a smack...

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Draft version June 25, 2015 Preprint typeset using L A T E X style emulateapj v. 5/2/11 A SMACK MODEL OF COLLIDING PLANETESIMALS AND DUST IN THE β PICTORIS DEBRIS DISK: THERMAL RADIATION AND SCATTERED LIGHT Erika R. Nesvold Department of Physics, University of Maryland Baltimore County 1000 Hilltop Circle Baltimore, MD 21250 Marc J. Kuchner NASA Goddard Space Flight Center Exoplanets and Stellar Astrophysics Laboratory, Code 667 Greenbelt, MD 21230 Draft version June 25, 2015 ABSTRACT We present a new model of the β Pictoris disk-and-planet system that simulates both the plan- etesimal collisions and the dynamics of the resulting dust grains, allowing us to model features and asymmetries in both thermal and scattered light images of the disk. Given the observed inclination and eccentricities of the β Pictoris b planet, the model neatly ties together several features of the disk: the central hole in the submillimeter images, the two-disk “x”-pattern seen in scattered light, the “wing-tilt” asymmetry, and possibly even the clumpy gas seen by ALMA. We also find that most of the dust in the β Pictoris system is likely produced outside the ring at 60-100 AU. Instead of a birth ring, this disk has a “stirring ring” at 60-100 AU where the high-velocity collisions produced by the secular wave launched by the planet are concentrated. The two-disk x-pattern arises because collisions occur more frequently at the peaks and troughs of the secular wave. The perturbations of the disk in this region create an azimuthally and vertically asymmetric spatial distribution of collisions, which could yield an azimuthal clump of gas without invoking resonances or an additional planet. 1. INTRODUCTION The β Pictoris system harbors a debris disk whose an- gular size, brightness, and intriguing morphology make it one of the most-observed disks in the sky. Smith & Terrile (1984) made the first resolved images of the disk using a coronagraph on the du Pont 2.5-m telescope at the Las Campanas Observatory, revealing a nearly edge- on system. Eleven years later, more detailed images re- vealed asymmetries (Kalas & Jewitt 1995), including a pattern they referred to as a “warp” in the inner region of the disk (Burrows et al. 1995; Heap et al. 2000). In high- resolution scattered light images of the disk, the warp re- sembles two separate, inclined disks, forming a x-shaped pattern in the images (Golimowski et al. 2006; Ahmic et al. 2009; Apai et al. 2015). Mouillet et al. (1997) and later Augereau et al. (2001) were able to simulate a warp (but not the “x”-pattern) in numerical simulations by in- cluding a planet with an orbit inclined to the main disk. Lagrange et al. (2010) later discovered a planet, β Pic- toris b, orbiting between 8 and 15 AU from the star, and estimated its mass as 9 ± 3 M Jup , in agreement with the model predictions of Mouillet et al. (1997) and Augereau et al. (2001). Initial measurements of the planet’s orbit indicated that it might be misaligned with the second in- clined disk (Currie et al. 2011), but modeling indicated that, within the observational uncertainties, the observed planet could still be sufficiently inclined to the main disk to be responsible for the warp (Dawson et al. 2011). More recent observations have been able to constrain the orbit [email protected] [email protected] of β Pic b and confirm that the planet is inclined to the main disk (Chauvin et al. 2012; Nielsen et al. 2014). Kalas & Jewitt (1995) also observed that the posi- tion angles of the NE and SW midplanes of the com- posite disk (the combined primary and secondary disks), seen in scattered light, differ by 1.3 and named this phenomenon the “wing-tilt asymmetry”. This asymme- try was also observed with HST by Golimowski et al. (2006), who measured position angle differences sepa- rately in the primary and secondary disks (0.9 and 0.3 , respectively). Ahmic et al. (2009) also reproduced this asymmetry in their fits to the Golimowski et al. (2006) observations, but found different values for the position angle differences in the primary and secondary disks (0.1 and 2.2 , respectively). Ahmic et al. (2009) and Kalas & Jewitt (1995) attribute this asymmetry to the scatter- ing phase function of the dust and the inclination of the secondary disk to the line-of-sight. Submillimeter observations of the disk show a broad belt of planetesimals orbiting at 94 ± 8 AU (Wilner et al. 2011; Dent et al. 2014), orbiting on the same plane as the extended main disk of smaller dust grains observed in scattered light (Golimowski et al. 2006, etc.). Sub- millimeter and infrared images of the β Pic disk have revealed a variety of additional asymmetries in the β Pictoris disk. Wahhaj et al. (2003) and Telesco et al. (2005) observed a bright clump of emission in the mid- infrared in the SW region of the disk at a radius of 50 AU. Li et al. (2012) confirmed this mid-infrared clump at 52 AU, but noted a spatial displacement be- tween the two epochs, which they attributed to Keple- rian motion. Using ALMA, Dent et al. (2014) observed arXiv:1506.07187v1 [astro-ph.EP] 23 Jun 2015

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Page 1: A smack model_of_colliding_planetesimals_and_dust_in_the_beta_pictoris_debris_disk_thermal_radiation_and_scattered_light

Draft version June 25, 2015Preprint typeset using LATEX style emulateapj v. 5/2/11

A SMACK MODEL OF COLLIDING PLANETESIMALS AND DUST IN THE β PICTORIS DEBRIS DISK:THERMAL RADIATION AND SCATTERED LIGHT

Erika R. NesvoldDepartment of Physics, University of Maryland Baltimore County

1000 Hilltop CircleBaltimore, MD 21250

Marc J. KuchnerNASA Goddard Space Flight Center

Exoplanets and Stellar Astrophysics Laboratory, Code 667Greenbelt, MD 21230

Draft version June 25, 2015

ABSTRACT

We present a new model of the β Pictoris disk-and-planet system that simulates both the plan-etesimal collisions and the dynamics of the resulting dust grains, allowing us to model features andasymmetries in both thermal and scattered light images of the disk. Given the observed inclinationand eccentricities of the β Pictoris b planet, the model neatly ties together several features of thedisk: the central hole in the submillimeter images, the two-disk “x”-pattern seen in scattered light,the “wing-tilt” asymmetry, and possibly even the clumpy gas seen by ALMA. We also find that mostof the dust in the β Pictoris system is likely produced outside the ring at 60-100 AU. Instead of a birthring, this disk has a “stirring ring” at 60-100 AU where the high-velocity collisions produced by thesecular wave launched by the planet are concentrated. The two-disk x-pattern arises because collisionsoccur more frequently at the peaks and troughs of the secular wave. The perturbations of the diskin this region create an azimuthally and vertically asymmetric spatial distribution of collisions, whichcould yield an azimuthal clump of gas without invoking resonances or an additional planet.

1. INTRODUCTION

The β Pictoris system harbors a debris disk whose an-gular size, brightness, and intriguing morphology makeit one of the most-observed disks in the sky. Smith &Terrile (1984) made the first resolved images of the diskusing a coronagraph on the du Pont 2.5-m telescope atthe Las Campanas Observatory, revealing a nearly edge-on system. Eleven years later, more detailed images re-vealed asymmetries (Kalas & Jewitt 1995), including apattern they referred to as a “warp” in the inner region ofthe disk (Burrows et al. 1995; Heap et al. 2000). In high-resolution scattered light images of the disk, the warp re-sembles two separate, inclined disks, forming a x-shapedpattern in the images (Golimowski et al. 2006; Ahmicet al. 2009; Apai et al. 2015). Mouillet et al. (1997) andlater Augereau et al. (2001) were able to simulate a warp(but not the “x”-pattern) in numerical simulations by in-cluding a planet with an orbit inclined to the main disk.

Lagrange et al. (2010) later discovered a planet, β Pic-toris b, orbiting between 8 and 15 AU from the star, andestimated its mass as 9± 3 MJup, in agreement with themodel predictions of Mouillet et al. (1997) and Augereauet al. (2001). Initial measurements of the planet’s orbitindicated that it might be misaligned with the second in-clined disk (Currie et al. 2011), but modeling indicatedthat, within the observational uncertainties, the observedplanet could still be sufficiently inclined to the main diskto be responsible for the warp (Dawson et al. 2011). Morerecent observations have been able to constrain the orbit

[email protected]@nasa.gov

of β Pic b and confirm that the planet is inclined to themain disk (Chauvin et al. 2012; Nielsen et al. 2014).

Kalas & Jewitt (1995) also observed that the posi-tion angles of the NE and SW midplanes of the com-posite disk (the combined primary and secondary disks),seen in scattered light, differ by ∼ 1.3 and named thisphenomenon the “wing-tilt asymmetry”. This asymme-try was also observed with HST by Golimowski et al.(2006), who measured position angle differences sepa-rately in the primary and secondary disks (0.9 and 0.3,respectively). Ahmic et al. (2009) also reproduced thisasymmetry in their fits to the Golimowski et al. (2006)observations, but found different values for the positionangle differences in the primary and secondary disks (0.1

and 2.2, respectively). Ahmic et al. (2009) and Kalas& Jewitt (1995) attribute this asymmetry to the scatter-ing phase function of the dust and the inclination of thesecondary disk to the line-of-sight.

Submillimeter observations of the disk show a broadbelt of planetesimals orbiting at 94±8 AU (Wilner et al.2011; Dent et al. 2014), orbiting on the same plane asthe extended main disk of smaller dust grains observedin scattered light (Golimowski et al. 2006, etc.). Sub-millimeter and infrared images of the β Pic disk haverevealed a variety of additional asymmetries in the βPictoris disk. Wahhaj et al. (2003) and Telesco et al.(2005) observed a bright clump of emission in the mid-infrared in the SW region of the disk at a radius of∼ 50 AU. Li et al. (2012) confirmed this mid-infraredclump at 52 AU, but noted a spatial displacement be-tween the two epochs, which they attributed to Keple-rian motion. Using ALMA, Dent et al. (2014) observed

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a clump of short-lived CO gas, also in the SW region,but orbiting at a circumstellar radius of ∼ 85 AU, in-dicating an azimuthally-asymmetric region of enhancedcollisions. Apai et al. (2015) compared HST/STIS ob-servations of the disk from epochs 15 years apart andwere unable to detect Keplerian motion from any pointsource contributing > 3% of the disk surface brightnessat projected separations between 3.′′0 and 5.′′0. However,this region of the disk does not directly probe the CO ormid-infrared clumps.

Interpreting these patterns in the β Pic disk and inother debris disks requires modeling collisions betweenplanetesimals (Stark & Kuchner 2009; Thebault 2012;Thebault et al. 2012; Charnoz & Taillifet 2012; Nesvoldet al. 2013; Kral et al. 2013; Nesvold & Kuchner 2015;Kral et al. 2015). The collisional lifetime of a planetes-imal orbiting in the β Pic disk at 10 AU is ∼ 2 × 104

yr (assuming a low optical depth of 10−4), while at 100AU, the collision time is ∼ 6 × 105 yr. This timescale isroughly two orders of magnitude less than the age of thesystem, 21 Myr (Binks & Jeffries 2014), indicating thatcollisions are occurring frequently enough to influence theevolution of the disk. Collisions clearly play some role inthe observed ALMA asymmetry (Dent et al. 2014), high-lighting the need for a model incorporating planetesimalcollisions. New models of the disk also need to show howthe recent best-fit planet orbits from Nielsen et al. (2014)influence the disk morphology.

Interpreting the scattered light images calls for newdynamical models of the dust in the β Pic disk (Mouil-let et al. 1997; Augereau et al. 2001; Dawson et al.2011). Existing models fail to reproduce the two-diskx-pattern seen in high-resolution scattered light images(e.g., Golimowski et al. 2006). The dynamics of the dustdiffer from the dynamics of the planetesimals, as the dustis subject to radiation pressure and Poynting-Robertsondrag (Robertson 1937). The distribution of the dust alsodepends on the spatial distribution and collision rate ofthe planetesimals that produce the dust via collisions.

To meet these needs, we present simulations of the βPictoris debris disk using our 3D collisional disk modelSMACK, which traces the location of dust-producing col-lisions within a disk simultaneously with the evolvingspatial distribution of larger planetesimals (> 1 mm).We combine these with a dynamical model of the dustto create the first physical model of both the planetesi-mal collisions and the dust dynamics in this system. InSection 2 we describe the SMACK simulations and theparameters used. In Section 3 we measure the level of col-lisional relaxation within the disk to determine whethercollisional damping is significant. In Section 4 we discusstwo kinds of spiral structures created by the slightly-eccentric β Pic b, and its implications for observationsof asymmetries in the disk. We also discuss the origin ofthe observed central hole in the distribution of mm-sizedplanetesimals. In Section 5 we discuss the 3D distribu-tion of collisions in the disk. In Section 6 we present amodel of the dust in the β Pic disk, created by integratingthe orbits of the dust grains produced in SMACK colli-sions, and in Section 7 we present simulated scatteredlight images of the disk created using our dust models.Finally, in Section 8, we summarize our results and pro-pose future work with this model.

2. SMACK SIMULATIONS OF COLLIDINGPLANETESIMALS

Modeling the dynamical and collisional evolution of theparent bodies in a disk is essential for accurately calcu-lating the effects of collisional damping and collisionalerosion on the morphology of a disk (Nesvold et al. 2013;Nesvold & Kuchner 2015). Our debris disk simulator, theSuperparticle-Method Algorithm for Collisions in KuiperBelts (SMACK), simulates the evolution of the dynamicsand the spatially-dependent size distribution of a disk ofplanetesimals under the influence of one or more planets.SMACK uses the N-body integrator REBOUND (Rein& Liu 2012), but each body in the integrator represents asuperparticle, a cloud of planetesimals with the same po-sition and trajectory, but a range of sizes from 1 mm-10cm. Each superparticle is characterized by a size distri-bution. When an overlap is detected between two su-perparticles, SMACK adjusts the size distributions andtrajectories of the colliding superparticles to representthe outcome of fragmenting planetesimal collisions. Dataoutput by SMACK include the time-dependent 3D den-sity map of the planetesimals and a 3D map of the dust-producing collisions throughout the simulation time.

We simulated the evolution of the β Pictoris systemwith SMACK using 100,000 superparticles for a simula-tion time of 21 Myr, the age of β Pic as measured byBinks & Jeffries (2014). We inserted a planet of mass9 MJup, using the best-fit orbit of Nielsen et al. (2014),who used the Gemini/NICI and Magellan/MagAO in-struments to measure the position of β Pic b relativeto its star with greater accuracy than previous observa-tions (e.g., Chauvin et al. 2012; Bonnefoy et al. 2013).We distributed the initial superparticle orbits uniformlyin semi-major axis, eccentricity, inclination, longitudeof ascending node, argument of pericenter, and meananomaly, and assigned their size distributions to bepower laws with index −3.5, normalized such that theinitial optical depth of the disk at 100 AU was 10−2.The initial parameters of the superparticles and planetare listed in Table 1. We selected a superparticle sizeof rsp = 10−1.5 AU. The finite superparticle size lim-its the size of the features that SMACK can resolve.At the location of the planet, the superparticle size wechose corresponds to an inclination and an eccentricityof rsp/a ≈ 0.18 and rsp/a ≈ 0.003, respectively. Thesimulation ran for ∼ 270 hr of wall clock time on theNASA Center for Climate Simulation’s Discover cluster,using a hybrid OpenMP/MPI parallelization on 48 cores.

Parameter Superparticle Range Planet ValueSemi-Major Axis 5-200 9.1Eccentricity 0.0-0.01 0.08Inclination 0.0-0.005 2.59

Long. of Ascending Node 0-2π 0Argument of Periapsis 0-2π 6.5

Mean Anomaly 0-2π 0

TABLE 1Initial conditions of the superparticles and planet for

the simulation described in Section 2.

Although we ran the simulation for 21 Myr, the mostrecent estimate for the age of the star (Binks & Jeffries2014), the planet may have been formed or scattered ontoits current inclined orbit more recently than the star was

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β Pictoris 3

formed (Dawson et al. 2011; Currie et al. 2013), so 21Myr does not necessarily represent the age of the planet-star system in its current configuration. Mouillet et al.(1997) derived a relationship between the radial extentof the warp, rw, and the system age, tage,

rw = 6.31

[mpl

M∗

( rpl10 AU

)2 tage5.2 yr

]0.29, (1)

where mpl is the planet’s mass, M∗ is the star’s mass,and rpl is the planet’s orbital radius. For example, fora system with our parameters, in which we place theplanet on its inclined orbit at time t = 0, the warp shouldextend out to ∼ 86 AU after 10 Myr, while the maximumextent at 21 Myr would be ∼ 107 AU. Observations of thedisk in scattered light and submillimeter emission tendto place the extent of the warp in the planetesimals anddust between ∼ 85 − 95 AU, so we synthesized imagesfrom our simulations after 10 Myr of evolution ratherthan 21 Myr.

Figure 1 shows a simulated image of the β Pictorisdisk at 10 Myr at a wavelength of 850 µm. To simulatethe image, we assumed that each planetesimal within asuperparticle emits thermally as a spherical blackbody,and iteratively calculated the temperature of each plan-etesimal as a function of its distance from the star. Wesummed the contribution from each superparticle in agiven pixel, and averaged over 50 output timesteps, or5 × 105 yr, to reduce the Poisson noise in the image.Separate models of the dust dynamics are not needed tocompare to ALMA images because the dust grains aretoo small to efficiently couple to mm-wavelength pho-tons.

Our simulations reproduce the general morphology ofprevious models of the warp created by the inclined orbitof the planet (e.g., Augereau et al. 2001; Dawson et al.2011). At 10 Myr after the addition of an inclined planet,the warp has propagated out to ∼ 100 AU. The shapeof the disk as simulated with SMACK is similar to theresults from collisionless simulations (e.g., Figure 1 inDawson et al. 2011), except that our simulated imageexhibits a deficit of the material represented by orangedots in Dawson et al. (2011). We discuss this inner clear-ing in more detail in Section 4.2.

Figure 2 shows the same simulated image at 850 µm,with a linear scaling and no vertical stretch. We con-volved this image with a Gaussian beam with a full-widthhalf-max (FWHM) of 12 AU to simulate an ALMA ob-servation. Clearly, our simulations reproduce the basicmorphology seen in the ALMA image, but there are someconsequential differences between the model and obser-vations. Comparing Figure 2 to the continuum image ofthe disk with ALMA (Dent et al. 2014), we note thatthe brightness peaks correspond to the radial extent ofthe warp, which is ∼ 95 AU at 10 Myr in our simulation(rather than the ∼ 86 AU predicted by Mouillet et al.1997). In the ALMA observation, the surface brightnesspeaks at ∼ 60 AU on either side of the star, indicatingthat β Pictoris b reached its current inclined orbit < 10Myr ago, assuming the planet mass is ≥ 9MJup. Whilethe simulated disk exhibits a small brightness asymmetryin this viewing orientation, our simulation is unable toreproduce the ∼ 15% brightness asymmetry between theSW and NE halves of the disk seen in the ALMA image

(see Section 4 for a further discussion of the brightnessasymmetry).

3. COLLISIONAL RELAXATION

SMACK models have the novel ability to numericallyexplore the process of collisional relaxation in debrisdisks. This process, the gradual removal of free eccen-tricity and free inclination from planetesimal orbits, hasbeen invoked in influential models of debris disks (e.g.,Quillen 2007; Chiang et al. 2009), but not demonstratednumerically. We describe here what our simulations of βPic disk show about this process.

A planet on an orbit inclined to a disk will force aninclination on the disk’s planetesimals. A planetesimal’sinclination, i, and longitude of ascending node, Ω, canbe written together as a vector with components

p = i sin Ω

q = i cos Ω.(2)

The planetesimal’s inclination i will precess about theforced inclination iforced induced by the planet, such that

i = ifree + iforced, (3)

where ifree is the free or proper inclination of the plan-etesimal (Murray & Dermott 1999). Inelastic collisionswill tend to damp the free inclinations of the superpar-ticles.

Plotting the inclination vectors of the planetesimals inp-q space can help illustrate this relationship and pro-cess. In Figure 3, we plot the p-q diagrams of the su-perparticles at various times during the simulation. Theblack arrow represented the forced inclination vector ofthe planet. At t = 0, the superparticles have inclina-tions uniformly distributed in a small range of values,and are uniformly distributed in Ω from 0 − 2π. Theyare represented on the p-q diagram as a small grey cir-cle. As the system begins to evolve, the superparticles’inclination precess about the planet’s forced inclinationat different rates, spreading the superparticles into a ringin p-q space. The magnitude of the superparticles’ incli-nation vectors oscillates between their initial inclinationsnear ∼ 0 and twice the planet’s inclination.

In a completely damped system, the superparticleswill have zero free inclination, so by Equation (3), theirinclinations will all equal the forced inclination of theplanet. On a p-q diagram, the points representing com-pletely damped superparticles would appear clustered atthe planet’s inclination vector. We can see from theright panel in Figure 3 that the superparticles are barelydamped at all after 21 Myr. Some superparticles (rep-resenting 3% of the mass of the system) have moved in-wards towards the planet’s inclination vector due to col-lisional damping of their free inclinations; others (1% ofthe system by mass) have been collisionally scattered intohigher inclinations. However, most (96% of the mass inthe simulation) remains within the annulus defined bythe range of their initial inclinations. This lack of damp-ing is consistent with recent measurements of the orbit ofβ Pic b relative to the disk with Gemini/NICI and Mag-ellan/MagAO, which indicate that the planet’s positionangle lies in between the position angles of the main diskand the inner warp on the sky (Nielsen et al. 2014).

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−200 −100 0 100 200y (AU)

−20

−10

0

10

20

z (

AU

)

−10

−9

−8

−7

−6

−5

log

Bri

gh

tne

ss

(m

Jy

/pix

el)

Fig. 1.— Simulated SMACK image of the β Pictoris disk at 850 µm after 10 Myr. The location of the star is indicated with the white x.The vertical axis has been stretched to emphasize the warp structure. The simulated disk after 10 Myr resembles the collisionless simulationof Dawson et al. (2011), but our simulated disk is missing most of the planetesimals that have completed a full secular oscillation.

−200 −100 0 100 200y (AU)

−50

0

50

z (

AU

)

0

1×10−5

2×10−5

3×10−5

4×10−5

5×10−5

Bri

gh

tne

ss

(m

Jy

/pix

el)

Fig. 2.— Simulated SMACK image of the β Pictoris disk shown in Figure 1, with linear color scaling and no vertical stretch. Thisimage has been convolved with a Gaussian beam with a FWHM of 12 AU. Comparing this simulated image with the ALMA continuumobservation by Dent et al. (2014), we find that we reproduce the basic morphology of the ALMA image, but our brightness peaks arelocated farther out and our simulation does not reproduce the brightness asymmetry.

Following the example of Dawson et al. (2011), we plot-ted the inclination vs. semi-major axis of the superpar-ticles at 10 Myr in the top panel of Figure 4. Again, theresults are similar to the collisionless results of Dawsonet al. (2011). The forced inclination from the planet, if ,is independent of radial distance to the planet and simplyequals the planet’s inclination, if = ipl. The superparti-cles orbiting at & 170 AU are still at low inclination, thesuperparticles within ∼ 100 − 150 AU have just reachedtheir maximum inclination of 2if for the first time, andthe superparticles inside ∼ 100 AU are oscillating be-tween the inclination of the main disk (around 0) and2if . However, in the SMACK models, we see the effectsof collisions in the superparticles that have been colli-sionally scattered to both lower- and higher-inclination( 2if ) orbits. Again, there is little evidence of collisionaldamping, which would manifest as a clustering of super-particles at the planet’s inclination if , beginning with thesuperparticles farthest in, closest to the planet’s orbit.

4. SPIRAL STRUCTURE

4.1. Simulation Results

Just as the planet’s inclination can create a warp ina debris disk, secular perturbations from an eccentricplanet will perturb the orbits of the particles in the disk.This effect will create a spiral structure in an initiallyaxisymmetric disk, as planetesimals at different distancesfrom the planet precess at different rates (Wyatt 2005a).Collisionless numerical simulations of the β Pictoris diskby Mouillet et al. (1997), Matthews et al. (2014) andApai et al. (2015) predict that an eccentric β Pic b willinduce a spiral in the disk via this mechanism.

As with inclination, the eccentricity of a particle canbe written as a vector using the particle’s longitude ofpericenter, $: e = (e sin$, e cos$). Laplace-Lagrangesecular theory provides an expression for a planetesimal’seccentricity vector e(t) perturbed by an eccentric planet.If we assume that the initial eccentricity of each planetes-imal is zero, e(0) = 0, the eccentricity of a planetesimal

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β Pictoris 5

t = 0 yr

0 2 4 6

i cos Ω

−4

−2

0

2

4

i sin

Ω

t = 2.1x107 yr

0 2 4 6

i cos Ω

−4

−2

0

2

4

i sin

Ω

Fig. 3.— The p-q diagrams of all of the superparticles at the beginning and end of the simulation described in Section (2). The blackarrow represents the forced inclination vector due to the planet. The dashed black lines illustrate the annulus in which the inclinationvectors of the superparticles would precess in the absence of collisions. Ninety-six percent of the mass in the simulation is within theannulus, indicating that minimal collisional damping has occurred by 2.1 × 107 yr.

if

2 if

0 50 100 150 200a (AU)

0

2

4

6

8

inc (

deg

)

Fig. 4.— Inclination vs. semi-major axis of the superparticles at 10 Myr. The red lines indicate the planet’s forced inclination, if , and2if . This figure strongly resembles the results of Dawson et al. (2011) for a & 59 AU, but at a < 59 AU in our simulations, collisionsdominate and scatter planetesimals to high inclination.

with semi-major axis a is

e(t) =

(b23/2(αpl)

b13/2(αpl)epl cos$pl(1 + cosAt),

b23/2(αpl)

b13/2(αpl)epl sin$pl(1 + sinAt)

),

(4)

where epl is the planet’s eccentricity, $pl is the planet’slongitude of pericenter, b23/2 and b13/2 are Laplace coef-

ficients, and αpl = a/apl for a > aapl, where apl is thesemi-major axis of the planet. The precession rate A ofthe planetesimal is given by

A =n

4

mpl

M∗αplαplb

13/2(αpl), (5)

where n is the planetesimal’s mean motion, mpl is themass of the planet, M∗ is the mass of the star, andαpl = 1 for a > apl. To first order, the planetesimal’sinclination vector i(t), perturbed by an inclined planet,

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6

will precess with the same precession rate A, and theforced inclination is simply equal to the inclination ofthe planet, ipl:

i(t) = (ipl cos Ωpl(1 + cosAt), ipl sin Ωpl(1 + sinAt)) ,(6)

where Ωpl is the longitude of the ascending node of theplanet.

To better understand the geometry of the perturbeddisk, we used Equations 4-6 to analytically calculate theeffects of the forced eccentricity and inclination from βPic b on the orbits after 10 Myr using the planet param-eters listed in Table 1 and plotted the resulting orbitsin Figure 5. We considered the orbits of planetesimalson initially circular orbits with semi-major axes rangingfrom 11 AU to 155 AU. As described in Wyatt (2005a),the orbits precess at different rates, forming a spiral den-sity wave extending radially outward to ∼ 100 AU. Inte-rior to ∼ 59 AU, the planetesimals have completely morethan one precession period and their orbits have becomephase mixed, while exterior to ∼ 100 AU, the planetesi-mals are still orbiting with very low eccentricities.

As we described in Section 3, the inclinations of theplanetesimals precess around the forced eccentricity ina similar manner: the planetesimals interior to ∼ 59AU have completed at least one precession period, whileplanetesimals exterior to ∼ 100 AU still have very low in-clinations (Figure 4). Rather than a spiral density wave,however, the secular effects of the planet’s inclination in-duce a vertical displacement wave in the planetesimals.In Figure 5 we mark the ascending and descending nodesof the orbits with blue and red x’s, respectively. Thenodes form a double-armed spiral, indicating that thevertical displacement wave varies azimuthally. We willrefer to the spiral density wave created by the planet’seccentricity as the “eccentricity spiral” and the verticaldisplacement wave created by the planet’s inclination asthe “inclination spiral”.

Although collisions can eventually destroy an eccen-tricity spiral in a disk (Nesvold et al. 2013), the eccen-tricity spiral induced in the β Pictoris disk survives to21 Myr. Figure 6 shows a simulated image of the face-onβ Pictoris disk at 10 Myr, the same simulation shown inFigure 1. A spiral structure is evident, extending radi-ally outward to ∼ 100 AU, in good agreement with theanalytically-predicted spiral shown in Figure 5.

The outermost portion of the spiral in Figure 6 cor-responds to the planetesimals that have completed halfa precession period, and have reached their eccentricitymaximum. Since the precession rate for inclinations andeccentricity are equal to first order, these planetesimalshave also reached their maximum inclination, so the out-ermost spiral in the β Pic disk is roughly co-located withthe maximum radial extent of the warp.

Wilner et al. (2011) observed the β Pictoris disk at withthe Submillimeter Array and detected two peaks in 1.3mm emission along the disk plane, which they interpretedas a ring or belt of larger, dust-producing planetesimalsat 94±8 AU, with a deficit of mm-sized grains interior tothe belt. Our simulation results suggest that if the warpin the β Pic disk extends to ∼ 85 AU (Golimowski et al.2006; Heap et al. 2000), a spiral structure created by βPic b’s eccentricity will also extend out to ∼ 85 AU. The“ring” of planetesimals observed by Wilner et al. (2011)

may, in fact, be this spiral structure.Apai et al. (2015) proposed that the spiral structure

could contribute a small brightness asymmetry to thedisk. Wilner et al. (2011) noted that the SW side ofthe disk appeared brighter in their SMA observations,but the difference was below the noise level. Dent et al.(2014) observed the β Pic disk at 870 µm and found thatthe continuum emission from the SW side of the disk is15% brighter, on average, than from the NE side.

To test whether the spiral structure could explain thisasymmetry, we calculated the simulated emission fromthe NE and SW halves of the disk, observed edge-on,while rotating the disk about its axis. While we did findthat the brightness ratio of the NE and SW regions variedwith the orientation of the spiral, the maximum possiblebrightness excess of the SW disk versus the NE disk wasonly ∼ 2%. So we infer that the brightness asymmetrydue to the spiral may contribute to the observed bright-ness excess in the SW disk, but probably cannot be solelyresponsible for it. Note that the spiral structure prop-agates outwards with time and does not orbit the star,indicating that the brightness asymmetry due to the spi-ral will not move to the opposite side of the star, but willinstead move radially outwards with time.

4.2. Central Clearing

The brightness of the disk shown in Figure 6 dropsoff interior to the ring (at radii . 59 AU) by roughlyan order of magnitude compared to regions of the diskexterior to the spiral. Wilner et al. (2011) observed adeficit of mm-sized grains at radii . 94 AU. Dent et al.(2014) also observed that the mm-sized grains lie in abelt with a central clearing. However, the mechanismfor clearing these larger grains from the interior regionof the disk is not immediately obvious. The planet willclear a gap around its orbit via resonance overlap, butthis gap will only extend out to ∼ 14.5 AU in 10 Myreven accounting for the effects of collisions, which tendto widen the gap (Nesvold & Kuchner 2015).

Our model suggests that a different mechanism is pro-ducing the central clearing of planetesimals in the β Picdisk. The top panel of Figure 7 shows the normalizedradial surface brightness for the simulated disk shown inFigure 6, compared with two new simulations, each runwith 10,000 superparticles for 10 Myr. In one simula-tion, we set the eccentricity of the planet to zero. In theother, we kept the planet’s eccentricity set to e = 0.08,but we turned off the collision simulation and let the sys-tem evolve with only dynamical perturbations. Only thesystem with an eccentric planet in a disk experiencingcollisions shows a brightness deficit interior to ∼ 59 AU.

Mustill & Wyatt (2009) showed that a planet’s sec-ular perturbations will stir a disk of planetesimals andplace them on intersecting orbits. Where planetesimalorbits cross, collisions are more frequent, and collidingbodies will be gradually eroded and removed from thesystem. They derived an expression for tcross, the timerequired for a planet’s secular perturbations to causetwo neighboring planetesimal orbits to intersect. For theplanet and star parameters in our SMACK simulation,tcross ≈ 10 Myr at a semi-major axis of 60 AU in thedisk. This is illustrated further in the bottom panel ofFigure 7, where we plot the minimum orbit intersectiondistance (MOID) versus semi-major axis for pairs of ad-

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β Pictoris 7

−100 −50 0 50 100x (AU)

−100

−50

0

50

100

y (

AU

)

Fig. 5.— Face-on diagram of the analytically-derived orbits of planetesimals perturbed by a planet with the orbit and mass of β Pic bafter 10 Myr. The black x marks the location of the star. Orbits at different semi-major axes precess at different rates, forming a spiraldensity wave where they approach adjacent orbits. Blue and red x’s mark the locations where the derived orbits cross the z = 0 plane,indicating the ascending nodes and descending nodes of each orbit, respectively. They trace a double-armed spiral that intersects the spiraldensity wave (see Section 5).

jacent orbits from Figure 5, calculated using the methoddescribed in Wisniowski & Rickman (2013). The lastzero of the MOID appears to be at ∼ 59 AU. Exterior to59 AU, the MOID increases, then asymptotes to a valueof 1.1 AU, which was the initial separation of the circularorbits of the planetesimals. The orbit crossings depictedby the green curve must cause the collisions that createthe central clearing depicted by the black curve.

The two methods of disk clearing discussed in thissection, resonance overlap (Nesvold & Kuchner 2015)and secular excitation (Mustill & Wyatt 2009), invoketwo different sets of initial conditions for the disk. TheNesvold & Kuchner (2015) model assumes that the plan-etesimals in the disk have some initial eccentricity dis-tribution, such that planetesimals orbiting just outsidethe resonance overlap region of the planet (the planet’s“chaotic zone”) will collide frequently and widen the gap.The Mustill & Wyatt (2009) mechanism assumes thatthe disk is initially cold, and depends on secular per-turbations from the planet to excite collisions betweenplanetesimals. Our simulation results for the β Pictorissystem suggest that the mechanism described by Mustill& Wyatt (2009) dominates in this system, widening thecleared inner region to much greater radial distances thanthe resonance clearing mechanism can reach in the ageof the system. Future simulations should explore the dif-ferences between these “hot-start” and “cold-start” diskmodels, and investigate what observed disk clearing cantell us about the initial conditions of the disk.

5. SPATIAL DISTRIBUTION OF COLLISIONS

The left panel of Figure 8 shows a face-on map of thedistribution of collisions in the β Pic disk between 10and 10.5 Myr. A spiral structure is evident in the colli-sion rate, roughly corresponding to the spiral seen in thesimulated disk image in Figure 6. However, the spiralin the collision distribution in Figure 8 exhibits several“breaks” in its azimuthal structure that are not seen inFigure 6.

These breaks can be understood by examining the 3Dstructure of the disk, specifically the interaction betweenthe eccentricity spiral (the spiral density wave induced bythe planet’s eccentricity) and the inclination spiral (thevertical displacement wave induced by the planet’s incli-nation). In the edge-on image of the simulated disk inFigure 1, the inclination spiral appears as a vertical oscil-lation of the planetesimals with radius from the star. Theinclination spiral also appears in Figure 5, where blueand red x’s mark the ascending and descending nodesfor each orbit. The orbital nodes form a double-armedspiral, extending out to ∼ 100 AU. As Figure 5 shows,this inclination spiral intersects with the eccentricity spi-ral as several azimuthal locations.

What happens where these two kinds of spiral inter-sect? In the right panel of Figure 8, we plot the sameface-on collision map shown in the left panel, but maskout the pixels within 0.2 AU of the plane of the planet’sorbit at radii > 75 AU. The right panel shows a two-armed spiral, like the pattern the ascending and descend-

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−200 −100 0 100 200

−200

−100

0

100

200

y (

AU

)

−10

−9

−8

−7

−6

−5

log

Bri

gh

tness (

mJy/p

ixel)

Fig. 6.— Face-on simulated image of the β Pictoris disk at 850 µm after 10 Myr (seen edge-on in Figure 1). The white x indicates thelocation of the star. The disk at . 60 AU is roughly an order of magnitude fainter than the disk exterior to the spiral structure becausecollisions have destroyed the planetesimals in this central clearing.

ing nodes illustrated in Figure 5. By comparison with theleft panel of Figure 8, we can see that deficits in the col-lision rate occur where the inclination spirals intersectthe eccentricity spiral in the plane of the planet’s orbit.Two such locations are indicated with right arrows inthe right panel of Figure 8, while at least two additional,unmarked intersection points are also visible further in.

One reason the collision rate drops in the plane of theplanet’s orbit is that the density drops in this plane – notthe surface density, but the mass density. This densitydrop is shown in Figure 9, where we plot a cut throughthe mass density of the simulated β Pic disk at 10 Myr inthe the x = 0 plane. We also show a cut through the col-lision rate in the x = 0 plane. The density and collisionrate are enhanced at the vertical peaks of the inclinationwave and the troughs in midplane of the initial planetes-imals disk, and minimized where the planetesimals crossthe plane of the planet’s orbit. When the disk is viewedface-on (as in Figure 6), projection effects mask this ef-fect, and the spiral density wave appears continuous.

ALMA observations show an azimuthally-asymmetricdistribution of short-lived CO gas at ∼ 85 AU (Dentet al. 2014). The deprojected distribution of this gas ap-pears as either two clumps of gas orbiting on either sideof the star, or a single clump with a long tail. Two ma-jor hypotheses have been suggested for the origin of theALMA asymmetry: dust production from larger plan-etesimals trapped in a resonant orbit by a hypotheticalsecond planet, producing the two-clump distribution, orthe recent breakup of a massive body, corresponding tothe single-clump distribution (Telesco et al. 2005; Dentet al. 2014). This second scenario was modeled by Jack-

son et al. (2014), who found that the timescale of theobservable signature of such a breakup could be as longas 1 Myr, but that the resulting clump would be station-ary in the system.

Dent et al. (2014) concluded that the mass loss rate ofcolliding grains required to maintain the gas-productionrate is ∼ 1.4 × 1019 kg/yr. The total mass of grainsinvolved in catastrophic collisions in our SMACK sim-ulations is only 8.77 × 1013 kg/yr. However, our sim-ulations only track dust-producing parent bodies up to10 cm in diameter. Including parent bodies up to 1 min size may produce enough dust to reproduce the ob-servations of Dent et al. (2014), but future simulationsare needed to test this hypothesis. Czechowski & Mann(2007) described the production of gas in the β Pic diskvia high-velocity collisions, based on a simple 2D empiri-cal description of the collision rate. We hope to improveupon this model in a future paper with SMACK, whichcan simulate both the 3D distribution of collisions andthe collisional velocities in the β Pic disk.

6. INTEGRATING THE DUST ORBITS

The SMACK models described above predict wheredust grains are generated in the β Pictoris disk and theirinitial orbits. To understand the distribution of this dustand to model images of the disk in scattered light, wefed the output of SMACK into a second N-body inte-grator to track the dust orbits. Then, using a populartechnique (e.g., Dermott et al. 1999; Liou & Zook 1999;Wilner et al. 2002; Moran et al. 2004; Stark & Kuch-ner 2008; Debes et al. 2009; Kuchner & Stark 2010), werecorded the output of the integrator in a 3D histogram

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β Pictoris 9

0 50 100 150 2000.0

0.2

0.4

0.6

0.8

1.0N

orm

alized

Su

rface B

rig

htn

ess

0 50 100 150 200Radius (AU)

0.0

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0.4

0.6

0.8

1.0

MO

ID (

AU

)

Eccentric Planet with Collisions

Circular Planet with Collisions

Eccentric Planet with No Collisions

Fig. 7.— Top panel: Normalized 850 µm surface brightness vs. radius at 10 Myr for three simulations: the disk with an eccentric planet(shown in Figure 6), the same simulation with the planet’s eccentricity set to zero, and the eccentric planet simulation with the collisionsturned off. An eccentric planet in a collisional disk (black curve) creates a central brightness deficit in mm-sized bodies. Bottom panel:Minimum orbit intersection distance (MOID) versus semi-major axis for pairs of adjacent orbits in Figure 5. Orbits in the interior regionof the disk (. 59 AU) intersect, increasing the collision rate and creating the brightness deficit see in the black curve in the top panel.

to simulate the density distribution of the dust cloud.We first generated a list of all the mass production

events that took place in the simulation between 10.00and 10.01 Myr. Every superparticle collision found bySMACK yields two such mass production events, gen-erally producing different amounts of mass with differ-ent initial orbits. We recorded the semi-major axis, ec-centricity, inclination, longitude of ascending node, ar-gument of pericenter, mean anomaly, and total mass ofdust produced for each event. The mass produced rep-resents the mass of the fragments between 1 µm and 1mm, distributed in a power law with an index of -2.8 forincremental logarithmic bins.

Figure 10 illustrates the radial distribution of the mostsignificant mass production events, binned according tomass production into bins containing collisions with massproduction from 10−18 to > 10−17, 10−17 to 10−16 and> 10−16 solar masses. This figure reveals an interestingresult. The biggest mass production events are mostlylocated in a ring roughly from 59 AU to 100 AU from thestar, but other collisions are spread over a wider rangeof circumstellar distance, and even dominate the massproduction at some radii. Our results indicate that only46% of the dust, by mass, is produced in the planetesimalbelt (60-90 AU). In other words, attempting to applythe “birth ring” approximation to describe the β Pictorisdisk (Strubbe & Chiang 2006) would miss major sourcesof dust outside this ring.

As a next step, we chose a subset of these mass produc-

tion events to feed to the second N-body integrator. Weselected all 960 mass production events creating > 10−16

solar masses of dust. We selected a similar number ofmass production events in the range 10−18 to > 10−16

solar masses of dust produced, by choosing at random3.5% of all the events in this range.

It is thought that the grains that dominate images ofcollision dominated disks like the β Pictoris disk are thosenear the blowout size, i.e. with β ∼ 1/2 where β isthe magnitude of the force on the grain from radiationpressure divided by the magnitude of the force on thegrain from stellar gravity (Strubbe & Chiang 2006). Wetherefore sampled 6 different values of β, logarithmicallydistributed, near β = 1/2. These values are listed in Ta-ble 2. The grain radius corresponding to a given β valuedepends on assumptions about the grain’s shape, compo-sition, and optical parameters. To illustrate the range ofpossibilities, we list three different cases in Table 2: thesimple geometric case with solid spherical grains with 0%porosity, and two scenarios modeled by Augereau et al.(2001): 4% ice with 98% porosity and 10% ice with 95%porosity. For each case, we assumed a density of 3 g/cm3

for the rock component.Table 2 lists values for the corresponding grain radii

under three possible assumptions about the grain prop-erties, as well as the fraction of the total dust surface arearepresented by each radius bin. In total, we integratedthe orbits of 11,520 grains, assuming the simplified caseof solid spherical grains. Since the dynamics of grains

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−200 −100 0 100 200x (AU)

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100

200y (

AU

)

−200 −100 0 100 200x (AU)

−200

−100

0

100

200

y (

AU

)−5.5 −5.0 −4.5 −4.0 −3.5

log Collisions (year−1

pixel−1

)

Fig. 8.— Left panel: Face-on map of superparticle collision rate between 10 and 10.5 Myr. The collision rate map contains a brokenspiral structure corresponding roughly to the spiral density wave in Figure 6 (the “eccentricity spiral”). Right panel: The same map,with the collisions in the planet’s orbital plane masked out. The mask creates a two-armed black spiral pattern (the “inclination spiral”)corresponding to the red and blue x’s in Figure 5. The collision rate drops where the eccentricity spiral meets the inclination spiral, creatingthe complex azimuthal structure. Two such breaks in the collision rate spiral are shown by the white arrows.

−200 −100 0 100 200y (AU)

−20

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0

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AU

)

−200 −100 0 100 200y (AU)

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)

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nsit

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14 k

g/p

ixel)

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1.0

1.5

2.0

2.5

3.0

Co

llis

ion

s (

10

−3 y

ear−

1 p

ixel−

1)

Fig. 9.— Mass density (upper panel) and collision rate (lower panel) of the simulated β Pic disk at 10 Myr, cut through the x = 0 plane.The density and collision rate are enhanced in the crests and troughs of the secular wave.

with β < 0.1 are similar, we weighted the β = 0.09 grainsmore to represent all grains up to a size of 1 mm.

Our numerical integrator utilizes a symplectic N -bodymap (Wisdom & Holman 1991), modified to includeterms for radiation pressure and Poynting-Robertson

drag (Wilner et al. 2002; Moran et al. 2004). It integratesthe equation of motion for a dust grain (Robertson 1937),

d2r

dt2= −GM?(1 − β)

r3r− GM?

r3β

c[rr + rv], (7)

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β Pictoris 11

Fig. 10.— Radial distribution of the mass of dust produced by collisions in the SMACK simulation, binned according to mass. The mostmassive dust production events are confined mostly to a ring between 59-100 AU, while other collisions are spread radially through thedisk.

(a) (b) (c)

β s m fA s m fA s m fA0.09 21 14 0.26 260 4400 0.42 110 740 0.380.14 13 3.6 0.057 160 1100 0.045 67 190 0.0480.23 8.3 0.90 0.082 100 280 0.065 42 47 0.0690.36 5.2 0.23 0.12 65 69 0.093 27 12 0.0990.57 3.3 0.057 0.17 41 17 0.13 17 3.0 0.140.90 2.1 0.014 0.25 26 4.4 0.19 11 0.74 0.21

TABLE 2The β values we sampled for our dust orbit integrations,with the corresponding radii (s, in µm), masses (m, in 10−9

g), and the fraction of the total dust surface arearepresented by each bin (fA) for three different cases:(a) Geometric optics, solid spherical grains, 0% porosity,(b) Augereau et al. (2001) case for 4% ice, 98% porosity,(c) Augereau et al. (2001) case for 10% ice, 95% porosity.Note that the β = 0.09 bin includes grains up to 1 mm in

diameter, resulting in much larger values of fA.

where r and v are the circumstellar position and velocityof the grain. To the right-hand side of this equation, weadded the gravitational force from the orbiting planet,though we saw no evidence of planet-dust interactions(see Section 7.1), probably because the planet is locatedso far interior to most of the dust. We chose a timestep of one year and set the integrator to output thepositions of the dust grains every ∼ 211.56 yr (i.e., 10.25planet orbits). When the grains are created, their initialorbits conserve their birth velocities, resulting in highinitial eccentricities, e ≈ β/(1 − β), for grains created incollisions between two bodies on low eccentricity orbits.As expected (Wyatt 2005b; Strubbe & Chiang 2006), wesaw little Poynting-Robertson evolution of the dust grainorbits during the simulation of this collision-dominateddisk, except for the largest values of β. Since β Pictoris isan A-type star, we neglected the effects of stellar winds.

We ran the integration for the grain-grain collision time(Wyatt et al. 1999), under the simplifying assumptionthat grains of this size are either vaporized during acollision or broken into daughter grains that contributenegligibly to the optical depth because they are quicklyremoved by radiation pressure. We approximated thelocal grain-grain collision time as a function of circum-stellar distance by calculating the local optical depth ofthe material tracked by the superparticles, and extrap-olating down 1 µm to approximate the contribution ofsmaller grains to the local optical depth. We collectedthe output coordinates into three histograms matchingthe bins used for the SMACK simulated images: one forthe face-on image with 2 AU by 2 AU bins, and two fortwo orthogonal views of the disk edge-on, with 2 AU by0.5 AU bins. We summed together the histograms rep-resenting dust from the biggest mass production eventswith the histograms representing the dust from the massproduction events in the range 10−18 to > 10−16 solarmasses, weighting the latter histograms by 1.0/0.035 tocompensate for our sparse sampling of these latter massproduction events. We also applied the weightings fAlisted in case (a) shown in Table 2.

7. SIMULATED SCATTERED LIGHT IMAGES

To compare our simulated dust distributions with im-ages of the disk in scattered light, we synthesized im-ages from our dust density histograms assuming thatthe dust is illuminated by 0.5 µm light from β Pictoris,and scatters the light via a Henyey-Greenstein scatter-ing phase function. Ahmic et al. (2009) modeled im-ages the β Pictoris disk taken by the Advanced Cam-era for Surveys (ACS) on the Hubble Space Telescope(Golimowski et al. 2006) using a model consisting of apair of thin intersecting disks, with two different values

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12

of the Henyey-Greenstein parameter, g. In our physicalmodel there is no a priori physical distinction betweenthe dust in one “disk” or another, so we used a singlevalue for all the dust, the mean of the values in Ahmicet al. (2009): g = 0.743. Figure 11 shows the threesimulated images, showing the normalized histograms ofthe optical depth of the dust from three different view-ing orientations. Each image has been multiplied by thecircumstellar distance squared to highlight the faint fea-tures toward the outer edge of the disk.

The face-on distribution of the dust grains exhibits aspiral structure, reminiscent of the spiral structure inthe planetesimal density distribution (Figure 6). How-ever, unlike in the planetesimal distribution, which hasa deficit of larger grains in the inner region of the disk(. 59 AU), there is an enhancement of dust grains inthe inner region. This enhancement does not representmaterial falling inwards towards the star. It arises fromthe enhanced collision rate in the . 59 AU region of thedisk. The planetesimals remaining in this region collideviolently and often, because they have been stirred by theplanet. Our dust simulations indicate that this enhanceddust production is observable despite the enhanced colli-sional destruction of small grains that should also occurin this region of the disk.

By comparing the bottom panel of Figure 11 with theedge-on simulated image of the planetesimals in Figure1, we note that the dust grains on inclined orbits thatmake up the warp extend to greater circumstellar dis-tances than the planetesimals as they are pushed out-wards by radiation pressure. Figure 11 shows that whilea significant amount of the dust orbiting inclined to themain disk is within the radial extent of the planetesi-mal warp (i.e., . 95 AU), there are inclined dust grainsorbiting out to ∼ 200 AU.

Figure 12 offers a more quantitative comparison of themodels to scattered-light observations of the β Pictorisdisk. It shows an edge-on view of the model seen inthe bottom panel of Figure 11 (solid black curve) com-pared to the surface brightness of the disk measured bythe ACS with arbitrary scaling. The observational data(dashed curve) are the power law fits to the radial surfacebrightness profile listed in Table 2 of Apai et al. (2015).Models and data represent the vertically-averaged sur-face brightness, and the models depict the side of thedisk that is along the direction of the planet’s apocenter.We stretched the model in the radial direction by a fac-tor of 1.35 to better fit the data. Equation 1 indicatesthat a planet with the mass and orbit used in our simu-lation would require 28 Myr to perturb the disk to thatradius, but a larger planet mass or semimajor axis wouldincrease the extent of the secular perturbations withinthe age of the system.

The top panel of Figure 12 shows the slope of the modelvs. radius from the star, compared with the power lawindices fit by Apai et al. (2015). We found that thisstretching produces a good agreement for the location ofthe breaks in the surface brightness profile between ourmodel and the observations (as shown in Figure 12). Ourchoice to plot the dust produced at 10 Myr was only arough estimate for the age of β Pic b, so we can improvethe fit of our model by radially stretching the model dataas an approximation for simulating a later time.

Although our dust model seems to provide a reason-

ably good explanation for the gross characteristics of theobserved scattered light profile shown in Figure 12, it ap-pears to be missing a significant dust component. Thereremains a deficit in surface brightness in our model past∼ 20 AU, and a glance at the ACS images of the β Pic-toris disk shows that our models contain too little dustin the disk midplane. The likely reason for these mis-matches is that our SMACK models used the wrong ini-tial radial distribution of planetesimals. For example, toconserve computing time, we only included planetesimalsout to a circumstellar distance of 200 AU.

Since the external region of the disk has not yet beenstirred by the planet, we can remedy this deficit byadding an additional disk of dust to the model that picksup where the SMACK models leave off. To illustrate thispoint, we calculated a synthetic image of a dust cloudaround β Pictoris with inner radius 80 AU and outer ra-dius 500 AU using the ZODIPIC tool (Moran et al. 2004)and added it to our physical model. For the ZODIPICimage, we selected the same Henyey-Greenstein scatter-ing phase function (g = 0.743) and a dust density distri-bution with a radial power law of −1.5.

Figure 13 shows the sum of our physical model and thisempirical dust cloud model. The model is multiplied bycircumstellar distance squared, and masked in the centerto simulate observations made with a coronagraph. Asin Figure 12, we have radially stretched this image by afactor of 1.35 to simulate the observed radial extent ofthe warp in the dust distribution. This image (Figure13) is only meant to be an illustrative model, but it doesdemonstrate that we can replicate the gross morphologyseen in the ACS images. In particular, our simulatedscattered light image in Figure 13 exhibits the two-disk x-shaped pattern seen in the high-resolution scattered lightobservations of Golimowski et al. (2006); Ahmic et al.(2009); Apai et al. (2015). This feature has not beenreproduced by previous dynamical models of the dust inβ Pic (e.g., Mouillet et al. 1997; Augereau et al. 2001;Dawson et al. 2011), as these models did not capturethe 3D spatial distribution of dust-producing collisions.Our SMACK simulations reveal that dust is producedin larger quantities at the maximum vertical extent ofthe warp and in the plane of the main disk (Figure 9),creating an x-pattern as that dust is pushed outwards byradiation pressure.

In the lower panel of Figure 13, we fit straight linesto the spines of the NE and SW extensions of the in-clined disk. These lines do not intersect at the positionof the star, and their position angles differ by ∼ 1.1,indicating that we have reproduced the “wing-tilt asym-metry” first reported by Kalas & Jewitt (1995) and laterobserved by Heap et al. (2000) and Golimowski et al.(2006). This asymmetry has been attributed to forwardscattering from a disk of material inclined to the ob-server’s line-of-sight (Kalas & Jewitt 1995; Ahmic et al.2009). In our simulated images, we view the system ex-actly edge-on, with no inclination to the line-of-sight.However, the longitude of pericenter of the planet’s or-bit is non-zero (see Table 1), so the planet’s orbit, andtherefore the inclined disk, are slightly inclined to the ob-server. The difference between the position angles of theinclined parts of our simulated disk (∼ 1.1) agrees rea-sonably well with the wing-tilt asymmetry observed byKalas & Jewitt (1995), who measured a 1.3 difference in

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β Pictoris 13

Fig. 11.— Simulated images of the β Pictoris disk in scattered light, multiplied by the distance to the star squared. The dust does notshow the same deficit in the inner region of the disk as the mm-sized grains. The edge-on view in the bottom panel reveals the two-disk“x”-pattern seen in HST images.

the position angles of the combined primary and inclineddisks, and Ahmic et al. (2009), who found a 2.2 differ-ence between the position angles of the inclined disk.

7.1. No Structures Orbiting With The Planet’s MeanMotion

By choosing an output time step for the integrator ofprecisely 10.25 planet orbits, we can easily use our mod-els to search for disk structures that evolve on the timescale of the planet’s orbit, or at harmonics of the planet’sorbit (see Wilner et al. 2002; Moran et al. 2004). We con-structed four new “stroboscopic” dust density histogramsfor this purpose: one from coordinates output when theplanet’s mean anomaly was 0, one from coordinates out-put when the planet’s mean anomaly was 0.25, and soon. These histograms, like four frames of a movie, canreveal blobs that are otherwise smeared out in other rep-resentations.

When we constructed these histograms and differencedthem to search for moving patterns, we found nothingbut Poisson noise. The histograms are subject to Poissonnoise from the finite number of particles per pixel; with11520 particles and typically 700 output steps per each

stroboscopic histogram, the pixels have Poisson noise atthe level of about 1-σ = 7% per 2 AU by 2 AU pixel inthe face-on view. So, given the limitations of our model,we predict that there is probably no structure rotatingwith planet’s mean motion that is stronger than about21%, even at the planet’s semi-major axis.

This result stands in contrast to Apai et al. (2015), whoreported a marginal detection of a difference between im-ages of β Pictoris taken with the Hubble Space Tele-scope/Space Telescope Imaging Spectrograph in 1997(Heap et al. 2000) and images taken with the same in-strument in 2012. Apai et al. (2015) interpret this dif-ference, seen at 3′′ − 6′′ (58-117 AU) as a possible 50%perturbation to the surface density. Future observationsof this system at smaller inner working angles and futuredynamical models that focus more on the inner region ofthe disk should resolve this issue.

8. DISCUSSION AND SUMMARY

We have developed the first dynamical model for the βPictoris disk combining the colliding planetesimals anddust grains and the best-fit orbital parameters for theplanet β Pic b. Here, we discuss several features observedof the β Pic disk and consider the possible origins of

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Fig. 12.— Radial surface brightness profile of our dust model (solid black line) compared with the power-law fits (dashed line) of Apaiet al. (2015). The grey line shows the surface brightness profile of our model plus the material we added to the outer disk. The modelradial surface brightness profiles have been radially stretched by a factor of 1.35. The upper panel shows the slope of the radial surfacebrightness profile for our models compared with the Apai et al. (2015) fits. The locations of the breaks in the power law agree well betweenour models and the observations.

these features by comparing observations to our SMACKsimulations and analysis.

1. Using SMACK’s ability to model the dynamical ef-fects of collisions, we showed that the free incli-nations of the planetesimals are not damped sig-nificantly by collisions in the age of the system(Figures 3 and 4). Our SMACK simulations repro-duced the warp seen in both submillimeter (Dentet al. 2014) and scattered light observations (Bur-rows et al. 1995; Heap et al. 2000; Golimowski et al.2006, etc.) of the disk, confirming that this struc-ture can be induced by the planet’s inclination, inagreement with numerical simulations by Mouilletet al. (1997), Augereau et al. (2001), and Dawsonet al. (2011).

2. We showed that the spiral density wave in the mm-sized planetesimals induced by the planet’s eccen-tricity extends out to roughly the same radial ex-tent as the warp (Figures 1 and 6), and could beinterpreted as a ring or belt in edge-on observations(Wilner et al. 2011; Dent et al. 2014).

3. Our SMACK simulations also demonstrated thatthe deficit in mm-sized grains interior to this belt

(Wilner et al. 2011; Dent et al. 2014) is createdvia the mechanism described by Mustill & Wyatt(2009): collisions are excited by the planet’s eccen-tricity interior to the spiral density wave, erodingand removing material from the inner region of thedisk and creating an observable deficit (Figure 7).

4. We propose a new mechanism for producingazimuthally-asymmetric dust and gas (Wahhajet al. 2003; Telesco et al. 2005; Li et al. 2012; Dentet al. 2014) via collisions, without invoking MMRsor massive collisions (Telesco et al. 2005; Dentet al. 2014; Jackson et al. 2014): an azimuthally-asymmetric collision rate along the spiral densitywave (Figure 8), created by the interaction betweenthe spiral density wave (induced by the planet’s ec-centricity) and the vertical displacement wave (in-duced by the planet’s inclination). Though thecollisions among the bodies we modeled (< 10 cmin size) do not themselves produce enough CO tomatch the ALMA observations, collisions amongslightly larger parent bodies int he same disk struc-ture probably could.

5. We predict that there are no dust structures or-

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Fig. 13.— Upper panel: Simulated scattered light image including an additional midplane dust component beyond 80 AU from the star.Lower panel: The same image, with white lines drawn along the spines of the inclined components of the dust distribution. The two linesdo not intersect at the star and have different position angles relative to the main disk, exhibiting the “wing-tilt asymmetry” first describedby Kalas & Jewitt (1995).

biting with the planet that are detectable abovethe Poisson noise of the simulation, supporting thesuggestion that the 50% perturbation measured byApai et al. (2015) may have been produced by arecent massive collision, since such collisions werenot included in our simulations.

6. Our SMACK simulations of the dust production inthe disk revealed that only 46% of the dust is pro-duced in the planetesimal belt (60-90 AU), indicat-ing that the “birth ring” approximation (Strubbe& Chiang 2006) fails to account for over 50% ofthe mass of dust produced via collision in the disk.Instead of a “birth ring” at this location, the βPictoris system appears to have a “stirring ring”where only the high-velocity planetesimal collisionsare concentrated (Figure 12).

7. Our scattered light images (Figures 11 and13) reproduced the x-shaped pattern seen inhigh-resolution images of the edge-on disk byGolimowski et al. (2006); Ahmic et al. (2009); Apaiet al. (2015). This pattern arises because more dustis produced in the peaks and troughs of the secu-lar wave (Figure 9). It has not been reproducedby previous dynamical models of the dust (Mouil-let et al. 1997; Augereau et al. 2001; Dawson et al.2011).

8. Our scattered light image also reproduces the“wing-tilt” asymmetry (Figure 13) first observedby Kalas & Jewitt (1995) and later by Heap et al.(2000) and Golimowski et al. (2006). Previousmodelers and observers (Kalas & Jewitt 1995; Heapet al. 2000; Ahmic et al. 2009) attributed this phe-nomenon to a slight inclination of the disk; our sim-ulation assumes that the disk has zero inclinationfrom an edge-on geometry. The position angles ofthe spines of the secondary disks differ by ∼ 1.1

in our simulations, in reasonable agreement withobservations.

Though our model appears to have explained severalsalient features of the β Pictoris disk, many open ques-tions remain about the physics of this complicated plan-etary system.When and how was the planet scattered into its cur-

rent orbit? Possible mechanisms suggested by Dawsonet al. (2011) include scattering by a second planet. Thetimescale for this scattering could be estimated by mod-eling the speed at which the secular perturbations of theplanet propagate outwards through the disk. However,the observational evidence for the radial extent of the sec-ular perturbations appears contradictory. Wilner et al.(2002) and Dent et al. (2014) measured brightness peaksin the submillimeter observations, which they interpretedas the planetesimal belt. But Wilner et al. (2002) placed

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this belt at a radius of ∼ 94 AU while Dent et al. (2014)measured it at ∼ 60 AU. Scattered light observations ofthe disk indicate that the warp in the dust distributionextends out to ∼ 85 AU (Heap et al. 2000; Golimowskiet al. 2006). An inverse model that includes the variousmeasurements of the warp and stirring belt could usethe radial extent of these secular perturbations from theplanet to predict when the planet was scattered onto itscurrent orbit.What is the role of gas in shaping the dust in the disk?

Thebault & Augereau (2005) argued that the long-termeffects of gas drag alone probably have had a negligibleeffect on the dust in the β Pictoris disk. However, Lyra &Kuchner (2013) demonstrated that instabilities involvinggas drag, photoelectric heating and streaming effects cancause clumps and rings to form in debris disks like βPictoris.What is the role of large planetesimals in shaping the

disk? Our simulations did not track bodies larger than10 cm in size, but the rare collisions between these mas-sive bodies could yield important contributions to theobservable debris (e.g., Jackson et al. 2014; Stark et al.2014; Kral et al. 2015).What is the origin of the NE-SW brightness asymme-

try? The difference in surface brightness between theNE and SW wings of the disk has been observed at vari-ous wavelengths, including the optical, infrared, and sub-millimeter (Apai et al. 2015). But perhaps the planet’sorbit may be more eccentric than current measurementsindicate, or perhaps there are additional planets in thesystem creating this asymmetry.

The 3D planetesimal and dust model we used for theβ Pic disk could also be applied to other interesting de-bris disk systems. AU Microscopii, for example, is an-other edge-on disk with a planetesimal belt observed inthe submillimeter (MacGregor et al. 2013) and small-scale structure in the dust disk (Fitzgerald et al. 2007).A combined planetesimal and dust model are needed toconnect the submillimeter and scattered light observa-tions, by understanding how the dynamics of the plan-etesimals in the presence of a hypothetical planet affect

the locations of dust production events. Previous mod-elers have also applied the “birth ring” approximation tothe AU Mic disk (Fitzgerald et al. 2007; MacGregor et al.2013), but we have shown that the birth ring approxima-tion can miss over half of the sources of dust in a diskwith a planetesimal belt, indicating that observations ofAU Mic should be revisited with a combined planetesi-mal and dust model to account for these sources.

The HD 15115 debris disk, often referred to as the“blue needle”, exhibits a brightness asymmetry despiteits morphological symmetry, and was recently found tohave a large central cavity (Mazoyer et al. 2014). Colli-sional clearing may be responsible for the central clear-ing, and the asymmetric collision effect could create abrightness asymmetric in the absence of an offset be-tween the disk and star. The edge-on HD 32297 alsoexhibits a brightness asymmetry and a central clearing(Currie et al. 2012), and should be studied further witha collisional model.

The asymmetric collision effect may also be observ-able in disks that are not viewed edge-on. For exam-ple, high-resolution near-IR imaging of the HD 141569Adisk has revealed an inner spiral feature inclined to theouter ring, and a cleared inner region (Biller et al. 2015).A planet on an inclined and eccentric orbit could beresponsible for such a disk morphology, and may alsoproduce observable dust clumps via to asymmetric colli-sions. High-resolution submillimeter observations of HD141569A (for example, with ALMA) are needed to searchfor azimuthally-asymmetric gas distributions, and simu-lations like the SMACK and dust models described inthis work could constrain the mass and orbit of a possi-ble exoplanet orbiting in the disk.

Erika Nesvold and Marc Kuchner are supported inpart by NASA Planetary Geology and Geophysics grantPGG11-0032. Erika Nesvold is supported in part bythe ALMA Student Observing Support Program throughNRAO. Marc Kuchner is supported in part by the NASAAstrobiology Institute through the Goddard Center forAstrobiology.

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