What is the Mass of a Gap-Opening Planet?
WWhat is the Mass of a Gap-Opening Planet?
Ruobing Dong & Jeffrey Fung , ABSTRACT
High contrast imaging instruments such as GPI and SPHERE are discoveringgap structures in protoplanetary disks at an ever faster pace. Some of thesegaps may be opened by planets forming in the disks. In order to constrainplanet formation models using disk observations, it is crucial to find a robustway to quantitatively back out the properties of the gap-opening planets, inparticular their masses, from the observed gap properties, such as their depthsand widths. Combing 2D and 3D hydrodynamics simulations with 3D radiativetransfer simulations, we investigate the morphology of planet-opened gaps innear-infrared scattered light images. Quantitatively, we obtain correlations thatdirectly link intrinsic gap depths and widths in the gas surface density to observeddepths and widths in images of disks at modest inclinations under finite angularresolution. Subsequently, the properties of the surface density gaps enable usto derive the disk scale height at the location of the gap h , and to constrainthe quantity M /α , where M p is the mass of the gap-opening planet and α characterizes the viscosity in the gap. As examples, we examine the gaps recentlyimaged by VLT/SPHERE, Gemini/GPI, and Subaru/HiCIAO in HD 97048, TWHya, HD 169142, LkCa 15, and RX J1615.3-3255. Scale heights of the disks andpossible masses of the gap-opening planets are derived assuming each gap isopened by a single planet. Assuming α = 10 − , the derived planet mass in allcases are roughly between 0.1–1 M J . Subject headings: protoplanetary disks — planets and satellites: formation —circumstellar matter — planet-disk interactions — stars: variables: T Tauri,Herbig Ae/Be — stars: individual (HD 169142, TW Hya, HD 97048, LkCa 15,RX J1615.3-3255) Steward Observatory, University of Arizona, [email protected] Department of Astronomy, University of California, Berkeley NASA Sagan Fellow a r X i v : . [ a s t r o - ph . E P ] J u l
1. Introduction
In the past few years, high angular resolution observations of protoplanetary disks haverevolutionized our understanding of planet formation. Rich structures have been identified indisk images in near infrared (NIR) and in mm/cm dust continuum and gas emission. Amongthem, a particularly intriguing class of features is narrow gaps, found in systems such as TWHya (Debes et al. 2013; Rapson et al. 2015b; Akiyama et al. 2015; Andrews et al. 2016),HD 169142 (Quanz et al. 2013; Momose et al. 2015), HD 97048 (Ginski et al. 2016; Walshet al. 2016; van der Plas et al. 2016), AB Aur (Hashimoto et al. 2011), V 4046 Sgr (Rapsonet al. 2015a), HD 141569 (Weinberger et al. 1999; Mouillet et al. 2001; Konishi et al. 2016;Perrot et al. 2016), RX J1615.3-3255 (de Boer et al. 2016), and HL Tau (ALMA Partnershipet al. 2015; Yen et al. 2016; Carrasco-Gonz´alez et al. 2016). These gaps have both theirinner and outer edges revealed in images, enabling a full view of the gap structure andparticularly, a measurement of the gap width. Numerical studies of observational signaturesof planet-induced structures have suggested that gaps may be opened by planets (e.g. Wolf& D’Angelo 2005; Jang-Condell & Turner 2012, 2013; Fouchet et al. 2010; Gonzalez et al.2012; Ruge et al. 2013; Pinilla et al. 2015b,a; Dong et al. 2015; Jin et al. 2016; Dong et al.2016). While giant gaps, found in systems such as transitional disks (e.g. Hashimoto et al.2012; Follette et al. 2015; Stolker et al. 2016b), may be common gaps opened by multipleplanets (Zhu et al. 2011; Dodson-Robinson & Salyk 2011; Duffell & Dong 2015), the abovementioned narrow gaps are more likely to be the products of single planets.The masses and locations of planets still forming in disks are key quantities in the studyof planet formation. Different formation scenarios, such as “core accretion” and “gravita-tional instability” models, predict planets to form at different locations, with different finalmasses. Measuring the masses of gap-opening planets ( M p ) will thus help us distinguish dif-ferent planet formation scenarios. To do this, quantitative connections between M p and gapproperties that are directly measurable from images must be established. At the moment, ageneral scheme for this purpose is lacking. Rosotti et al. (2016) pioneered in this directionby connecting M p with gap width at both infrared and millimeter (mm) wavelengths. Theyexplored a limited parameter space (e.g., no variation on disk viscosity), and it is unclearwhether synthetic observations based on 2D hydro simulations employed by Rosotti et al.(2016) generate the same results as 3D hydro simulations.In this work, we propose to advance this field by connecting M p with gap propertiesin NIR scattered light. Our goal is to establish direct and quantitative relations betweentwo observables — the gap depth ( δ I ; the contrast) and width ( w I ; the radial extent) —with M p , and two disk parameters, the aspect ratio h/r and α -viscosity (Shakura & Sunyaev1973). This effort is newly motivated by recent advances in NIR polarized scattered light 3 –imaging driven by Subaru/HiCIAO (Tamura et al. 2006), Gemini/GPI (Macintosh et al.2008), VLT/NACO (Lenzen et al. 2003) and SPHERE(Beuzit et al. 2008). At the momentthese high resolution imaging instruments are discovering gaps at an ever faster pace, whichurgently necessitates the effort presented here.This [ δ I , w I ] − [ M p , h/r, α ] connection can be segmented into two parts,1. [ δ I , w I ] − [ δ Σ , w Σ ]: connect gap depth and width in images to the physical gap depthand width in the gas distribution, δ Σ and w Σ , measured from the disk surface densityΣ;2. [ δ Σ , w Σ ] − [ M p , h/r, α ]: connect δ Σ and w Σ to M p , h/r , and α .The second part has been well studied both analytically and numerically. Analytically,Kanagawa et al. (2015, see also Fung et al. 2014; Duffell 2015) have shown that based ontorque balancing one can derive a simple scaling relation δ Σ − Σ min − ∝ q ( h/r ) α , (1)where Σ min and Σ are the depleted and initial (undepleted) surface density in the gap, and q = M p /M (cid:63) . For the gap width, Goodman & Rafikov (2001) calculated the propagationof planet-induced density waves, and showed that in the weakly-nonlinear regime the gapwidth, the waves dissipate and deposit angular momentum over the length scale “shockinglength”, which is proportional to the disk scale height. It is therefore likely that the widthof the gap will also be proportional to the scale height: w Σ = r out − r in ∝ h, (2)where r out and r in are the radius of the outer and inner gap edges, respectively. This relationwill be tested and confirmed with our models. The weakly-nonlinear regime applies to planetswith masses comparable or smaller than the thermal mass M thermal , M p (cid:46) M thermal = M (cid:63) (cid:18) hr (cid:19) . (3)Several groups have carried out numerical simulations to fit δ Σ and w Σ as power laws of M p , h/r , and α (Duffell & MacFadyen 2013; Fung et al. 2014; Duffell 2015; Kanagawa et al.2015, 2016a). These empirical correlations were synthesized from 2D hydro simulations, andFung & Chiang (2016) confirmed that gap opening in 3D produces the same gap profiles asin 2D. The exact functional forms vary somewhat in the literature, mostly due to variations 4 –in the definitions of w Σ and δ Σ . For the gap depth, Fung et al. (2014) found, for deepgaps with δ Σ exceeding 10, up to 10 , Equation 1 agrees with simulations to within a factorof 2. For the gap width, hydro simulations by Muto et al. (2010), Dong et al. (2011b,a),Zhu et al. (2013), and Duffell & MacFadyen (2013) have verified the results of Goodman &Rafikov (2001), and showed that gap opening indeed initially occurs at a fixed number ofscale heights away from the planet’s orbit for a given planet mass. By defining the gap widthas the radial distance between the two edges where the surface density drops to 50% of theinitial value, Kanagawa et al. (2016a) found w Σ ∝ q / ( h/r ) − / α − / , which not only hasdependencies on M p and α , but also predicts an inverse relation with h/r . Whether thesedifferences reflect some unknown physics of of gap opening, or simply due to the differentdefinitions of w Σ , is unclear.The first step, finding out how a density gap looks like in scattered light images, is theaim of this paper. We address this question by carrying out 2D and 3D hydro and 3D radia-tive transfer simulations to study synthetic images of planet-opened gaps with parametrized h/r and α . This step is affected by properties of the disk and observational conditions, inparticular, the angular resolution η (a known parameter for a specific observation) and theinclination of the system i . The dependence i can be eliminated by only measuring thescattered light radial profile at ∼ ◦ scattering angle (Section 3.5), such as in face-on disksand along the major axis of inclined disks. A few other factors, including the total disk mass M disk , wavelength λ , and grain properties, may affect the conversion as well; however, withinreasonable ranges in the parameter space assumed in this paper (optically thick, gravita-tionally stable disks; 0 . µ m ≤ λ ≤ µ m; close to 90 ◦ scattering angle) they are generallyunimportant (e.g., Dong et al. 2012). For extremely faint gaps in which the gap bottomreaches the noise level in images, the sensitivity of the observations (i.e., detection limit)will affect the measurement of δ I ; in this work we focus on relatively low mass planets andshallow gaps and do not consider such situation.We focus on narrow gaps (∆ gap < . r p ) opened by a single relatively low mass giantplanet about Neptune ( M N ) and Saturn ( M S ) mass, up to one Jupiter ( M J ) mass. Themotivation is mainly threefold. First, such planets are probably more common than theirmore massive siblings (Cumming et al. 2008; Brandt et al. 2014), and therefore it may bemore likely to see gaps opened by such planets. Bowler (2016) concluded that only ∼ . . − M (cid:12) stars have 5 − M J planets at orbital distances of 30 −
300 AU. Althoughlittle is known about the statistics of sub-Jovian planet at tens of AU, observations such asHL Tau hints at their existence (Jin et al. 2016). Second, massive planets ( (cid:38) M J ) may openeccentric gaps (e.g., Kley & Dirksen 2006; D’Angelo et al. 2006), and vortices may form atthe gap edge excited by the Rossby wave instability (e.g., Li et al. 2000; Lin & Papaloizou2010; Zhu et al. 2014). These features complicate the interpretation of gaps in observations. 5 –Third, wide gaps may be common gaps opened by multiple planets, in which case thereis a degeneracy between the number of planets and their masses (Dong & Dawson 2016),preventing unique solutions on planets masses. The lower limit of M p in our work is set sothat its gap may be visible in scattered light (e.g., Rosotti et al. 2016).
2. Simulation Setup
In this section, we briefly introduce the numerical methods employed to produce syn-thetic images of gaps. In short, we use the hydrodynamics code
PEnGUIn (Fung 2015) tocalculate the density structures of planet-opened gaps in 2D and 3D, then “translate” theresulting density structures to scattered light images using radiative transfer simulations bythe Monte Carlo Whitney et al. (2013) code. For hydro simulations we follow the proceduresdescribed in Fung et al. (2014), while for radiative transfer simulations we adopt the methodsin Dong et al. (2015, 2016). Below we briefly summarize the key processes.
The 2D simulations performed in this paper have an identical setup as the one used inFung et al. (2014), and the 3D one is identical to Fung & Chiang (2016). The only differencesare that we choose the parameters q , h/r , and α from a different parameter space, focusingmore on smaller planets and lower disk viscosities, and that all results here are obtainedafter 6000 orbits, equivalent to about 1 Myr at 30 AU around a solar-mass star. This timeis sufficient for models with α ≥ − to reach a quasi-steady state. For the models withthe lowest α values, the gaps have not yet fully settled, but they are still relevant modelssince 1 Myr is already a significant fraction of the typical protoplanetary disk lifetime. Werecapitulate some important features here, and refer the reader to Fung et al. (2014) andFung & Chiang (2016) for the details.The initial disk profile assumes the following surface density and sound speed profiles:Σ = Σ p (cid:18) rr p (cid:19) − , (4) c s = c p (cid:18) rr p (cid:19) − , (5) 6 –where we set Σ p = 1, , and c p is a parameter we vary to obtain different h/r values. Sincethe disk scale height h is c s / Ω k with Ω k being the Keperlian frequency, h/r is constant inradius. In 3D hydrostatic equilibrium, the initial density structure is: ρ = ρ p (cid:18) rr p (cid:19) − exp (cid:18) GMc (cid:20) √ r + z − r (cid:21)(cid:19) , (6)where ρ p = Σ p / (cid:112) πh , with h p being the scale height at r = r p , is the initial gas densityat the location of the planet. Density and sound speed are related by a locally isothermalequation of state. The kinematic viscosity in the disk is ν = αc s h , where α is constant inradius. The simulation grid spans 0 . r p to 2 r p in radius, which corresponds to 12 AU to60 AU, if the planet is placed at 30 AU.Table 1 lists the parameters and results for all models. For radiative transfer calculations, we assume a central source of 2.325 R (cid:12) and 4350 K,appropriate for a 1 M (cid:12) star at 1 Myr (Baraffe et al. 1998). For 3D hydro simulations, wedirectly feed the density grid into radiative transfer simulations; for 2D hydro simulations,we puff up the disk surface density in the vertical direction z by a Gaussian profile, ρ ( z ) = Σ h √ π e − z / h , (7)using the same scale height h as in the hydro simulations. The MCRT simulation domainspans ± ◦ from the disk midplane, which is about a factor of 2 higher than the scatteredlight surface. A 1 AU radius circumplanetary region centered on the planet is excised (exceptin Section 2.3 and Figure 2, see below), for reasons laid out in Section 2.3.The hydro simulations are gas only, while scattered light comes from the dust. Weconvert gas into dust by assuming the two are well mixed, and assume interstellar mediumgrains (Kim et al. 1994) for our dust model. These grains are sub- µ m in size, and theirscattering properties are calculated using Mie theory. They are small enough that theirstopping time is very short compared to their orbital timescale and vertical stirring timescale(characterized by the viscous parameter α ) in our simulations, so they are dynamically wellcoupled with gas (e.g., Zhu et al. 2012). Since we do not consider the self-gravity of the disk, this normalization has no impact on our results. ∼ (cid:48)(cid:48) . r sub , where dust reachestemperature of 1600 K; between r sub and the inner boundary of hydro simulations (12 AU),we assume an axisymmetric inner disk smoothly joining the outer hydro disk with a constantsurface density. The details of the inner disk does not matter as long as it is axisymmetricand does not cast significant shadows on the other disk. We scale the hydro disk so thatthe initial condition Σ ( r ) = 10( r/ − / g cm − , resulting in a final total dust diskmass between 5 × − to 10 − M (cid:12) inside 60 AU (depending on how deep the gap is). Wenote that gap width and contrast do not depend on the assumed disk mass within one orderof magnitude from the chosen value. Radiative transfer simulations produce full resolutionpolarized intensity (PI) images at H -band, which are convolved by a Gaussian point spreadfunction to achieve a desired angular resolutions. Fiducially we choose η = 0 (cid:48)(cid:48) .
04, comparableto what is achievable by SPHERE and GPI, unless otherwise noted.As examples, Figure 1 shows the dust surface density and scattered light images at bothface on and i = 45 ◦ inclination for Model 1 M S h α H -band polarized intensity I , although we note that our results generallyapply to total intensity images as well (see below). In panel (e) and (f), we scale the imagesby r to compensate for the distance from the star, where r is the distance from the star(deprojected in inclined cases). Unless otherwise noted, all H -band images and their radialprofiles below have been scaled by r . Fung & Chiang (2016) have shown that the surface density of planet-opened gaps in 3Dhydro simulations are nearly the same as gaps in 2D simulations. In this section we showthat synthetic images are also practically identical between a 3D model and a model with2D surface density manually puffed up using Equation 7.Figure 2 shows two face-on synthetic images and their azimuthally averaged radialprofile for the model 1 M S h α h/r profile as in the 3D hydro calculation. Note that unlike all other images in the paper, inboth panels the circumplanetary region is not excised. The gap profiles (c) from the twoimages are essentially identical. This result means vertical kinematic support is unimportantin the vertical distribution of material in the gap region, and we can safely puff up 2D hydro 8 –surface density structures in MCRT simulations to study gaps in scattered light.The differences between the two images are in the local structures. First, the spiralarms are clearly more prominent in (a) than in (b), due to vertical kinematic support inplanet-induced density waves (Zhu et al. 2015, Dong & Fung, submitted). Second, thecircumplanetary region sticks out in (b) while it is not noticeable in (a). This is becausewhen puffing up 2D structures into 3D, the circumplanetary region, which has a highersurface density than the surroundings, is artificially expanded in the vertical direction bythe same h/r as the rest of the disk. In reality though the gravity in the circumplanetaryregion is dominated by the planet so the actual h/r of the circumplanetary region is muchsmaller. This leads to artificially brightening of the region in scattered light in (b). For thisreason, and to avoid the artificial shadow in the outer disk produced by this structure (e.g.,Jang-Condell 2009), we excise the circumplanetary region in 2D surface density maps whenpuffing it up in MCRT simulations in the rest of the paper.
3. The Conversions
In this section, we explore the connections between gap properties in surface density todisk and planet parameters h/r , α , and M p , and the correlations between gaps in scatteredlight and gaps in surface density. Figure 3 shows the radial profiles of the surface density, and both full resolution andconvolved images at face-on viewing angle for three representative models with α = 10 − , h/r = 0 .
05, and M p = 2 M N , 1 M S , and 1 M J . The images have been scaled by r . Qual-itatively, in all three cases, each with very different gap depths, the radial profiles of thefull resolution images closely traces the variations in the surface density profiles. This gen-eral behavior has been found by Muto (2011) using analytical calculations of scattered lightgrazing angles in disks with narrow gaps. The two deviate near the gap bottom, where thescattered light profile does not reach as deep as the surface density. Also, the outer edge ofthe scattered light gap lies slightly inside the surface density gap.These observations can be qualitatively explained. Approximately, scattered light comesfrom a surface with optical depth of unity to the star (Takami et al. 2014). For disks withconstant h/r , if the surface density variation is smooth, this surface is flat at roughly aconstant polar angle θ surface ∝ h/r (i.e., not concave-up as in flared disks). In this case, at 9 – i = 0 the r -scaled intensity of the scattered light at each radius is roughly proportional to thesurface density of the dust above the surface, I ∝ Σ θ>θ surface ( r ). Because Σ θ>θ surface ( r ) ∝ Σ( r )for disks with constant h/r , it follows that I ∝ Σ( r ), which is what we find. This relationbreaks when the surface density fluctuation is so large that the disk surface is no longerat roughly a constant θ surface , which happens when the gap depletion is high, and at thegap edges. Also, this analysis does not formally hold if the disk surface is flared (i.e., h/r increases outward). However, gaps are local structures, and the global flareness of the diskmay not significantly affect the width and depth of local structures. We will come back tothis point later in Section 3.6.Convolving the full resolution image by a finite PSF smooths structures. Convolved im-ages (dotted line) thus closely follow full resolution images outside the gap, as the variationsoccur on spatial scale much larger than the angular resolution ( ∼ η = 0 (cid:48)(cid:48) .
04 and d =140 pc). Inside the gaps, convolved images have less steep edges and shallower depths. A planet-opened gap is a perturbation on top of an initial “background” disk with noplanets. The definition and characterization of a gap are relative to this background. Whilein simulations a planetless background can be well defined, in real observations of actualsystems it is nearly impossible to do so. In addition, a complete global disk profile is oftennot accessible in real observations with finite inner and outer working angles. In fact, basedon scattered light alone it is usually difficult to verdict whether a “gap-like” structure isindeed a planet-opened gap or not. To mimic actual observations and to maximize theapplicability of the correlations derived in this work, below we will assume gaps as localstructures so that its parametrization only involves its immediate neighborhood, and we willcharacterize an observed gap in such a way that does not require prior knowledge of theunderlying planetless background disk. We emphasize this is an necessary assumption inorder to construct useful empirical relations to quickly and quantitatively convert observedgap properties to planet and disk properties. For individual actual systems, whether thisassumption is good or not may be indicated by the reasonableness of the fitting. Moreaccurate assessments of the planet and disk properties require detailed modeling of individualsystems with specifically designed simulations to match the 1D (or even 2D) gap profiles,instead of using parameterized gap depth and width.The definitions of gap depth and width in surface density vary somewhat in the litera-ture. For the former, the common practice is to define it as the ratio between the minimumΣ inside the gap (or averaged over a finite radial range) to a fiducial “undepleted” value 10 –(Σ ). What counts as Σ varies. For the gap width, the difficulty lies in defining the twogap edges, r out and r in (such that w Σ = r out − r in ). The common practice is to define themas where Σ reaches a certain threshold fraction of Σ (e.g., Kanagawa et al. 2016a). Thisclass of width definitions does not work for shallow gaps in which the depletion is less thanthe threshold fraction. With this fixed-depth-width definition, w Σ depends on M p and α inaddition to h/r (e.g., Kanagawa et al. 2016a,b).In this paper, we adopt dynamical definitions of the gap depth and width in both surfacedensity and images, as illustrated in Figure 4. Our definitions work for of gaps with a widerange of depth and width. For Σ, we first find the location in the gap where Σ reachesthe minimum, r min , Σ . The initial value of Σ at r min , Σ , Σ ( r min ), is regarded as the fiducialundepleted value. Thus δ Σ is defined as δ Σ ≡ Σ ( r min , Σ )Σ( r min , Σ ) . (8)We then regard the inner and outer gap edges, r in , Σ and r out , Σ , as where surface densityreaches the geometric mean of the minimum and the undepleted value,Σ edge = Σ( r in , Σ ) = Σ( r out , Σ ) = (cid:113) Σ ( r min , Σ )Σ( r min , Σ ); (9)therefore the gap width w Σ is w Σ ≡ r in , Σ − r out , Σ , (10)and the normalized gap width ∆ Σ is defined as∆ Σ ≡ w Σ r min , Σ . (11)We note w Σ has a physical unit (e.g., AU or arcsec), while ∆ Σ is dimensionless.The definitions of gap width and gaps in scattered light are similar. We first scalethe image by r , then find the location in the gap where I reaches the minimum, r min , I (a gap cannot be defined if the r -scaled intensity monotonically varies with radius). Wefind that in general r min , I ∼ r min , Σ ∼ r p , with small model-by-model differences. We thenread the intensity at r = 2 r min , I / r = 3 r min , I /
2. These two locations are far awayfrom the gap region so they are considered to be outside the influence of the gap. A fiducial“background” I at r min is subsequently defined as the geometric mean of the two points (notethat this choice assumes the underlying “planetless” background profile follows a power law), I ( r min , I ) ≡ (cid:112) I ( r ) I ( r ), and the gap depth is δ I ≡ I ( r min , I ) /I ( r min , I ) . (12) 11 –The inner and outer edges of the gap, r in , I and r in , I , are found at where I reaches the geometricmean of I and I at r min , I . I edge = I ( r in , I ) = I ( r out , I ) = (cid:113) I ( r min , I ) I ( r min , I ) . (13)Correspondingly, the width of the gap w I is defined as w I ≡ r in , I − r out , I , (14)and the normalized gap width ∆ I is ∆ I ≡ w I r min , I . (15)The gap parameters in surface density and in face-on images for all models are listed inTable 1.Finally, we note that all image-related quantities, such as δ I , w I , and ∆ I , are functionsof both angular resolution η , and inclination i , i.e., δ I ( η, i ), w I ( η, i ), and ∆ I ( η, i ). We omitthe variables when the context is clear. [ α, h/r, M p ] and [∆ Σ , δ Σ ]Previous numerical studies have fit empirical relations to express w Σ and δ Σ as functionsof [ α, h/r, M p ]. Here we derive our version based on the new definitions of depth and widthin Figure 4. As discussed in Section 1, analytical weakly-nonlinear theory and simple scalingrelations based on torque balancing have motivated Equations 1 and 2 as possible functionalforms for w Σ and δ Σ . We are able to achieve good fit for both correlations, as shown inFigure 5, by varying only the constant proportionality factor. For gap width we have∆ Σ = 5 . hr . (16)The standard deviation σ is 9% for all models. If we restrict to only the models that havebeen run for longer than viscous timescale ( α = 10 − and α = 3 × − , the black points),we again recover Eqn (16) while σ shrinks to 6%. We note that if we adopt the Kanagawaet al. (2016a) definition of the gap width, we do recover their ∆ Σ = 0 . q / ( h/r ) − / α − / correlation (with a standard deviation of 17%). However we prefer our dynamical definitionof the gap width over defining the width as the distance between two points on the edges witha fixed depletion factor for a few reasons. First, the latter does not work for shallow gapsopened by relatively low mass planets, and in real observations such gaps are more common 12 –than deeper gaps opened by more massive planets, due to the higher occurrence rate ofsmaller planets than their more massive siblings (Cumming et al. 2008). Second, adoptingthe latter definition returns a less robust correlation between the gap width in images andthe gap width in surface density (not shown). Third, while the Kanagawa et al. (2016a)correlation is a pure empirical correlation synthesized from simulations within a confinedparameter space, our Eqn (16) has some theoretical footing from gap opening theory, thusmore robust when applying to real systems that may lie outside the parameter space exploredby our models.For gap depth, we have δ Σ − . q (cid:18) hr (cid:19) − α − , (17)with σ = 17% for all but the three M p = M J and h/r = 0 .
05 models, which have thelargest thermal masses in our models, M p = 8 M thermal , and so their gap opening processis in the strongly nonlinear regime and the analytical torque balancing calculations break.We have experimented with freeing the power law indexes in the fit; overall no significantimprovement is found.We thus conclude that the gap width and depth definitions in Figure 4 successfullycapture the physics of gap opening as argued by analytical theories. One thing to note is that h/r can now be directly constrained based on measured gap properties. Assuming hydrostaticequilibrium in the vertical direction, scale height h/r is related to disk temperature T ateach radius as hr = c s v k = √ kT /µv k , (18)where v k is the Keplerian speed, and µ is the mean molecular weight. Thus, our constraintson h/r based on measured gap properties can be used to infer the midplane temperature ofthe disk at the location of the gap. At i = 0, we measure 4 quantities for each model for a given η : ∆ Σ , δ Σ , ∆ I ( η ), and δ I ( η ). Figure 6 compares ∆ Σ with ∆ I ( η ), and δ Σ with δ I ( η ), for both η = 0 and η = 0 (cid:48)(cid:48) . I (0) = ∆ Σ , with a standarddeviation of 7% for all gaps. For the depth, δ I (0) is close to δ Σ for shallow gaps ( δ Σ (cid:46) δ Σ for deeper gaps. Once convolved, we expect the gap to become shallowerand narrower, and we expect the effect of PSF smearing to depend on the ratio of β = η/w I , (19)since with higher angular resolution (smaller η ) and/or wider gaps (bigger w I ), the intrinsicgap profile at η = 0 is better preserved; and vice versa. This is confirmed in the figure — forwider gaps ∆ I ( η ) well correlates with ∆ Σ (as well as ∆ I (0)), while narrower gaps becomesnoticeably wider with finite η . For the depth, PSF smearing makes the gap shallower in allcases.Quantitatively, we find ∆ I ( η ) is correlated to ∆ Σ ( η ), β , and δ I in the following way,∆ I = (cid:115) ∆ + 0 . β δ I , (20)with σ = 9%. In the high angular resolution limit ( β → I → ∆ Σ . For the gap depth,we find δ I = (cid:18) δ Σ . δ Σ (cid:19) . − . β , (21)with σ = 20%. In the high angular resolution ( β →
0) and shallow gap limit ( δ Σ → δ I → δ Σ . Both equations are invertible and it is straight forward to obtain δ Σ and w Σ , given δ I , w I , and η . These correlations are synthesized for the parameter space of 0 . (cid:46) ∆ I (cid:46) . (cid:46) δ I (cid:46)
50; we caution their applications to the parameter space beyond.
In this section, we compare the radial profile of face-on images to the radial profilealong the major axis of disks at modest inclinations ( i (cid:46) ◦ ). In the latter, the radialprofiles are averaged over a wedge of ± ◦ from the major axis centered on the central star,as indicated in Figure 8(c). We note that in inclined disks, equal distance points from thecenter approximately fall on ellipses not centered on the star (shifted along the minor axis),because these ellipses are at non-zero height (Fig. 1 in Stolker et al. 2016a). Therefore,strictly speaking the major axis going through the star is not the major axis of the gapellipse (the two are parallel to each other). Nevertheless, measuring the radial profile alongthe major axis centered on the star is common practice in the field (e.g., Ginski et al. 2016; deBoer et al. 2016); to facilitate the applications of our results, we do the same. The scatteringangle along the major axis is close to 90 ◦ , the same as in face-on images. Therefore, therelative radial profile along the major axis is minimally affected by dust scattering properties. 14 –Major axis radial profiles at modest inclinations closely follow their face-on counterparts.Figure 8 compares the two for three representative models with α = 10 − , h/r = 0 .
05, and M p = 2 M N , 1 M S , and 1 M J . In all cases the gap region in the inclined disk is slightly shiftedinward comparing with the face-on disk, due to the the inclination effect discussed above.The outer gap edge is shifted slightly more than the inner edge. Figure 9 shows the gapprofile at i = 0, 15 ◦ , 30 ◦ , and 45 ◦ for both the full resolution and convolved images of model1 M S h α i . When i approaches 0, the differences diminish.Quantitatively, the gap width in inclined disks agree with face-on disks very well,∆ I (45 ◦ ) = ∆ I (0) , (22)with a standard deviation of 5% between the two. On the other hand, inclined images havesystematically shallower gaps, δ I ( i ) = δ I (0)(1 − . × − i ) , (23)with a standard deviation of 9% ( i is in the unit of degree). The effect of inclination on δ I ( i )is weak — with i = 45 ◦ , the gap depth is only reduced by 10%. Figure 10 compares ∆ I (45 ◦ ) vs ∆ I (0), and δ I (45 ◦ ) vs δ I (0) for 6 representative models. So far, h/r is taken to be a constant in our models (i.e., no radial dependence). Thischoice results in a flat disk surface, which simplifies the problem in our qualitative analysisof the physical picture, and enables us to obtain a straightforward intuition to why scatteredlight features closely follow the structures in surface density (Section 3.1). However, realdisks passively heated by the central star may be flared, with h/r ∝ r γ and γ > γ = 0 . ± . h/r variation across the gap under a reasonable γ (e.g., γ (cid:46) .
25) may be toosmall to produce any significant effect. To test this hypothesis, we carry out experimentsand compare three models with different γ ’s, but otherwise the same parameters, to examine 15 –the dependence of gap width and depth on γ . The results are shown in Figure 11. With h/r = 0 .
05 at r p = 30 AU, α = 10 − , and M p = 1 M S , the three models with γ = 0 (flat disk),0.1, and 0.25 (flare disks) produce very similar gap properties in both the surface density andscattered light (the global scattered light profile in these models are of course different —flared disks have a brighter outer disk — as expected). Specifically, for ∆ Σ and ∆ I , all threemodels return essentially the same values; for δ Σ , the two flared models have measurementshigher than the flat model by ∼ δ I ( η | γ (cid:46) . [∆ I , δ I ] with [ M p , h/r, α ]Fitted correlations 16–23 compose a complete set of equations to link [∆ I , δ I ] with[ M p , h/r, α ]. Once a narrow gap (∆ I ( η, i ) (cid:46) .
5) is identified in scattered light of a disk at amodest inclination ( i (cid:46) ◦ ) with both the inner and outer gap edges clearly revealed, onecan follow the following steps to link observed gap properties to planet and disk parameters[ M p , h/r, α ]:I Use Equations 22–23 to eliminate the effect of inclination: δ I ( η, i ) → δ I ( η, I ( η, i ) → ∆ I ( η, δ I ( η, → δ Σ , ∆ I ( η, → ∆ Σ ;III Finally, Equations 16 and 17 link the gap properties in surface density to [ M p , h/r, α ].Note that normally h/r can be constrained directly from the gap width, and it leavesthe quantity q /α to be constrained from the gap depth.We note that these correlations are derived for polarized intensity at H -band, but theyalso apply to total intensity images, and/or images in other spectral bands within a factorof ∼ H -band (as long as the signals are dust scatteringdominated), because both depth and width are measured in a relative sense, and face-onradial profiles and major axis radial profiles in inclined images minimize the dependenceon scattering angles. Finally, we emphasize that a key precondition when applying thesecorrelations to actual systems is that the gap bottom needs to be detected (i.e., the gap 16 –bottom in images should be above the detection threshold). If not, the gap depth becomesilly defined. In simulating the gap images and deriving the correlations, we have made a numberof assumptions in disk structures and modeling. Here we comment on the effects of theseassumptions, and caution the readers about the caveats when applying our results to realdisks.1. Jang-Condell & Turner (2012, Figure 2) highlighted that the outer gap edge mayreceive extra stellar radiation due to the depletion of material inside the gap, leadingto higher temperature thus higher h/r (see also Jang-Condell 2008; Isella & Turner2016). However, whether this effect can increase the contrast of the gap at the outeredge need additional investigations. Fung & Chiang (2016, Figure 6) showed thatgas inside the gap and around the gap edges circulate, therefore the heating at theouter edge is redistributed to a much larger region in the disk, weakening this thermalfeedback. Future coupled hydro-radiation-transfer simulations are needed to quantifythis effect.2. Ribas et al. (2014, see also Hern´andez et al. 2008) showed that the fraction of stars withprotoplanetary disks, indicated by their infrared excesses, drops on a time scale of about3 Myr. The timescale that a planet takes to fully open a gap with a gap width of 6 h (i.e.,our gap width) is approximately the viscous timescale to cross 6 h , τ ν = 6 / ( α ( h/r )Ω K ).With α ( h/r ) / ( r p / . (cid:46) − , τ ν is longer than 3Myr. Therefore, in low viscositydisks, very young systems, and large separations from the center, gaps may not reachtheir final depth and width, which invalids the [ M p , h/r, α ] − [∆ Σ , δ Σ ] correlationssynthesized in this work and in others. This gap opening timescale issue is howeverirrelevant to the [ δ I , ∆ I ] − [ δ Σ , w Σ ] conversions, and we expect them to hold even whenthe gap is only partially opened.3. In our MCRT simulations, no noise is added into the images (apart from the intrinsicPoisson noise due to finite number of photons), thus the gap bottom reach their theo-retically minimum. If in real observations the gap bottom reaches the noise level, themeasured gap depth, and the derived planet mass, will be their lower limits, while themeasured gap width, and the derived disk scale height, will be their upper limits. 17 –
4. Applications to the Gaps in HD 169142, TW Hya, HD 97048, LkCa 15,and RX J1615.3-3255
As examples, in this section we apply our results to a few gaps imaged in actual sys-tems (Figure 12; “gaps” in these systems refer to the regions around local minimums onthe r -scaled scattered light profile). The gap bottom in these disks are robustly detected.Assuming each gap is opened by a single planet, we derive h/r and q /α at the gap location,as well as the planet mass for several assumed α . The results are summarized in Table 2. Westress that the derived disk and planet properties in the table should be taken as suggestivevalues only, as real disks may not obey the assumptions in our models outlined in Section 3.8,and some of the gaps observed so far may or may not be opened by (a single) planet. Actualhydro+MCRT modeling of individual disks with planet-disk interaction models quantita-tively fitting the observations is encouraged to more accurately recover the disk and planetproperties in specific systems. HD 97048 is a ∼ . M (cid:12) Herbig Ae/Be star located at ∼
158 pc (van den Ancker et al.1998; van Leeuwen 2007) surrounded by a protoplanetary disk several hundreds of AU insize. The disk is inclined at ∼ ◦ based on mm observations (Walsh et al. 2016; van derPlas et al. 2016). Maaskant et al. (2013) first suggested that this disk may be gap/cavityharboring based on SED modeling. Very recently, VLT/SPHERE J -band imaging in bothpolarized and total intensity found multiple rings and gaps in this system (Ginski et al.2016). In addition, the mm dust continuum counterparts of some of these structures mayhave been found by van der Plas et al. (2016), suggesting the scattered light gaps and ringsare likely to be physical density structures instead of shadow effects.Here we focus on the “gap 2” in the J -band VLT/SPHERE dataset (Ginski et al. 2016).Comparing with the other gaps, gap 2 is well detected in both polarized and total intensity,with inner and outer edges clearly resolved. We obtain h/r = 0 .
06 at the location of the gap(0 (cid:48)(cid:48) .
67; 106 AU), and constrain the planet mass to be between 0.4–4 M J for 10 − (cid:46) α (cid:46) − .By analyzing the offset of the center of the gap ellipse from the star, Ginski et al. (2016)determined that at the gap location the disk’s surface is at θ ∼ . M J at the gap location (assuming the BT-SETTLisochrones; Allard et al. 2011). Comparing with our constraints on the planet mass, this 18 –suggests the viscosity in the gap may be low, α (cid:46) − . TW Hya is a ∼ . M (cid:12) K6 star located at 54 pc (Torres et al. 2006, 2008). Thenearly face-on disk ( i ∼ ◦ ; Qi et al. 2004) has been imaged by HST (Debes et al. 2013,2016), Gemini/GPI (Rapson et al. 2015b), Subaru/HiCIAO (Akiyama et al. 2015), andVLT/SPHERE (van Boekel et al. 2016), and multiple gaps have been identified in scatteredlight: one at ∼
80 AU, one at ∼
20 AU, and one at (cid:46) ∼
20, and the one at ∼
80 AU, in the H -band VLT/SPHERE dataset van Boekel et al. (2016, Fig. 3). Due to the low inclination,the azimuthally averaged radial profile is used. For the inner gap, h/r at the gap location(0 (cid:48)(cid:48) .
37; 20 AU) is 0.068, and the planet mass is between 0.05–0.5 M J for 10 − (cid:46) α (cid:46) − ;for the outer gap, h/r at the gap location (1 (cid:48)(cid:48) .
5; 81 AU) is 0.055, and the planet mass isbetween 0.03–0.3 M J for 10 − (cid:46) α (cid:46) − .Using hydro+MCRT simulations, Rapson et al. (2015b) tentatively fit the observed gapprofile at ∼
20 AU in the Gemini/GPI dataset using a 0 . M J planet at 21 AU assuming α = 10 − and h/r = 0 . h/r = 0 .
05, while at 80 AU theyobtained h/r = 0 .
08. Subsequently, assuming α = 2 × − , they derived the masses of theplanets at 20 and 80 AU to be 0.05 M J and 0.11 M J , based on the a similar version of Eqn 17in Duffell (2015). While their h/r and M p at the inner gap is broadly consistent with ourresults, our estimated h/r for the outer gap is significantly lower (and also lower than ourestimated h/r at 20 AU), resulting in a lower estimate for M p as well. This may indicate thatthe 80 AU gap is not opened by a planet; alternatively, this may be because our assumptionof the gap structure being local is no longer valid for the 80 AU gap, which may lie in theshadow created by the inner disk thus the underlying “background” is no longer smooth. HD 169142 is a ∼ M (cid:12) Herbig Ae star located at 145 pc (Raman et al. 2006; Sylvesteret al. 1996). It has a prominent protoplanetary disk at an inclination of 13 ◦ , determined 19 –from gas kinematics (Raman et al. 2006; Pani´c et al. 2008). A narrow gap around 50 AUwas first discovered in H -band VLT/NACO polarized light imaging (Quanz et al. 2013), andsubsequently found by Subaru/HiCIAO (Momose et al. 2015) and in dust thermal emissionat 7mm by VLA (Osorio et al. 2014). The radius-varying disk profile was interpreted as twopower laws in the inner and outer disks joined by a transition region in between by Momoseet al. (2015). Osorio et al. (2014) also discovered a blob in the 7mm VLA dataset residingright inside the gap, with an estimated total mass of 6 × − M (cid:12) in dust. Through radiativetransfer modeling, Wagner et al. (2015) determined that this gap cannot be explained byshadow effects caused by the inner disk, leaving the planet-sculpting scenario as a favoritehypothesis. By comparing the apparent gap depth in 7mm observations with their M p − δ Σ relation, Kanagawa et al. (2015) estimated the mass of the putative planet to be M p (cid:38) . M J .As Rosotti et al. (2016) pointed out, such estimate is risky as mm observations trace mm-sized grains, which can have substantially different spatial distribution from the gas due todust/gas coupling effects. The disk has another inner gap at ∼
25 AU (Honda et al. 2012;Quanz et al. 2013), and a companion candidate at the edge of the inner gap (Reggiani et al.2014, see also Biller et al. 2014).Here we examine the gap at 40–70 AU in the Subaru/HiCIAO H -band dataset (Momoseet al. 2015). Due to the low inclination, we adopt azimuthally averaged radial profile afterdeprojecting the disk Momose et al. (2015, Fig. 2; assuming i = 13 ◦ and position angle =5 ◦ ). We derive h/r at the gap location (0 (cid:48)(cid:48) .
35; 51 AU) to be 0.08, and the planet mass isbetween 0.2–2.1 M J for 10 − (cid:46) α (cid:46) − . LkCa 15 is a ∼ M (cid:12) K3 star located at 140 pc (Pi´etu et al. 2007; Simon et al. 2000).Using Spitzer IRS spectrum modeling, Espaillat et al. (2010) identified the system as a tran-sitional disk with an outer disk, an inner disk, and a gap in between. SMA mm observationsresolved the gap and determined its outer radius to be 50 AU in dust continuum emission(Andrews et al. 2011). The gap has since been resolved in scattered light by Subaru/HiCIAO(Thalmann et al. 2010), Gemini/NICI (Thalmann et al. 2014), and VLT/SPHERE (Thal-mann et al. 2015, 2016) in both total and polarized intensity. The latest VLT/SPHEREobservations by Thalmann et al. (2016) at multiple optical to NIR bands clearly showedthat the system has an substantial inner disk, and the gap is narrow enough (∆ I < .
5) towarrant the hypothesis that it may be opened by a single planet. The inclination of the diskis ∼ ◦ based on mm observations (Andrews et al. 2011; Pi´etu et al. 2007; van der Marelet al. 2015). 20 –Here we examine the gap profile along the major axis of the disk in the VLT/SPHERE H -band polarized intensity dataset presented in Thalmann et al. (2016). We derive h/r atthe gap location (0 (cid:48)(cid:48) .
26; 36 AU) to be 0.07, and the planet mass is between 0.15–1.5 M J for10 − (cid:46) α (cid:46) − .LkCa has several detected planet candidates. Using non-redundant aperture maskinginterferometry on Keck, Kraus & Ireland (2012) identified a point source at a deprojecteddistance of ∼
20 AU from the star. Recently, Sallum et al. (2015) reported the detectionsof two additional point sources in the system with LBT/LBTI and Magellan/MagAO. Thethree planet candidates in Sallum et al. (2015) are located between 15–19 AU. We notethat these planet candidates are unlikely to be the one responsible for maintaining the gapedge in scattered light at ∼
50 AU (i.e., the one whose properties we are inferring here), asthey are too far away from the gap edge. The radial profile of the gap (Figure 12d) is notinconsistent with the gap being opened by just one planet. Additional planets at r (cid:46)
20 AUwith sufficiently low mass will not significantly affect the gap opened by the outer planet.
RX J1615.3-3255 (hereafter J1615) is a 1 . M (cid:12) K5 star at 185 pc (Wichmann et al. 1997;Wahhaj et al. 2010; Andrews et al. 2011). The system has a gap with an outer radius of ∼ −
30 AU, revealed in mm dust continuum emission first by SMA (Andrews et al. 2011)and subsequently by ALMA (van der Marel et al. 2015). The disk has a modest inclination, i ∼ ◦ . Recently, de Boer et al. (2016) resolved this system in scattered light at multipleoptical-to-NIR wavelengths using VLT/SPHERE, and identified multiple gaps and rings inthe system.Here we examine the the major axis profile of the gap at ∼
90 AU in the J -bandVLT/SPHERE dataset presented by de Boer et al. (2016, marked “G” in Fig. 1). We derive h/r at the gap location (0 (cid:48)(cid:48) .
52; 96 AU) to be 0.06, and the planet mass is between 0.07–0.7 M J for 10 − (cid:46) α (cid:46) − .By analyzing the alternating bright/dark pattern on the rings, de Boer et al. (2016)found tentative evidence to suggest that the rings in J1615 might be caused by shadows(i.e., variation in scale height instead of surface density). At the moment the evidence isinclusive. 21 –
5. Summary
Combing 2D and 3D hydro simulations with 3D radiative transfer simulations, we ex-amine the morphology of planet-opened gaps in near-infrared scattered light images. Quan-titatively, we obtain correlations between gap depth and width in inclined disks with finiteangular resolution, to the intrinsic gap depth and width in face-on images with infinite res-olution. The latter is subsequently quantitatively linked to the gap depth and width in disksurface density assuming parametrized h/r profile across the gap region, which can be usedto constraints the mass of the gap-opening planet mass M p , the disk scale height at thelocation of the gap h/r , and disk viscosity α . The main take aways are:1. 2D hydro simulations, puffed up using an assumed midplane temperature profile, pro-duce the same gap profile in scattered light images as 3D simulations.2. With our definition illustrated in Figure 4, the aspect ratio h/r in the gap region canbe directly backed out from the gap width. This can be used to constrain the midplanetemperature in disks.3. Equations 16–23 compose a complete set of correlations to link observed [∆ I , δ I ] to[ M p , h/r, α ] for narrow gaps (∆ I (cid:46) .
5) in disks with modest inclinations ( i (cid:46) ◦ )and flareness ( h/r ∝ r (cid:46) . ). Once such a gap is identified in scattered light with boththe inner and outer gap edges clearly revealed, one can follow the steps outlined inSection 3.7 to constrain fundamental planet and disk parameters M p , h/r , and α .4. We apply our results to the gaps imaged in scattered light in HD 97048, TW Hya, HD169142, LkCa 15, and RX J1615.3-3255, to derive h/r and M /α at the locations oftheir gaps. The results are listed in Table 2 (see also Figure 12). Assuming α = 10 − ,the masses of all gap-opening planets are roughly between 0.1–1 M J . Acknowledgments
We thank Eugene Chiang, Roman Rafikov, Sascha Quanz, Avenhaus Henning, andTomas Stolker for motivating this work, Munetake Momose for sharing with us the SEEDSimages of HD 169142, Sascha Quanz for sharing with us the VLT/NACO image of HD 169142,Joel Kastner and Valerie Rapson for sharing with us the Gemini/GPI image of TW Hya,Christian Ginski and Tomas Stolker for sharing with us the VLT/SPHERE data of HD 97048,Christian Thalmann for sharing with us the VLT/SPHERE LkCa 15 data on HD 97048,Jos de Boer for sharing with us the VLT/SPHERE data on RX J1615.3-3255, and Andrew 22 –Youdin for insightful discussions. We also thank the referee, Takayuki Muto, for constructivesuggestions that largely improved the quality of the paper. This project is supported byNASA through Hubble Fellowship grant HST-HF-51320.01-A (R. D.) awarded by the SpaceTelescope Science Institute, which is operated by the Association of Universities for Researchin Astronomy, Inc., for NASA, under contract NAS 5-26555. J.F. gratefully acknowledgessupport from the Natural Sciences and Engineering Research Council of Canada, the Centerfor Integrative Planetary Science at the University of California, Berkeley, and the SaganFellowship Program funded by NASA under contract with the Jet Propulsion Laboratory(JPL) and executed by the NASA Exoplanet Science Institution.
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This preprint was prepared with the AAS L A TEX macros v5.2.
28 – M o d e l q h / r α δ Σ w Σ ∆ Σ δ I ( η | ) w I ( η | ) ∆ I ( η | ) δ I ( η | (cid:48)(cid:48) . ) w I ( η | (cid:48)(cid:48) . ) ∆ I ( η | (cid:48)(cid:48) . ) AU a r c s ec a r c s ec M J h α e - − . × − . . . . . . . . .
34 1 M J h α e - − . − . . . . . . . . .
30 1 M J h α e - − . × − . . . . . . . . .
30 1 M J h α e - − . × − . . . . . . . . .
33 1 M J h α e - − . − . . . . . . . . .
34 1 M J h α e - − . × − . . . . . . . . .
36 1 M S h α e - . × − . × − . . . . . . . . .
27 1 M S h α e - . × − . − . . . . . . . . .
26 1 M S h α e - . × − . × − . . . . . . . . .
28 1 M S h α e - . × − . × − . . . . . . . . .
27 1 M S h α e - . × − . − . . . . . . . . .
29 1 M S h α e - . × − . × − . . . . . . . . .
29 1 M S h α e - . × − . × − . . . . . . . . .
35 1 M S h α e - . × − . − . . . . . . . . .
36 1 M S h α e - . × − . × − . . . . . . . . .
37 2 M N h α e - − . × − . . . . . . . . .
25 2 M N h α e - − . − . . . . . . . . .
27 2 M N h α e - − . × − . . . . . . . . .
28 2 M N h α e - − . × − . . . . . . . . .
28 2 M N h α e - − . − . . . . . . . . .
31 2 M N h α e - − . × − . . . . . . N / A ( a ) N / AN / A M N h α e - − . × − . . . . . . . . .
39 1 M N h α e - . × − . × − . . . . . . . . .
25 1 M N h α e - . × − . − . . . . . . . . .
28 1 M N h α e - . × − . × − . . . . . . N / AN / AN / A T a b l e : M o d e l p a r a m e t e r s a nd ga pp r o p e rt i e s i n s u r f a ce d e n s i t y a nd i n f a ce - o n i m ag e s . S ee t h e d e fin i t i o n o f p a r a m e t e r s i nS ec t i o n . . ( a ) G a pund e fin e db y o u r d e fin i t i o n s ( F i g u r e ) .
29 – H D ( a ) T W H y a ( b ) T W H y a ( c ) H D ( d ) L k C a15 R X J ( e ) G i n s k i e t a l. ( ) v a n B o e k e l e t a l. ( ) v a n B o e k e l e t a l. ( ) M o m o s ee t a l. ( ) T h a l m a nn e t a l. ( ) d e B o e r e t a l. ( ) B a nd J HHH JJ R a d i a l P r o fi l e ( f ) M a j o r A x i s A z i m - A v e r ag e d A z i m - A v e r ag e d A z i m - A v e r ag e d M a j o r A x i s M a j o r A x i s d p c p c p c p c p c p c i ◦ ◦ ◦ ◦ ◦ ◦ M (cid:63) ( M (cid:12) ) . . . . r m i n , I (cid:48)(cid:48) . ( AU ) (cid:48)(cid:48) . ( AU ) (cid:48)(cid:48) . ( AU ) (cid:48)(cid:48) . ( AU ) (cid:48)(cid:48) . ( AU ) (cid:48)(cid:48) . ( AU ) w I (cid:48)(cid:48) . (cid:48)(cid:48) . (cid:48)(cid:48) . (cid:48)(cid:48) . (cid:48)(cid:48) . (cid:48)(cid:48) . ∆ I . . . . . . δ I . . . . . . η ( g ) (cid:48)(cid:48) . (cid:48)(cid:48) . (cid:48)(cid:48) . (cid:48)(cid:48) . (cid:48)(cid:48) . (cid:48)(cid:48) . β . . . . . . ∆ I ( i | ) . . . . . . δ I ( i | ) . . . . . . ∆ Σ . . . . . . δ Σ . . . . . h / r ( h ) . . . . . . q / α . × − . × − . × − . × − . × − . × − M p / M J ( α = − ) ( i ) . . . . . . M p / M J ( α = − ) . . . . . . M p / M J ( α = − ) . . . . . . T a b l e : A pp l y i n go u rr e s u l t s t o s i x ga p s d e t ec t e d i n s c a tt e r e d li g h t . T h e r a d i a l p r o fi l e s o f t h e ga p s a r e s h o w n i n F i g u r e . ( a ) T h e “ga p i n G i n s k i e t a l. ( , F i g . ) . ( b ) T h e ∼ p i n v a n B o e k e l e t a l. ( ) . ( c ) T h e ∼ p i n v a n B o e k e l e t a l. ( ) . ( d ) T h e AU ga p i n M o m o s ee t a l. ( ) . ( e ) T h e ga p s tr u c t u r e “ G ” i nd e B o e r e t a l. ( , F i g . ) . ( f ) T h e s o u r ce o f t h e r a d i a l p r o fi l e . F o r H D w e u s e t h e m a j o r a x i s r a d i a l p r o fi l e o n t h e n o rt h s i d e , a s o n t h e s o u t h s i d e t h e p o l a r i ze d i n t e n s i t y i n s i d e t h e ga p m a y b e n o i s e d o m i n a t e d ( G i n s k i e t a l. , F i g . ) . F o r T W H y a , a s t h e d i s k i s c l o s e t o f a ce - o n , w e m a c h i n e - r e a d t h e a z i m u t h a ll y a v e r ag e d H - b a nd ga pp r o fi l e i n v a n B o e k e l e t a l. ( , F i g . ) . F o r H D , a s t h e d i s k i s c l o s e t o f a ce - o n , w e u s e a z i m u t h a ll y a v e r ag e d r a d i a l p r o fi l e a f t e r d e p r o j ec t i n g t h e d i s k a s i n M o m o s ee t a l. ( , F i g . ; a ss u m i n g i = ◦ a ndp o s i t i o n a n g l e = ◦ ) . F o r L k C a15a nd J w e u s e t h e m a j o r a x i s r a d i a l p r o fi l e a v e r ag e d o v e rt h e t w o s i d e s i n T h a l m a nn e t a l. ( , F i g . ; D ee p ) a ndd e B o e r e t a l. ( , F i g . ) . ( g ) A n g u l a rr e s o l u t i o n o f t h e o b s e r v a t i o n a s t h e F W H M o f t h e P S F i n t h e a c t u a l d a t a . ( h ) D e r i v e d h / r a tt h e l o c a t i o n o f t h e ga p . ( i ) P l a n e t m a ss a ss u m i n g t h e d i s kv i s c o s i t y i n t h e p a r e n t h e s i s . S ee S ec t i o n f o r d e t a il s .
30 – x (AU) -50 0 50 y ( AU ) -60-40-200204060 ' dust (g cm ! ) 1 M S h , (b) x (AU) -50 0 50 y ( AU ) -60-40-200204060 0.01 0.1 1 (d) -50 0 50 H -band Polarized Intensity (mJy arcsec ! ) (f) -50 0 500 0.01 0.02 F a ce O n i = / (a) Full Resolution -60-40-200204060 (c)
Convolved (e)
Convolved r Fig. 1.— Dust surface density (top) and H -band polarized intensity images at face-on(a,c,d) and i = 45 ◦ inclinations (b,d,f) of the 1 M S h α η = 0 (cid:48)(cid:48) .
04 (c,d), and convolved images scaledby r (e,f; r is deprojected distance from the star). See Section 2.2 for details. 31 – (b)From Hydro Surface Density -50 0 50 (a)From 3D Hydro Simulation x (AU) -50 0 50 y ( AU ) -60-40-200204060 Full Resolution H -band PI r (mJy arcsec ! ) CPDCPD r (AU)
20 30 40 50 r - S c a l e d H - b a nd P I -2 -1 (c)(a)(b) Fig. 2.—
Left:
Face-on images of a disk with h/r = 0 . α = 10 − , and a 0 . M J planetat 30 AU. Image (a) is from a 3D hydro simulation, while image (b) is produced by puffingup the surface density of the model by the same h/r profile as in the 3D hydro calculation.Note that in both panels the circumplanetary region (CPD) is not excised (indicated bythe arrow). Panel (c): radial profile of the two images. The gap in the two images areessentially identical. See Section 2.3 for details. 32 –
20 30 40 50 g c m ! o r m J y a r s ec ! -2 -1 M N h , e -3 Surface DensityFace-on Full Resolution SB r Face-on Convolved SB r r (AU)
20 30 40 50 M S h , e -3
20 30 40 50 M J h , e -3 Fig. 3.— Radial profiles of the surface density (solid) and H -band polarized intensity surfacebrightness (circles: full resolution; dotted line: convolved; both scaled by r ) of three modelswith h/r = 0 . α = 10 − , and M p = 2 M N , 1 M S , and 1 M J , from left to right . Boththe full resolution and convolved surface brightness curves have been scaled by the sameconstant factor so that they roughly meet the surface density curves at r = 15 AU. 33 – r (AU)
20 30 40 Su r f a ce D e n s i t y ( g c m ! ) p ' ( r min ; I ) ' ( r min ; I ) / ga p ; ' w gap ; ' (a) r min ; I ' ( r min ; I ) '' r (AU)
20 30 40 50 r - S c a l e d P I ( m J y a r c s ec ! ) -3 -2 / ga p ; I w gap ; I p I ( r min ; I ) I ( r min ; I ) (b) r min ; I r r I ( r min ; I ) Fig. 4.— Definitions of w Σ and δ Σ in surface density (a), and w I and δ I in scattered lightimage (b). The gap depth (the vertical dashed segment) is defined as the ratio between an“undepleted background” and the minimum value of the quantity inside the gap at radius r min ( r min , Σ or r min , I ; ≈ r p =30 AU): δ Σ = Σ ( r min , Σ ) / Σ( r min , Σ ) and δ I = I ( r min , I ) /I ( r min , I )( I is the r -scaled image surface brightness). Σ ( r min , Σ ) is the initial Σ (dash-dotted line) at r min , Σ ; I ( r min , I ) is taken to be the geometric mean of I at r = (2 / r min , I and r = (3 / r min , I (marked in (b)). r and r are far away from the gap, and for narrow gaps, planet-inducedperturbation should be small at these distances. The gap width (horizontal dashed segment)is defined as the distance between the inner and outer edges of the gap (two vertical dottedlines), at which radius the quantities reaches the geometric mean of the background and thegap minimum (i.e., (cid:112) Σ ( r min , Σ )Σ( r min , Σ ) and (cid:112) I ( r min , I ) I ( r min , I )). 34 – -1 -1 Fig. 5.— The correlations between gap width (a) and depth (b) in surface density and thethree planet and disk parameters — q = M p /M (cid:63) , h/r , and α — as described in Equations 16and 17. See Section 3.3 for details. Fig. 6.— Gap width in scattered light vs gap width in surface density, and gap depth inscattered light vs gap depth in surface density (b). See Section 3.4 for details. 35 – Fig. 7.— The effect of PSF smearing in gap width and depth, showing the correlationsdescribed by Equations 20 and 21. See Section 3.4 for details. 36 – r (AU)
20 30 40 50 r - S c a l e d P I ( A r b i t r a r y U n i t) -4 -3 -2 -1 Solid: i = 0Dashed: i = 45 / (a)Full Resolution M J h , M S h , M N h ,
20 30 40 50 (b)Convolved (c) x (AU) -50 0 50 y ( AU ) -60-40-200204060 Fig. 8.— Radial profiles of three representative models with h/r = 0 . α = 10 − , and M p = 1 M J , 1 M S , and 2 M N (red, green, and blue curves, respectively). Solid curves areazimuthally averaged radial profiles for face-on images; dashed curves are radial profilesalong the major axis at i = 45 ◦ , averaged over a wedge with an opening angle of 20 ◦ , asindicated in panel (c). Panels (a) and (b) are for full resolution and convolved images,respectively. All i = 45 ◦ radial profiles have been scaled by the same constant factor to meetthe solid curves at the same point at 12 and 55 AU. Radial profiles at 45 ◦ are very similarto radial profiles at face-on. See Section 3.5 for details. 37 – r (AU)
20 30 40 50 r - S c a l e d P I ( A r b i t r a r y U n i t) -4 -3 -2 -1 M S h , i = 0 / i = 15 / i = 30 / i = 45 /
20 30 40 50
Convolved
Fig. 9.— Radial profiles of the 1 M S h α (left) and with η = 0 (cid:48)(cid:48) . (right) at 4 different inclinations. For i = 0 radial profiles are azimuthally averaged, whileat finite i radial profiles are averaged over a wedge with an opening angle of 20 ◦ (Figure 8c).The curves with i (cid:54) = 0 have been scaled so that they meet the i = 0 profile at 12 and 55 AU.The gap depth and width at i ≤ ◦ depend on inclinations only weakly. See Section 3.5 fordetails. 38 – Fig. 10.— Gap width and depth in inclined disks. Panel (a) shows the gap width at i = 45 ◦ vs i = 0 for 6 representative models for both full resolution (black points) and convolvedimages (gray points). The models are 1 M J h α M J h α M J h α M S h α M N h α M N h α i = 45 ◦ vs i = 0 for thesemodels. Correlations 22 and 23 are overplotted. 39 –
20 30 40 50 g c m ! o r m J y a r s ec ! -1 (a) / ' = 11 : ; " ' = 0 : / I ( j
0) = 6 : ; " I ( j
0) = 0 : / I ( j :
04) = 4 : ; " I ( j :
04) = 0 : M S h , e -3 with h=r =constant Surface DensityFace-on Full Resolution SB r Face-on Convolved SB r r (AU)
20 30 40 50 (b) / ' = 11 : ; " ' = 0 : / I ( j
0) = 7 : ; " I ( j
0) = 0 : / I ( j :
04) = 4 : ; " I ( j :
04) = 0 : M S h , e -3 with h=r / r :
20 30 40 50 (c) / I ( j
0) = 11 : ; " I ( j
0) = 0 : / I ( j
0) = 6 : ; " I ( j
0) = 0 : / I ( j :
04) = 4 : ; " I ( j :
04) = 0 : M S h , e -3 with h=r / r : Fig. 11.— Radial profiles of the surface density (solid) and H -band polarized intensitysurface brightness (circles: full resolution; dotted line: convolved; both scaled by r ) ofthree models with h/r = 0 .
05 at r p = 30 AU, α = 10 − , M p = 1 M S , and different radiusdependence of h/r : (a) h/r is a constant (flat disk; our fiducial setting), (b) h/r ∝ r . , (c) h/r ∝ r . . The disk in (b) and (c) is flared. Both the full resolution and convolved surfacebrightness curves in each panel have been scaled by the same constant factor so that theyroughly meet the surface density curves at r = 15 AU. As the disk becomes more flared, theouter disk becomes brighter, as expected. However, the width and depth of the gap in boththe surface density and scattered light (measurements printed in each panel) stay roughlythe same, as the gap is approximately a local structure, thus not significantly affected bythe global flareness of the disk. See Section 3.6 for details. 40 – r (AU)
50 100 150 200
HD 97048 (SPHERE J -band)Ginski et al. 2016 r (AU)
20 40 60 80 100
TW Hya (SPHERE H -band)van Boekel et al. 2016 r (AU)
40 60 80 100
HD 169142 (HiCIAO H -band)Momose et al. 2015 r (AU)
20 40 60 80
LkCa 15 (SPHERE J -band)Thalmann et al. 2016 r (AU)
50 100 150
J1615 (SPHERE J -band)de Boer et al. 2016 Su r f a ce B r i g h t n e ss r ( A r b i t r a r y U n i t) -5 -4 -3 -1 r (arcsec) -6 -5 r peak Fig. 12.— Radial profiles of r -scaled polarized intensity (arbitrary unit) of HD 97048 (alongthe major axis on the north), TW Hya (azimuthally averaged), HD 169142 (azimuthallyaveraged after deprojecting the disk as in Momose et al. 2015), LkCa 15 (major axis; averagedover the two sides), and J1615 (major axis; averaged over the two sides). In each panel, thevertical dashed-line indicates r min , I ; the two horizontal dotted lines indicate I and I min ; andthe two vertical dotted lines indicate r out and r in (in TW Hya, we use two horizontal dash-dotted lines and two vertical dash-dotted lines to indicate I , I min , r out , and r in for the outergap). In HD 97048, the outer disk point “ r ” in the gap definition (Figure 4) is significantlyoutside the radius of the peak in the outer disk ( r peak = 0 (cid:48)(cid:48) .
95; indicated by the arrow); wethus fix r = r peak as the “outside the gap” point. r out , r in , I , and I minmin