0.0 2 µ¯ n o i t a

Using this model, we can calculate the tidal energy transfer p i 0.2 s

rate and torque via equations (1). s i

The material and structural properties of a small body D are unknown. Because extrasolar small bodies have mostly 0.4 rocky compositions (Xu et al. 2019b), we have some recourse 0.0 0.5 1.0 1.5 2.0 to geophysical constraints and laboratory measurements. If Ωs/Ωp we adopt the appropriate rigidity for silicate rock or solid iron at low temperatures and pressures, the dimensionless rigidity is e = 0.99 µ ρ −2 s −2 µ¯ = 1.2 × 109 . 0.2

−3 Z

500 kbar 2.5 g cm 1 km n

(23) o i

t 0.1 Rocky bodies with sizes s . 1000 km satisfyµ ¯ 1. In that c n limit, equation (22) reduces to u F 0.0 3 ω¯η¯ n ˜ o =(k2) ' . (24) i 2 t

2 1 + (¯ωη¯) a p

i 0.1

Evidently, the tidal dissipation in a small body is deter- s s i

mined primarily by the dimensionless viscosityη ¯ (see also D Efroimsky 2015). A nominal value for this quantity is 0.2

η ρ −2 s −2 0.0 0.5 1.0 1.5 2.0 η¯ ≈ 3.8 × 109 1 bar yr 1 g cm−3 1 km Ωs/Ωp 1 − e −3/2 P −1 × −3 , (25) 10 1 yr Figure 1: Upper panel: Tidal dissipation functions Z1 where P is the orbital period and where η = 1 bar yr is a (dashed curves) and Z2 (solid), as defined in equations1 characteristic viscosity for water ice. Geophysical viscosities and9, plotted with respect to the dimensionless rotation vary over many orders of magnitude, depending on the com- rate Ωs/Ωp for an orbital eccentricity of e = 0.3. Black position and state of the material in question. curves show the result of the Maxwell viscoelastic model We note that the tidal theory for a Maxwell body re- withµ ¯ 1 andη ¯ = 1; magenta curves show the result of duces to the weak friction theory when the series in equa- weak tidal friction with k2/Q = 1. Lower panel: The same, tions (1) comprise only terms where the combinationω ¯η¯ but with an orbital eccentricity e = 0.99. 1. In that case, we have =(k˜2) ∝ ω as before. is not the case for the Maxwell model: in the limit where viscous stresses overwhelm a body’s self-gravity (¯η 1, 2.2 Behavior of the Dissipation Functions in our notation), the behaviour of these functions with re- We now illustrate the properties of the dissipation functions spect to Ωs/Ωp is quite rich. There are two main qualitative Z1 and Z2 (defined through equations1 and9) for both of behaviours of Z1 and Z2 in this limit, to which we refer our tidal models. In Figure1, we show Z1 and Z2 with re- as “jagged” and “smooth.” Jagged behaviour is character- spect to Ωs/Ωp for cases of moderate (e = 0.3) and high ized by numerous zero-crossings, both stable and unstable; (0.99) orbital eccentricities. For these examples, we have while smooth behaviour is characterized by a single, sta- chosenη ¯ = 1 and the value of k2/Q so that the functions ble pseudosynchronous state. We explain the emergence of have similar magnitude. We see that, in this instance, the both behaviours in terms of the microphysics of dissipation Maxwell model and weak friction yield qualitatively similar in AppendixA. For the present discussion, it suffices to results. The functions Z2 both admit a stable, pseudosyn- say that jagged behaviour occurs for dimensionless viscosity −3/2 chronous state for Ωs/Ωp > 0, albeit at slightly different η¯ & Ωp/n ∼ (1 − e) . rotation rates. The complexity of the dissipation functions in the high- Figure2 shows Z1 and Z2 for a wide range of eccentric- viscosity limit makes an analytic prescription for rotation ities and dissipation parameters. The qualitative behaviour elusive: the pseudosynchronous value of Ωs/Ωp cannot be of weak tidal friction is independent of k2/Q because the expressed in closed form as a function ofη ¯ and e. Conse- pseudosynchronous rotation rate is a function of eccentricity quently, we must take a simpler tack to estimate the value only and the pseudosynchronous state is always stable. This of Z1. In Figure3, we display Z1 as a function ofη ¯ for a

MNRAS 000,1–17 (2020) Planetesimals around polluted WDs 5

e = 0.30 e = 0.90 0.2 0.010 Maxwell Z Z

Weak n n o o

i 0.1 i 0.005 t t c c n n u u F F 0.0 0.000 n n o o i i t t a a p p i i

s 0.1 s 0.005 s s i i D D

0.2 0.010 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0

Ωs/Ωp Ωs/Ωp

e = 0.75 e = 0.99 0.02 0.010 Z Z

n n o o

i 0.01 i 0.005 t t c c n n u u F F 0.00 0.000 n n o o i i t t a a p p i i

s 0.01 s 0.005 s s i i D D

0.02 0.010 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0

Ωs/Ωp Ωs/Ωp

Figure 2: Each panel is similar to those of Fig.1, but with diﬀerent choices of eccentricity and, for the Maxwell dissipation model, a higher dimensionless viscosityη ¯ = 102. Note the transition from “jagged” behaviour at moderate eccentricities to “smooth” behaviour as e → 1.

fixed eccentricity e = 0.99 and several rotation rates. For 2.3 Constraints on Tidal Migration −3/2 viscosityη ¯ . (1 − e) , Z1 is qualitatively insensitive to the assumed rotation rate. We identify three main regimes: Having established the properties and behaviour of the dissipation function Z1, we are now able to evaluate the conditions under which tidal friction can mediate the high- eccentricity migration of planetesimals, such as those ob- (i) Forη ¯ 1, we find Z1 ∝ η¯; this is consistent with served by Vanderburg et al.(2015) and Manser et al.(2019). convergence to weak tidal friction. Given a tidal theory with specified values of the dissipation

(ii) A maximum Z1 ∼ 0.1 occurs atη ¯ ∼ 1. parameters (e.g., k2/Q for weak friction,µ ¯ andη ¯ for viscoelastic friction), we assess the effectiveness of tidal friction (iii) For 1 η¯ (1 − e)−3/2, we find Z ∝ η¯−1. This . . 1 by asking whether the WD’s cooling age (t ) is longer or corresponds to the quasi-elastic response of a Maxwell ma- WD shorter than the tidal circularization time-scale (t ). We terial. circ define tcirc as the time required for tidal evolution from an initial eccentricity e → 1 to the value e = 0.1 (our results are insensitive to the exact choice of the lower eccentricity −3/2 threshold). We have determined through numerical integra- For extremely high viscosityη ¯ & (1 − e) , the behaviour of Z1 is jagged and therefore highly spin-dependent. For tion of the tidal evolution equations for a variety of initial some relatively rapid rotation rates, Z1 abruptly shifts from conditions that the circularization time is typically between positive to negative, meaning that energy is transferred from 2.2ta0 and 2.6ta0, where ta0 is the value of equations (11) for the spin to the orbit. For relatively slow rotation, Z1 remains the initial condition. Figure4 depicts three such examples. −1 positive. Overall, the scaling |Z1| ∝ η¯ persists regardless Hereafter, we adopt the estimate tcirc ≈ 2.5ta0. of rotation. Thus, we conclude that tidal friction is most The time-scale ta depends strongly on the pericentre efficient when the dimensionless viscosityη ¯ is of the order distance rp. To avoid tidal disruption, however, we require of unity. that rp be greater than the critical radius for tidal disrup-

MNRAS 000,1–17 (2020) 6 O’Connor & Lai

1 Maxwell model, e = 0.99 ] 0 10 10 U a0 = 1 AU A [ a0 = 3 AU a 0 s 10 10-1 i a0 = 10 AU x A

r o -2 j 10-1

10 a M - i 1 Z m -3 -2

e 10

n 10 S o i t c

n -4 0.0 0.5 1.0 1.5 2.0 2.5 3.0

u 10 F 1.0 g n

i -5

c 10 r 0.8 e o

F y t i

c 0.6 0 i r

Ωs/Ωp = (Ωs/Ωp)WF t n

Ωs/Ωp = 0.75 e 0.4 c -10-5 c Ωs/Ωp = 0.65 E 0.2 Ωs/Ωp = 0.55 -10-4 0.0 10-4 10-2 100 102 104 0.0 0.5 1.0 1.5 2.0 2.5 3.0

Viscosity η¯ Time t/ta0

Figure 3: A bi-symmetric logarithmic plot of Z1 versusη ¯ in Figure 4: Three examples of the tidal evolution of planetesi- the Maxwell model (withµ ¯ 1) for e = 0.99 and various mals. The upper panel shows the evolution of the semi-major rotation rates. axis a and the lower panel the eccentricity e. The time t is normalized by ta0, the timescale ta (see equation 11) evalu- tion. For a body bound by self-gravity only, the tidal dis- ated at the initial condition. All three examples start with ruption (or Roche) radius is the same pericentre distance rp = 0.58R , but diﬀerent initial a (as labelled). We use the weak-friction theory in these 1/3 ρ∗ calculations, but similar results can be obtained using the rdis = KR∗ . (26) ρ Maxwell dissipation model. These examples show that 2.5ta0 provides a good estimate for the tidal circularization time- where R∗ and ρ∗ are the WD’s radius and mean density. The 1/2 numerical factor K ∼ 2, depending on the body’s shape, scale tcirc (deﬁned in the text). Note that ta0 ∝ (a0) ; thus internal structure, and rotation. For a body with tensile the true circularization time-scale is longer for a0 = 10 au strength γ in addition to self-gravity, the generalized tidal than for a0 = 1 au. disruption radius is

2 1/3 0 Gρ∗ρs (given M∗ = 0.6M ). Following Rappaport et al.(2016), we r = K R , (27) 23 dis ∗ γ + β take its mass to be 10 g, roughly 10 per cent that of Ceres. We also assume a bulk density ρ = 3 g cm−3, which yields a 2 where again β = G(ρs) is a measure of self-gravity and size s = 320 km. 0 where K is analogous to K. Self-gravity is the dominant Pseudosynchronous tidal migration conserves orbital binding force whenγ ¯ ≡ γ/β 1. For the typical strength angular momentum, meaning that the quantity a(1 − e2) = of terrestrial rock or iron, rp(1 + e) would have remained constant. This implies that −2 −2 the object’s initial pericentre distance was half of its present 6 γ ρ s γ¯ ≈ 1.5 × 10 . (28) semi-major axis, or about 0.58R . To avoid disruption at 1 kbar 1 g cm−3 1 km this distance, the object’s dimensionless tensile strength Thus, the disruption radius of a cohesive object can be much must beγ ¯ & 7; given the assumed size and density, its true smaller than that of a rubble pile of the same size and bulk tensile strength was greater than ∼ 4 kbar, which we note to density. Since the Roche radius is typically ∼ R , km-sized be somewhat higher than the measured strengths of silicate objects with high tensile strengths may withstand disruption meteorites (Petrovic 2001). This suggests that the original −2 near the surface of a typical white dwarf (∼ 10 R ). object was a monolith rather than a rubble pile. The cooling age of WD 1145+017 is some 200 Myr (Van- derburg et al. 2015; Izquierdo et al. 2018). We take this as 2.3.1 WD 1145+017 the upper limit of tcirc and therefore find ta0 . 80 Myr. As- WD1145 is transited by solid debris with orbital periods suming initial values a = 5.0 au and rp = 0.58R , as well between 4.5 and 4.9 h. The strongest transit signal occurs as the values of ρ, s, and M∗ mentioned above, we obtain every 4.5 h and exhibits highly variable depths (e.g., Van- a lower limit on the dissipation function Z1 during migra- −6 derburg et al. 2015; Gänsicke et al. 2016; Rappaport et al. tion, namely Z1 & 2.0 × 10 . For weak tidal friction, this −5 2016, 2018; Croll et al. 2017; Gary et al. 2017). This has corresponds to the quality factor k2/Q & 1.3 × 10 by the been attributed to a disintegrating asteroid or planetesimal result Z1 ' 0.15(k2/Q) for e → 1. For Maxwell viscoelastic on a near-circular orbit with a = 1.16R around the WD friction, we first note that the rigidity µ is presumably at

MNRAS 000,1–17 (2020) Planetesimals around polluted WDs 7 least of the order of the minimal tensile strength required to for weak friction and avoid disruption, γ ≈ 4 kbar. This provides a lower bound −2 −2 ρ s ofµ ¯ 60 on the dimensionless rigidity, which indicates that η¯ 2 × 10 , & & 8 g cm−3 10 km the approximationµ ¯ 1 is reasonable. To constrain the −1 viscosity, we use the asymptotic expressions we obtained for ρ s 2 η¯ . 50 −3 µ¯ 1 and e → 1. In the low-viscosity limit, Z1 ' 0.1¯η and 8 g cm 10 km −5 we findη ¯ & 2 × 10 . In the high-viscosity limit, Z1 ' 0.1/η¯ 4 for viscoelastic friction. If the planetesimal’s current orbit and thusη ¯ . 5 × 10 . is eccentric, then the conserved quantity a(1 − e2) would The constraint on Z1 is modified if we relax our assumed be smaller than a circular orbit would imply and so the knowledge of one or more properties of the system. Because initial pericentre distance would have been smaller as well. rp is well constrained by conservation of angular momentum, In effect, this loosens the constraint on the planetesimal’s M∗ is known from spectroscopic analysis of the WD, and 2 6 6 Z1 by a factor of (1 − e ) because ta ∝ rp. a has a relatively muted effect on ta, the most important uncertainties are the values of ρ and s. By treating these quantities as free parameters, we would find a less specific constraint 3 MIGRATION FROM DISC DRAG

ρ s −2 The presence of accreting gaseous or dusty discs around Z 4.2 × 10−5 . 1 > 3 g cm−3 320 km some polluted WDs presents another possibility for high- eccentricity migration. A planetesimal would experience a For a larger or a less compact planetesimal, the correspond- drag force while passing through the circumstellar material, ing constraint on Z1 is weaker. resulting in shrinkage and circularization of the orbit. Sim- ilar effects have been studied previously in different astrophysical contexts, such as active galactic nuclei (e.g., Rauch 1995; Subrˇ & Karas 1999; MacLeod & Lin 2020) and proto- 2.3.2 SDSS J1228+1040 planetary discs (Rein 2012). Indeed, Grishin & Veras(2019) have discussed the possible relationship between compact The candidate planetesimal around J1228 was discovered accretion discs and close-in planetesimals around WDs. In through spectroscopic rather than photometric observations. this section, we present a simplified calculation of the disc Manser et al.(2019) used theoretical arguments to constrain migration scenario. Our formulation allows us to evaluate the size of the hypothesized object within the range this migration mechanism efficiently for a variety of conditions and parameters. 4 km . s . 600 km. Consider a small body moving with velocity v through a disc with local density ρ . The disc itself rotates with bulk In brief, the lower size limit follows from the requirement d velocity v about the central WD. It exerts a drag force on that radiation-driven sublimation of the body’s surface is d the body through ram pressure: sufficient to provide the observed metal accretion rate. The upper limit follows from the requirement that the body with- C F = − σρ |v − v |(v − v ), (29) stand tidal disruption on its current orbit through internal 2 d d d strength alone. These calculations assumed the object’s den- where σ is the body’s cross-sectional area and C is a dimen- −3 sity to be ρ ≈ 8 g cm , consistent with an iron-rich com- sionless drag coefficient of order unity. As before, we define position. However, the composition of the object is not con- the body’s size s and bulk density ρ so that its cross-section strained by existing observations, so as in Section 2.3.1 we 2 3 is σ = s and its mass is Mb = ρs . Equation (29) is valid treat both the size and density as free parameters. for supersonic motion, when ∆v = |v − vd| is much larger Manser et al.(2019) calculated the planetesimal’s or- than the local sound speed or the velocity dispersion of dust bital semi-major axis to be a ≈ 0.73R . They also specu- particles in the disc; or motion with high Reynolds number, late that the orbit is moderately eccentric in order to explain Re = s∆v/ν 1, where ν is the kinematic viscosity of the the system’s long-term variability relativistic precession; the gas. These conditions are satisfied for thin discs (aspect ra- required eccentricity for this is e ≈ 0.54. We will consider tio h/r 1). We consider a Keplerian disc with constant the constraints on tidal migration that we obtain with and aspect ratio h/r 1 and a surface mass distribution Σd(r). without the supposed eccentricity. We neglect any disc evolution due to accretion by the WD In the case without eccentricity, the reasoning used to or perturbations by the encroaching body. constrain the value of Z1 is the same as for WD1145. Using The body should experience a significant drag force only J1228’s cooling age of 100 Myr (Gänsicke et al. 2006), we while it is within an altitude z ∼ h of the disc’s midplane. find For simplicity, we replace the local disc density ρd with its

−2 vertically averaged value Σd/2h and stipulate zero drag force −4 ρ s Z1 2 × 10 for |z| > h. Thus the drag force per unit mass acting on the & 8 g cm−3 10 km body while |z| 6 h is is required for tidal circularization. The corresponding con- F C Σd straints on the parameters of the tidal theory are f = = − |∆v|∆v, (30) Mb 4h ρs k ρ s −2 where ∆v = v − vd. 2 1.4 × 10−2 Q & 8 g cm−3 10 km The speciﬁc energy E and speciﬁc angular momentum `

MNRAS 000,1–17 (2020) 8 O’Connor & Lai of the small body are related to the orbital elements a and and the unit vector `ˆ changes direction by e by 2 ˆ δ` δ` ˆ CΣd r0 ˆ 1 δ` = − ` = − |∆v||vd|(uˆ × `), (37) E = − n2a2, (31a) ` ` 2ρs ` 2 where r0 = r0uˆ. 2 21/2 ` = na 1 − e , (31b) Finally, the change of the body’s eccentricity vector can

3 1/2 be calculated as where n = (GM∗/a ) is the mean motion. We denote the ˆ v × ` (δv) × ` + v × (δ`) specific angular momentum vector ` = `` and the eccen- δe = δ − uˆ = , (38) tricity vector e = eeˆ. The relative orientation of the orbit GM∗ GM∗ and the disc is specified by the inclination angle I and the where we have neglected the change of uˆ = r0/r0 according apsidal angle $, defined such that to the impulse approximation. As with the angular momen- nˆ = `ˆcos I + sin I (−eˆ sin $ + qˆ cos $) , (32) tum, we can isolate the perturbation of the orientation of the unit vector eˆ as where qˆ = `ˆ× eˆ and nˆ is the unit vector normal to the disc δe δe plane. We call $ the “apsidal” angle in the sense that, when δeˆ = − eˆ (39) e e the body intersects the disc midplane such that v · nˆ > 0, it makes an angle $ with respect to the reference direction eˆ. where δe ≡ δe · eˆ. If the orbit is inclined with respect to the disc midplane The scalars δE and δ` and the vectors δ`ˆ and δeˆ are by an angle I h/r, then the body experiences a nonzero sufficient to determine the evolution of the four orbital ele- drag force in short intervals near the line of nodes. In this ments. As with tidal friction, the changes of a and e during case, the duration of disc-crossing is much shorter than the a single disc passage are given by orbital period; thus, for orbits with I h/r we may use δa δE = , (40a) the impulse approximation to compute the orbital evolution. a |E| The aspect ratio of a WD’s accretion disc is small, typically 2 ∼ 10−3 for a gaseous disc and orders of magnitude smaller δe 1 − e δE δ` = 2 − 2 . (40b) for dust grains (Melis et al. 2010). Therefore, the impulse e 2e |E| ` approximation is valid even for orbits with inclinations as ˆ ◦ Using the identity cos I = ` · nˆ and noting that the disc small as I ∼ 0. 5. We treat the orbital evolution for I . h/r, normal nˆ is fixed (recall that we ignore the body’s effect on the “coplanar” limit, in AppendixB. the disc), we find the inclination angle changes by δ`ˆ· nˆ δI = − (40c) 3.1 Single Disc Passage sin I Consider a single passage of the body through the disc. We in a single passage through the disc. Indeed, equation (37) approximate the its trajectory as a straight line traversed at describes a small rotation of `ˆ by δI about uˆ. Similarly, constant velocity v. We consider the region of the disc that the change of the apsidal angle follows from the identity the body crosses to be a homogeneous slab of vertical thick- cos $ = eˆ · uˆ, which implies ness 2h and density Σd/2h that moves uniformly with the δeˆ · uˆ local Keplerian velocity v directed in the plane normal to δ$ = − (40d) d sin $ nˆ. The local properties of the disc and both velocity vectors For a Keplerian disc, we have are evaluated at the position r0, the point where the orbit intersects the disc midplane. δE CΣ F (e, I, $) = − d E , (41a) Under these assumptions, the duration of the particle’s |E| ρs (1 − e2)(1 + e cos $) sin I passage through the disc is δ` CΣ F (e, I, $) = − d ` , (41b) 2h ` 2ρs (1 + e cos $) sin I ∆t = . (33) |v · nˆ| CΣ F (e, I, $) δI = − d I , (41c) The work done by the drag force on the orbit during a single 2ρs (1 + e cos $)3/2 passage is CΣd F$(e, I, $) δ$ = − (41d) ρs e(1 + e cos $) sin I CΣ |∆v| δE = (f · v)∆t = − d (v · ∆v). (34) 2ρs |v · nˆ| In equations (41), the functions FE , F`, FI , and F$ are dimensionless functions defined so as to be of order unity Note that δE does not depend explicitly on the disc’s scale for all eccentricities and inclinations; explicit expressions for height h. these functions can be obtained from equations (34) through Similarly, the change of the body’s angular momentum (41d). In Figure5, we display these functions over the full due to drag is range of $ for various inclinations and a high eccentricity CΣd |∆v| (e = 0.995). δ` = (r0 × f)∆t = − (r0 × ∆v), (35) 2ρs |v · nˆ| Therefore, for a highly eccentric orbit (where a/r0 1), we find that |δE/E| |δ`/`| and |δI| ∼ |δ`/`|. This which is also independent of h. The magnitude of the specific implies that drag has a similar effect on the orbit to tidal angular momentum vector ` changes by friction in the high-eccentricity limit. The semi-major axis δ` = δ` · `ˆ (36) and eccentricity decrease with each orbit, the relative change

MNRAS 000,1–17 (2020) Planetesimals around polluted WDs 9

e = 0.995, I = 5 deg I0=5.0 deg, 0=0.0 deg, Λ=7.0e+03 2 104 ` I a E 1 F F F F r 3 1 / 10 a(1 e) r

s 2

e a(1 e ) 2 0 c 10 − n a t s i 101

1 D

π π/2 0 π/2 π 100 − − 0 5000 10000 15000 20000 25000 e = 0.995, I = 45 deg a=1.54R , e=9.1e-03, I=0.06 deg 3 ¯ 100

I e 2 n i

s sin I

, -1

e 10 1 s t n 0 e 10-2 m e l Auxiliary Functions 1 E π π/2 0 π/2 π 10-3 − − 0 5000 10000 15000 20000 25000 e = 0.995, I = 85 deg 10.0 2π 7.5 3π/2 5.0

e l π g

2.5 n A π/2 0.0 π π/2 0 π/2 π − − Angle 0 0 5000 10000 15000 20000 25000

Time t/Pin

Figure 5: Auxiliary functions FE , F`, FI , and F$ (see legend in top panel) from equations (41), plotted over the apsidal Figure 6: Secular evolution of a small body due to drag angle $ for eccentricity e = 0.995 and inclinations of I = forces from repeated disc crossings. The migration param- 5 deg (top panel), 45 deg (middle), and 85 deg (bottom). eter is Λ ≈ 7.0 × 103 and the initial orbital elements are ◦ a = 5.0 au, rp = R , I = 5 , and $ = 0. Time t = ΛPin is marked with a vertical red line. Top panel: Characteristic orbital distances a (solid black curve), a(1±e) (dashed), and a(1 − e2) (dotted) expressed in units of the disc’s inner radius r . The shaded region indicates the radial extent of the of e being slower than that of a by a factor ∼ (1 − e2). The 1 circumstellar disc. Middle panel: Orbital elements e (solid inclination of the orbit decreases on a time-scale similar to black) and sin I (dashed). The terminal values of a, e, and I that of eccentricity damping. The revolution of the line of are listed above this panel. Bottom panel: Apsidal angle $. apsides can be prograde or retrograde, depending on the value of F$; however, we will show in Section 3.2 that drag forces are not the dominant perturbation on $ over many orbits and that the total precession rate is almost always the Keplerian period P = 2π/n. Thus, negative. da δa + δa = + − , (42a) dt P de δe + δe = + − , (42b) dt P dI δI + δI = + − , (42c) dt P d$ δ$ + δ$ 3.2 Secular Evolution = + − , (42d) dt P It is simple to compute the secular evolution of the disc– orbit system under the impulse approximation. For a quasi- where the subscripts (+) and (−) refer to the ascending and Keplerian orbit, we simply divide the total change accrued descending nodes, respectively. by the orbital elements over a possible two disc passages by From equations (40a), (41a), and (42a), we ﬁnd that the

MNRAS 000,1–17 (2020) 10 O’Connor & Lai

I0=45.0 deg, 0=0.0 deg, Λ=7.0e+03 I0=85.0 deg, 0=0.0 deg, Λ=7.0e+03 104 104 a a 1 1

r 3 r 3 / 10 a(1 e) / 10 a(1 e) r r

± ±

s 2 s 2

e a(1 e ) e a(1 e ) 2 2 c 10 − c 10 − n n a a t t s s i 101 i 101 D D

100 100 0 5000 10000 15000 20000 25000 0 5000 10000 15000 20000 25000 a=1.20R , e=1.0e-03, I=0.06 deg a=0.54R , e=1.0e-03, I=0.06 deg ¯ ¯ 100 100

I e I e n n i i

s sin I s sin I

, -1 , -1

e 10 e 10

s s t t n n e 10-2 e 10-2 m m e e l l E E 10-3 10-3 0 5000 10000 15000 20000 25000 0 5000 10000 15000 20000 25000

2π 2π

3π/2 3π/2

e e l π l π g g n n A A π/2 π/2

0 0 0 5000 10000 15000 20000 25000 0 5000 10000 15000 20000 25000

Time t/Pin Time t/Pin

Figure 7: Same as Fig.6 but with initial I = 45◦. Figure 8: Same as Fig.6 but with initial I = 85◦.

characteristic time-scale of migration due to disc drag is sublimation of solid grains at r = r1 (Metzger, Rafikov & ρs Bochkarev 2012). The sublimation radius rsub ∼ r1 is related tdrag = (1 − ein)Pin, (43) CΣ0 to both the properties of the WD and the composition of the grains (Rafikov 2011a,b): where Pin and ein are the initial period and eccentricity of 2 the orbit and Σ0 ≡ Σd(r0). The characteristic number of R∗ T∗ orbits over which migration and circularization occur is rsub = , (46a) 2 Tsub ρs 2 −2 Λ ≡ (1 − ein), (44a) R∗ T∗ Tsub CΣ0 = 0.50R −2 4 3 , (46b) 10 R 10 K 10 K 100 ρ s Σ −1 1 − e = 0 in . C 1 g cm−3 1 km 1 g cm−2 10−3 where Tsub is the sublimation temperature of grains (∼ (44b) 1500 K for silicates). Observations of real debris discs show that r2 ∼ 10r1 in most cases (Jura, Farihi & Zuckerman The examples depicted in Figures6 through 10 illustrate the 2009). In our examples, we fix the properties of the central 4 qualitative features of the orbital evolution and the reliabil- WD to be M∗ = 0.6M , R∗ = 1.4R⊕, T∗ = 10 K. For ity of our estimated circularization time-scale. Note that we the disc’s solid component, we let Tsub = 1500 K, so that have included leading-order general-relativistic corrections r1 = 0.28R . We also fix the following initial conditions to the secular equations as described later in this section. of the orbit: ain = 5.0 au and rp,in = R (Pin = 14.4 yr, −4 For all of our examples, we adopt a truncated power-law 1 − ein = 9.3 × 10 ). We vary the initial values of I and model for the disc’s surface density profile: $, which specify the initial orientation of the orbit; and Λ, −γ which fixes the ratio ρs/CΣ at r = r . We halt the orbital r d 0 Σ = Σ , r r r , (45) evolution when the inclination becomes less than or equal d 1 r 1 6 6 2 1 to the disc’s aspect ratio (h/r = 10−3) because the impulse where Σ1 is the disc’s density at r = r1. We choose γ = 2 approximation is invalid for I . h/r. in imitation of a viscous, gaseous accretion disc fed by the In Figs.6 through 10, we display the evolution of the

MNRAS 000,1–17 (2020) Planetesimals around polluted WDs 11

I0=45.0 deg, 0=45.0 deg, Λ=9.6e+03 I0=85.0 deg, 0=90.0 deg, Λ=2.8e+04 104 104 a 1 1

r 3 r 3 / 10 a(1 e) / 10 r r

± a

s 2 s

e a(1 e ) e 2 2 c 10 − c 10 a(1 e)

n n ±

a a 2

t t a(1 e ) s s

i i − 101 101 D D

100 100 0 5000 10000 15000 20000 25000 0 5000 10000 15000 20000 25000 a=1.21R , e=1.0e-03, I=0.06 deg a=0.54R , e=1.0e-03, I=0.06 deg ¯ ¯ 100 100

I e I n n i i

s sin I s

, -1 , -1

e 10 e 10

s s t t n n e 10-2 e 10-2 m m e e e l l

E E sin I 10-3 10-3 0 5000 10000 15000 20000 25000 0 5000 10000 15000 20000 25000

2π 2π

3π/2 3π/2

e e l π l π g g n n A A π/2 π/2

0 0 0 5000 10000 15000 20000 25000 0 5000 10000 15000 20000 25000

Time t/Pin Time t/Pin

Figure 9: Same as Fig.7 but with initial $ = 45◦ and Λ ≈ Figure 10: Same as Fig.8 but with initial $ = 90◦ and 1 × 104. Λ ≈ 3 × 104. orbital elements a, e, I (as sin I), and $, as well as the char- and 10, we see that changing $ can change the true circu- acteristic orbital distances a(1 + e), a(1 − e), and a(1 − e2). larization time-scale by factors of a few. We also see that By inspecting the evolution of a(1 − e2) (which is propor- the estimated circularization time can be either less than or tional to the square of the orbital angular momentum), we greater than the true time-scale. The initial value of I has see that there are two distinct phases of the evolution: Dur- a more pronounced effect on the orbital evolution, but on ing the first phase, the orbit evolves essentially at constant the whole equation (43) provides a robust estimate for the angular momentum; neither e, I, nor a(1 − e2) changes by a migration time-scale. discernible amount during this phase. This reflects the factor A notable trend in these examples is that the radius ∼ (1 − e2) that appears in the expression for energy dissi- of a circularized orbit varies systematically with the initial pation (equation 41a) but not in that for the torque (equa- inclination of the orbit but not the initial apsidal angle. For tions 41b and 41c). During the second phase, when (1 − e2) low-to-moderate initial inclinations (Figs.6,7, and9), the is of order unity, the torque exerted on the orbit by the disc final orbital radius tends to be somewhat larger than the ini- becomes substantial. The quantity a(1 − e2) shrinks mono- tial pericentre distance. For high initial inclinations (Figs.8 tonically throughout this phase; meanwhile, both e and I and 10), the final orbital distance can be barely half the decrease precipitously from their initial values. If we denote initial pericentre distance. This reflects a significant loss of by tI the time elapsed between t = 0 and the instant that orbital angular momentum to the disc over the course of I = h/r (so that the integration is halted artificially), then migration. It also presents the possibility of delivering plan- the duration of the first phase is between 80 and 90 per cent etesimals well inside of the Roche limit (as apparently is the of tI in all cases we examined. case with J1228), even if their initial pericentre distances By comparing each of Figs.6 through 10, we see that were outside it. changing the initial orientation of the orbit (I or $) changes Given the body’s close proximity to the WD at peri- both the time-scale of evolution and the ultimate orbit of the centre, one must ask whether general-relativistic corrections body. For instance, by comparing Figs.7 and9 or Figs.8 to the equations of motion are important. The leading-order

MNRAS 000,1–17 (2020) 12 O’Connor & Lai post-Newtonian eﬀect is prograde apsidal precession of the 3 orbit at a rate Viscosity-Size Constraints, ρ = 3 g cm− 1012 3GM∗ $˙ GR = n, c2a(1 − e2) 109 Earth crust/asthenosphere 2π M r −1 P −1 ' ∗ p , (47) 5 106 3 × 10 yr M R 1 yr ] r where in the second expression we have taken the limit y 3

r 10 WD 1145+017 e → 1. Due to the deﬁnition of $ in terms of the relative a b [

SDSS J1228+1040 orientation between the disc and the orbit (equation 32), the η 0 10 y

GR contribution to (d$/dt) is negative: t i s o

d$ δ$+ + δ$− c 10-3 s = − $˙ GR (48) i dt P V 10-6 Equation (47) suggests that, for an initial orbit with P & 1 yr, the precession period is comparable to the disc life- time (see Section 3.3); this would mean that GR does not 10-9 qualitatively affect the early evolution of the orbit. After lava flows the orbit has shrunk somewhat, however, the precession pe- 10-12 0 1 2 3 riod becomes shorter than the migration time-scale. Thus, 10 10 10 10 Size s [km] we have included GR effects in our examples for accurate computation of the orbital evolution. We have found that, indeed, the contribution to precession due to GR tends to Figure 11: Maxwell viscosities inferred for WD candidate overwhelm that due to drag forces in determining the overall planetesimals as functions of their assumed physical sizes. precession rate. The magenta and cyan shaded regions refer to the objects in orbit around WD1145 and J1228, respectively. The black point with error bars shows the result for WD1145 using the 3.3 Constraints on Disc-Driven Migration mass calculated by Rappaport et al.(2016). The grey shaded If the candidate planetesimals around WD1145 and J1228 regions show the range of viscosity values for Earth’s crust achieved their current orbits via inclined disc-driven migra- and asthenosphere (as a proxy for solid rock) and lava flows. tion, what constraints do they imply on the properties of the We have fixed the bulk densities at ρ = 3 g cm−2 because this disc–planetesimal system? Clearly, we require the migration quantity varies by a factor of a few at most while the size time-scale to be less than the disc life-time. Observations can vary over several orders of magnitude. place the disc life-time between a few 104 and a few 106 yr on average (Girven et al. 2012), in keeping with theoretical es- in these systems. Nonetheless, they illustrate the minimal timates for various accretion mechanisms (Rafikov 2011a,b; amount of circumstellar material required in order for disc Metzger et al. 2012). We therefore adopt a fiducial disc life- migration to be viable. We discuss the implications of this 5 time of tdisc = 4 × 10 yr as the appropriate upper limit on constraint in Section 4.2. the migration time-scale tdrag in this scenario. Thus, we require the dimensionless migration parameter Λ (the number of orbits over which migration occurs; see equation 44) to satisfy 4 DISCUSSION

tdisc 4.1 Viscosity from Tidal Migration Λ . (49) Pin In Section 2.3, we obtained constraints on the internal vis- in order for migration to occur. Unfortunately, referring to cosity of the planetesimals in orbit around WD 1145+017 equations (44), we see that the properties of the migrating and SDSS J1228+1040 under the assumption that their planetesimal (ρs) and the accretion disc (Σ0) are degenerate present orbits are the result of tidal evolution with Maxwell with the initial orbital parameters (Pin and ein). For the viscoelastic dissipation. Although these constraints are im- remainder of this discussion, we will assume an initial orbital precise and are somewhat degenerate with the objects’ sizes period of 10 yr and an initial eccentricity of 0.999. and bulk densities, they are interesting from a geophysical In Section 2.3, we took the (main) object in orbit around perspective. 7 −2 4 WD1145 to have ρs ≈ 9×10 g cm . By taking Λ . 4×10 In Figure 11, we display the range of viscosity values −3 and 1−ein = 10 , we ﬁnd that disc migration for this object consistent with tidal migration as a function of the assumed −2 −3 requires a typical surface density Σ & 2.5 g cm . The same size of the object (setting ρ = 3 g cm for both). We also reasoning applied to the J1228 system yields show the typical range of viscosity values that have been measured or inferred for Earth’s crust and upper mantle ρs Σ 0.2 g cm−2 , (Mitrovica & Forte 2004) and for terrestrial lava ﬂows of & 8 × 106 g cm−2 various compositions at typical temperatures of 1300–1400 K where the quoted value of ρs corresponds to an iron body (as compiled by Chevrel et al. 2019). We see that the plan- 10 km in size. These results are only as certain as exist- etesimals around WD1145 and J1228 have bulk viscosity ing constraints on the size and mass of the orbiting objects consistent with being molten in whole or in part. On the

MNRAS 000,1–17 (2020) Planetesimals around polluted WDs 13 other hand, both are inconsistent with having the same bulk Section 3.3, and adopting a fiducial outer disc radius of 2R , viscosity as Earth’s crust by orders of magnitude. we estimate the characteristic disc mass around WD1145 to A molten interior would be consistent with the hypoth- be at least esis of tidal migration, due to the large amount of orbital ρs 1.5 × 1023 g energy that would have to be dissipated, provided that the 9 × 107 g cm−3 time-scale of tidal heating be shorter than the body’s internal cooling time-scale. In a small, rocky body, the dominant and around J1228 to be at least cooling mechanism is thermal conduction. Using a thermal ρs 1.2 × 1022 g . diffusivity similar to terrestrial rock (κ ∼ 10−2 cm2 s−1; Tur- 8 × 106 g cm−2 cotte & Schubert 2002, Hartlieb et al. 2016), the cooling time These estimates are consistent with those given by the of a planetesimal is sources cited above. s2 s 2 t ∼ ≈ 3 × 104 yr . (50) cool κ 1 km 4.3 Caveats To estimate the rate of tidal heating, we use equation (9a) with |Z1| ∼ 1. For rock with a nominal melting point 4.3.1 Viscoelastic Rheology temperature T ∼ 103 K and specific heat capacity c ∼ m P The validity of the constraints we have derived on the vis- 107 erg g−1 K−1 (Turcotte & Schubert 2002; Hartlieb et al. cosity of small bodies (Section 2.3) during tidal migration 2016), the body is heated to the melting point on a time- extends only as far as the validity of the Maxwell rheology scale of in describing their response to tidal forcing. The Maxwell 3 6 3 1/2 cP Tmρs cP Tmρrp a model is the simplest viscoelastic theory, and thus it fails to theat ≡ ∼ 2 3 5 , (51a) |E˙ | s G M∗ capture certain aspects of material physics. The sensitivity of tidal evolution to one’s choice of rheological model has 6 −2 4 cP Tm rp s been discussed elsewhere (e.g., Makarov & Efroimsky 2013; ≈ 7 × 10 yr 10 −1 10 erg g R 1 km Renaud & Henning 2018). The complex Love number k˜2(ω) −5/2 3/2 ρ M∗ a can be calculated for other, more sophisticated rheologies × −3 . (51b) 1 g cm M 1 au (Renaud & Henning 2018); it may be possible to extract constraints on bulk properties within these alternative models 3 Similarly, the heat required to melt the body is Hf ρs , where by arguments analogous to those we have used. 10 −1 Hf ≈ 10 erg g is the latent heat of fusion (Turcotte & Schubert 2002). Thus, the time-scale of internal melting is 4.3.2 Accretion Disc Properties 3 Hf ρs Hf tmelt ≡ = theat . (52) |E˙ | cP Tm Our calculations in Section3 assume a power-law density profile for the disc (see Metzger et al. 2012). This is phys- Since Hf ∼ cP Tm for common minerals, heating and melting ically reasonable if the gaseous and dusty disc components occur on roughly the same time-scale. coincide, as for SDSS J1228+1040 and others (Brinkworth Comparing equations (50) and (51), we see that the et al. 2009; Melis et al. 2010). For WD 1145+017, however, time-scale of heating and melting is shorter than the cooling the circumstellar gas is concentrated near the sublimation time for bodies larger than a few km. This is the case for radius (e.g., Cauley et al. 2018; Xu et al. 2019a; Fortin- the planetesimals around WD1145 and J1228, and thus a Archambault et al. 2020), while the dust mostly coincides molten interior is indeed plausible. with transiting debris (e.g., Vanderburg et al. 2015; Xu et al. 2019a). Moreover, polluted WDs with detectable circumstel- 4.2 Disc Masses from Disc Migration lar gas appear to be very much the minority: a recent analysis of a joint Gaia/SDSS WD catalogue (Gentile Fusillo Most accretion discs around polluted WDs are composed et al. 2019) found that less than 10 per cent of dusty de- primarily of dust rather than gas (Manser et al. 2020). Such bris systems have gaseous components (Manser et al. 2020). discs are detected mainly by their thermal emission at in- Thus, it is of interest to examine the sensitivity of the disc frared wavelengths (e.g., Zuckerman & Becklin 1987; Farihi migration scenario to the disc’s mass distribution. 2016); simple models of this emission can be constructed We consider two simple alternatives to the power-law in order to estimate their radial extent and optically thin disc model: an uniform (or top-hat) disc with Σd = constant dust mass (e.g., Jura et al. 2007a,b, 2009). However, the to- and a Gaussian disc with tal dust mass cannot be calculated from infrared emission (r − r )2 alone, since the disc’s infrared continuum emission may be Σ ∝ exp − ctr . (53) d (∆r)2 due to an opaque component. Sometimes, it is possible to estimate the total mass of accreted material in WD’s convec- We model the latter after Metzger et al.(2012), selecting tive zone (Jura 2006; Zuckerman et al. 2007), which provides rctr = 5rsub and ∆r = 0.5rsub. As before, we truncate these an indirect estimate of the disc’s mass budget (Jura et al. disc models at r < rsub and r > 10rsub. For the Gaussian 2009). profile, these nominal dimensions matter very little because The disc migration mechanism provides an independent the bulk of the disc’s mass is concentrated within a few ∆r estimate of the disc’s total mass because the time-scale of a of rctr. planetesimal’s orbital evolution is determined mainly by the We have computed the evolution of the body’s orbit for column-density ratio ρs/Σd. According to the constraints of the same initial conditions as shown in Figs.6 through 10

MNRAS 000,1–17 (2020) 14 O’Connor & Lai using these alternative disc models. Our results for the uni- ρs3 is form mass distribution are qualitatively and quantitatively 3 2 similar to those for the power-law distribution: For suffi- ρs ρsHs rp −3/2 tsub ≡ ∼ 4 (1 − e) , (56a) ciently small values of Λ, the orbit can circularize and align hM˙ i ςT∗ R∗ with the accretion disc within the latter’s fiducial life-time. ρ s H We note in particular that the product ΛP remains a rea- ∼ 104 yr s in 1 g cm−3 1 km 1010 erg g−1 sonable estimate of the migration time-scale; if anything, it 2 −3/2 −4 −2 is slightly more of an overestimate. rp 1 − e T∗ R∗ × −3 4 −2 . We obtain somewhat different results for the disc with a R 10 10 K 10 R Gaussian mass distribution. For some orbital configurations, (56b) the orbit evolves in much the same way as for the power-law and uniform disc models. For others, migration begins nor- When sublimation is taken into account, successful mally but slows to an effective standstill while the eccentric- high-eccentricity migration requires tcirc < tsub for the mi- ity and inclination remain large. For still others, migration gration mechanism in question. For tidal migration, this does not occur at all within the nominal disc life-time. This can present a more stringent constraint on a planetesimal’s is mainly because a Gaussian disc can be much narrower in properties than the condition tcirc < tWD. For instance, the effect than the radial extent r1 6 r 6 r2 of the mass dis- life-time of the object in orbit around WD1145 (assuming −3 tribution would na¨ıvely suggest. For ∆r (r2 − r1), a disc ρ = 3 g cm and s = 320 km) would be between 1 and 10 11 −1 presents a much smaller ‘target’ for the orbit to intersect, 10 Myr for Hs between 10 and 10 erg g , significantly and thus the space of orbital configurations for which disc shorter than the WD cooling age of ∼ 200 Myr. Under this migration is efficient is much smaller. These differences dis- alternative condition, we calculate a more stringent con- −4 appear when ∆r is increased by a factor of a few. Thus, we straint of Z1 & 8.4 × 10 . conclude that the success or failure of disc migration is sensi- On the other hand, the sublimation time-scale for rocky tive mainly to the width of the disc and its total mass (which bodies at least a few km in size is at least as long as the together determine the typical surface density). We reiterate fiducial life-time of accretion discs around polluted WDs, a that many dusty discs around polluted WDs are observed few 105 yr. Moreover, the life-time of such an object can be to be fairly broad (Jura et al. 2009), which is favourable for extended by the disc’s shielding effect if the object’s orbit disc migration (but see Li et al. 2017 for an exception). becomes embedded within it. These considerations suggest that our earlier constraints on disc migration are mostly insensitive to this additional effect. Planetesimals undergoing inclined disc migration might 4.3.3 Sublimation and Ablation also lose significant mass by ablation, owing to the super- Many polluted WDs have effective temperatures in excess sonic relative velocity between the body and the disc. We es- of 10 000 K, with WD1145 and J1228 among them. Objects timate the ablation time-scale following Jura(2008). During orbiting in close proximity to the WD are strongly irradi- each pericenter passage, we suppose that ablation removes ated by UV light and thus may lose mass by sublimation. a layer from the planetsimal’s surface of thickness This effect provides an additional constraint on the origins Σ y and properties on close-orbiting planetesimals because their δs ∼ d , (57) ρ sin I hypothetical high-eccentricity migration must occur before the objects vaporize completely. where y ≈ 0.1 is the sputtering yield. The ablation time- We estimate the mass-loss rate of a small body under scale is then the assumption that all of the WD’s radiation that is inci- dent on the body’s geometric cross-section during a short s ρs sin I tabl ≡ P ∼ P. (58) time interval is converted to the heat required to sublime a δs Σd y thin layer at the surface. The resulting mass-loss rate is Comparing this expression to the characteristic circulariza- 4 2 2 ˙ ςT∗ R∗ s tion time under drag forces (equation 43), we have M ∼ 2 , (54) Hs r tabl sin I where ς is the Stefan–Boltzmann constant, T and R are ∼ . (59) ∗ ∗ t y(1 − e) the WD’s effective temperature and radius, r is the radial drag distance from the WD’s center, and Hs is the body’s latent Since we take the initial orbit to be highly eccentric and heat of sublimation. For a body on a highly eccentric orbit, moderately-to-highly inclined, we see that tabl tdrag. This mass loss by sublimation takes place mostly near pericentre indicates that ablation is not important in the early stages ˙ and thus the mass lost over a single orbit is ∼ Mptp, where of orbital evolution. As the orbit circularizes and becomes ˙ tp = 1/Ωp is the duration of pericentre passage and Mp is aligned with the disc over time, the ablation time-scale may the value of equation (54) at r = rp. The mass-loss rate become comparable to the drag time-scale. averaged over one orbit is then On the whole, it is plausible that the objects orbiting WD1145 and J1228 have lost mass in the course of their mi- M˙ t hM˙ i ∼ p p . (55) gration by either sublimation or ablation. If so, a consistent P theory of their origin must take these effects into account Thus, the sublimation time-scale of a body with initial mass (see also Veras, Eggl & Gänsicke 2015b).

MNRAS 000,1–17 (2020) Planetesimals around polluted WDs 15

5 SUMMARY ACKNOWLEDGEMENTS We have studied the high-eccentricity migration of the We thank Yubo Su, Michelle Vick, Dimitri Veras, and Brian candidate planetesimal companions to WD 1145+017 and Metzger for helpful discussions. We also thank the referee, SDSS J1228+1040 from an initial orbit at several au to their Alexander Mustill, for insightful comments that improved present locations near or within the tidal disruption radius. the manuscript. DL thanks the Department of Astronomy We have shown that either internal tidal dissipation or drag and the Miller Institute for Basic Science at the University forces from an accretion disc could be responsible for the of California, Berkeley, for hospitality while part of this work circularization on an appropriate time-scale. We have pre- was carried out. This work has been supported in part by sented general analytical expressions for the migration rates NSF grant AST-1715246 and NASA grant 80NSSC19K0444. that can be easily rescaled to various situations and param- Software: matplotlib (Hunter 2007), SciPy (Virtanen eters. Our principal conclusions regarding each mechanism et al. 2020) are as follows:

• If tidal friction is responsible for the migration of the candidate planetesimals, then it is possible to constrain both their tensile strength and their internal viscosity. The requi- REFERENCES site tensile strength to avoid disruption can be slightly high Alexander M. E., 1973, Ap&SS, 23, 459 compared to those of silicate meteorites. Under a Maxwell Barber S. D., Patterson A. J., Kilic M., Leggett S. K., Dufour P., rheology, the range of viscosity that allows tidal migration Bloom J. S., Starr D. L., 2012, ApJ, 760, 26 within the WD cooling age is inconsistent with the viscos- Bonsor A., Veras D., 2015, MNRAS, 454, 53 ity of Earth’s crust or mantle, but includes a wide range of Bonsor A., Carter P. J., Hollands M., Gänsicke B. T., Leinhardt measured values for molten rock. We speculate that the ob- Z., Harrison J. H. D., 2020, MNRAS, 492, 2683 jects may have been partly or totally molten due to internal Brinkworth C. S., Gänsicke B. T., Marsh T. R., Hoard D. W., heating during their migration. Tappert C., 2009, ApJ, 696, 1402 • If drag forces from an accretion disc around the WD is Cauley P. W., Farihi J., Redfield S., Bachman S., Parsons S. G., the main migration mechanism, then it is possible constrain Gänsicke B. T., 2018, ApJ, 852, L22 the column density of the disc relative to the density and Chevrel M. O., Pinkerton H., Harris A. J. L., 2019, Earth Sci. Rev., 196, 102852 size of the migrating body. Accordingly, we have obtained Croll B., et al., 2017, ApJ, 836, 82 the characteristic total disc mass required for complete mi- 5 Debes J. H., Sigurdsson S., 2002, ApJ, 572, 556 gration within a disc life-time of several 10 yr. The results Efroimsky M., 2015,AJ, 150, 98 for WD1145 and J1228 are consistent with previous esti- Farihi J., 2016, New Astron. Rev., 71, 9 mates of the total mass of metals accreted by other polluted Farihi J., Jura M., Zuckerman B., 2009, ApJ, 694, 805 WDs. Fortin-Archambault M., Dufour P., Xu S., 2020, ApJ, 888, 47 Frewen S. F. N., Hansen B. M. S., 2014, MNRAS, 439, 2442 Is either scenario favoured over the other in explaining Gänsicke B. T., Marsh T. R., Southworth J., Rebassa-Mansergas the origin of the candidate planetesimals? Tidal migration A., 2006, Science, 314, 1908 seems the most natural explanation by default, given that Gänsicke B. T., et al., 2016, ApJ, 818, L7 debris from tidal disruption of asteroids seems to be ubiq- Gänsicke B. T., Schreiber M. R., Toloza O., Fusillo N. P. G., uitous around polluted WDs – clearly, it is possible to de- Koester D., Manser C. J., 2019, Nature, 576, 61 Gary B. L., Rappaport S., Kaye T. G., Alonso R., Hambschs F. J., liver planetesimals to pericentre distances near or inside the 2017, MNRAS, 465, 3267 Roche radius. In that case, the most likely reason that the Gentile Fusillo N. P., et al., 2019, MNRAS, 482, 4570 observed objects were not fully disrupted is that they are Girven J., Brinkworth C. S., Farihi J., Gänsicke B. T., Hoard monoliths, rather than rubble piles, and therefore have a D. W., Marsh T. R., Koester D., 2012, ApJ, 749, 154 moderate amount of internal strength. Grishin E., Veras D., 2019, MNRAS, 489, 168 While the disc migration scenario is somewhat non- Hartlieb P., Toifl M., Kuchar F., Meisels R., Antretter T., 2016, standard, it is physically well motivated in that it relies on Minerals Engineering, 91, 34 little more than the presence of a sufficiently massive accre- Hunter J. D., 2007, CSE, 9, 90 tion disc when a ‘fresh’ planetesimal approaches the tidal Hut P., 1981, A&A, 99, 126 radius for the first time. Because massive discs are relatively Izquierdo P., et al., 2018, MNRAS, 481, 703 rare and last for finite time, the limiting factor is the rate at Jura M., 2003, ApJ, 584, L91 Jura M., 2006, ApJ, 653, 613 which small bodies are excited onto highly eccentric orbits, Jura M., 2008,AJ, 135, 1785 which depends on the presence and orbital configuration of Jura M., Farihi J., Zuckerman B., Becklin E. E., 2007a,AJ, 133, outer planets or binary stellar companions. 1927 Jura M., Farihi J., Zuckerman B., 2007b, ApJ, 663, 1285 Jura M., Farihi J., Zuckerman B., 2009,AJ, 137, 3191 Klein B., Jura M., Koester D., Zuckerman B., 2011, ApJ, 741, 64 Koester D., Gänsicke B. T., Farihi J., 2014, A&A, 566, A34 Li L., Zhang F., Kong X., Han Q., Li J., 2017, ApJ, 836, 71 Data availability MacLeod M., Lin D. N. C., 2020, ApJ, 889, 94 Makarov V. V., Efroimsky M., 2013, ApJ, 764, 27 The data underlying this article will be shared on reasonable Manser C. J., et al., 2016, MNRAS, 455, 4467 request to the corresponding author. Manser C. J., et al., 2019, Science, 364, 66

MNRAS 000,1–17 (2020) 16 O’Connor & Lai

Manser C. J., Gänsicke B. T., Gentile Fusillo N. P., Ashley R., µ¯ 1, e = 0.5, m = 0 Breedt E., Hollands M., Izquierdo P., Pelisoli I., 2020, MN- 1.00 À RAS, 2 η¯= 10− Melis C., Jura M., Albert L., Klein B., Zuckerman B., 2010, ApJ, η¯= 1 722, 1078 0.75 η¯= 102 Metzger B. D., Rafikov R. R., Bochkarev K. V., 2012, MNRAS, 423, 505 0.50 Mitrovica J. X., Forte A. M., 2004, Earth Planet. Sci. Lett., 225, 177 0.25 Mustill A. J., Villaver E., Veras D., Gänsicke B. T., Bonsor A., )

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MNRAS 000,1–17 (2020) Planetesimals around polluted WDs 17 andη ¯ = 10−2, the resonant ‘humps’ are suﬃciently broad to encompass multiple forcing frequencies; as a result, their Coplanar, Λ = 1.3e+06, h/r = 1.0e-03 forcing functions are smooth with respect to Ωs/Ωp. On the 2 other hand, in the caseη ¯ = 10 the proﬁle of =(k˜2) is so nar- row that the forcing frequencies “ignore” or “pass by” the 103

resonant region. A nonzero rotation rate would shift the set 1 r /

of forcing frequencies (for m = ±2 modes), so that overlap r

a s with the resonant region becomes possible. e 2

c 10 a(1 e)

n ±

a 2

t a(1 e ) s

i −

APPENDIX B: COPLANAR DISC MIGRATION D 101 In Section 3.1, we calculated the orbital evolution of a planetesimal intersecting a circumstellar disc at a relatively high inclination (I & h/r), in which case the impulse approxima- 100 0 1 2 3 4 tion is valid. In this Appendix, we discuss the case I . h/r, 10 10 10 10 10 where the planetesimal’s orbit and the disc are nearly coplanar. 100 As before, we consider the planetesimal’s orbit to be Keplerian and we take the drag force per unit mass to be e

C Σd

f = − |∆v|∆v (B1) y t

4h ρs i c i r where the notations are the same as in the main text (see t n e

equation 30). This expression is valid when the relative ve- c -1 c 10 locity is supersonic or has a high Reynolds number and when E s < h. The changes of the orbital energy and angular momentum per orbit are given by C I Σ (r) δE = − d |∆v|(v · ∆v) dt, (B2a) 100 101 102 103 104 4ρs h(r) Time t/Pin C I Σ (r) h i δ` = − d |∆v| (r × ∆v) · `ˆ dt, (B2b) 4ρs h(r) Figure B1: Orbital evolution of a planetesimal due to drag where Σ and h(r) are the disc’s density profile and vertical d forces in a coplanar disc. The migration parameter is Λ ≈ scale-height evaluated at the (time-dependent) position of 1.3×106 and the disc aspect ratio is h/r = 10−3. The epoch the planetesimal. t = 7.5Λ(h/r)P is marked with a vertical red line. Upper In Figure B1, we show an example of coplanar orbital in panel: Characteristic orbital distances a (solid black curve), evolution from a similar initial condition (a, e) to the cases a(1 ± e) (dashed), and a(1 − e2) (dotted) expressed in units shown in Figs.6 through 10 and with the power-law mass of the disc’s inner radius r . The shaded region indicates the distribution from Section3. As then, we find that the cir- 1 radial extent of the circumstellar disc. Lower panel: Eccen- cularization time-scale can be estimated simply for initial tricity e. orbits with e → 1: h tdrag ≈ 7.5Λ Pin, (B3) body has a greater tangential velocity than the disc (when r the body feels a headwind) and positive in the opposite case where Λ is the same as in equation (44) except that the col- (a tailwind). Since the disc and body are assumed to move umn density Σd is evaluated at r = rp. Comparing equations (locally) on circular and eccentric Keplerian orbits, respec- (B3) and (43), we see that coplanar disc migration can be tively, it can be shown that the sign of the instantaneous orders of magnitude faster than inclined disc migration for torque is a thin disc, all else being equal.  2  +1, r > a(1 − e ); Several features of coplanar orbital evolution differ qual- sgn(N ) = −1, r < a(1 − e2); (B5) itatively from the inclined case. The most important of these  0, r = a(1 − e2). is that, depending on the relative sizes of the disc and orbit, the torque exerted by drag can be positive. This aspect can The sign of the total torque is determined by the definite be explained in terms of the kinematics of Keplerian orbits. integral in equation (B2b). The sign of the torque exerted by drag on the orbit at a distance r from the star is determined by h i N ∝ − (v − vd) · φˆ , (B4) where φˆ is the unit vector in the azimuthal direction. This implies that the instantaneous torque is negative when the

MNRAS 000,1–17 (2020)