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Accepted for Publication in ApJ: November 15, 2010Preprint typeset using LATEX style emulateapj v. 04/20/08

EMPIRICAL CONSTRAINTS ON TURBULENCE IN PROTOPLANETARY ACCRETION DISKS

A. Meredith Hughes1, David J. Wilner1, Sean M. Andrews1, Chunhua Qi1, Michiel R. Hogerheijde2

Accepted for Publication in ApJ: November 15, 2010

ABSTRACT

We present arcsecond-scale Submillimeter Array observations of the CO(3-2) line emission fromthe disks around the young stars HD 163296 and TW Hya at a spectral resolution of 44m s−1.These observations probe below the ∼100m s−1 turbulent linewidth inferred from lower-resolutionobservations, and allow us to place constraints on the turbulent linewidth in the disk atmospheres.We reproduce the observed CO(3-2) emission using two physical models of disk structure: (1) a power-law temperature distribution with a tapered density distribution following a simple functional formfor an evolving accretion disk, and (2) the radiative transfer models developed by D’Alessio et al. thatcan reproduce the dust emission probed by the spectral energy distribution. Both types of modelsyield a low upper limit on the turbulent linewidth (Doppler b-parameter) in the TW Hya system(.40m s−1), and a tentative (3σ) detection of a ∼300m s−1 turbulent linewidth in the upper layersof the HD 163296 disk. These correspond to roughly ≤10% and 40% of the sound speed at size scalescommensurate with the resolution of the data. The derived linewidths imply a turbulent viscositycoefficient, α, of order 0.01 and provide observational support for theoretical predictions of subsonicturbulence in protoplanetary accretion disks.Subject headings: circumstellar matter — planetary systems: protoplanetary disks — stars: individual

(HD 163296, TW Hydrae)

1. INTRODUCTION

The accretion disks around young stars provide the rawmaterial and physical conditions for the planet formationprocess. The viscous transport of angular momentumdrives the evolution of protoplanetary disks (Lynden-Bell & Pringle 1974; Hartmann et al. 1998), determin-ing when, where, and how much material is availablefor planet formation. An understanding of the physi-cal mechanisms behind the viscous transport process istherefore central to constraining planet formation theory.The source of viscosity is uncertain, since molecular vis-cosity implies a disk evolution timescale far longer thanthe observed 1-10Myr. The classic result from Shakura& Sunyaev (1973) sets up a framework for describing theaction of turbulent viscosity in accretion disks, whichcan to account for disk evolution on the appropriatetimescales. However, while turbulence is commonly in-voked as the source of viscosity in disks, its physical ori-gin, magnitude, and spatial distribution are not well con-strained.The mechanism most commonly invoked as the source

of this turbulence in disks around young stars is themagnetorotational instability (MRI), in which magneticinteractions between fluid elements in the disk couplewith an outwardly decreasing velocity field to producetorques that transfer angular momentum from the innerdisk outward (Balbus & Hawley 1991, 1998). Models ofdisk structure indicate that the conditions for the MRIare likely satisfied over much of the extent of a typi-cal circumstellar disk, with the possible exception of anannular dead zone (Gammie & Johnson 2005, and ref-

1 Harvard-Smithsonian Center for Astrophysics, 60 Gar-den Street, Cambridge, MA 02138; mhughes, dwilner, cqi,sandrews@cfa.harvard.edu

2 Leiden Observatory, Leiden University, P.O. Box 9513, 2300RA, Leiden, The Netherlands; michiel@strw.leidenuniv.nl

erences therein). MRI turbulence has also been invokedto address a wide array of problems in planet formationtheory. For example, it has been proposed to regulatethe settling of dust particles (e.g., Ciesla 2007), to ex-plain mixing in meteoritic composition (e.g., Boss 2004),to form planetesimals (e.g., Johansen et al. 2007), and toslow planet migration (e.g., Nelson & Papaloizou 2003).Measurements that constrain the magnitude and physi-cal origin of disk turbulence therefore promise to provideimportant insight into the physics of planet formation ona variety of physical and temporal scales.The only directly observable manifestation of turbu-

lence is the non-thermal broadening of spectral lines. Todate, no lines have been detected in disks that would al-low an independent determination of temperature andnon-thermal broadening, similar to NH3 in molecularcloud cores (e.g., Ho & Townes 1983, and referencestherein). Previous interferometric observations of molec-ular line emission from several disks show gas in Keple-rian rotation around the star with inferred subsonic tur-bulent velocity widths, close to the scale of the spectralresolution of (∼200m s−1, e.g. Pietu et al. 2007). Spec-troscopic observations of infrared CO overtone band-head emission originating from smaller disk radii indi-cate larger, approximately transonic, local line broaden-ing that may be associated with turbulence (Carr et al.2004), although those observations of optically thick linescan only probe far above the midplane. In combinationwith the millimeter data, this may indicate variations ofturbulent velocity with radius. However, this interpreta-tion is uncertain: it is important to exercise caution whenderiving information about velocity fluctuations on scalessmaller than the spectral resolution of the data (as notedby, e.g., Pietu et al. 2007). The advent of a high spectralresolution mode of the Submillimeter Array (SMA) cor-relator, capable of resolving well below the ∼200m s−1

2

linewidths in the low-J transitions of CO derived fromlower-resolution observations, permits access to turbu-lent linewidth measurements in the cold, outer regions ofmolecular gas disks around young stars.In this paper, we conduct high spectral resolution

(44m s−1) observations of the CO(3-2) emission from thedisks around two nearby young stars, HD 163296 andTW Hya. These systems were selected on the basis oftheir bright CO(3-2) line emission (e.g., Kastner et al.1997; Dent et al. 2005), to ensure adequate sensitivityfor high-resolution spectroscopy. They are also partic-ularly well-studied using spatially-resolved observationsat millimeter wavelengths, so that excellent models ofthe temperature and density structure of the gas anddust disks are already available (Calvet et al. 2002; Isellaet al. 2007, 2009; Hughes et al. 2008; Qi et al. 2004,2006, 2008). Both exhibit CO(3-2) emission that is con-sistent with Keplerian rotation about the central star,and neither suffers from significant cloud contamination.TW Hya is a K7 star with an age of∼10Myr (Webb et al.1999), located at a distance of only 51± 4 pc (Mamajek2005). It hosts a nearly face-on “transition” disk, withan optically thin inner cavity of radius ∼4AU indicatedby the SED (Calvet et al. 2002) and interferometricallyresolved at wavelengths of 7mm (Hughes et al. 2007) and10µm (Ratzka et al. 2007). The low spectral resolutionCO(3-2) line emission from the disk around TW Hyawas modeled with a 50m s−1 turbulent linewidth by (Qiet al. 2004). HD 163296 is a Herbig Ae star with a massof 2.3M⊙, located at a distance of 122pc (van den An-cker et al. 1998). Its massive, gas-rich disk extends toat least 500AU (Grady et al. 2000) and is viewed at anintermediate inclination angle of 45◦ (Isella et al. 2007).We describe the high spectral resolution SMA obser-

vations of the CO(3-2) line emission from TW Hya andHD 163296 in Section 2 and present the results in Sec-tion 3 (with full channel maps provided in Appendix A).In Section 4 we outline the standard procedures thatwe use to model the temperature, density, and veloc-ity structure of the disk, including the fixed parametersand assumptions about how the turbulent linewidth isspatially distributed. Section 4.3 presents the best-fitmodels, and the degeneracies between parameters arediscussed in Section 4.4. We compare our results to the-oretical predictions of the magnitude and spatial distri-bution of turbulence in Section 5, and describe the im-plications for planet formation. A summary is providedin Section 6.

2. OBSERVATIONS

The SMA observations of TW Hya took place on 2008March 2 in the compact configuration, with baselinelengths of 16-77m, and on 2008 February 20 during themove from compact to extended configuration, with base-line lengths of 16-182m. The weather was good bothnights, with stable atmospheric phases. Precipitable wa-ter vapor levels were extremely low on February 20, with225GHz atmospheric opacities less than 0.05 through-out the night, while the March 2 levels were somewhathigher, rising smoothly from 0.08 to 0.11. In order tocalibrate the atmospheric and instrumental gain varia-tions, observations of TW Hya were interleaved with thenearby quasar J1037-295. To test the efficacy of phasetransfer, observations of 3c279 were also included in the

observing loop. Flux calibration was carried out usingobservations of Callisto; the derived fluxes of 3c111 were0.76 and 0.78 Jy on February 20 and March 2, respec-tively.The observations of HD 163296 were carried out in

the compact-north configuration on 2009 May 6, withbaseline lengths of 16 to 139m, and in the extended con-figuration on 2009 August 23, with baseline lengths of44 to 226m. Atmospheric phases were stable on bothnights, and the 225GHz opacities were 0.05 on May 6and 0.10 on August 23. The observing loop includedJ1733-130 for gain calibration and J1924-292 for testingthe phase transfer. Callisto again served as the flux cal-ibrator, yielding derived fluxes of 1.17 and 1.30 Jy for1733-130 on May 6 and August 23, respectively.For all observations, the correlator was configured to

divide a single 104MHz-wide chunk of the correlator into2048 channels. This high-resolution chunk was centeredon the 345.796GHz frequency of the CO(3-2) line, yield-ing a spectral resolution of 44.1m s−1 across the line.Because this used up a large portion of the available cor-relator capacity, only 1.3GHz of the 2GHz bandwidthin each sideband was available for continuum observa-tions. The bandpass response was calibrated using ex-tended observations of 3c273, 3c279, and Saturn for theTW Hya tracks and 3c454.3, Callisto, and 1924-292 forthe HD 163296 tracks. Since the sidebands are separatedby 10GHz and the CO(3-2) line was located in the up-per sideband for the HD 163296 observations and in thelower sideband for the TW Hya observations, the contin-uum observations are at frequencies of 340GHz for HD163296 and 350GHz for TW Hya.Routine calibration tasks were carried out using the

MIR software package3, and imaging and deconvolutionwere accomplished with MIRIAD. The observational pa-rameters, including the rms noise for both the line andcontinuum data, are given in Table 1. Note that dueto weather the compact observations of HD 163296 weresubstantially more sensitive than the extended data andso the combined data set is dominated by the compactdata; the reverse is true for TW Hya.

3. RESULTS

We detect CO(3-2) emission at 44m s−1 resolutionfrom both TW Hya and HD 163296 in the compact andextended configurations. Figures 1 and 2 present theline emission from HD 163296 and TW Hya, respectively.The upper left panel shows the full line profile summedwithin a 6 arcsec square box (neglecting emission withinthe range ±2σ), with emission detected across ∼50 chan-nels for TW Hya and ∼200 for HD 163296. Beneath theline profiles are the spatially resolved channel maps fora subset of the data, indicated by the gray box aroundthe line peak. In the upper right are the zeroth (con-tours) and first (colors) moment maps: these are thevelocity-integrated intensity and intensity-weighted ve-locity of the emission, respectively. The line emissionis regular, symmetric, and consistent with material inKeplerian rotation around the central star viewed at aninclination to our line of sight.The peak and integrated fluxes for each of the four

tracks and the combined data sets are listed in Table 1.

3 See http://cfa-www.harvard.edu/∼cqi/mircook.html.

3

TABLE 1Observational Parametersa

HD 163296 TW HyaCompact-N Extended C+E Compact Extended C+E

Parameter 2009 May 6 2009 August 23 2008 March 2 2008 February 20

CO(3-2) Line

Beam Size (FWHM) 2.′′1×1.′′4 0.′′9×0.′′7 1.′′7×1.′′3 1.′′0×0.′′8 1.′′0×0.′′7 1.′′0×0.′′8P.A. 50◦ 8◦ 47◦ 5◦ -17◦ -16◦

RMS Noise (Jy beam−1) 0.51 0.97 0.49 0.35 0.52 0.40Peak Flux Density (Jy beam−1) 8.9 3.1 8.1 4.8 4.0 4.8Integrated Fluxb (Jy kms−1) 76 14 76 19 4.8 24

340GHz Continuum 350GHz ContinuumBeam Size (FWHM) 2.′′1×1.′′4 0.′′9×0.′′7 1.′′7×1.′′3 1.′′0×0.′′9 1.′′0×0.′′7 1.′′0×0.′′8

P.A. 52◦ 9◦ 47◦ 8◦ -21◦ -21◦

RMS Noise (mJy beam−1) 7.0 10 7.0 16 10 8.5Peak Flux Density (Jy beam−1) 1.14 0.6 1.05 1.21 0.47 0.51Integrated Fluxc (Jy) 1.78 1.72 1.75 1.67 1.49 1.57

aAll quoted values assume natural weighting.bThe integrated line flux is calculated by integrating the zerothmoment map inside the 3σ brightness contours using the MIRIADtask cgcurs.cThe integrated continuum flux is calculated using the MIRIAD

task uvfit, assuming an elliptical Gaussian brightness profile.

Appendix A presents the full channel maps of the com-bined (compact and extended configuration) data set foreach source.

4. ANALYSIS

In order to constrain the turbulent linewidth in thedisks around TW Hya and HD 163296, we fit models ofthe temperature, density, and velocity structure to thehigh spectral resolution CO(3-2) line data. For the ini-tial modeling effort presented here, we use two well-testedphysical models of disk structure: (1) power-law mod-els of the disk temperature structure combined with ta-pered surface density profiles corresponding to the func-tional form predicted for a simple viscous accretion disk(Lynden-Bell & Pringle 1974; Hartmann et al. 1998),and (2) the 1+1D radiative transfer models developedby D’Alessio et al. to reproduce the dust emission repre-sented in the SEDs of young systems. These models aredescribed in more detail in Section 4.1.We use these two classes of models because they are

well established and have been successful in describingthe observed structure of circumstellar disks across awide range of wavelengths, particularly in the submil-limeter (see, e.g., Calvet et al. 2002, 2005; Andrews et al.2009). However, each class of models has limitations.The similarity solution models have a large number offree parameters, some with significant degeneracies (seediscussion in Andrews et al. 2009). By fitting only theCO(3-2) emission, these models also neglect potential in-formation provided by dust emission, including strongerconstraints on the disk density. However, the neglect ofdust emission avoids complications due to heating pro-cesses and chemistry that affect gas differently than dust.The D’Alessio et al. models of dust emission includeonly stellar irradiation and viscous dissipation as heatingsources, and do not take into account the additional heat-ing processes that may affect molecular line strengths inthe upper layers of circumstellar disks (Qi et al. 2006).While the constraints from the dust continuum reducethe number of free parameters in this class of models,

they also have the disadvantage of an unrealistic treat-ment of the density structure at the disk outer edge: sincethey are simply truncated at a particular outer radius,they are not capable of simultaneously reproducing theextent of gas and dust emission in these systems (Hugheset al. 2008). The similarity solution models are verticallyisothermal, which is an unrealistic assumption. Withonly one molecular line included in the model, this lim-itation will not affect the results of the study presentedhere, although caution should be exercised when apply-ing the best-fit model parameters to other lines.The primary reason for using the two types of mod-

els, however, is that they differ substantially in theirtreatment of the disk temperature structures. For theD’Alessio et al. models, the temperature structure isfixed by the dust continuum. The similarity solutionmodels, by contrast, allow the temperature to vary tobest match the data. There are a few independent con-straints on temperature: it should increase with heightabove the midplane, due to surface heating by the starand low viscous heating in the midplane, and the dustwill generally not be hotter than the gas, since the gas issubject to additional heating processes beyond the stel-lar irradiation that determines dust temperature. Thetemperature structure in the disk is the single factormost closely tied to the derived value of the turbulentlinewidth (see discussion in Section 4.4), which will bemodel-dependent. We therefore fit both classes of mod-els to the data, in order to compare the model-dependentconclusions about turbulent linewidth for two distincttypes of models with very different treatments of gastemperature. The spatial dynamic range of the data isinsufficient to investigate radial variations in turbulentlinewidth. We therefore assume a global value, ξ, thatwill apply to size scales commensurate with the spatialresolution of the data.

4.1. Description of Models

4.1.1. D’Alessio et al. Models

4

Fig. 1.— CO(3-2) emission from the disk around HD 163296 observed with the SMA at a spectral resolution of 44m s−1. Top plot showsthe line profile, summed within a 6 arcsec box using the MIRIAD task imspec (neglecting emission with absolute values between ±2σ).Channel maps across the bottom show the segment of the line indicated by the shaded gray box at its full spatial and spectral resolution,imaged with a 1.′′0 taper to bring out the large-scale emission (complete channel maps are provided in Appendix A). LSR velocity isindicated by the numbers in the upper right of each channel. Contours are [3,6,9,...]×0.55Jy beam−1 (the rms noise). Inset in the upperright corner is a zeroth (contours) and first (colors) moment map of the CO(3-2) line emission. The 2.′′0×1.′′7 beam is indicated in the lowerright of the inset. Note that while the colors in the channel and moment maps both represent LSR velocity (blue is low; red is high), thescales are different for the two representations: the moment map contains the full line data, while the channel maps span only a subset ofthe line.

The D’Alessio et al. models are described in detail inD’Alessio et al. (1998, 1999, 2001, 2006). Here we providea general outline of the model properties and discuss theparticular models used in this paper.The D’Alessio et al. models were developed to repro-

duce the unresolved SEDs arising from warm dust orbit-ing young stars, although they have also been demon-strated to be successful at reproducing spatially resolveddust continuum emission at millimeter wavelengths (see,e.g., Calvet et al. 2002; Hughes et al. 2007, 2009a) aswell as spatially-resolved molecular line emission (see,e.g., Qi et al. 2004, 2006). The models include heat-ing from the central star and viscous dissipation withinthe disk, although they tend to be dominated by stellarirradiation. The structure is solved iteratively to pro-vide consistency between the irradiation heating and thevertical structure. The mass accretion rate is assumedto be constant throughout the disk. The assumed dustproperties are described by Calvet et al. (2002), and themodel includes provisions for changing dust properties,dust growth, and settling. We allow the outer radius ofthe model to vary to best reproduce the extent of themolecular line observations.We use the structure model for TW Hya that was de-

veloped by Calvet et al. (2002) and successfully comparedto molecular line emission by Qi et al. (2004, 2006). ForHD 163296, we use a comparable model that reproducesthe spatially unresolved SED and is designed to repro-duce the integrated line strengths of several CO transi-tions as well as other molecules (Qi et al., in prep).Since the D’Alessio et al. models were developed pri-

marily to reproduce the dust emission from the SED, weare required to fit several parameters to match the ob-served CO(3-2) emission using the SED-based models.

We fit the structural parameters {RD, XCO} (the diskouter radius and CO abundance, respectively), the ge-ometrical parameters {i, PA} (the disk inclination andposition angle), and the turbulent linewidth, {ξ}.

4.1.2. Viscous Disk Similarity Solutions

We also fit the observations using a power-law temper-ature distribution and surface density profile that followsthe class of similarity solutions for evolving viscous accre-tion disks described by Lynden-Bell & Pringle (1974) andHartmann et al. (1998). This particular method of pa-rameterizing circumstellar disk structure has a long his-tory of success in reproducing observational diagnostics,although with limitations. Theoretical predictions of thepower-law dependence of temperature for accretion disksaround young stars were first made by Adams & Shu(1986), and power-law parameterizations of temperatureand surface density have been used by many studies sincethen (e.g. Beckwith et al. 1990; Beckwith & Sargent 1991;Mundy et al. 1993; Dutrey et al. 1994; Lay et al. 1994;Andrews & Williams 2007). The similarity solutions areequivalent to a power-law surface density description inthe inner disk, but with an exponentially tapered outeredge, which was shown by Hughes et al. (2008) to betterreproduce the extent of gas and dust emission than tra-ditional power-law descriptions with abruptly truncatedouter edges. Recent high spatial resolution studies haveused this class of models to reproduce successfully the ex-tent of gas and dust emission in circumstellar disks fromseveral nearby star-forming regions (e.g. Andrews et al.2009; Isella et al. 2009).The temperatures and surface densities of these models

5

Fig. 2.— Same as Figure 1 but for TW Hya. The channel maps were imaged with a 1.′′2 Gaussian taper to emphasize the emission onlarger scales, and the contours are [4,8,12,...]×0.55Jy beam−1 (the rms noise). For the full set of channel maps, see Appendix A.

are parameterized as follows:

T (R) = T100

(

R

100AU

)−q

(1)

Σ(R) =c1Rγ

exp

[

−(

R

Rc

)2−γ]

(2)

where R is the radial distance from the star in AU, T100 isthe temperature indicated by the CO(3-2) line at 100AUfrom the star, q describes how the temperature decreaseswith distance from the star, c1 is a constant describingthe surface density normalization, Rc is a constant re-lated to the radial scale on which the exponential taperdecreases the disk density, and γ describes how surfacedensity falls with radius in the inner disk regions (com-parable to the parameter p in typical power-law descrip-tions of surface density; see e.g. Dutrey et al. 1994, fora description of the power-law model parameters). Be-cause the high optical depth of the CO(3-2) line is a poortracer of the radial dependence of Σ, we fix γ at a value of1 for both systems, which is consistent with theoreticalpredictions for a constant-α accretion disk (Hartmannet al. 1998), as well as observations of young disks inOphiuchus (Andrews et al. 2009) and previous studiesof the continuum emission from these systems (Hugheset al. 2008). We therefore fit the high spectral resolutionCO(3-2) observations using four structural parameters,{T100, q, c1, Rc}, two geometric parameters, {i, PA},and the turbulent linewidth, {ξ}.

4.2. Modeling Procedure

We assume that the disks have a Keplerian velocityfield, using stellar masses and distances from the litera-ture to model the rotation pattern (0.6M⊙ at 51 pc forTW Hya and 2.3M⊙ at 122pc for HD 163296; see Cal-vet et al. 2002; Webb et al. 1999; Mamajek 2005; vanden Ancker et al. 1998). Gas and dust are assumed to

be well-mixed in both models; the gas-to-dust mass ra-tio is fixed at 100 while the CO abundance is allowed tovary in the D’Alessio et al. models in order to reproducethe CO emission while maintaining consistency with thedust continuum emission from which the model was de-rived. Since we don’t include continuum emission in thefits, we fix the CO abundance at 10−4 for the similar-ity solution models because there is no constraint on therelative content of gas and dust. In regions of the diskwhere the temperature drops below 20K, the CO abun-dance is reduced by a factor of 104 to simulate the effectsof freeze-out onto dust grains. We assume a global, spa-tially uniform value of the turbulent linewidth, which isimplemented as an addition to the thermal linewidth.In order to compare the models with the data, the

systemic velocity, or central velocity of the line, mustbe determined. We calculated the visibility phases forthe spatially integrated CO(3-2) emission and fit a lineto the central few channels (45 for HD 163296; 12 forTW Hya) to determine the systemic velocity. Figure 3presents a cartoon of this method, which is largely model-independent, requiring only the assumption that the fluxdistribution is axisymmetric. The visibility phases en-code information about the spatial symmetry of the lineemission in each channel: for a Keplerian disk, it is mostsymmetric, and therefore the phase is zero, at line center.The channel maps are most asymmetric about disk centernear the line peaks, with opposite spatial offsets at thetwo line peaks. The phase therefore reverses sign betweenone peak and the other, with an approximately linear re-lationship near the line center. The linear fit thereforeuses the integrated emission from the central few chan-nels to pinpoint the precise location where the phase iszero and the line is most symmetric, i.e., at the systemicvelocity (it should be noted that if there were large-scaleasymmetries in the CO data this method would not pro-duce a reliable systemic velocity, but we see no evidenceof such asymmetries in the data). We derive a systemic

6

Fig. 3.— Cartoon showing the amplitude and phase of a spectralline as a function of velocity and how they correspond to the spatialdistribution of flux in the disk. The line wings originate from smallareas of fast-moving material close to the star, resulting in lowamplitude and small phase offsets. The line peaks arise from a largesurface area of intermediate-velocity material with a large spatialoffset from the star, resulting in high amplitudes and large visibilityphases. The line center is symmetric and therefore has zero phase.We determined the systemic velocity by assuming an axisymmetricflux distribution and fitting a line to the central channels of thevisibility phases (similar to the dotted line in the figure). Thesystemic velocity is the point at which the asymmetry changessign and the line crosses zero.

velocity of 5.79 km s−1 for HD 163296 and 2.86 km s−1

for TW Hya. In order to generate model emission withthe appropriate velocity offset, we create a model im-age at higher spectral resolution than the data, and thenre-bin it at the appropriate velocity sampling using theMIRIAD task regrid.Each disk model specifies a particular density, temper-

ature, and velocity structure. We use the Monte-Carloradiative transfer code RATRAN (Hogerheijde & van derTak 2000) to calculate equilibrium populations for eachrotational level of the CO molecule and generate a sky-projected image of the CO(3-2) line emission at a givenviewing geometry {i,PA} for each model. We comparethese simulated models directly to the data in the Fourierdomain. In order to sample the model images at theappropriate spatial frequencies for comparison with theSMA data, we use the MIRIAD task uvmodel. We thencompute the χ2 statistic for each model compared withthe data using the real and imaginary simulated visibil-ities. Due to the high computational intensity of themolecular line radiative transfer, it is prohibitively time-intensive to generate very large and well-sampled gridsof models for the χ2 comparison. Instead, we move fromcoarsely-sampled grids that cover large regions of param-eter space to progressively more refined (but still small)grids to avoid landing at a local minimum. However,this has the result that the degeneracies of the parame-ter space are poorly characterized. A discussion of thesedegeneracies is included in Section 4.4 below.

4.3. Best-fit Models

The best-fit parameters for both types of models arepresented in Table 2. Their temperature and densitystructures are plotted in Figure 6. Note that the mid-

plane temperatures for the similarity solution models arelikely much lower than indicated; the power-law temper-ature representation parameterizes only the radial de-pendence of temperature in the upper disk layers fromwhich the optically thick CO(3-2) emission arises.The χ2 values for the similarity solutions are lower

than for the D’Alessio et al. models; this may be dueto gradients between gas and dust properties that influ-ence the D’Alessio et al. model fit but not the similaritysolution models. A sample of channel maps comparingthe data with the two classes of models are presented inFigures 4 and 5. From the residuals in the D’Alessio etal. model of TW Hya, there is evidence of a mismatchin the temperature gradient between the model and thedata: the residuals are systematically more positive nearthe disk center and negative farther from the star. ForHD 163296, the difference is more subtle: the residu-als are small and apparently spatially random (with theexception of some positive emission seen near a veloc-ity of 4.7m s−1 but not near the corresponding positionin the mirror half of the line). It should be noted thatthe CO abundance derived for this source is extremelylow, nearly three orders of magnitude below the stan-dard value of 10−4. The reason for this is most likelyan overestimate of the temperature in the upper disklayers. In the absence of better information, this SEDmodel was created with a very low turbulent linewidth(∼50m s−1) and correspondingly little stirring of largedust grains above the midplane. The addition of a tur-bulent linewidth comparable to the best-fit value for theCO lines would substantially reduce settling and lowerthe temperature of the upper disk layers as more of themass is placed in large dust grains. Such a model is un-der development by Qi et al. (in prep), and can alsoaid in explaining the spatial distribution of multiple COtransitions and CO isotopologue emission from this disk.One important outcome of the modeling process is the

consistency in the measurement of turbulent linewidthin each source for the two types of models, despite thedifferences in their treatment of temperature. In bothtypes of models for HD 163296, the best-fit model withturbulence fits the data better than a comparable modelwithout turbulence at the ∼ 3σ level. If the turbulentlinewidth is fixed at 0m s−1 in the similarity solutionand the temperature allowed to vary to compensate, theparameter T100 must increase to 77K; even then, the χ2

for a model with higher temperature and no turbulenceis a poorer fit than the best-fit model with turbulence atthe ∼ 3σ level (the line profile for this model is plottedin Figure 4). The TW Hya data are consistent with noturbulent linewidth whatsoever.

4.4. Parameter Degeneracies

In order to better characterize our ability to measureturbulent linewidth, it is important to understand its re-lationship to the other parameters. The interdependencebetween parameters other than the turbulent linewidthhas been explored at length in previous papers (see, e.g.,discussion of similarity solution parameters in Andrewset al. 2009), so here we focus on the relationships and de-generacies specific to the turbulent linewidth. There arefour main categories of line broadening in circumstellardisks that are relevant to our investigation: rotational,thermal, turbulent, and optical depth. These types of

7

Fig. 4.— Comparison of CO(3-2) emission from HD 163296 between the data and best-fit models for a subset of the data. The top rowshows the same subset of channels as in Figure 1. The central set of channel maps shows the corresponding channels of the best-fit similaritysolution model and the residuals (subtracted in the visibility domain). The bottom set of channel maps shows the best-fit D’Alessio et al.model and residuals. Contour levels, beam sizes, and imaging parameters are identical to those in Figure 1. An additional line profile forthe best-fit similarity solution without turbulence is overplotted in gray.

TABLE 2Best-Fit Model Parameters

Parameter HD 163296 TW Hya

Similarity SolutionT100 (K) 60 40q 0.5 0.4c1 (cm−2) 1.0× 1012 1.0× 1011

Rc (AU) 150 50ξ (m s−1) 300 . 40i (◦) 40 6.0PA (◦) 131 155Reduced χ2 2.642 2.106

D’Alessio et al. Model

RD (AU) 525 155XCO 5× 10−7 1.5× 10−5

ξ (m s−1) 300 . 40i (◦) 40 5PA (◦) 138 155Reduced χ2 2.885 2.108

line broadening are all incorporated in detail into the ray-tracing portion of the RATRAN radiative transfer code,and will be handled appropriately for a given disk struc-ture. The goal is to understand how to distinguish thedistinct contributions of each of these different sources ofline broadening and their relationships to the parametersof our disk structure models.As discussed above, a detailed characterization of the

multi-dimensional parameter space is prohibitively com-putationally expensive. We therefore investigate param-eter relationships by letting the two-dimensional χ2 val-ues generated in Section 4.2 guide an investigation using

a toy model of an optically thick spectral line profile tohighlight the distinct contribution of each related param-eter to the observable properties.The χ2 values indicate that for the similarity solution

models, the parameters that are most strongly degen-erate with the turbulent linewidth are the temperature(T100 and q) and inclination (i). This is unsurprising,given the obvious relationship between temperature andthermal broadening and between inclination and rota-tional broadening. The optically thick CO(3-2) line re-sponds only weakly to variations in density, and the outerradius and position angle of emission should intuitivelybe unrelated to line broadening, hence the independenceof turbulent linewidth from c1, Rc, and PA. For theD’Alessio et al. models, inclination and CO abundance(i and XCO) have the strongest relationships with tur-bulent linewidth. The contribution of the CO abundancein this case can be understood as a thermal broadeningeffect: because of the vertical temperature gradient (seeFigure 6), the CO abundance controls the location of theτ=1 surface and therefore the apparent temperature ofthe CO(3-2) line emission.To characterize the effects of these variables on the

observable properties of the CO(3-2) emission, we inves-tigate their influence on a toy model of optically thickline emission. We assume a power-law temperature dis-tribution for a geometrically flat, optically thick, az-imuthally symmetric circumstellar disk. In the Rayleigh-Jeans approximation, the brightness of the line at agiven frequency will be directly proportional to the tem-perature. We include two sources of line broadening,thermal and turbulent, implemented by the relation-ship ∆v(r) =

√

2kBT (r)/m+ ξ2, where ∆v is the total

8

Fig. 5.— Same as Figure 4 but for TW Hya. The channels, contour levels, and imaging parameters are identical to those in Figure 2.

linewidth, ξ is the turbulent linewidth, and the thermallinewidth is

√

2kBT/m where kB is Boltzmann’s con-stant, T is the local temperature in the disk, and m isthe average mass per particle. Rotational broadeningis implicitly included in the assumed Keplerian rotationpattern of the material, so that in polar coordinates, thecentral frequency of the line as a function of position isgiven by ν(r, θ) = ν0/c

√

GM∗/r sin i cos θ, where ν0 isthe line frequency and M∗ is the stellar mass. The lineprofile at any point in the disk is then given by

φ(ν, r) ∝ T (r)

∆v(r)ν0/cexp

[ −(ν − ν0)2

(∆v(r)ν0/c)2

]

(3)

where ν0 is the frequency of the line center and c is thespeed of light. We project the disk onto the sky accordingto its inclination, and integrate over the r and θ coordi-nates across the extent of the elliptical disk to calculatethe line profile as a function of frequency. We investigatethe contribution of the different sources of broadeningby varying ξ, T (r), and i to correspond to the turbu-lent, thermal, and rotational broadening effects. Herewe discuss the ways in which turbulent broadening canbe distinguished from the other two effects in the contextof our toy model.Temperature — The left panel of Figure 7 shows

model line profiles that illustrate the relationship be-tween broadening due to temperature and broadeningdue to a global turbulent linewidth. The solid line is themodel line profile calculated using values from the best-fitsimilarity solution: T100=60K, q=0.5, and ξ=300m s−1.The dotted line represents the profile for a model withthe same parameters but no turbulent linewidth. To cre-ate the dashed line profile, the temperature was varieduntil the peak line flux of the model without turbulencematched the peak flux of the original model with turbu-

lence. This sequence of line profiles illustrates how ther-mal and turbulent broadening produce distinct effects onthe observable properties of the data, rather than beingfully interchangeable. Since increases in temperature in-crease the line flux throughout the disk while turbulencesimply redistributes the flux in frequency space, turbu-lence tends to shift flux from the line peaks to the center,changing the shape of the line. As a result, the peak-to-center line flux ratios will be different for line profileswith comparable widths and peak fluxes, depending onthe relative contributions of turbulence and temperatureto line broadening. The difference between the best-fit300m s−1 turbulent linewidth in HD 163296 and a com-parable model without turbulence corresponds to a 30%change in peak-to-center line flux in the context of thistoy model, which should be easily distinguishable giventhe quality of our data.As mentioned above, the temperature power law in-

dex q is also found to be degenerate with the turbu-lent linewidth. The effects of this parameter are similarto those of the temperature normalization, illustrated inthe left panel of Figure 7: varying q also alters the peak-to-center ratio of the line. However, since the emissionfrom both the peaks and the center is dominated by spa-tial scales of order 100 AU, the relevant spatial dynamicrange is not large enough for variations of q (within theuncertainties) to account for broadening on the scale ofthe derived turbulent linewidth.Inclination — The right panel of Figure 7 shows model

line profiles that illustrate the relationship between tur-bulent broadening and rotational broadening due to in-clination. As in the left panel, we first generate a modelwith no turbulent broadening (dotted line), and thenadjust the inclination until the peak flux matches theoriginal line model (dashed line). In this case, the ab-

9

Fig. 6.— Comparison between the temperature and density structures of the similarity solution and D’Alessio et al. models for HD163296 (left) and TW Hya (right). The bar across the top shows the temperature scale represented by the colors, while the white contoursrepresent density, marked with the base 10 logarithm of the total (gas+dust) mass density in units of g cm−3. Note that while the D’Alessioet al. model of HD 163296 appears to be two orders of magnitude more dense than the similarity solution model, the CO abundance ismore than two orders of magnitude lower, making the CO mass densities comparable. The CO abundance of the TW Hya model is oneorder of magnitude lower than for the similarity solution. The abscissae of the plots are scaled to match the radial extent of the power-lawdisk model (excising the central 10AU, which are not accessible with the data), although the similarity solution models will extend farther.

sence of turbulence once again alters the peak-to-centerline ratio. But another obvious difference between theturbulent model and the more inclined model withoutturbulence is that rotational broadening through incli-nation alters the separation between the line peaks invelocity space. The sequence of toy models shows thatthe contribution of the 300m s−1 turbulent linewidth inthe HD 163296 model corresponds to a 6◦ change in in-clination, in addition to the large depression of flux atthe line center. Since spatially resolved spectra of theKeplerian rotation patterns in molecular lines allow forvery precise determination of circumstellar disk inclina-tion (to within a degree or less, given the stellar massand assuming axisymmetry; see discussion in Qi et al.2004), turbulent line broadening at the level detectedin HD 163296 should be distinguishable from rotationalbroadening. It should be noted that rotational broad-ening includes the combined effects of inclination andstellar mass since the two variables are almost preciselydegenerate to within ∼10◦ (or tens of percent in mass):we cannot disentangle their effects. Nevertheless, sincewe allow inclination to vary for the assumed stellar mass,and have demonstrated that inclination can be distin-guished from turbulent broadening in the toy model lineprofiles, the uncertainty in stellar mass will not stronglyaffect our determination of the turbulent linewidth.We have focused this discussion on HD 163296 to in-

vestigate the robustness of the turbulent linewidth mea-surement rather than the upper limit for TW Hya. Thesituation is somewhat different for TW Hya, since the

face-on disk orientation does not result in splitting ofthe line profile, so diagnostics like peak-to-trough ratiosand peak separations do not apply. Nevertheless, the Ke-plerian rotation pattern – the combination of inclinationand stellar mass – is still very precisely determined fromthe channel maps (Qi et al. 2004), and the temperature iswell constrained by the brightness of the optically thickline. As a result, we achieve a strong constraint on theturbulent linewidth for this system. It should be notedthat a turbulent linewidth comparable to that derived forHD 163296 would have a dramatic effect on the line pro-file for this system, since a 300m/s turbulent linewidthwould represent roughly a 50% perturbation on the ob-served 700m/s FWHM of the CO(3-2) line.

5. DISCUSSION

5.1. Comparison with Theory

The problem of accurately modeling and predicting themagnitude of velocity fluctuations arising from magneto-hydrodynamic (MHD) turbulence has proven somewhatintractable, but a few general features seem to be agreedupon. The difficulty hinges largely on derived values ofthe parameter α and how they relate to the expectedmagnitude of the turbulent linewidth. The dimensionlessparameter α was defined by Shakura & Sunyaev (1973) todescribe the effective viscosity in terms of a proportional-ity constant multiplied by the largest velocity and lengthscales on which turbulence may act: the scale height andthe sound speed. In mathematical terms, νeff = αcsH ,where νeff is the effective viscosity, cs is the sound speed,

10

Fig. 7.— Model spectral line profiles for a geometrically flat, optically thick, azimuthally symmetric circumstellar disk (for details, seetoy model description in Section 4.4). The left panel explores the relationship between thermal and turbulent line broadening, while theright panel explores the relationship between turbulent broadening and rotational broadening through changes in inclination. The solid linein each panel is generated using values from the best-fit similarity solution model (T100=60K, q=0.5, and ξ=300m s−1), while the dottedlines are the same model without turbulence, and the dashed lines seek to compensate for the lack of turbulence through the alternativeline broadening mechanisms, scaled so that the peak flux is the same as the original model. The parameters for each line are given in thelegend. The obvious differences between the solid and dashed lines in each panel (e.g., peak-to-center flux ratios and velocity separationbetween peaks) illustrate how the effects of rotational and thermal broadening differ from those of turbulent broadening in the context ofthis toy model.

H is the local scale height, and α is then an efficiencyfactor with a value ≤1. The magnitude of turbulent ve-locity fluctuations depends both on the value of α andon how this efficiency factor is apportioned between thesound speed and the scale height. If, for example, themajority of the power in turbulent fluctuations occursat the length of the scale height, the velocity fluctua-tions can be as small as αcs (perhaps augmented by ageometric factor of a few). On the other hand, if theproportionality factor α applies evenly to the length andvelocity scales in the problem, the turbulent fluctuationscould be as large as

√αcs (again possibly modified by a

geometric factor). Since there is no evidence of shocksthat would point to sonic or supersonic turbulence in cir-cumstellar disks, it is unlikely that the turbulent velocityfluctuations would be much larger than

√αcs.

It is clear that shearing box simulations of MHD tur-bulence with zero net magnetic field flux do not give reli-able values of the viscosity parameter α due to numericaldissipation, which results in values of α that depend onresolution (Pessah et al. 2007; Fromang & Papaloizou2007). This is mitigated by the use of more realistic sim-ulation conditions including vertical stratification (e.g.,Stone et al. 1996; Fleming & Stone 2003; Davis et al.2010; Flaig et al. 2010), but the resulting dependence ofα and therefore turbulent velocity fluctuations on themagnitude of the magnetic field is troublesome, sincemagnetic field strengths and geometries in circumstel-lar disks are unconstrained by observations (e.g., Hugheset al. 2009b). The inclusion of vertical gravity has alsobeen shown to lead to convergence (e.g. Davis et al. 2010;Flaig et al. 2010). It is perhaps instructive to investigatehow αmay be related to the observed turbulent linewidthin the context of shearing-box simulations. Quantitiestypically reported for shearing boxes include the shearstress −BxBy/4π, where Bx and By are the magneticfield along the x and y directions, respectively; Reynoldsstress ρvxδvy, where ρ is density, vx is the x compo-nent of velocity, and δvy is the y component of velocitywithout shear; and the magnitude of velocity fluctuations

along the different dimensions of the simulation ρv2x,y,z/2.Each of these quantities is routinely normalized by theinitial midplane pressure, P0 (see, e.g., Stone et al. 1996;Davis et al. 2010). The total stress T is then the sum ofthe Maxwell and Reynolds stresses, and is directly pro-portional to the viscosity parameter α, according to therelationship T = αρc2s, where cs is the sound speed. It isthen straightforward to relate α to the turbulent velocityas a function of sound speed:

α =T

ρv2turb

v2turbc2s

(4)

α =T/P0

2(ρv2z cos2 i+ ρv2x,y sin

2 i)/2P0

v2turbc2s

(5)

where vturb is the measured turbulent linewidth (typ-ically the FWHM calculated from ξ, which is the 1/ehalf-width) and i is the disk inclination. Using reportedvalues of the relevant simulation parameters from Daviset al. (2010), α ∼ 3(vturb/cs)

2 for a face-on disk, andα ∼ 1.5(vturb/cs)

2 for an edge-on disk. For the valueof α ∼ 0.01 derived in Davis et al. (2010), this predictsa midplane turbulent linewidth of roughly 6-8% of thesound speed, depending on the source geometry.Observational attempts at constraining the value of α

in circumstellar disks, while generally quite uncertain,seem to cluster near values of 10−2, with large scatter(e.g., Hartmann et al. 1998; Andrews et al. 2009). If thevelocity fluctuations are estimated as

√αcs, this result

would imply velocity fluctuations up to 10% of the soundspeed near the midplane, although this global value islikely to vary widely depending on local conditions thataffect the ionization fraction and the coupling of ions andneutrals. A theoretical comparison between circumstel-lar disks and the Taylor-Couette flow by Hersant et al.(2005) finds that the 100m s−1 turbulent linewidth (oforder .30% of the sound speed) derived from low spec-tral resolution observations of DM Tau by Guilloteau &Dutrey (1998) are on par with expectations from labo-

11

ratory measurements by Dubrulle et al. (2005). There issome evidence, both from the study of FU Ori objects(Hartmann et al. 2004) and global MHD simulations ofstratified disks (e.g. Fromang & Nelson 2006), that theturbulent linewidth may be larger at several scale heightsabove the midplane of the disk, perhaps up to 40% ofthe sound speed. While these estimates affect the localvalue of α, it should also be noted that there are globaldisk features that may also affect the measured turbulentlinewidth. In the dense inner disk, dead zones may formnear the midplane where the ionization fraction is toolow to support the MRI (e.g. Sano et al. 2000). However,these should not be relevant for our observations since thepredicted extent of a dead zone in these disks is interiorto a radius of 5-25AU for TW Hya and 10-63AU for HD163296, depending on the assumed grain properties (N.Turner 2010, private communication); these size scalesare significantly smaller than the linear scale probed byour observations.In this context, HD 163296 seems to fit in fairly seam-

lessly, with a turbulent linewidth of 300m s−1 corre-sponding to about 40% of the sound speed at the sizescales probed by our data (if 1.′′5 corresponds to ∼180AUat a distance of 122pc). If the turbulent linewidth dropsby a factor of a few between the upper layers probedby the CO(3-2) line and the midplane (as predicted by,e.g., Fromang & Nelson 2006), this implies a turbulentlinewidth of ∼0.1 cs near the midplane, consistent withα ∼ 0.01. The lower turbulent linewidth in TW Hya,.10% of the local sound speed at the scale of our ob-servations (80AU at 51pc), could be associated with alower global value of α, although the reason for such adifference is unclear. Following Andrews et al. (2009),the best-fit similarity solution model parameters imply α= 0.03 for TW Hya and 0.009 for HD 163296, althoughthese numbers are quite uncertain given the reliance onthe optically thick CO(3-2) tracer for the determinationof density. The D’Alessio et al. models use α = 0.0018for TW Hya and 0.003 for HD 163296. Perhaps the onlyconclusion that should be drawn from this estimate isthat our observations are consistent with estimates thatplace α in the range of 10−2-10−3. With a sample of onlytwo sources and with a wide range of theoretical resultsand approaches to the study of turbulence, it can be dif-ficult to compare our results to theoretical predictions ina detailed way. Nevertheless, the detection of a turbu-lent linewidth in the HD 163296 system and the upperlimit on turbulence in the TW Hya system are sugges-tive. There are several potential explanations for thisdifference rooted in theoretical studies of protoplanetarydisk turbulence.Inclination and Vertical Structure — In general, the

CO(3-2) line emission from these systems is expected tobe optically thick, and therefore will arise from the tenu-ous upper disk layers several scale heights above the mid-plane. However, this simple picture can be complicatedby geometry and velocity: the combination of inclinationand rotational line broadening in HD 163296 will permitthe escape of radiation from deeper layers of the disk. Asa result, a different vertical height may be probed in theHD 163296 system than in TW Hya. Naively, it would beexpected that the turbulent linewidth in TW Hya shouldbe larger as a result, since turbulence is predicted to be-come stronger farther from the midplane. It is difficult

to calculate accurately the height above the midplanefrom which the CO(3-2) emission originates, both be-cause it varies by position across the disk and velocity off-set across the line, and because it depends on very poorlyconstrained quantities like the vertical gradients in gastemperature and density. Nevertheless, a rough estimateat line center for the D’Alessio et al. models places theorigin of the emission at approximately 2 scale heightsabove the midplane for TW Hya and 3 for HD 163296 atthe radii of 80 and 180 AU corresponding to the respec-tive linear resolution of the observations for each source.This suggests that the difference in turbulent linewidthfor the two sources could be due to the different physi-cal regions probed by the line. But given the uncertain-ties, the height of formation is not constrained stringentlyenough to definitively demonstrate that this is the case.It is also possible that poor coupling of ions to neutrals inthe low-density uppermost disk layers could inhibit thedetection of turbulence even if it is present. This mayalso depend on the relative amounts of small dust grainsin the upper layers of the two disks, since small grainsare more adept at absorbing free electrons.Stellar Mass and Ionizing Flux — One of the factors

determining the extent of turbulent regions in circum-stellar disks is the magnitude of ionizing X-ray flux fromthe star (e.g., Gammie 1996; Igea & Glassgold 1999).With stellar masses differing by a factor of 4, the radi-ation (and therefore ionization) properties of TW Hyaand HD 163296 are likely to be quite different. Despitethe tendency for X-ray flux to be lower for intermediate-mass than low-mass stars of comparable ages, HD 163296has the largest X-ray flux of the sample of 13 Herbig Aestars studied by Hubrig et al. (2009), comparable to thatof lower-mass T Tauri stars. Nevertheless, its measuredX-ray luminosity of 4.0 × 1029 erg s−1 is still lower thanthat of TW Hya, which is estimated as 2.0× 1030 erg s−1

by Kastner et al. (1999). Given the comparable diskdensities, by X-ray luminosity alone, TW Hya should bethe more active disk. However, since we are observingthese disks on relatively large spatial scales (∼80AU forTW Hya and ∼180AU for HD 163296, with 1.′′5 reso-lution viewed from their respective 51 and 122pc dis-tances), the ionization at these distances may instead bedominated by cosmic rays. It is extremely difficult todetermine how the cosmic ray environment of these twosources might compare; if HD 163296 were located in aregion of greater cosmic ray activity, that could accountfor the greater turbulent linewidth observed for this sys-tem.Evolutionary State — One of the most obvious differ-

ences between TW Hya and HD 163296 is their respec-tive evolutionary states. TW Hya is a 10Myr-old tran-sition disk with an inner cavity of ∼4AU radius, whileHD 163296 is a primordial disk with an inner radius con-sistent with the expected ∼0.4AU extent of the dust de-struction zone (see, e.g., Isella et al. 2007). It is possiblethat X-ray ionization and the MRI might operate differ-ently at different stages of evolution; one example is theinside-out MRI clearing proposed by Chiang & Murray-Clay (2007) to explain the cavities in transition disks. Intheir scenario, the mass accretion rate onto the star canbe explained entirely by the MRI operating on the diskinner rim, and requires no resupply from the outer disk;they therefore require little to no turbulent viscosity in

12

the outer disk to explain the observed accretion ratesin transitional systems. However, they note that theirtheory cannot be readily applied to primordial systems,leaving the viscous transport mechanism responsible forlarge accretion rates at earlier stages unexplained. Thereis no reason to expect MRI turbulence in the outer diskto “shut off” when a gap is opened, so while our observa-tion of a small turbulent linewidth in the TWHya systemis consistent with the Chiang & Murray-Clay (2007) hy-pothesis, it is still surprising that the turbulent linewidthin HD 163296 should be so much larger. Another pos-sibility unrelated to the MRI is that HD 163296 is stillexperiencing infall onto the disk from a remnant enve-lope. Since the high optical depth of the CO(3-2) lineimplies that most of the emission arises from the upperlayers of the disk, such a scenario could mimic the signa-ture of a turbulent linewidth that we observe; however,there is no observational evidence for an envelope in thissystem. We plan to address this possibility more thor-oughly in a follow-up paper modeling multiple lines withlower optical depths that probe deeper towards the diskmidplane.

5.2. Implications for Planet Formation

The presence of subsonic turbulence in protoplanetaryaccretion disks – likely substantially subsonic in the mid-plane – is consistent with the observations presented inthis study. Subsonic turbulence has important implica-tions for the formation and evolution of young plane-tary systems. One series of papers (Papaloizou & Nel-son 2003; Papaloizou et al. 2004; Nelson & Papaloizou2003, 2004) explores in detail the effects of turbulenceon planet-forming disks. Their cylindrical models of tur-bulent disks have an average α in the range of 10−2-10−3,but they demonstrate that the realistic implementationof turbulence results in different effects than are seenin laminar disk simulations with comparable values ofα incorporated as an anomalous Navier-Stokes viscos-ity. They show that for massive planets, turbulence canwiden and deepen the gap opened by massive protoplan-ets, and may reduce the accretion rate onto the proto-planet. For the case of migrating low-mass planet cores,the presence of turbulence in the disk can slow or evenreverse the migration rate, converting the monotonic in-ward motion of the planet into a random walk. The pres-ence of dead zones in the radial direction may also actto halt migration and encourage the survival and growthof protoplanets (e.g. Matsumura et al. 2009). Anotherimportant proposed effect of subsonic turbulence is toaid in concentrating planetesimals to allow them to col-lapse gravitationally (Johansen et al. 2007). MHD tur-bulence on these scales can also reduce the strength ofthe gravitational instability and reduce disk fragmenta-tion (Fromang 2005). There is also substantial literatureon the effects of turbulence on dust settling and graingrowth (e.g. Johansen & Klahr 2005; Carballido et al.2006; Ciesla 2007; Balsara et al. 2009; Fromang & Nel-son 2009).Although it is difficult to compare the properties of the

simulations directly with our observations, the genericfeatures of these models (α=10−2-10−3, subsonic turbu-lence even in the upper disk layers) are globally consis-tent with the derived properties of turbulence in the disksaround HD 163296 and TW Hya, indicating that these

effects are likely to play a role in planet formation.

5.3. Future Directions

The most obvious improvement to our method wouldbe to include additional spectral lines from different tran-sitions or isotopologues of the CO molecule in order toprovide independent constraints on the gas temperature.While this would necessarily introduce additional param-eters into the model (i.e., to describe the vertical distri-bution of temperature and turbulence, as well as a consis-tent density distribution to properly account for the lineopacity), the addition of several lines that are resolvedin the spectral and spatial domains would more firmlyconstrain the models. It might also provide direct mea-surements of the vertical profile of the turbulent velocitystructure. Dartois et al. (2003) and Panic et al. (2008)provide examples of studies that use multiple molecularlines to study the vertical structure of density and tem-perature in circumstellar disks; these techniques could beextended to constrain the turbulent linewidths in similarsystems.Another possibility is to observe ions rather than neu-

tral species. This would eliminate complications intro-duced by the interaction between ions and neutrals, andwould more directly probe the turbulent motions of thecharged gas.Even with the current set of observations, greater

sensitivity would be extremely valuable in constrainingthe turbulent linewidth, since the distinctions betweenturbulent and thermal broadening are subtle (see Sec-tion 4.4). The vast improvements in sensitivity providedby the Atacama Large Millimeter Array will permit sig-nificantly better modeling of the velocity structure ofyoung disks. Such data will also allow us to firmly ruleout deviations from perfect Keplerian rotation that couldcomplicate the derivation of turbulent linewidth. In ad-dition, higher sensitivity combined with a greater spatialdynamic range will allow for the investigation of radialvariations in the turbulent linewidth.

6. SUMMARY AND CONCLUSIONS

We have obtained the first spatially resolved observa-tions of molecular line emission from two nearby circum-stellar disks with spectral resolution finer than the ex-pected turbulent linewidth. We fit these high spectralresolution observations of the CO(3-2) line emission us-ing two well-tested models of circumstellar disk struc-ture, and derive a turbulent linewidth of ∼300m s−1 forthe disk around HD 163296 and .40m s−1 for the diskaround TW Hya. The results are consistent for the twomodel classes despite their different treatments of thetemperature structure of the disk, which is significantsince thermal broadening is closely related to turbulentbroadening. The magnitude of turbulent velocity fluctu-ations implied by these results – up to tens of percent ofthe sound speed – is broadly consistent with theoreticalpredictions for MRI turbulence in the upper layers of cir-cumstellar disks, although it is unclear why the linewidthshould be lower for TW Hya than for HD 163296.These results demonstrate the potential of this method

for constraining theories of the magnitude and spa-tial distribution of turbulence. Future observationswith greater sensitivity, perhaps incorporating differentmolecular line species and isotopologues of the same

13

Fig. 8.— Channel maps of the CO(3-2) line emission from the disk around HD 163296. The LSR velocity is indicated in the upper rightof each channel, while the synthesized beam size and orientation (2.′′0×1.′′7 at a position angle of 41◦) is indicated in the lower left panel.

The contour levels start at 2σ and increase by factors of√2, where σ is the rms noise of 0.6 Jy beam−1. The star symbol indicates the disk

center while the dark solid line indicates the disk position angle as determined by CO fitting in Section 4.

molecule, have the potential to vastly improve our abilityto characterize turbulence in these systems.

The authors thank Shin-Yi Lin for collaboration on col-lection of the extended TW Hya observations. We alsothank Ramesh Narayan for helpful conversations aboutcomparing the data with theoretical simulations, as well

as Shane Davis for providing several quantities that werenot tabulated in his paper. Partial support for this workwas provided by NASA Origins of Solar Systems Pro-gram Grant NAG5-11777. A. M. H. acknowledges sup-port from a National Science Foundation Graduate Re-search Fellowship.

APPENDIX

A. CHANNEL MAPS

Figures 8 and 9 show the full channel maps for the high spectral resolution observations of the CO(3-2) emissionfrom the disks around TW Hya and HD 163296. The line overlaid on the emission indicates the disk major axis. TheTW Hya and HD 163296 maps have been imaged with Gaussian tapers of 1.′′2 and 1.′′0, respectively, to bring out thelarge-scale emission.

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Fig. 9.— Same as Figure 8 above, but for TW Hya. The beam size is 1.′′9×1.′′4 at a position angle of 21◦ and the contour levels are thesame as in Figure 8.