C: - Isella2010p9983Astrophys

The Astrophysical Journal

, 714:1746–1761, 2010 May 10

doi:

10.1088/0004-637X/714/2/1746

INVESTIGATING PLANET FORMATION IN CIRCUMSTELLAR DISKS: CARMA OBSERVATIONS OF

RY Tau AND DG Tau

Andrea Isella, John M. Carpenter, and Anneila I. Sargent

Division of Physics, Mathematics and Astronomy, California Institute of Technology, MC 249-17, Pasadena, CA 91125, USA;

isella@astro.caltech.edu

Received 2009 August 12; accepted 2010 March 22; published 2010 April 22

ABSTRACT

We present CARMA observations of the thermal dust emission from the circumstellar disks around the young stars

RY Tau and DG Tau at wavelengths of 1.3 mm and 2.8 mm. The angular resolution of the maps is as high as

15, or 20 AU at the distance of the Taurus cloud, which is a factor of 2 higher than has been achieved to date

at these wavelengths. The unprecedented detail of the resulting disk images enables us to address three important

questions related to the formation of planets. (1) What is the radial distribution of the circumstellar dust? (2)

Does the dust emission show any indication of gaps that might signify the presence of (proto-)planets? (3) Do the

dust properties depend on the orbital radius? We find that modeling the disk surface density in terms of either a

classical power law or the similarity solution for viscous disk evolution reproduces the observations well. Both

models constrain the surface density between 15 and 50 AU to within 30% for a given dust opacity. Outside this

range, the densities inferred from the two models differ by almost an order of magnitude. The 1.3 mm image

from RY Tau shows two peaks separated by 0

2 with a decline in the dust emission toward the stellar position,

which is significant at about 2

–4

. For both RY Tau and DG Tau, the dust emission at radii larger than 15 AU

displays no significant deviation from an unperturbed viscous disk model. In particular, no radial gaps in the dust

distribution are detected. Under reasonable assumptions, we exclude the presence of planets more massive than 5

orbiting either star at distances between about 10 and 60 AU, unless such a planet is so young that there has

been insufficient time to open a gap in the disk surface density. The radial variation of the dust opacity slope,

was investigated by comparing the 1.3 mm and 2.8 mm observations. We find mean values of

of 0.5 and 0.7 for

DG Tau and RY Tau, respectively. Variations in

are smaller than

7 between 20 and 70 AU. These results

confirm that the circumstellar dust throughout these disks differs significantly from dust in the interstellar medium.

Key words:

accretion, accretion disks – stars: pre-main sequence – stars: variables: T Tauri, Herbig Ae

Be –

submillimeter: planetary systems – techniques: interferometric

Online-only material:

color figures

1. INTRODUCTION

Resolved images of circumstellar disks around young stars

provide the most direct tool for investigating the formation of

planets. At millimeter wavelengths, the thermal dust emission

is generally optically thin and measures the radial distribution

of circumstellar dust (Beckwith et al.

1990

). However, since

circumstellar disks in nearby star-forming regions typically

have radii between 100 and 500 AU, sub-arcsecond angular

resolution is required to spatially resolve the dust emission, even

in nearby star-forming clouds. Millimeter-wave interferometers

are essential for such studies.

Since sub-arcsecond observations at millimeter wavelengths

require both high sensitivity and high dynamical range, only a

small number of bright disks have been observed at resolutions

of 0

4–1

to date (Brown et al.

2008

; Guilloteau et al.

1999

;

Isella et al.

2007

;Pi

etu et al.

2005

2006

2007

; Simon et al.

2000

; Testi et al.

2003

; Wilner et al.

2000

). The Combined

Array for Research in Millimeter-wave Astronomy (CARMA)

and the new extended configuration of the Sub-millimeter Array

are rapidly enabling more extensive high-resolution surveys of

circumstellar disks, particularly in the Taurus and Ophiuchus

star-forming regions (Andrews et al.

2009

; Hughes et al.

2009

;

Isella et al.

2009

, hereafter Paper I).

The highest angular resolution achieved so far by millimeter-

wave interferometers is 0

3–0

4, corresponding to spatial scales

of 40–50 AU at the distance of Taurus and Ophiuchus. In most

cases, the dust density appears to increase smoothly inward

down to the orbital radius resolved by the observations, typically

∼

25 AU. However, central cavities in the dust distribution are

revealed in a number of disks (Andrews et al.

2009

; Hughes

et al.

2009

). It remains a matter of debate whether these cavities

are caused by dynamical interactions, inside-out disk dispersal

mechanisms, dust opacity variations, or viscous evolution (e.g.,

Alexander et al.

2006

;Calvetetal.

2005

; Chiang & Murray-Clay

2007

; Dullemond & Dominik

2005

; Paper I).

Nevertheless, these observations still lack the angular reso-

lution required to resolve the innermost part of the disk where

the density of the circumstellar material is highest and the for-

mation of planets is more probable. Here we describe CARMA

observations of the thermal dust emission toward the young

stars DG Tau and RY Tau at an angular resolution of 0

15 at

1.3 mm and 0

3 at 2.8 mm. At the distance of Taurus (140 pc),

15 corresponds to spatial scales of 20 AU, such that emission

on orbital scales comparable to Saturn can be resolved. This is

more than a factor of 2 improvement over previous observations

of circumstellar disks at these wavelengths.

DG Tau and RY Tau are classical T Tauri stars of spectral

type M0 and K1, respectively (Muzerolle et al.

1998

; Kenyon &

Hartmann

1995

). Stellar ages inferred from stellar evolutionary

models are less than 1 Myr (see Paper I for more details and

references). The relative youth of both systems is confirmed by

the presence of large amounts of gas and dust extending to 0.1 pc

and by associated stellar jets and outflows (see, e.g., McGroarty

&Ray

2004

; St-Onge & Bastien

2008

). From near-infrared to

millimeter wavelengths, both objects exhibit strong emission in

1746

No. 2, 2010

INVESTIGATING PLANET FORMATION IN CIRCUMSTELLAR DISKS

1747

Table 1

Summary of CARMA Continuum Observations

Object

Date

Array

Baseline

Beam

Flux

Seeing

Noise

(UTC)

Configuration

Range (m)

FWHM (

), PA (

◦

)

(mJy)

(

)

(mJy beam

−

)

Observations at 1.3 mm

DG Tau

A+B+C

0.17

0.15, 103

367

0.03

0.96

···

2007 Oct 8

21–279

1.04

0.81, 112

···

2007 Dec 14

81–937

0.33

0.29, 116

···

2009 Jan 31

130–1884

0.15

0.13, 102

RY Tau

A+B+C

0.17

0.14, 81

227

0.90

···

2007 Oct 22

16–280

1.24

0.78, 102

···

2008 Dec 30

82–935

0.40

0.31, 109

···

2009 Jan 19

139–1884

0.15

0.13, 82

Observation at 2.7 mm

DG Tau

A+B

0.45

0.38, 131

0.07

0.45

···

2008 Jan 14

80–798

1.12

0.60, 125

···

2009 Feb 5

136–1678

0.34

0.32, 163

RY Tau

A+B

0.36

0.30, 82

0.07

0.28

···

2008 Feb 1

82–945

0.77

0.59, 95

···

2009 Feb 10

123–1884

0.35

0.29, 82

excess of that from the stellar photospheres. This is attributed to

rotating disks with radii of few hundred AU that first absorb and

then re-emit radiation from the central stars (Koerner & Sargent

1995

; Testi et al.

2002

). Our earlier CARMA observations of

1.3 mm thermal dust emission from these disks, at a resolution of

7, suggested disk masses between 5% and 150% of the stellar

mass for both sources (see Paper I). These high disk masses

and the youth of RY Tau and DG Tau make these prime targets

to investigate the earliest stages of planet formation. Our new

observations of RY Tau and DG Tau have a factor of 5 better

angular resolution and a factor of 3 better sensitivity than the

previous data.

This paper investigates three main questions related to the

formation of planets in young circumstellar disks. (1) What is

the surface density distribution in the observed disks down to

an orbital radius of 10 AU? (2) Are there any signatures of

planet formation contained in the dust distribution? Finally, (3),

do the dust properties vary with orbital radius? A qualitative

answer to the first two questions is proposed in Section

where

we present the observations and discuss the morphology of the

dust emission. A quantitative analysis is described in Section

where we compare the observations with theoretical models of

disk emission. Implications of these results for disk structure,

for the possible presence of planets, and for the radial variation

of the dust opacity are considered in Section

. The conclusions

are presented in Section

2. OBSERVATIONS AND DATA REDUCTION

We observed thermal dust emission from the RY Tau and

DG Tau circumstellar disks using CARMA in the A, B, and

C configurations. The date of observation, array configurations

used, baseline range, sizes and orientations of the synthesized

beams, integrated fluxes, seeing, and noise levels are summa-

rized in Table

. The C-configuration observations were pre-

sented in Paper I.

The observations were obtained at local oscillator frequencies

of 228.1 GHz (

3 mm) and 106.2 GHz (

8 mm). The

CARMA correlator at the time of the observations contained

three bands, each of which was configured to 468 MHz band-

width to provide maximum continuum sensitivity. The bandpass

shape was calibrated by observing 3C273; flux calibration was

set by observing Uranus and 3C84. The radio galaxy 3C111

was observed every 9 minutes to correct for atmospheric and

instrumental effects. Variations of the atmospheric conditions

on timescales shorter than 9 minutes are not corrected, in effect

resulting in seeing. We quantified the atmospheric seeing by

measuring the size of the phase calibrator image; if the seeing is

negligible, the phase calibrator appears as a point source. Oth-

erwise, the seeing produces a Gaussian smoothing that can be

quantified through the full width at half-maximum (FWHM) of

the resulting image. We find that at 1.3 mm the effect of seeing

is negligible for RY Tau but produces an FWHM of 0

03 for

DG Tau. Atmospheric conditions were slightly worse during the

2.8 mm observations, resulting in seeing of 0

07 for both ob-

jects. These seeing estimates do not account for variations in the

atmospheric conditions on angular scales of 10

◦

, corresponding

to the separation between the source and the calibrator. Values

for the atmospheric seeing are summarized in Table

and are

adopted in the model fitting described in Section

The raw data were reduced using the MIRIAD software

package. The maps of the continuum emission shown in Figure

were derived using GILDAS software. Corresponding complex

visibilities are shown in Figure

. At 1.3 mm, natural weighting

of the A, B, and C configuration observations produced an

FWHM synthesized beam size of

∼

15. The noise levels

are 0.96 mJy beam

−

and 0.90 mJy beam

−

, respectively, for

DG Tau and RY Tau. Dust emission at 2.8 mm was observed

in the A and B configurations at angular resolution of

∼

and noise levels of 0.45 mJy beam

−

and 0.28 mJy beam

−

for

DG Tau and RY Tau, respectively.

3. MORPHOLOGY OF THE DUST EMISSION

In Figure

, the dust emission in both disks is clearly resolved

and characterized by a smooth and centrally symmetric radial

profile. DG Tau intensity contours are almost circular suggesting

a disk inclination smaller than 30

◦

. For RY Tau, the intensity

contours are elongated in the northeast direction suggesting a

disk position angle of about 24

◦

measured east from north and

a disk inclination of at least 65

◦

. For both sources, the disk

orientations agree with those found in Paper I.

1748

ISELLA, CARPENTER, & SARGENT

Vol. 714

Figure 1.

Maps of the dust thermal emission observed at a wavelength of 1.3 mm (upper panels) and 2.8 mm (lower panels) toward RY Tau (left panels) and DG Tau

(right panels). The color scale shows the surface brightness starting from the 3

level, with contours plotted every 4

.The1

noise level and the size of the synthesized

beam are given in Table

. The inset in the upper left panel shows the central 0

4 region of the RY Tau disk where contours start at 28

with increments of 1

The surface brightness is characterized by two peaks separated by

∼

(A color version of this figure is available in the online journal.)

3.1. RY Tau Disk Morphology

The 1.3 mm dust continuum emission from the RY Tau disk

shows two spatially resolved peaks separated by about 0

(28 AU), and oriented along the apparent major axis of the

disk. Details of the central 0

4 region are displayed in the

inset in the upper left panel of Figure

, and the radial profile

of the surface density along the disk major axis is shown in

Figure

. The intensity at both peaks is 29 mJy beam

−

, which is

2 mJy beam

−

(i.e., 2.2

) higher than the intensity at the

center of the disk. We also estimated the expected central

surface brightness by fitting a Gaussian to the surface brightness

distribution at angular distances larger than 0

15. The fitted

Gaussian is shown as the solid curve in Figure

. A Gaussian

function was chosen since it provides a reasonable parametric

representation of the dust emission. Interpolating this Gaussian

fit to the center of the disk suggests an expected central surface

brightness of 31 mJy beam

−

, which is 4

higher than the

measured value. The significance level of the two intensity

peaks, the fact that they appear in the map before cleaning,

their orientation along the disk major axis, and the symmetry

with respect to the central star, suggests that they are real and,

therefore, that the dust emission decreases inside an orbital

radius of about 14 AU. This is analogous to the situation in

“transitional” disks, where the inner gaps observed in the dust

emission are attributed to dusty depleted inner regions (see, e.g.,

Hughes et al.

2009

; Brown et al.

2008

2009

At a first sight, this interpretation is incompatible with

RY Tau’s large near- and mid-infrared excesses, which suggest

the presence of warm dust within 10 AU of the star (Robitaille

et al.

2007

). If, however, the inner disk is only partially depleted

and dust emission remains optically thick in the infrared,

the observed double intensity peak and the spectral energy

distribution (SED) can be reconciled. A number of physical

mechanisms could reduce the dust density in the inner region

of circumstellar disks. For example, planets less massive than

Jupiter may carve partially depleted gaps in the surface density

distribution by tidal interaction with the surrounding material

(Bryden et al.

1999

). This possibility is discussed in more detail

in Section

5.3

. In Paper I, we also proposed that a surface density

profile that gradually decreases toward the star may originate

naturally from the viscous evolution of a disk if viscosity

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INVESTIGATING PLANET FORMATION IN CIRCUMSTELLAR DISKS

1749

0.1

0.2

RY Tau - 1.3 mm

-0.05

0.05

300

600

900

1200

1500

0.1

0.2

0.3

0.4

DG Tau - 1.3 mm

-0.05

0.05

300

600

900

1200

1500

0.02

0.04

RY Tau - 2.8 mm

Real (Jy)

-0.01

0.01

200

400

600

)

Imag (Jy)

0.02

0.04

0.06

DG Tau - 2.8 mm

-0.01

0.01

200

400

600

Figure 2.

Correlated flux measured at 1.3 mm and 2.8 mm toward RY Tau (left panels) and DG Tau (right panels) as a function of the baseline length

expressed

in k

(A color version of this figure is available in the online journal.)

decreases with radius. Finally, it is also possible that the decrease

in dust emission may originate from a lowering of opacity due

to the growth of dust grains to centimeter sizes (Dullemond

& Dominik

2005

). Unambiguously disentangling these models

requires even higher angular resolution observations than are

yet available.

We must note that radial velocity studies (Herbig & Bell

1988

) and

Hipparcos

observations of the variability of the pho-

tocenter (Bertout et al.

1999

) suggest that RY Tau is a binary.

The

Hipparcos

data imply a minimum projected separation of

23.6 mas and a position angle of 304

◦

, almost perpendic-

ular to the position angle of the disk inferred from our obser-

vations. Assuming that the binary and the disk have the same

inclination, the spatial separation between the binary compo-

nents is 6–9 AU, and could explain the double peak in the

dust continuum emission. Indeed, the presence of a stellar mass

companion orbiting at a radius of 6–9 AU would push the inner

radius of the circumstellar disk to a distance of 9–13 AU by

tidal interactions (Wolf et al.

2007

). However, the binary na-

ture of RY Tau has been rendered questionable by near-infrared

interferometric observations that suggest an inner disk radius

at 0.1 AU from the central star and exclude the presence of

a stellar mass companion between 0.35 AU and 4 AU down

to a stellar flux ratio of 0.05 (Akeson et al.

2005

;Pottetal.

2010

). A stellar companion was also undetected in recent spec-

troscopic and aperture masking observations (D. C. Nguyen &

A. Kraus 2009, private communication). As discussed above,

the SED is also inconsistent with the existence of a large inner

gap completely depleted of gas and dust as could be expected

for a stellar companion (Robitaille et al.

2007

). These results

suggest that RY Tau is indeed a single star, and the variability

observed by

Hipparcos

and the radial velocity variations may

be attributed to brightness changes in the circumstellar environ-

ment (see the discussion in Schegerer et al.

2008

, and references

therein).

A notable characteristic of our images of the dust emission

is the high degree of central symmetry and, with the exception

of the innermost region, the almost complete absence of fea-

tures in the surface brightness distribution. If the emission is

optically thin (we will examine this assumption is Section

5.3

this translates to a smooth radial profile for the dust. The degree

of symmetry of the emission can be quantified by analyzing

the imaginary part of the correlated flux, plotted in Figure

as a function of the angular frequency

. Point-symmetric

emission will have a zero imaginary part at all spatial fre-

quencies. For RY Tau, the deviations from zero are compa-

rable to the noise in the observations (see the left panels of

Figure

3.2. DG Tau Disk Morphology

The surface brightness distribution for the DG Tau disk has

a central peak coincident with the stellar position and smoothly

decreases outward to reach the noise level at an angular distance

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ISELLA, CARPENTER, & SARGENT

Vol. 714

-1

-0.5

0.5

Surface brightness (mJy/ beam)

dR (arcsec)

Figure 3.

Radial profile of the 1.3 mm surface brightness in the RY Tau disk

measured along the major axis of the disk at a position angle of 24

◦

east from

north. The intensity error bars (red points) correspond to the noise level of

0.9 mJy beam

−

. The solid curve shows a Gaussian fit, while the dashed line

indicates the zero intensity level. The bracket in the upper left indicates the

angular resolution.

(A color version of this figure is available in the online journal.)

of about 0

5. At both 1.3 and 2.8 mm the emission appears fairly

symmetric, and indeed the imaginary part of the correlated flux

is zero for

300 k

(see the upper right panel of Figure

The imaginary part increases up to 50 mJy at shorter spatial

frequencies, suggesting that the emission may be asymmetric

on angular scales larger than 1

∼

7. Comparing the real

and imaginary parts of the correlated flux at the shortest spatial

frequencies, we find that the asymmetric part of the emission

contributes about 14% to the total flux.

As already noted, the high angular resolution observations of

RY Tau and DG Tau allow us to examine disk properties that

bear on planet formation. In particular, the radial density profile

of the circumstellar material is critical to understanding where

planet formation may occur, or where it has occurred. With

observations at more than one wavelength, we may also consider

radial variations of the grain properties. As we describe below,

measurements of the dust radial profile and the variation of the

dust properties with radius are best undertaken by comparing the

observations with theoretical disk models in the Fourier domain.

There, the effects of different angular resolutions, sensitivity,

and atmospheric seeing can be more easily taken into account.

Moreover, comparison with theoretical models is the only way to

quantify the contribution from optically thick emission, leading

to an improved estimate of the dust mass.

4. DISK AND DUST MODELS

To investigate the dust radial distribution around DG Tau

and RY Tau, we consider two different models for the disk

surface density. The first model consists of the classical power-

law parameterization

(

)

(

40 AU

)

−

for

<R<R

out

(1)

where

is the disk surface density at a radius of 40 AU.

and

out

are the inner and outer disk radii, respectively. The second

model is the similarity solution for the evolution of a viscous

Keplerian disk (Lynden-Bell & Pringle

1974

; Hartmann et al.

1998

). As discussed in Paper I, this has the form

(

R, t

)

(

)

exp

{

−

2(2

−

)

[

(

)

−

)

−

]}

(2)

where

is the surface density at radius

, sometimes called

the

transition radius

.For

, the surface density has a

power-law profile characterized by the slope

, while at larger

radii the surface density falls exponentially.

These two different parameterizations are used to calculate

the dust emission by solving the structure of an hydrostatic

disk heated by the stellar radiation (Dullemond et al.

2001

). The

temperature on the disk mid-plane is self-consistently calculated

by adopting the

two-layer

approximation of Chiang & Goldreich

(

1997

). The disk temperature, which depends mainly on the disk

optical depth at optical and infrared wavelengths, is roughly

proportional to

−

for both surface density profiles (see Paper

I and references therein for a detailed discussion on the disk

temperature radial profile).

Fundamental to any disk model is the prescription adopted

for the dust opacity. Although the dust contributes only about

1% to the total disk mass, it dominates the disk opacity in the

wavelength range where most of the stellar and disk radiation

is emitted. We assume that the dust size distribution follows

apowerlaw

(

)

∝

−

, where

is the radius of a dust

grain. The assumptions on the slope

, on the minimum and

maximum grain sizes, on the dust chemical composition and on

the grain structure define the frequency dependence of the dust

opacity

, and, ultimately, the disk emission. The dust opacity

is calculated for compact spheres composed of astronomical

silicates and organic carbonates (Weingartner & Draine

2001

;

Zubko et al.

1996

). We assume a mass ratio of 1 between silicates

and organics, which leads to grain density of 2.5 g cm

−

The dust opacity averaged over the grain size distribution is

calculated by fixing the minimum grain size to 0.005

m. The

maximum grain size

max

and the slope

are set to reproduce

the observed slope of the SED as discussed below.

At millimeter wavelengths, the dust opacity can be approxi-

matedbyapowerlaw

(

ν/ν

)

(Beckwith & Sargent

1991

). If the dust emission is optically thin and the

Rayleigh–Jeans approximation is satisfied, the slope

of the

dust opacity is related to the spectral index

of the observed

disk emission

(

∝

) by the relation

. This re-

lation is only approximate if the dust emission is optically thick

at some radii. In Paper I, we derived values for

of 0.5 and 0.7

for DG Tau and RY Tau, respectively, from an analysis of the

SED, taking into account the optically thick contribution to the

total dust emission. For the assumed dust composition and struc-

ture, these values of

can be reproduced with different choices

of the maximum grain size

max

and the grain size slope

(see the

Appendix

). To investigate how the assumptions on the

grain size distribution affect the model fitting, we adopt two

different dust models that correspond to the extreme cases of

low (

) and high (

) opacity. The corresponding dust opacities

at both 1.3 mm and 2.8 mm are given in Table

Finally, we assume that the dust opacity is constant through-

out the disk. This is indeed one of the main assumptions we

want to test by modeling the observed dust emission at 1.3 mm

and 2.8 mm and will be discussed in detail in Section

5.4

5. RESULTS AND DISCUSSION

Models and observations are compared in Fourier space to

avoid the nonlinear effects introduced by the cleaning process.

The best-fit models are found by

minimization with five free

No. 2, 2010

INVESTIGATING PLANET FORMATION IN CIRCUMSTELLAR DISKS

1751

Table 2

Properties of the Adopted Dust Models

Object

max

3mm

8mm

(cm)

(cm

−

)(cm

−

)

High dust opacity model,

DG Tau

0.075

0.082

0.056

RY Tau

0.035

0.131

0.075

Low dust opacity model,

DG Tau

5.0

3.5

0.012

0.0078

RY Tau

5.0

3.7

0.026

0.015

parameters: the disk inclination

, the disk position angle PA,

out

, and

for the power-law surface density (Equation (

)),

and

,PA,

, and

for the similarity solution (Equation (

)).

The disk inner radius

is fixed at 0.1 AU. For both surface

density models, we find best-fit solutions for both the high (

)

and low (

) dust opacity models. The 1.3 mm and 2.8 mm data

are fitted independently.

To minimize

and evaluate the constraints on the model

parameters, we use a Bayesian approach that adopts uniform

prior probability distributions. In practice, we sample the

probability distribution by varying the free parameters using the

Markov Chain Monte Carlo method described in Paper I.

Once a best-fit solution is found, we confirm that this indeed

corresponds to an absolute minimum of

, as opposed to a

local minimum, by running multiple Monte Carlo simulations

with random initializations and verifying that they all converge

to the same solution. Each parameter is allowed to vary in a

large range: 0

◦

–80

◦

for the inclination,

◦

for the position

angle, 10–1000 AU for

and

out

4for

and

, and 0

1–

1000 g cm

−

for

and

The best-fit disk models found for high and low dust opacities

are listed in Tables

and

, respectively. Each table lists the

parameters for the similarity solution disk model in the upper

part, and for the power-law disk model in the lower part. The

probability distributions for each free parameter are shown in

Figures

and

for RY Tau in the case of the similarity solution

and power law, respectively. The same quantities for DG Tau

are shown in Figures

and

. In these figures, the black and

red histograms indicate the probability distributions derived by

fitting the 1.3 mm and 2.8 mm observations, respectively; solid

and dashed curves represent the

and

dust opacity models. For

each parameter, we derive the uncertainty range that corresponds

to a likelihood of 99.7% (3

) by fitting a normal distribution to

the probabilities.

Finally, Figure

shows comparisons between the observed

real part of the correlated flux (filled squares with error bars),

the best-fit models for the similarity solution (solid curve), and

a power-law surface density (dashed curve).

5.1. Dependence on the Dust Opacity and Implications on the

Disk Masses

The best-fit solutions for the

and

dust opacity models

are shown in Figures

and

with solid and dashed curves,

respectively. In all cases,

and

models lead to very similar

values for the disk position angle, the disk inclination and the

radial profiles of the surface density defined by

and

out

in the

case of the power-law models, and

and

for the similarity

solution models. As discussed in Paper I, these parameters

are essentially independent of the dust opacity. This is mainly

because the disk mid-plane temperature

(

) varies by only a

few percent between the different dust models, as long as the disk

is optically thick to the stellar radiation. Since

(

)

∝

(

)

−

the radial profile of the surface density varies by only small

fraction when different dust models are assumed.

By contrast, the surface density normalization (

and

)

varies with the dust opacity so that the product

remains

almost constant if the emission is optically thin. Consequently,

a lower dust opacity requires a higher dust mass in order to emit

the same amount of radiation at millimeter wavelengths. The

ratio

is then approximately equal to the ratio between

the dust opacities listed in Table

From the analysis of the surface density of the best-fit model,

we find that the RY Tau emission is always optically thin at both

1.3 and 2.8 mm. However, DG Tau emission is optically thick

within 20 AU at 1.3 mm for both the similarity solution and the

power-law models. The 1.3 mm flux emitted within this region

is about 25% of the total flux. At 2.8 mm the emission is always

optically thin in the case of the similarity solution, while it is

optically thick within 6 AU in the power-law case. In this case,

the optically thick contribution is 5% of the total flux.

Different dust opacities lead to different values for the total

mass of dust in the disks. For DG Tau, we obtain total dust

masses of about 33 and 233 Earth masses (

⊕

) in the case

of the high and low opacity dust models, respectively. Disk

masses of

∼

10 and 50

⊕

are found for RY Tau. Massive disks

can also be obtained by extending the grain size distribution

larger than 5 cm. For example, in Paper I we derived total dust

Table 3

Best-fit Parameters Assuming the

Dust Model

Similarity solution

Object

(mm)

(

◦

)PA(

◦

)

(AU)

(g cm

−

)

DG Tau

1.3

119

23.4

1.8

0.33

0.15

10.9

1.5

1.0608

···

2.8

144

27.7

3.0

0.10

0.24

7.5

1.3

1.0629

RYTau

1.3

224

26.7

1.2

−

0.54

0.18

2.6

0.2

1.0896

···

2.8

620

26.5

2.7

−

0.08

0.54

2.6

0.5

1.1894

Power law

Object

(mm)

(

◦

)PA(

◦

)

out

(AU)

(g cm

−

)

DG Tau

1.3

120

72.6

6.3

1.00

0.15

5.6

1.5

1.0611

···

2.8

144

82.2

10.5

0.74

0.24

4.5

1.6

1.0629

RYTau

1.3

224

70.6

3.9

0.12

0.15

1.9

0.6

1.0897

···

2.8

620

76.9

12.0

0.64

0.45

1.6

1.0

1.1894

Note.

The uncertainties correspond to a likelihood of 99.7% (i.e., 3

) for the normal distributions shown in Figures

and

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0.2

62 64 66 68 70 72 74 76 78

Probability

Inclination (deg)

0.1

0.2

Position Angle (deg)

0.1

0.2

22 23 24 25 26 27 28 29 30

Transition Radius, R

(AU)

0.1

0.2

0.3

-1

-0.8 -0.6 -0.4 -0.2

0.2 0.4

0.1

0.2

0.3

2.5

10 11 12 13 14 15 16

) (g cm

-2

)

Figure 4.

Probability distribution of the disk parameters obtained by fitting the RY Tau observations at 1.3 mm and 2.8 mm with the similarity solution for the sur

face

density distribution. The results obtained by fitting the 1.3 mm data only are shown by the black curves, while the red curves indicate the results obtai

ned by fitting

the 2.8 mm only. Solid curves show the probability distribution obtained assuming the dust opacity model

, and dashed curves correspond to the dust opacity model

(see Table

(A color version of this figure is available in the online journal.)

Table 4

Best-fit Parameters Assuming the

Dust Model

Similarity solution

Object

(mm)

(

◦

)PA(

◦

)

(AU)

(g cm

−

)

DG Tau

1.3

119

22.5

1.8

0.28

0.15

74.4

9.9

1.0608

···

2.8

144

26.4

2.7

0.07

0.27

55.4

8.9

1.0629

RYTau

1.3

224

25.6

1.2

−

0.58

0.18

13.6

1.2

1.0896

···

2.8

620

25.1

2.4

−

0.10

0.57

14.3

2.3

1.1893

Power law

Object

(mm)

(

◦

)PA(

◦

)

out

(AU)

(g cm

−

)

DG Tau

1.3

120

72.3

4.0

1.06

0.18

35.7

3.6

1.0611

···

2.8

144

81.8

9.3

0.74

0.24

32.1

4.5

1.0629

RYTau

1.3

224

70.5

3.9

0.11

0.18

9.7

1.2

1.0897

···

2.8

520

76.7

12.6

0.68

0.51

8.3

2.3

1.1893

Note.

The uncertainties correspond to a likelihood of 99.7% (i.e., 3

) for the normal distributions shown in Figures

and

masses of about 1331 and 216

⊕

for DG Tau and RY Tau,

respectively, by assuming a maximum grain size of 10 cm and

a slightly different grain composition. Additional uncertainties

in the disk mass come from the dust chemical composition. As

discussed in the

Appendix

, the presence of ice or vacuum in the

grains leads to smaller dust opacities at millimeter wavelengths

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INVESTIGATING PLANET FORMATION IN CIRCUMSTELLAR DISKS

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0.1

0.2

0.3

Probability

Inclination (deg)

0.1

0.2

Position Angle (deg)

0.1

0.2

Outer Radius, R

out

(AU)

0.1

0.2

0.3

-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4

0.1

0.2

0.3

0.4

1 1.2 1.4 1.6 1.8 2 2.2

9 10 11

(40AU) (g cm

-2

)

Figure 5.

Probability distribution of the disk parameters obtained by fitting the RY Tau observations at 1.3 mm and 2.8 mm with a power-law surface density

distribution. Colors, solid curves, and dashed curves are the same as in Figure

(A color version of this figure is available in the online journal.)

and consequently produces higher disk masses. We therefore

estimate that the circumstellar disks around DG Tau and RY Tau

contain a minimum mass of dust of 30 and 10

⊕

, respectively,

while the upper limit is not constrained due to the uncertainties

on the grain size distribution. Assuming the standard dust

gas

ratio of 0.01, these values correspond to minimum disk masses

of 0.009 and 0.003

for the DG Tau and RY Tau, respectively.

5.2. Constraints on the Surface Density: Similarity Solution

versus Power Law

Figure

shows the comparison between models and observa-

tions in terms of the real part of the correlated flux as a function

of the baseline length. To correct for the disk inclination, we

deprojected the baseline assuming the inclinations and position

angles listed in Table

. In this figure, the results for

and

dust

models lead to indistinguishable curves. Similarity solution and

power-law models are represented with solid and dashed curves,

respectively, and the observations are shown by black dots with

error bars. It is clear that both the similarity solution and power-

law disk models provide satisfactory fits to the observations.

The similarity solution model provides smaller values of

(see Tables

and

) and, in the case of DG Tau, a better fit

to the observations between 400 and 800 k

. In this range of

spatial frequencies, the power-law solution is characterized by

a wiggle due to the sharp truncation of the dust emission at

72 AU. On the other hand, the exponential tapering of the sim-

ilarity solution leads to a smooth visibility profile that matches

extremely well the observations. The same behavior is present

in the lower panel which compares the model and the obser-

vations at 2.8 mm. However, in this case the observations at

400 k

are too to distinguish between the two models.

Although not conclusive, this result make the similarity solu-

tion model a more appealing explanation for the dust emission

in circumstellar disks, confirming the conclusions of Hughes

et al. (

2008

Figure

shows the surface density derived from the 1.3 mm

observations for both the power law and the similarity solution

model in the case of high dust opacity. The two models lead to

similar values of

(

) in the region where most of the 1.3 mm

flux is emitted, namely between

∼

15 and 50 AU. In this region,

the surface density in the RY Tau disk is almost constant with the

radius, while it decreases roughly as 1

in the case of DG Tau.

Inside 15 AU and outside 50 AU, the observations lack both

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0.2

5 10 15 20 25 30 35 40 45

Probability

Inclination (deg)

0.1

0.2

-90 -80 -70 -60 -50 -40 -30 -20 -10 0

Position Angle (deg)

0.1

0.2

20 21 22 23 24 25 26 27 28 29 30 31

Transition Radius, R

(AU)

0.1

0.2

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

0.1

8 10 12 14

) (g cm

-2

)

Figure 6.

Probability distribution of the disk parameters obtained by fitting the DG Tau observations at 1.3 mm and 2.8 mm with the similarity solution for the sur

face

density distribution. Colors, solid curves, and dashed curves are the same as in Figure

(A color version of this figure is available in the online journal.)

the angular resolution and the sensitivity required to directly

constrain the surface density. As a consequence, the values of

(

) strongly depend on the assumed model and can differ by

an order of magnitude at the disk inner radius.

5.3. Surface Density and Implication on the Existence of

Planets

In this section, we discuss the implications of the inferred

surface density on the presence of planets. The analysis is limited

to surface density profiles obtained by fitting the observations

at 1.3 mm, which have the highest angular resolution.

5.3.1. DG Tau

For the similarity solution model, the surface density has a

radial profile characterized by

∼

18 and

∼

2 AU. The transition radius

agrees well with our earlier

observations (21

3) but

is significantly larger than the value

−

0.5

0.6 from Paper I. The discrepancy is probably due

to the fact that the earlier observations were taken in poorer

weather conditions and the model fitting did not account for

the atmospheric seeing. Figure

shows the residuals after

subtracting the best-fit model to the new observations. Note

that the power-law model gives very similar residuals. The

residuals are as high as 3

–6

and are found at angular scales

larger than 0

7 where the emission is slightly asymmetric (see

Section

). In this outermost disk region, the surface density may

deviate significantly from the symmetric radial profile assumed

in the model. We calculate that variations of

10–30 g cm

−

with respect to the best-fit surface density profile over a spatial

region comparable with the beam size may produce the observed

residuals. Larger variations of the surface density on smaller

angular scales are also possible.

The residuals do not show global deviations from the smooth

surface density profile, apparently excluding the possibility of

gaps in the dust distribution that might be produced by a planet.

Of course, low-mass planets may not produce any discernible

gap and may still exist in the DG Tau disk. The formation

of a gap is possible only if the efficiency in removing the

material close to the planet orbital radius via tidal torques is

larger than the mass accretion rate due to the disk viscosity (see,

e.g., Lin & Papaloizou

1993

). If we assume the

prescription

for the disk viscosity (Shakura & Sunyaev

1973

), and call

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INVESTIGATING PLANET FORMATION IN CIRCUMSTELLAR DISKS

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0.1

0.2

Probability

Inclination (deg)

0.1

0.2

-90 -80 -70 -60 -50 -40 -30 -20 -10 0

Position Angle (deg)

0.1

0.2

95 100

Outer Radius, R

out

(AU)

0.1

0.2

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3

0.1

0.2

3.5 4 4.5 5 5.5 6 6.5

26 28 30 32 34 36 38 40

(40AU) (g cm

-2

)

Figure 7.

Probability distribution of the disk parameters obtained by fitting the DG Tau observations at 1.3 mm and 2.8 mm with a power-law surface density

distribution. Colors, solid curves, and dashed curves are the same as in Figure

(A color version of this figure is available in the online journal.)

the pressure scale height of the disk, a planet orbiting at radius

can open a radial gap in the disk surface density only if

(

h/R

)

(Lin & Papaloizou

1993

; Bryden et al.

1999

). Moreover, for a disk in hydrostatic equilibrium with

the gravitational field of the central star, the pressure scale

is proportional to

(

)

−

(Chiang & Goldreich

1997

). Since the temperature is

(

)

∝

−

(see Paper I for

more details), the formation of a gap requires

α,

(3)

where

is expressed in Jupiter masses,

in AU, and the

numerical constant is calculated for a disk temperature of 194 K

at 1 AU as determined from our disk model. Typical values of

are in the range 10

−

to 10

−

, and imply that a planet can

open a gap at 1 AU only if its mass is larger than about 0.1

To open a gap at 30 AU, the mass must be larger than about

0.5

To investigate the effects that a planet more massive than

0.1

might have on the observations of the dust continuum

emission, we simulated the presence of a planet in the DG Tau

disk by opening a gap in the surface density distribution

corresponding to the best-fit models discussed above. For

simplicity, we assumed that the planet describes a circular

orbit and that the gap can be represented by a circular ring.

To be compatible with numerical simulations of planet–disk

interaction, the half-width of the ring

is assumed to be equal

to twice the Hill radius

√

) (e.g., Bryden

et al.

1999

;Wolfetal.

2007

). In the region between

,the

surface density is depleted by a fraction

that depends on the

mass of the planet and on the disk viscosity. For

−

,we

can assume

0 for planet masses

1for

17 for

, and

6for

(Wolf et al.

2007

). Therefore, only planets more

massive than 1

will produce completely cleaned gaps.

We simulated gaps corresponding to planets in the mass range

0.3–5

and with orbital radii between 1 and 90 AU. For each

model, we calculated the residuals as the difference between

the observations of DG Tau at 1.3 mm and the model image.

If the gap is too small compared to our angular resolution, or

too faint compared with our sensitivity, the residuals will be

similar to the case without gaps shown in Figure

.Inthis

case, we say that the gap is not detected. On the other hand, a

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ISELLA, CARPENTER, & SARGENT

Vol. 714

0.1

0.2

400

800

1200

RY Tau - 1.3 mm

0.1

0.2

0.3

0.4

400

800

1200

DG Tau - 1.3 mm

0.02

0.04

400

800

1200

deprojected B

)

RY Tau - 2.8 mm

Real (Jy)

0.02

0.04

0.06

100 200 300 400 500 600 700

DG Tau - 2.8 mm

Figure 8.

Real part of the correlated flux as a function of the deprojected baseline. The solid curves show the best-fit models where the surface density is describ

ed by

the similarity solution for viscous disk evolution. The dashed curves show the results for a model with a power-law surface density. The models were co

mputed using

the high dust opacity, but identical results were obtained for the low dust opacity case.

(A color version of this figure is available in the online journal.)

large and deep gap will produce a bright ring in the residual.

To quantify how reliable the detection of a gap is, we define a

signal-to-noise ratio of the gap (gap S

N) in the following way.

First, we deproject the residual for the inclination and position

angle of the DG Tau disk. Then we take the radial average of

the residuals at the distance corresponding to the orbital radius

of the planet adopting a radial bin width equal to the FWHM of

the synthesized beam (i.e., 0

17). We define the gap S

Nasthe

mean residual in the radial bin divided by the uncertainty in the

mean. In this way, detected gaps correspond to gap S

The results are summarized in the upper panel of Figure

Planets with masses and radii that lead to gap S

3 produce

detectable gaps. No gaps are detected in our observations of the

DG Tau disk at more than 3

(see the red curve). This enables

us to constrain the masses and orbital radii of any planets that

may be present. In particular, we can exclude that planets more

massive than Jupiter exist between 5 and 40 AU, or that planets

with masses slightly smaller than Jupiter exists between 10 and

25 AU. The observations lack both the angular resolution and

the sensitivity required to detect gaps produced by planets with

a mass smaller than about 0.5

An important caveat is that a planet may exist but may not

have had enough time to completely open a gap in the disk. The

gap formation timescale

results from the tradeoff between the

efficiency of the tidal torque exercised by the planet in removing

angular momentum, and the accretion of new material coming

from larger radii in the gap due to the disk viscosity. A lower

limit of the gap formation timescale is obtained in the zero

viscosity limit. In this case, an analytic formulation is provided

by Bryden et al. (

1999

)intheform

min

(

)

(4)

where

is the orbital period,

, and

defined above. Assuming Keplerian rotation, we can rewrite the

timescale for the gap formation as

min

1Myr

(

)

(

)

(

)

−

(

)

(5)

The upper panel of Figure

shows the calculated values of

min

for the stellar mass of DG Tau (0.3

). In the case of a

planet with a mass between 0.3 and 0.5

orbiting at a radius

larger than 40 AU, the minimum timescale for the gap formation

is comparable with the age of the system (0.1 Myr). For more

massive planets, or for closer radii, the minimum gaps timescale

is a small fraction of the age of the system.

We conclude that, for DG Tau, the observations lack the

sensitivity and angular resolution required to investigate the

presence of planets less massive than about 0.5

at any orbital

radius. Our analysis indicates that no planets more massive than

Jupiter are present between 5 and 50 AU, unless they are younger

than 10

years.

5.3.2. RY Tau

The similarity solution for the disk surface density is charac-

terized by

=−

18 and

∼

3 AU. As shown in

Figure

, the surface density increases roughly as

√

from the

inner radius at 0.1 AU up to about 26 AU and then decreases ex-

ponentially outward. This supports the suggestion in Section

that the RY Tau inner disk might be partially dust depleted with

respect to power-law disk models. We note that this surface den-

sity profile may provide an explanation for both the double peak

intensity at 1.3 mm and the disk excess at infrared wavelengths.

Indeed, within 10 AU the model disk remains optically thick at

optical and infrared wavelengths, exhibiting the infrared excess

typical of classical disks.

No. 2, 2010

INVESTIGATING PLANET FORMATION IN CIRCUMSTELLAR DISKS

1757

100

1000

90 100

u

rface density (

g

-2

)

0.1

90 100

u

rface density (

g

-2

)

Radi

u

s (AU)

Figure 9.

Surface density for DG Tau (top panel) and RY Tau (lower) disks.

The red and blue curves show the best-fit solutions for the power law and the

similarity solution models, respectively, in the low opacity case

. The dashed

curves show the 3

uncertainty range for the surface density. The surface

densities for the high opacity case will have similar shapes, but will be about a

factor of 6 lower.

(A color version of this figure is available in the online journal.)

Figure 10.

Black curves show the residuals for the 1.3 mm DG Tau observations

after subtracting the best-fit model for the similarity solution. Contours start at

and are spaced by 1

. The thin red curves and the color scale show the

observed dust emission, with contours spaced by 3

(A color version of this figure is available in the online journal.)

At larger radii, the surface density in the RY Tau disk

decreases smoothly and the residuals calculated by subtracting

Gap SNR

Orbital radius (AU)

0.5

Observations

Gap SNR

Orbital radius (AU)

Observations

Figure 11.

N of the detection of a gap generated by a planet as a function

of the orbital radius

and the planet mass. The different curves correspond

to masses between 0.5 and 10

as labeled. The thick red curves indicate the

N measured from the 1.3 mm images after subtracting the best-fit model for

the similarity solution. For gap S

3, planets should produce a detectable

gap. The upper and lower panels refer to the case of DG Tau and RY Tau,

respectively.

(A color version of this figure is available in the online journal.)

the best-fit models to the 1.3 mm dust emission map do not show

any structure at more than 3

. This excludes strong deviations

from an unperturbed viscous disk profile.

The lower panel of Figure

shows the S

N of the detection

of a gap generated by planets of 1, 5, and 10

as a function of

the orbital radius. Due to the higher disk inclination and stellar

mass, a planet orbiting around RY Tau would produce a less

visible gap. In particular, our observations seem to exclude the

presence of planets more massive than 5

between 10 and

60 AU. Given the higher stellar mass of RY Tau, the minimum

timescale for the formation of gaps is an order of magnitude

larger than the case of DG Tau (see the lower panel of Figure

This implies that planets less massive than Jupiter orbiting at

more than about 30 AU may not have had enough time to form

a gap in the disk.

5.4. Radial Dependence of the Dust Properties

A comparison of the best-fit solutions obtained for the

wavelengths of 1.3 mm and 2.8 mm enables us to investigate

the dependence of dust opacity on the orbital radius. If the dust

opacity is constant throughout the disk as assumed in Section

the model fitting necessarily leads to the same surface density

profile for observations at two different wavelengths. Otherwise,

different

(

) would suggest a radial variation in the relative

dust opacities at the observed wavelengths. To understand this

point, we assume that the dust emission is optically thin. In this

case the observations constrain the product

(

)

, where