Multiple Disk Gaps and Rings Generated by a Single Super-Earth: Ii. Spacings, Depths, and Number of Gaps, with Application to Real Systems

Draft version September 18, 2018 Preprint typeset using LATEX style emulateapj v. 12/16/11 MULTIPLE DISK GAPS AND RINGS GENERATED BY A SINGLE SUPER-EARTH: II. SPACINGS, DEPTHS, AND NUMBER OF GAPS, WITH APPLICATION TO REAL SYSTEMS Ruobing Dong1,2, Shengtai Li3, Eugene Chiang4, & Hui Li3 Draft version September 18, 2018 ABSTRACT ALMA has found multiple dust gaps and rings in a number of protoplanetary disks in continuum emission at millimeter wavelengths. The origin of such structures is in debate. Recently, we doc- umented how one super-Earth planet can open multiple (up to five) dust gaps in a disk with low −4 viscosity (α . 10 ). In this paper, we examine how the positions, depths, and total number of gaps opened by one planet depend on input parameters, and apply our results to real systems. Gap locations (equivalently, spacings) are the easiest metric to use when making comparisons between theory and observations, as positions can be robustly measured. We fit the locations of gaps empirically as functions of planet mass and disk aspect ratio. We find that the locations of the double gaps in HL Tau and TW Hya, and of all three gaps in HD 163296, are consistent with being opened by a sub-Saturn mass planet. This scenario predicts the locations of other gaps in HL Tau and TW Hya, some of which appear consistent with current observations. We also show how the Rossby wave instability may develop at the edges of several gaps and result in multiple dusty vortices, all caused by one planet. A planet as low in mass as Mars may produce multiple dust gaps in the terrestrial planet forming region. Subject headings: protoplanetary disks | planets and satellites: formation | planet-disk interac- tions | stars: variables: T Tauri, Herbig Ae/Be | stars: pre-main sequence | hydrodynamics 1. INTRODUCTION et al. 2016; Dipierro et al. 2016; Dipierro & Laibe 2017). The Atacama Large Millimeter Array (ALMA) has dis- At first glance, multiple gaps seem to imply multiple covered multiple gaps and rings in a number of proto- planets, with one planet embedded within each gap. Re- planetary disks in dust continuum emission at millime- cently, however, it has been appreciated that a single ter (mm) wavelengths at ∼10{100 AU. Examples include planet can generate multiple gaps. Using gas+dust sim- the disk around HL Tau (ALMA Partnership et al. 2015), ulations, Dong et al. (2017) showed that one super-Earth TW Hya (Andrews et al. 2016; Tsukagoshi et al. 2016), planet (with mass between Earth and Neptune) can pro- HD 163296 (Isella et al. 2016), AS 209 (Fedele et al. duce up to five dust gaps in a nearly inviscid disk, and 2018), AA Tau (Loomis et al. 2017), Elias 24 (Cieza that these gaps would be detectable by ALMA (see also et al. 2017; Cox et al. 2017; Dipierro et al. 2018), GY Muto et al. 2010; Duffell & MacFadyen 2012; Zhu et al. 91 (Sheehan & Eisner 2018), and V1094 Sco (Ansdell 2013; Bae et al. 2017; Chen & Lin 2018; Ricci et al. 2018). et al. 2018; van Terwisga et al. 2018). The origins of In particular, a pair of closely spaced narrow gaps (a these gaps are being debated. Several hypotheses have \double gap") sandwiches the planet's orbit. This struc- been put forward, including disk-planet interaction (e.g., ture is created by the launching, shocking, and dissipa- Dong et al. 2015; Dipierro et al. 2015; Pinilla et al. 2015; tion of two primary density waves, as shown in Goodman Jin et al. 2016), zonal flows (Pinilla et al. 2012), secu- & Rafikov (2001, see also Rafikov 2002a,b). Additional lar gravitational instability (e.g., Takahashi & Inutsuka gaps interior to the planet's orbit may also be present, 2014), self-induced dust pile-ups (Gonzalez et al. 2015), possibly opened by the dissipation of secondary and ter- radially variable magnetic disk winds (Suriano et al. tiary density waves excited by the planet (Bae & Zhu 2017a,b), and dust sintering and evolution at snowlines 2018a,b). arXiv:1808.06613v2 [astro-ph.EP] 14 Sep 2018 (Zhang et al. 2015; Okuzumi et al. 2016). In nearly all Motivated by the boom of multi-gap structures discov- mechanisms, perturbations in gas are followed by radial ered by ALMA, and the mounting evidence of extremely drift and concentration of mm-sized dust (e.g., Weiden- low levels of turbulence at the disk midplane (from di- schilling 1977; Birnstiel et al. 2010; Zhu et al. 2012). In rect gas line observations, e.g., Flaherty et al. 2015, 2017; planet-disk interaction scenarios, even super-Earth plan- Teague et al. 2018, and from evidence for dust settling, ets may open dust gaps (e.g., Zhu et al. 2014; Rosotti e.g., Pinte et al. 2016; Stephens et al. 2017), we investi- gate here what properties of the planet and disk can be inferred from real-life gap observations. We study sys- 1 Steward Observatory, University of Arizona, Tucson, AZ, tematically how gap properties depend on planet and 85719; [email protected] disk parameters, and propose generic guidelines con- 2 Department of Physics & Astronomy, University of Victoria, Victoria, BC, V8P 1A1, Canada necting ALMA observations with disk-planet interaction 3 Theoretical Division, Los Alamos National Laboratory, Los models. We introduce our models in x2, and present our Alamos, NM 87545 main parameter survey of gap properties in x3. These 4 Department of Astronomy, University of California at Berke- ley, Berkeley, CA 94720 results are applied to three actual disks in x4, discussed 2 in x5, and summarized in x6. The disk aspect ratio h=r is assumed radially invariant. We adopt a constant instead of a flared h=r profile to sup- 2. SIMULATIONS press the dependences of gap properties on the h=r gra- dient, and to increase the timestep at the inner bound- We use two-fluid (gas + dust) 2D hydrodynamics sim- ary. We note that fractional gap spacings, which we will ulations in cylindrical coordinates (radius r and azimuth use as our primary diagnostic of models and observations φ) to follow the evolution of gas and dust in a disk con- (see our equation 8), depend only weakly on radial gra- taining one planet. The code used is LA-COMPASS (Li dients in h=r,Σ , and the slope of the background et al. 2005, 2009). The numerical setup, largely adopted gas;0 Σgas(r) profile. Dong et al. (2017, Fig. 8) showed that from Dong et al. (2017), is summarized here (more de- 0 tails can be found in Fu et al. 2014 and Miranda et al. as the background Σgas(r) profile varies from Σgas / r −1:5 2017). to Σgas / r , all gaps and rings shift inward, while We adopt a locally isothermal equation of state: pres- their relative separations are unchanged at the level of a 2 few percent. sure P = Σgascs , where Σgas and cs are the surface density and sound speed of the gas. Dust is modeled as a The dimensionless stopping time of dust | the particle pressureless second fluid. The following Navier-Stokes momentum stopping time normalized to the dynamical equations are solved: time | is, in the Epstein drag regime, πsρparticle @Σgas τ = ; (7) + r · (Σ v ) = 0; (1) stop 2Σ @t gas gas gas @Σ −3 dust + r · (Σ v ) = where ρparticle = 1:2 g cm is the internal density of @t dust dust dust particles, and s = 0:2 mm is the particle size. Such Σdust particles are probed by continuum emission at mm wave- r · ΣgasDdustr ; (2) lengths (e.g., Kataoka et al. 2017). At r = 0:2{1 rp, Σgas τstop is on the order of 0.01. We note that for mm-to- @tΣgasvgas cm sized particles often traced by VLA cm continuum + r(vgas · Σgasvgas) + rP = @t observations (possibly having τstop ∼ 0:1 − 1), gaps and −ΣgasrΦG + Σgasfν − Σdustfdrag; (3) rings similar to those in 0.2 mm sized dust are generated. Qualitatively, as particle size increases, gaps widen @Σdustvdust + r(vdust · Σdustvdust) = and deepen with their locations largely unchanged; the @t amplitude of the middle ring (MR) co-orbital with the −ΣdustrΦG + Σdustfdrag; (4) planet drops; particle concentrations at the triangular Lagrange points L4 and L5 become stronger; and an az- where Ddust is the dust diffusivity defined as Ddust = 2 imuthal gap centered on the planet's location emerges ν=(1 + τstop) (Takeuchi & Lin 2002), ΦG is the gravita- and widens (Lyra et al. 2009; Zhu et al. 2014; Rosotti tional potential (including both the gravity of the planet et al. 2016; Ricci et al. 2018). and the self-gravity of the disk; a smoothing length The simulation domain runs from 0 to 2π in φ, and of 0.7h is applied to the gravitational potential of the 0.1 to 2.1rp in r. The initial gas disk mass is 0:015M . ΩK planet), fdrag = (vgas − vdust) is the drag force on We set up a wave damping zone at r = 0:1{0:2r as τstop p dust by gas, τstop is the dimensionless stopping time (see described in de Val-Borro et al. (2006), and an inflow below), and fν is the Shakura-Sunyaev viscous force with boundary condition at the outer boundary. The resolu- 2 tion is discussed in x2.1. ν = αh ΩK, h the disk scale height, ΩK the Keplerian angular frequency, and α < 1 a dimensionless constant. We summarize our models in Table 1. The model name Other symbols have their usual meanings. The last term indicates the values of h=r and Mp=Mth adopted (e.g., in Eqn.

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