arXiv:2007.05022v1 [astro-ph.EP] 9 Jul 2020 n hto h icmlntr ikcudaiefo the protoplanetar from turbulent arise a from could gas disk of circumplanetary accretion the stochastic of that and so ( that 2014 planets suggestions density to led low has very This hav signatures. syste would transit planetary systems deep extrasolar ring or In disks circumplanetary interest. misaligned of also is however, sepa orbital smaller at planets for (especially teractions 2020 Hamilton & goszinski gi- 2012 processes including al. late-time et ( misalignment, of regular be- impacts variety spin-orbit to The A ant planet close are produce spin. planets Neptune. the can these with and of aligned systems Uranus ing ring Saturn, and satellites for whic large obliquities, planetary are from comes disks cumplanetary 2020 bidelli 2002 Ward & w plane eccentricities; (those the of orbital satellites plane low equatorial regular the of to inclination birthplace Circumplanetary orbital has low the disk planet. also forming the are the of disks on orientation cir- impact the their significant from thus a accreted and been disks have may ( cumplanetary Jupiter forms as disk such giants circumplanetary 1999 a al. radius, et Hill the than 1996 Lubow & 2003 rrns( rings or iloln ayi 2019 Batygin & Millholland 2015 Nesvorn´y Vokrouhlick´y & bevtoa intrso omn lnt ( planets forming of signatures observational lntr ikoc t asruhyecesta fNeptune of that exceeds roughly ( mass its once disk planetary i aaozu1986 Papaloizou & Lin ml iainet ewe h lntr ria plane orbital planetary the between misalignments Small c misaligned of possibility the considering for Motivation rpittpstuigL using typeset Preprint D omn lnti bet ial pnagpi h proto- the in gap a open tidally to able is planet forming A atversion raft ih er-nepee spaeswt iainddisks misaligned with planets as re-interpreted be might ) ( gap the into flow to continues Material ). ujc headings: Subject in possible and dynamics. misa evolution, be spin planetary commonly di for may circumplanetary implications disk, detached circumstellar the that from imply accretion results Our orbits. ikwt ue setratio aspect outer ( with particle-based disk conditio linear using favorable a instability provide Using disks the dissipation. circumplanetary and of potential radii tidal the of nents io&Vsargd 2020 Vissapragada & Piro 1 ; ,si-ri eoacs( resonances spin-orbit ), crto ik nbnr ytm a xii itinstabi tilt a exhibit can systems binary in disks Accretion eateto hsc n srnm,Uiest fNevada, of University Astronomy, and Physics of Department ,admypoiesm ftems prominent most the of some provide may and ), ’neoe l 2002 al. et D’Angelo arnv1966 Safronov ; J uly .Snetesz ftepae smc smaller much is planet the of size the Since ). oqer srd 2003 Estrada & Mosqueira ATGOIGTL NTBLT FDTCE CIRCUMPLANETA DETACHED OF INSTABILITY TILT FAST-GROWING A 3 2020 13, A T E tl mltajv 12 v. emulateapj style X 1. INTRODUCTION ; 3 uie&Seesn1982 Stevenson & Lunine aua2014 Masuda ’neoe l 2002 al. et D’Angelo eateto hsc n srnm,SoyBokUniversit Brook Stony Astronomy, and Physics of Department n lntcruselrds in- disk planet-circumstellar and ) 2 lntr ytm tr:pemi sequence pre-main stars: – systems instabilitie planetary – – hydrodynamics – disks accretion accretion, etrfrCmuainlAtohsc,Faio Institut Flatiron Astrophysics, Computational for Center ; .Apioda contribution, primordial A ). eze l 1989 al. et Benz ; .Ms ftems fgas of mass the of Most ). rse e 2015 Lee & Brasser ; R ad&Hmlo 2004 Hamilton & Ward knam ta.2020 al. et Akinsanmi ebecca / H 16 / / 11 ; r otfHte tal. et Jontof-Hutter ≃ ; .M G. ayi Mor- & Batygin h 2015 Zhu 0 phantom . ,iiilymdrt it obeo iesaeo bu 15 about of scale time a on double tilts moderate initially 1, artin ; Artymowicz ; aee al. et Bate Morbidelli 1 rf eso uy1,2020 13, July version Draft ; Z , Lubow ). n rdbsd( grid-based and ) Canup ration; ; haohuan and t Ro- ABSTRACT ms, me ith ). ir- h y e ; a ea,40 ot ayadPrwy a ea,N 89154, NV Vegas, Las Parkway, Maryland South 4505 Vegas, Las Z eatosbtentetl ntblt n Kozai-Lidov and instability tilt the between teractions hu lntwt elra nua frequency angular Keplerian with planet radius at disk circumplanetary the in terial ii,tednmc famslge ikaedtrie by determined The are potential. disk tidal misaligned the of a components of two dynamics the limit, rnaine truncation disk. the of properties consider hr h ria rqec is frequency orbital the where ik( disk tr ik r ml n aealreds setrto we ratio, Section aspect In disk increase. to large tends a tilt their have that and here show small circumplan- are since disks However, etary alignment. coplanar is outcome by example circumplanetary for the onto (studied accretion interest of scales time inadpeesa oi oy( communica body disk wave-like solid a through The as together precess and itself disk. tion hold the to of able precession is nodal retrograde produces ae2009b Bate s s yrdnmcsmltos ecnld nSection in conclude We Section simulations. in hydrodynamic while use behaviour use the examine to methods analytic 2000 the while alignment, coplanar to leads component 1996 al. et Larwood 2014 ed otetl nraig( increasing tilt the to leads rdcsa oclaig oqewt eido afteor a the not half ( does of which period a period, with bital torque “oscillating” an produces emn h odtosudrwihasaltl ih grow. dis might first were tilt discs by small distorted a tidally in which instabilities Tilt under conditions the termine M sfrtl rwh eqatf h rwhrt of rate growth the quantify We growth. tilt for ns aze l 1982 al. et Katz 1 k,woeeouini o nieycnrle by controlled entirely not is evolution whose sks, nlss eso htteapc aisadouter and ratios aspect the that show we analysis, s h ieo icmlntr iki eemndb tidal by determined is disk circumplanetary a of size The ecnie lnto mass of planet a consider We , iy rsn rmteitrcinbtencompo- between interaction the from arising lity, Lubow and tobtlseparation orbital at indt h lntr ria ln.W discuss We plane. orbital planetary the to ligned ; ; rse ta.2013 al. et Gressel athena++ cui ta.2020 al. et Schulik aee l 2000 al. et Bate ,NwYr,N 01,UAand USA 10010, NY York, New e, P hilip ,SoyBok Y174 USA 11794, NY Brook, Stony y, ( ; ff 1992 ’neoe l 2002 al. et D’Angelo .A J. csdet h tr( star the to due ects yrdnmcsmltos o a For simulations. hydrodynamic ) rmitage .W aeoekysmlfiain nthe on simplification: key one make We ). .I h rsneo ispto the dissipation of presence the In ). paesadstlie:formation satellites: and –planets s 2. ; 2.1. NLTCESTIMATES ANALYTIC eqe 1998 Terquem .Frcruselrdss yial the typically disks, circumstellar For ). 2,3 a engetd nti “detached” this In neglected. be can ) a aiaae l 2012 al. et Tanigawa ieo h disk the of Size .Orga nthe in goal Our ). p ihobtlperiod orbital with uo 1992 Lubow ff Ω aaozu&Trum1995 Terquem & Papaloizou c h enpeeso rate precession mean the ect YDISKS RY b M .Acpaa circumplane- coplanar A ). aznk 1977 Paczynski = p .The ). q riigasa fmass of star a orbiting G = Ω ( ; 3 binary -30 M r uo Ogilvie & Lubow m m risaon the around orbits p ; Letter = = + q zlay tal. Szul´agyi et P USA component 2 component 0 M GM orb m ; s ) Ayli / = p 2 = st de- to is a / covered p 3 euse we r m 2 term 2 Ma- . 3 ff π/ We . disk = & e Ω 4 3 0 b - - ; . 2 tary disk is initially truncated at a radius of about rout = 0.4 rH where (Martin & Lubow 2011), where the Hill sphere radius GM r T = s b(1) Σr, 2 3/2 (8) 1/3 4ap " ap !# Mp rH = ap . (1) 3M ! (1) s and b3/2 is the Laplace coefficient. iωt For typical parameters the Hill sphere radius is We seek normal modes of the form lx, ly, Gx and Gy ∝ e where ω is a complex eigenvalue. We can write 1/3 ap Mp r = . . iωt −iω⋆t H 0 36 −3 au (2) W = W+e + W−e (9) 5.2au 10 Ms ! and The Hill sphere and therefore the disk size scale with the or- iωt −iω⋆t bital semi-major axis of the planet. We note that the tidal trun- G = G+e + G−e , (10) cation radius of a misaligned circumplanetary disk is larger where ω⋆ is the complex conjugate of the eigenvalue ω. We than that of a coplanar disk (Lubow et al. 2015; Miranda & follow Lubow & Ogilvie (2000) and expand the equations in Lai 2015). powers of the tidal potential such that 2.2. Viscous evolution timescale T = T (1) + T (2) + ... (11) The viscosity of the disk is (0) (1) (2) W+ = W+ + W+ + W+ + ... (12) 2 H 2 ν = αcsH = α r Ω, (3) W = W (1) + W (2) + ... (13)  r  − − − (1) (2) where α is the Shakura & Sunyaev (1973) viscosity parame- G+ = G+ + G+ + ... (14) ter, cs is the disk sound speed and H/r is the disk aspect ratio. (1) (2) The density of the disk falls off exponentially away from the G− = G− + G− + ... (15) midplane ρ ∝ exp[−(1/2)(z/H)2](Pringle 1981), where H is and the disk scale height. The disk aspect ratio of a circumplane- (0) (1) (2) tary disk can be significantly larger than that of a circumstellar ω = ω + ω + ω + ... (16) disk with typical values in the range 0.1 − 0.3(Ayliffe & Bate To lowest order, the rigid tilt mode has 2009a; Martin & Lubow 2011). The viscous timescale at the (0) (0) outer edge of the disk is ω = −Ωb and W+ = const. (17) r2 To first order equations (6)and (7) become t = out , (4) ν ν(r ) out 1 dG(1) iω(1)Σr2ΩW(0) = + − iT (1)W(0) (18) which for our typical parameters is + r dr + −1 −2 3/2 tν α H/r rout 1 dG(1) = 232 . (5) 2iΩ Σr2ΩW(1) = − − iT (1)W(0)⋆ (19) Porb 0.01 0.1 ! 0.4 rH ! b − r dr +

This viscous timescale is shown as a function of H/r and the 3 3 (1) (1) Ir Ω dW+ disk outer radius in the right panel of Figure 1 for α = 0.01. αΩG+ = (20) The viscous timescale is a measure of the lifetime of the disk 4 dr if there is no further accretion on to it. We do not include ac- and cretion on to the disk in our analysis of the evolution of the tilt 3 3 (1) (1) Ir Ω dW− of the disk. Accretion that occurs at close to zero inclination (2iΩb + αΩ)G− = . (21) will act to lower the tilt of the disk. 4 dr The precession rate to first ordercan be foundby integrating 2.3. Tilt evolution of the disk equation (18) over the whole disk to find

The angular momentum of each ring of the disk with sur- rout (1) (0) T W+ r dr face density Σ(r) is defined by the complex variable W = (1) rin ω = − r . (22) lx + ily, where l is a unit vector parallel to the disk angular R out 2 (0) Σr ΩW+ r dr momentum. The internal torque of the disk is represented rin R by the complex variable G. We begin with equations (37) We use the boundary conditions that the internal disk torque and (38) in Lubow & Ogilvie (2000) that describe the evo- vanishes at the boundaries lution of a disk in the binary frame that rotates with angular (1) (1) velocity Ωb = Ωbez. The equations are based on a linear G± (rin) = G± (rout) = 0. (23) theory that is valid for small warps and tilts. The equations We solve equations (18)to(21) for a fixed surface density are profile Σ ∝ r−3/2 and choose W(0) = 1 to find W(1) and G(1). 2 ∂W 1 ∂G ∗ + ± ± Σr Ω + iΩbW = − Ti(W + W ) (6) By considering the second order terms, the change to the in- ∂t ! r ∂r clination of the disk is determined by the tilt growth rate and rout 4α (1) 2 (1) 2 2 3 3 Ir4Ω2 |G+ | −|G− | r dr ∂G T ΣH r Ω ∂W ℑ(ω(2)) = rin (24) + iΩbG − iG + αΩG = (7) R  rout  2 (0) 2  ∂t Σr2Ω 4 ∂r Σr Ω|W+ | r dr rin R 3

Fig. 1.— Left: Contour plot of the log of the growth timescale of the tilt of a circumplanetary disk in units of the binary orbital period as a function of the disk aspect ratio H/r and the disk outer radius, rout, in units of the Hill sphere radius. The planet mass mass is Mp = 0.001 Ms. The surface density is fixed as −3/2 Σ ∝ r distributed between rin = 0.03 rH up to rout. The disk viscosity is α = 0.01. In the white region the disk tilt decreases. In the coloured regions the disk tilt increases. Right: Contour plot of the log of the disk viscous timescale in units of the binary orbital period. growth timescale as a function of the disk outer radius and the disk aspect ratio. The darkest region shows the location of the primary resonance. The resonance is where the frequency of the lowest order global bending mode in the disk matches the tidal forcing frequency, which is twice the binary orbital frequency (2Ωb). The trough becomes lower and narrower as α decreases (Lubow & Ogilvie 2000). Growth of the disk tilt occurs when the outer radius of the disk is close to the res- onance. Even in the case of zero viscosity, disc tilt growth may occur if the disk is within the resonant band (Lubow & Ogilvie 2000). Note that larger disk aspect ratios correspond to larger viscous torques, for which we expect larger disks. The white region shows where, for small H/r, the growth rate is positive, meaning that the disk tilt decays. The disk moves towards coplanar alignment with the orbital plane of the planet. However, for larger H/r, the growth rate is nega- tive and the disk inclination increases. For a fixed disk outer radius, there is a narrow range of values for H/r for which the Fig. 2.— Contour plot of the ratio of the growth timescale of the tilt of a circumplanetary disk to the viscous timescale as a function of the disk aspect growth rate is very large. ratio H/r and the disk outer radius, rout, in units of the Hill sphere radius. Figure 2 shows a contour plot of the ratio of the growth −3/2 The planet mass is Mp = 0.001 Ms. The surface density is fixed as Σ ∝ r timescale to the viscous timescale. The dark region shows distributed between rin = 0.03 rH up to rout. The disk viscosity is α = 0.01. where a disk is most unstable to the tilt instability. With a In the white region the disk tilt decreases. In the coloured regions the disk tilt increases. smaller α, the width of the unstable region is narrower (but at the same location) while the viscous timescale is longer. In (Lubow& Ogilvie 2000). The sign of this determines whether this case, whether the disk can tilt depends sensitively upon (1) the distribution of material within the disk. For larger α, the the disk tilt increases or decreases. The |G+ | term is caused by the m = 0 component of the potential and leads to damping viscous timescale decreases and so while the disk is more un- (1) stable to tilting, there may not be sufficient time for it to occur. while the |G− | term is caused by the m = 2 component and causes pure growth. 3. HYDRODYNAMIC SIMULATIONS

2.4. Tilt growth timescale The analytic estimates assume a fixed power law surface density and do not accurately model the truncation of the outer We calculate the tilt growth timescale as parts of the circumplanetary disk due to the tidal torques. To overcome these limitations we have investigated the tilt evo- tgrowth Ω = b . (2) (25) lution using both SPH and grid-based simulations. Porb 2π|ℑ(ω )| −3 3.1. SPH simulations We consider a planet with a mass of Mp = 10 Ms. The disk extends from rin = 0.03 rH up to rout, that we vary. Note We first use the smoothed particle hydrodynamics (SPH) that because we parameterise the disk size in terms of the code Phantom (Price & Federrath 2010; Lodato & Price 2010; Hill sphere radius, the results presented are independent of the Price et al. 2018) to model a misaligned circumplanetary disk. planet semi-major axis. The viscosity parameter is α = 0.01. Phantom has been used extensively to model misaligned The left panel of Figure 1 shows a contour plot of the tilt disks in binary systems (e.g. Nixon et al. 2013; Smallwood 4

Fig. 3.— Hydrodynamic simulations of a circumplanetary disk that is ini- tially misaligned by 10◦ showing the inclination of the disk (upper panel) and the nodal precession angle (lower panel). The angles are averaged by mass over the radial extent of the disk. The SPH simulations have 250, 000 parti- cles (red), 500, 000 particles (blue) and 106 particles (magenta) initially. The grid simulation results are shown as the black (low resolution) and cyan (high resolution) curves. The SPH and grid simulations both have an outer aspect ratio of H/r = 0.1, though they differ in details of the initial disk profile. et al. 2018; Franchini et al. 2019). The simulation has two sink particles, one representing the star with mass Ms and the = −3 other representing the planet with mass Mp 10 Ms. The ig thena++ accretion radius of the star is 1.4 r , while that of the planet F . 4.— The disk’s isodensity contours for the A low resolution H run at 3 Porb (the upper panel) and 19 Porb (the lower panel). The disk has is 0.03 rH. The simulation does not have any dependence on the same nodal precession angle at these two snapshots. the planet’s orbital separation since all lengths are scaled to the Hill radius. The planet is in a circular orbit. The disk is for the circumplanetary disk. The disk nodally precesses at initially tilted to the binary orbital plane by 10◦. roughly a constant rate as a result of the m = 0 tidal torque The surface density of the disk is initially distributed as a component. There are small scale oscillations on a timescale −3/2 power law Σ ∝ r between rin = 0.03 rH up to rout = 0.4 rH. Porb/2 as a result of the m = 2 tidal torque component. The We note that the initial truncation radii of the disk do not inclination of the disk increases in time meaning that the dis- significantly affect the evolution since the density evolves sipation of the m = 2 component dominates. This is in agree- quickly. We have tested both smaller and larger initial outer ment with the analytic predictions in the previous section that truncation radii. The mass of the disk does not affect the circumplanetary disks are unstable to tilting. We have also evolution since we do not include disk self-gravity, and we run simulations with varying disk aspect ratio and find that −6 growth occurs for H/r & 0.05. take the initial total disk mass to be 10 Ms. We consider three different initial SPH particle numbers, 250,000, 500,000 and 106. The disk is locally isothermal with sound speed 3.2. Grid simulations −3/4 cs ∝ r . This is chosen so that α and the smoothing length We have carried out independent grid-based simulations hhi /H are constant over the disk (Lodato & Pringle 2007). with Athena++ (Stone et al. 2020), using a planet-centered We take the aspect ratio at rin to be H/r = 0.2. This corre- spherical-polar co-ordinate system. The simulation domain sponds to H/r = 0.1 at the initial outer truncation radius. We extends from 0.0286 rH to rH in the radial direction, from 0.2 take the Shakura & Sunyaev (1973) α parameter to be 0.01. to π-0.2 in the θ direction, and a full 2π in the φ direction. This is the lowest value that can be physically resolved in One level of mesh-refinement is applied at θ=[0.885,2.256]. SPH simulations at the lowest resolution we employed (Price The third-order reconstruction scheme has been adopted. The et al. 2018). We implement the disk viscosity by adapting the circumplanetary disk’s density and temperature profiles are SPH artificial viscosity according to the procedure described the same as the SPH simulations in Section 3.1. The disk 2 in Lodato & Price (2010) with αAV = 0.36, 0.46, 0.58, in surface density has an exponential tail of exp(−(r/rcut) ) with order of increasing resolution, and βAV = 2. The circumplan- rcut = 0.3 rH. The detailed disk and grid setup is presented etary disk is initially resolved with shell-averaged smoothing in Zhu (2019). Two simulations with different resolutions length per scale height hhi /H = 0.28, 0.22 and 0.17 in order have been carried out. In the low resolution run, we have of increasing resolution. 72×56×128 grids at the base level in the r × θ × φ domain. Figure 3 shows the inclination and nodal precession angle With one level of mesh refinement, this is equivalent to 8 grid 5 cells per scale height at the innerboundaryand 5 grid cells per to the orbital plane while the planet obliquity may be non- scale height at rcut. Based on Zhu (2019), this is the minimum zero, the result is a warped non-precessing disk whose shape resolution required to study disk precession. To verify conver- follows the Laplace surface (Tremaine et al. 2009). In the gence, we have doubled the resolution in every direction and presence of tilt instability, the outer part of the disk will be carried out a high resolution simulation. Due to the compu- misaligned and precess on a relatively short time scale. Al- tational cost, we have only run the high resolution simulation though the torque that a warped disk exerts on the planet for ∼9 Porb. shortens the time scale on which the planet spin-axis changes Figure 3 shows the evolution of tilt in the grid-based sim- (compared to the accretion-only limit, see e.g. the analogous ulations. The high resolution run is almost identical to the black hole situation; Scheuer & Feiler 1996), the precession low resolution run, and displays similar but somewhat faster is likely to limit obliquity evolution. The combined effects of tilt growth than the corresponding SPH simulations. Figure 4 a warp and outer disk precession might restrict where in the shows the disk’s isodensity contours at 3 and 19 Porb from the disk regular satellites are able to form. low resolution simulation. Clear growth of the disk inclina- In future work we plan to investigate a range of circum- tion is observed at 19 Porb. We can also see that the disk is planetary disk parameters and study the long term behavior of significantly depleted with time due to accretion. disks that exhibit tilt instability. We expect sufficiently high inclinations to trigger the onset of Kozai–Lidov (Kozai 1962; 4. DISCUSSION AND CONCLUSIONS Lidov 1962) oscillations, which exchange inclination and ec- Fluid disks in binary systems can be linearly unstable to the centricity of the disk (Martin et al. 2014; Fu et al. 2015). The growth of disk tilt (Lubow 1992). In this Letter we have ar- critical inclination for a disk to become Kozai–Lidov unstable gued that the physical conditions of circumplanetary disks— depends upon the disk aspect ratio (Lubow & Ogilvie 2017; their aspect ratios and outer truncation radii—are favorable Zanazzi & Lai 2017). We also note that while a circumplane- for rapid tilt growth, at least in the limit where the disks are tary gas disk may be stable against Kozai–Lidov oscillations, detached and not accretion-dominated. Fast tilt growth is pre- any solid bodies that form within the gas disk may become dicted analytically, and recovered in SPH and grid-based sim- unstable once the gas disk has dissipated (e.g. Speedie & ulations that include physical effects (such as tidal truncation Zanazzi 2019). and spiral waves) that are not readily modeled analytically. ACKNOWLEDGEMENTS We have not included the effect of accretion from the circum- We thank Daniel Price for providing the phantom code stellar disk on to the circumplanetarydisk, which we expect to for SPH simulations. Computer support was provided by actasaneffective damping term. Nonetheless, given the rapid UNLV’s National Supercomputing Center. Athena++ sim- growth rates which we have found for H/r & 0.05, we expect ulations were carried out at the Texas Advanced Computing there to be a regime of parameter space for which circumplan- Center (TACC) at The University of Texas at Austin through etary disks commonly exhibit substantial tilts. If correct, there XSEDE grant TG-AST130002, and using resources from the will be implications both for the observability of circumplan- NASA High-End Computing (HEC) Program through the etary disks, and for the properties and dynamics of satellite NASA Advanced Supercomputing (NAS) Division at Ames formation within them. Research Center. We acknowledge support from NASA We have assumed that the planet is spherical in this work. TCAN award 80NSSC19K0639. Z. Z. acknowledges support The oblateness of a spinning planet would act to align the in- from the National Science Foundation under CAREER Grant ner parts of the circumplanetary disk to the spin of the planet. Number AST-1753168. In the classical scenario, where the disk at large radius aligns

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