accepted for publication in Astrophysical Journal

Investigation of the physical properties of protoplanetary disks around T Tauri stars by a one-arcsecond imaging survey: Evolution and diversity of the disks in their accretion stage1

Yoshimi Kitamura

Institute of Space and Astronautical Science, Yoshinodai, Sagamihara, Kanagawa, 229-8510, Japan.

[email protected]

Munetake Momose

Institute of Astrophysics and Planetary Sciences, Ibaraki University, Bunkyo 2-1-1, Mito, 310-8512, Japan.

[email protected]

Sozo Yokogawa, Ryohei Kawabe, and Motohide Tamura

National Astronomical Observatory, Mitaka, Tokyo, 181-8588, Japan.

[email protected], [email protected], [email protected]

and

Shigeru Ida

Department of Earth and Planetary Science, Tokyo Institute of Technology, Ookayama, Meguro-ku, Tokyo, 152-8551, Japan.

[email protected]

ABSTRACT

We present the results of an imaging survey of protoplanetary disks around single T Tauri stars in Taurus. Thermal emission at 2 mm from dust in the disks has been imaged with a maximum spatial resolution of one arcsecond by using the Nobeyama Millimeter Array (NMA). Disk images have been successfully –2–

obtained under almost uniform conditions for 13 T Tauri stars, two of which are thought to be embedded. We have derived the disk properties of outer radius, surface density distribution, mass, temperature distribution, and dust opacity coefficient, by analyzing both our images and the spectral energy distributions (SEDs) on the basis of two disk models: the usual power-law model and the standard model for viscous accretion disks. By examining correlations between the disk properties and disk clocks, we have found radial expansion of the disks with decreasing Hα line luminosity, a measure of disk evolution. This expansion can be interpreted as radial expansion of accretion disks due to outward transport of angular momentum with evolution. The increasing rate of the disk radius suggests that the viscosity has weak dependence on radius r and α ∼ 0.01 for the α parameterization of the viscosity. The power-law index p of the surface −p density distribution (Σ(r)=Σ0(r/r0) ) is 0 - 1 in most cases, which is smaller than 1.5 adopted in the Hayashi model for the origin of our solar system, while the surface density at 100 AU is 0.1 - 10 g cm−2, which is consistent with the extrapolated value in the Hayashi model. These facts may imply that in the disks of our sample it is very difficult to make planets like ours without redistribution of solids, if such low values for p hold even in the innermost regions.

Subject headings: circumstellar matter—stars: pre-main-sequence

1. Introduction

It has been revealed in last 15 years that low-mass pre-main-sequence stars (T Tauri stars) are commonly accompanied by circumstellar disks. Their physical properties have been derived mainly from analysis of spectral energy distributions (SEDs) under the assumptions that the disk is axisymmetric and its temperature and surface density distributions (T (r)and Σ(r)) have power-law dependence on radius r with inner and outer cutoffs (e.g., Beckwith et al. 1990). The analysis has shown that the disks contain gas and dust of (0.1 − 0.001)M within several hundreds AU in radius and the power-law index of T (r), q,is0.5−0.75. Since such characteristics of the disks are reminiscent of the “primordial solar nebula” assumed in standard theories of the formation of the solar system (e.g., Hayashi, Nakazawa, & Nakagawa

1Based on the long-term open use observations made at the Nobeyama Radio Observatory, which is a branch of the National Astronomical Observatory, an interuniversity research institute operated by the Ministry of Education, Science, Sports, Culture, and Technology. –3–

1985; Safronov & Ruzmaikina 1985), the disks are believed to be precursors of planetary systems, or “protoplanetary” disks (e.g., Beckwith & Sargent 1996). Dust particles in the disks emit optically thin thermal radiation, which traces the disk mass well at millimeter and submillimeter wavelengths. Gas molecules in the disks also emit thermal radiation, which provides us information about the disk kinematics. Therefore high-resolution imaging with interferometers at these wavelengths have played a crucial role in revealing various aspects of disk evolution in the course of star formation. Survey obser- vations of low-mass young stellar objects (YSOs) showed that the dust continuum emission around protostar candidates is more extended than that around T Tauri stars (Ohashi et al. 1991, 1996; Looney, Mundy, & Welch 2000), suggesting disks as well as central stars grow by accretion of matter caused by dynamical collapse of circumstellar envelopes in the protostar stage. Detailed velocity fields in several protostellar envelopes were obtained by aperture synthesis observations with molecular lines, showing that the typical mass accretion rate −6 −1 onto the central star/disk system is ∼ 5 × 10 M yr (e.g., Hayashi, Ohashi, & Miyama 1993; Ohashi et al. 1997; Momose et al. 1998). The timescale for disk persistence in later stages of star formation, on the other hand, has been investigated by systematic searches for dust and gas emission toward evolved T Tauri stars. For example, Duvert et al. (2000) made survey observations of T Tauri stars with a wide range of ages and found that all objects with no infrared excess do not have disks detectable in the dust continuum or molecular line emission at millimeter wavelengths. These results may imply that the entire disks disappear on almost the same timescale as that for disappearing of the innermost regions emitting infrared radiation (see Strom et al. 1989; Skrutskie et al. 1990). In spite of the above progress, understanding of the internal structure and evolution of the disks in the early T Tauri stage is still limited. Although the total mass and temperature distribution of the disks are derived from the analysis of their SEDs, the outer radius and the surface density distribution cannot be evaluated by this method (see Beckwith et al. 1990). This is easily understood because the SED data were obtained by flux measurements with large beams which provide no information about the spatial distribution of the emission. On the other hand, it has been revealed that, in the T Tauri stage, the mass accretion rate from the disk to the central star, which can be estimated from the amount of excess emission at ultraviolet and near-infrared wavelengths, becomes lower as the stellar age increases (Calvet, Hartmann, & Strom 2000). This trend is consistent with a possible evolutionary sequence from classical T Tauri stars (CTTSs) to weak-line T Tauri stars (WTTSs) (e.g., Strom et al. 1989) because these two categories are based on the equivalent width of the Hα emission line that must be tightly connected to the outflow activity, which is originally driven by the mass accretion activity (e.g., Edwards, Ray, & Mundt 1993). Owing to the lack of systematic studies of disk internal structures, however, it is still unclear how this evolutionary trend is –4– related to the internal structure of the disks themselves. Imaging at higher angular resolutions is crucial for studying the internal structure of the disks. One of the most important disk properties is the surface density distribution that dominates planet formation processes. High resolution images of several disks have been obtained in “silhouette” or in scattered light at optical and near infrared wavelengths, pro- viding us fairly firm information about the spatial extent of disk matter (e.g., McCaughrean & O’Dell 1996; McCaughrean et al. 1998; Padgett et al. 1999). In order to evaluate their surface density distributions as well as their outer radii, however, observations of thermal radiation at millimeter and submillimeter wavelengths are required. Detailed observations of some circumstellar or circumbinary disks were made at these wavelengths (e.g., Kawabe et al. 1993; Saito et al. 1995; Mundy et al. 1996; Guilloteau, Dutrey, & Simon 1999). Mundy et al. (1996) estimated, for the first time, the surface density distribution in the disk around HL Tau. Despite these case studies, a survey of a well-coordinated sample is required to reveal the evolutionary trend or diversity of the disk characteristics. We have carried out an imaging survey of protoplanetary disks associated with single T Tauri stars in the Taurus molecular cloud in dust continuum emission at 2 mm with the Nobeyama Millimeter Array (NMA). Physical properties of the disks, including the outer radius and the surface density distribution, have systematically been derived from the combination of SED analysis and image-based model fitting. Our survey is the first systematic study of the surface density distribution with the outer radius based on high- resolution images taken under almost uniform conditions. The images obtained by small synthesized beams (1 − 2), which can resolve the spatial extent of the disks, enable us to successfully estimate their internal structure. A sample of more than 10 sources allows us to extract some possible evolutionary trend and diversity of the disks, which would contribute to the understanding of diverse planetary systems. The outline of this paper is as follows. The sample selection is described in §2 and the details of the observations are in §3. Our observational results are presented in §4. In §5, our analysis to derive the disk physical parameters is described and their evolutionary trend or diversity is discussed.

2. Sample

We selected about 20 T Tauri stars by the following two criteria: 1) The star is known to be single and is located in the Taurus molecular cloud (d = 140 pc). We examined the multiplicity of the star on the basis of the catalogues of multiple T Tauri stars by Leinert et al. (1993), Ghez, Neugebauer, & Matthews (1993), Kohler & Leinert (1998), Richichi et al. (1994), Simon et al. (1992), Reipurth & Zinnecker (1993), and Mathieu (1994). 2) The –5– expected flux density of 2 mm dust continuum emission is larger than 20 mJy. The 2 mm flux density of each object is estimated from the surveys of 1.3 mm continuum emission (Beckwith et al. 1990; Osterloh & Beckwith 1995) and the measurements of the β index (Beckwith & Sargent 1991; Moriarty-Schieven et al. 1994), where we assume β = 1 for unknown cases. If the total flux density from a disk exceeds ∼ 20 mJy, simple calculations show that imaging of the disk with a radius of ∼ 100 AU becomes successful by achieving a rms noise level of 2 mJy beam−1. Since such a low noise level requires one or two day observations under good weather conditions for each source, we were able to complete the disk imaging for more than 10 objects in three winter seasons. In our survey we have succeeded in imaging the disks for 13 objects among the T Tauri stars selected prior to the survey. Table 1 lists the 13 objects. We mainly observed CTTSs and 8 objects are typical CTTSs. Haro 6-5B and HL Tau are now thought to be transient sources from protostars to CTTSs on the basis of the HST images, which show the presence of envelopes as reflection nebulae (Stapelfeldt et al. 1995; Krist et al. 1998). The remaining three sources, IQ Tau, DN Tau, and LkCa 15 are relatively older and are likely to be on the borderline between CTTSs and WTTSs, because these objects have equivalent widths of the Hα emission line as narrow as ∼ 10 A˚ (see Table 2).

3. Observations

Observations were carried out with the NMA, which consists of six 10 m antennas, over the three winter seasons of 1998 December to 1999 February, 1999 December to 2000 February, and 2000 November to 2001 February. We used all the array configurations, D, C, and AB, and their projected baseline lengths ranged from (5 - 40), (10 - 80), and (25 - 175) kλ, respectively. Dust continuum emission at 2 mm from the disks was detected with SIS receivers operated in double sideband (DSB) mode. System noise temperatures during the observations were typically 200 K in DSB at the zenith. For the back end, the digital spectral Ultra Wide Band Correlator, UWBC (Okumura et al. 2000) was employed. Visibility data of the continuum emission in both the lower (135±0.512 GHz) and upper (147±0.512 GHz) sidebands were obtained simultaneously with the phase-switching technique. To obtain a higher signal-to-noise ratio (S/N), the data of both the sidebands were equally added in a final image with center frequency 141 GHz. The field center was set on the position of each object and the size of the primary beam (i.e., the field of view) was about 46 FWHM. In the lowest-resolution D configuration, we used the source positions previously reported (e.g., Strom et al. 1989). In the higher- resolution AB and C configurations, the peak positions in the D configuration were used as –6– the center positions. Table 1 shows peak positions in the 2 mm continuum images obtained with the AB or AB+C configurations. The 13 objects in Table 1 were observed in the following way. First, we observed all the sources with the compact D configuration to detect the disk emission as point-like sources. In the D configuration, the size of the synthesized beam was ∼ 5 and the array was insensitive to structures extending more than 20 (∼ 2800 AU) FWHM (see Appendix in Wilner & Welch 1994). We can accurately measure the total flux density from a disk in this compact configuration. In contrast, the estimation of the total flux density over a resolved disk in a higher-resolution image is much likely to be affected by noise, because the determination of the disk periphery highly depends on the noise. Therefore, we used the disk total flux densities measured with the D configuration to check the depth of integration for successive higher-resolution images with the AB and AB+C configurations: we tried to integrate the disk image as deeply as possible in order that the total flux density over the entire disk area may reproduce the total flux density with the D configuration. In the highest-resolution AB configuration, the size of the synthesized beam was ∼ 1 and the array was insensitive to structures extending more than 4 (∼ 560 AU) FWHM. The beam shape was nearly circular, and thus the distortion of the disk image due to the beam pattern was minimized. In the AB+C configurations, the size of the synthesized beam was ∼ 2 and structures extending more than 10 (∼ 1400 AU) FWHM were probably resolved out. The response across the observed passband for each sideband was determined from 30- 40 minute observations of 3C454.3 or 3C273. A gain calibrator, 0446+112, 0507+179, or 0528+134 was observed every 30 minutes in the D configuration. In the AB and C config- urations, the gain calibrators were observed as frequently as possible (every 8-10 minutes) to minimize phase error in resultant images. The flux densities at 2 mm of 0446+112, 0507+179, and 0528+134 were derived to be (1.9 - 2.1) Jy, (1.7 - 2.2) Jy, and (1.5 - 2.5) Jy, respectively, by comparison with Uranus or Neptune (Griffin & Orton 1993). The overall uncertainty in the flux calibration was about 10%. After the calibrations, we made final images only from data taken under good weather conditions. Using the AIPS package developed at the NRAO, we CLEANed the continuum maps by natural weighting with no taper in the UV plane. The rms noise levels were (2 - 7) mJy beam−1 with ∼ 5 beam in the D configuration, about 2 mJy beam−1 with ∼ 1 beam in the AB configuration, and about 2 mJy beam−1 with ∼ 2 beam in the AB+C configurations. Positional accuracy was dominated by S/N and absolute positional errors were less than 0.3. Since source sizes were much smaller than the field of view, the primary beam attenuation was negligible. –7–

4. Results

4.1. Total flux densities with different array configurations

Total flux densities of the continuum emission in the maps obtained with the different array configurations are shown in Table 1. The spatial extent of each continuum emission with the D configuration is almost the same as the synthesized beam size (∼ 5), indicating that most emission originates from the region of r<350 AU from the central star and that the contribution of extended components such as an envelope is negligible. Detailed analyses of the SEDs of some T Tauri stars have revealed that continuum flux densities at frequencies higher than 100 GHz are attributed solely to dust thermal radiation, though the contribution of free-free emission from ionized gas should be taken into account at lower frequencies (e.g., Mundy et al. 1996; Wilner, Ho, & Rodriguez 1996). We therefore consider all the detected emission is from dust particles in circumstellar disks in the following. The spatial extent of the emission in the maps constructed from the data with the sparse configurations (AB and C) is more extended than the beamsize (1−2): detailed descriptions of the spatial distributions are presented in §4.2. Figure 1 shows a comparison of the detected total flux densities in the compact (D) and sparse (AB and AB+C) configurations. In the case of 10 sources, the total flux density detected by the D configuration, Fν(D), agrees with that by the AB configuration, Fν(AB), within uncertainties, suggesting all the disk emission is successfully detected and mapped with ∼ 1 beam. In the case of the other three sources

(AA Tau, IQ Tau, and LkCa 15), Fν(AB) is only (40 − 60) % of Fν(D). This is due to lower sensitivity to surface brightness and missing more extended components in the AB configuration. The total flux densities by the D configuration for these three sources are comparable to that for DM Tau or DN Tau, which shows Fν(D) = Fν(AB), suggesting that the disks around the three sources have larger radii, or lower surface brightness. Since our purpose is to reveal the whole disk structure, it is essential to recover Fν(D) as much as possible even when we try to obtain a higher-resolution image. We therefore add the data with the C configuration to improve the sensitivity to low-brightness and more extended components of the emission. The resultant flux density for the three sources, Fν(AB+C), becomes greater than 70 % of Fν(D) (see Table 1 and Figure 1). Although the sensitivity of the present observations is still insufficient to recover all the emission from the circumstellar disks around IQ Tau and LkCa 15, we use in our analysis the images constructed from the data with the AB+C configurations for these sources. –8–

4.2. Disk imaging and comparison with previous results

The high-resolution images of the disks are presented in Figure 2: the images obtained by the AB configuration for the ten objects whose Fν(AB) agrees with Fν(D), and those by the AB+C configurations for the other three sources (AA Tau, IQ Tau, and LkCa 15). The spatial extent of each continuum emission is more extended than the synthesized beam size, indicating that all the disks are spatially resolved. The continuum emission in Figure 2 clearly exhibits a source-to-source difference in the spatial extent or the contour spacing, suggesting there are varieties of disk characteristics such as the outer radius or the surface density distribution. Table 3 shows beam-deconvolved Gaussian sizes of the emission, which were derived from Gaussian fitting in regions where the intensity is stronger than half the peak intensity, giving us a rough estimate of the spatial extent of the disks. The synthesized beam sizes and estimated seeing sizes, which are described in detail in §4.3, are also listed in Table 3. The nearly circular synthesized beams and the fairly small seeing sizes in our imaging allow us to accurately derive disk physical parameters such as the outer radius and the surface density distribution (see §5). We compare our imaging results with previous ones for the nine sources described below, whose disk images were independently taken at other frequencies, to check the quality and reliability of our results. For our results we mainly use the beam-deconvolved Gaussian sizes of the disks traced by the dust continuum emission (i.e., major and minor axes and position angles), listed in Table 3. In some sources we use the radius and inclination angle of the disk calculated from the beam-deconvolved size by assuming a geometrically thin disk. If only line data are reported in the previous studies, we consider mainly the inclination and position angles. This is because the radius of a disk traced by line emission tends to be larger than that of the disk traced by continuum emission owing to large optical depths (τ>1) of the line emission. Furthermore, the disk mass from the line observations does not necessarily agree with that from the continuum observations (see Table 4 for our sources), because the disk mass from the line observations is likely to be underestimated owing to depletion of the molecular species used to trace the disk mass such as CO (Guilloteau & Dutrey 1994; Dutrey, Guilloteau, & Guelin 1997; Aikawa et al. 1996), and because the line and continuum emission often traces different regions.

4.2.1. Haro 6-5B

The source, a CTTS in the Herbig-Bell Catalogue (HBC; Herbig & Bell 1988), was recently imaged with the HST at visible and near-infrared wavelengths, and an envelope and a disk were revealed as a reflection nebula and a dark lane, respectively (Krist et al. 1998; –9–

Padgett et al. 1999). The central star is found to be obscured by the dark lane, suggesting that this source is an embedded source, i.e., a protostar candidate. The spatial extent of the dust emission in Figure 2 agrees well with that of the reflection nebula with the dark lane, as already reported by Yokogawa et al. (2001). The non-axisymmetric component extended to the south-west seen in Figure 2 would be a part of the envelope around the star.

4.2.2. HL Tau

This source was also classified as a CTTS in the HBC, but is now thought to be a protostar candidate. An infalling envelope around the source was found by Hayashi et al. (1993) with the NMA, and HST observations demonstrated that the source is really embedded in circumstellar matter (Stapelfeldt et al. 1995; Close et al. 1997). Furthermore, a compact dust disk with a radius of 70 AU was imaged in 2.7 mm dust continuum emission by Mundy et al. (1996) with the BIMA array. The beam-deconvolved Gaussian size of the disk was (1.0 ± 0.2) × (0.5 ± 0.2) at PA = 125◦ ± 10◦. On the other hand, our 2 mm image in Figure 2 clearly shows the centrally peaked dust disk with a weak ridge-like structure extended to the north, probably a part of the infalling envelope. The Gaussian size of the dust disk imaged at 2 mm is (1.04 ± 0.03) × (0.60 ± 0.04) at PA = 144◦ ± 2◦ and agrees with that at 2.7 mm within uncertainties. In addition, our peak position of the 2 mm image agrees with the position of the 3.6 cm continuum source observed by Rodriguez et al. (1994) with the VLA.

4.2.3. CY Tau

This source is a CTTS in the HBC and a rotating disk was imaged in CO J =2−1with the IRAM interferometer (Simon, Dutrey, & Guilloteau 2001). The radius of the gas disk was derived to be (270 ± 10) AU on the basis of a power-law disk model. The inclination and position angles of the disk were estimated to be 47◦ ± 8◦ and 124◦ ± 7◦, respectively, from the 1.3 mm continuum data. In our 2 mm observations, the radius, inclination angle and position angle of the dust disk are (63 ± 7) AU, 57◦ ± 8◦, and 68◦ ± 8◦, respectively. Both the inclination angles show agreement, while our position angle differs from PA at 1.3 mm by 56◦. In the CO channel maps, however, one can see an elongated feature along the direction at our PA. –10–

4.2.4. RY Tau

Koerner & Sargent (1995) detected the CO J =2− 1 emission toward the source with the OVRO millimeter interferometer. Although their S/N was not high, the profile of the CO emission seems to have double peaks suggesting a rotating disk. The radius, inclination angle, and position angle of the gas disk were estimated to be 110 AU, 25◦, and 48◦ ± 5◦, −5 respectively, by Gaussian fitting. They also derived the disk mass of 1 × 10 M from the line data and noted the mass is much smaller than that derived from continuum observations. From our 2 mm observations, the radius, inclination angle, and position angle of the dust disk are estimated to be (51 ± 4) AU, 59◦ ± 7◦,and27◦ ± 7◦, respectively. Furthermore, we −3 estimate the disk mass to be 6 × 10 M from our model fitting (see §5). The peak position of the 2 mm continuum emission agrees with that of the CO emission, while the inclination and position angles at 2 mm differ from those at 1.3 mm. The disagreement might be due to a difference between the gas and dust distributions.

4.2.5. DL Tau

A rotating disk was imaged in CO J =2−1 with millimeter interferometers. Koerner & Sargent (1995) found the disk with a radius of 250 AU with the OVRO array, and recently, Simon et al. (2001) revealed the more detailed velocity structure of the disk with the IRAM array. The radius of the gas disk was estimated to be (250 - 520) AU, and the inclination and position angles from the line and continuum observations were 12◦ -49◦ and 44◦ -84◦, −6 respectively. The disk mass was derived to be 1 × 10 M from the line observations, which is much smaller than the mass from continuum observations (Koerner & Sargent 1995). We have obtained from the 2 mm imaging that the radius, inclination angle, position angle, and ◦ ◦ ◦ ◦ −2 mass of the dust disk are (80 ± 4) AU, 47 ± 4 ,52 ± 6 ,and5× 10 M, respectively. Our inclination and position angles agree with the CO results, and are also consistent with the inclination angle of 49◦ ± 3◦ and the position angle of 44◦ ± 3◦ derived from the IRAM continuum observations.

4.2.6. DM Tau

Double-peaked profiles of 12CO and 13CO emission were first detected with the NRO 45 m and IRAM 30 m telescopes and the detection strongly suggested the presence of a rotating disk around the source (Handa et al. 1995; Guilloteau & Dutrey 1994). Subsequently, the disk was imaged with the NMA and the IRAM interferometer, and Keplerian rotation was –11– revealed (Saito et al. 1995; Guilloteau & Dutrey 1998). Furthermore, more detailed studies of disk properties were made with the IRAM interferometer (Dutrey et al. 1997; Simon et al. 2001). The radius of the gas disk was estimated to be (800 ± 5) AU on the basis of a power-law disk model, and its inclination and position angles were 32◦ -45◦ and 153◦ - 179◦, respectively, from both the line and continuum data. The disk mass was also derived to be (0.002 - 0.02) M. Our 2 mm continuum observations show that the radius, inclination angle, position angle, and mass of the dust disk are (172 ± 12) AU, 68◦ ± 4◦,134◦ ± 4◦,and (0.01 - 0.02) M, respectively. The inclination and position angles differ from the IRAM results owing to the presence of a new component elongated in the south-east direction in Figure 2, which was not detected by the IRAM interferometer. Since the component was seen at independent observing runs, it would suggest some non-axisymmetric distribution of the disk matter.

4.2.7. DO Tau

Koerner & Sargent (1995) found a disk with a radius of 350 AU around the source in CO J =2− 1 with the OVRO interferometer. The velocity structure of the disk, however, can not be described simply by Keplerian rotation, and outflow and infall motions seem to be required. Although their S/N was not high, the radius, inclination angle, position angle, ◦ ◦ ◦ −4 and mass of the gas disk were estimated to be 350 AU, 45 , 160 ± 5 ,and1× 10 M, respectively. Subsequently, Koerner, Chandler, & Sargent (1995) obtained a 0”.6 resolution image of 7 mm continuum emission from the disk with the VLA, and accurately estimated −2 the disk mass of 1×10 M, which is much larger than the above mass derived from the line data and is consistent with our estimated mass described below. From our map in Figure 2, the radius, inclination angle, position angle, and mass of the dust disk are estimated to ◦ ◦ ◦ ◦ be (71 ± 3) AU, 38 ± 5 ,67 ± 9 ,and0.004M, respectively. Our inclination angle is roughly consistent with the previous value. Our position angle, however, is significantly different from the previous value of 160◦: this fact means that the elongation of the dust disk in Figure 2 is parallel to the optical jet. Since it is not likely that the dust continuum emission traces the jet, there seems to exist misalignment between the disk and the jet as in the DG Tau case (Kitamura, Kawabe, & Saito 1996; Tamura et al. 1999). Note that the elongation at the 1.5σ level shown in Figure 2 seems to be vertical to the jet. This extended weak component does not seem to be an outer part of the disk and might be a part of an envelope remaining around the source. –12–

4.2.8. GM Aur

A Keplerian disk was found around the source in 13CO J =2− 1 by Koerner, Sargent, & Beckwith (1993) with the OVRO interferometer. Furthermore, higher-resolution 12CO J =2− 1 observations were made with the IRAM interferometer (Dutrey et al. 1998; Simon et al. 2001) and these studies revealed the detailed properties of the disk. The beam- deconvolved Gaussian sizes of the dust disk were (1.07 ± 0.05) × (0.63 ± 0.05) with PA = 57 ◦ ± 5◦ at 1.3 mm and (1.25 ± 0.20) × (0.65 ± 0.17) with PA = 58◦ ± 11◦ at 2.7 mm. The disk mass was derived to be 0.03 M from the 1.3 mm and 2.7 mm data. On the other hand, the beam-deconvolved Gaussian size of the 2 mm continuum image in     ◦ ◦ Figure 2 is (1 .5 ± 0 .1) × (0 .8 ± 0 .1) at PA = 57 ± 6 , and the disk mass is 0.04 M. Our derived parameters of the disk are consistent with the previous values, and the peak position at 2 mm agrees well with that at 1.3 mm. However, the elongated feature to the south seen in the 2 mm image has no counterpart at 1.3 mm. This feature would indicate non-axisymmetry of the disk as in the case of DM Tau.

4.2.9. LkCa 15

A rotating disk with a radius of 600 AU was imaged in HCO+ J =1− 0andCO J =2− 1 with the IRAM interferometer (Duvert et al. 2000; Simon et al. 2001). The beam- deconvolved Gaussian size of the dust disk was estimated to be (1.45 ± 0.08) × (1.20 ± 0.08) with PA = 64◦ ± 13◦ at 1.3 mm. Our 2 mm image shows that the Gaussian size of the dust disk is (2.1 ± 0.2) × (0.6 ± 0.4) at PA = 79◦ ± 5◦. Our disk size differs from the previous one, while the peak position at 2 mm agrees with that at 1.3 mm.

4.3. Estimation of atmospheric seeing

To estimate the size of the atmospheric seeing disk during our observations, we made CLEAN maps of the 141 GHz continuum emission from the gain calibrators 0446+112, 0507+179, and 0528+134. In the maps the continuum emission seems slightly extended compared with the synthesized beams, and non-zero beam-deconvolved Gaussian sizes of the calibrators are derived. Since the angular sizes of the calibrators are much smaller than the NMA beam sizes of ∼ 1”, the spatial extent of the calibrators in the CLEAN maps is probably attributed to the phase fluctuation due to the atmospheric turbulence (i.e., the seeing sizes toward the calibrators). In this paper we define the seeing size as the geometrical mean of the 1/2 lengths of the major and minor axes of the deconvolved Gaussian profile ((SmajorSminor) ), –13– assuming that the turbulence is isotropic and that the image distortion is described by a Gaussian profile. The estimated seeing size for each imaging is listed in Table 3. Since the seeing sizes are smaller than the source sizes, as shown in Table 3, it is not likely that the disk images in Figure 2 were seriously distorted by the atmospheric turbulence. For some sources, however, the estimation of disk inclination angles might be influenced by the seeing because the lengths of the source minor axes are comparable to the seeing sizes. We consider the seeing sizes toward the calibrators as those toward the target objects. This approximation would be valid as previously discussed in the case of Haro 6-5B by Yokogawa et al. (2001), although the angular distances between the calibrators and the sources are 10◦ − 20◦ and the integration time of the calibrators (2 - 3 minutes) was shorter than that of the sources (4 - 5 minutes). For Haro 6-5B they analyzed the bandpass calibrator as well as the gain calibrator and concluded that the seeing size did not sensitively depend on both the sky direction and the integration time at least under good weather conditions in Nobeyama.

5. Discussion

5.1. Analysis of the disk images and the SEDs based on disk models

To estimate the physical properties of the protoplanetary disks, we analyze the disk images obtained by this study together with the SEDs on the basis of two disk models.

The disk parameters to be determined are as follows: inner and outer radii (Rin, Rout), mass (Mdisk), inclination angle (i), position angle (PA), surface density distribution (Σ(r)), temperature distribution (T (r)), and dust opacity coefficient with the β index. Here we 12 β assume the dust opacity coefficient follows the usual power-law form of κν =0.1(ν/10 Hz) [cm2 g−1] (e.g., Beckwith et al. 1990). We adopt the following two disk models. The first model (model 1) is a power-law model that is most frequently used in data analysis of disk observations (e.g., Beckwith et al. 1990). In the model 1, the radial profiles of the surface density and temperature of a −p −q disk have the power-law forms of Σ (r)=Σ0(r/r0) and T (r)=T0(r/r0) , respectively, for Rin ≤ r ≤ Rout. In our paper the lower limit of the temperature is fixed at 8 K (Goldsmith & Langer 1978). It is to be noted that the model was used in the standard model for the origin of our solar system by Hayashi et al. (1985). The model, however, seems to have the unphysical nature that the mass distribution is sharply truncated at the outer radius. In contrast to the model 1, the second model (model 2) has a surface density distribution −p 2−p of Σ (r)=Σ0(r/r0) exp[−3(r/Rout) ], the same form as in a similarity solution for viscous –14– accretion disks (Lynden-Bell & Pringle 1974; Hartmann et al. 1998). This distribution does not have any sharp outer edge. Since the radial profile is extended to the infinite, we define the disk outer radius as the radius of a region which contains 95% of the total disk mass (see Appendix). In the similarity solution, the power-law index p of the surface density distribution is equivalent to the index γ of the power-law function of the viscosity γ (ν(r)=ν0(r/r0) ). The temperature distribution has the same form as in the model 1 with the free parameter of q, although the viscous accretion disks are predicted to have 3/4 for q in a steady state. We take account of the distortion of disk images due to the radio seeing in applying the disk models to the disk images. Prior to the model fitting, we convolved the synthesized beam patterns, which are approximately Gaussian, with the Gaussian profiles of the radio seeing obtained in § 4.3, and we used the new effective beam patterns in calculating contour maps for the model disks. The seeing does not seem to seriously affect the estimation of the disk parameters, because the seeing sizes are smaller than both the beam and source sizes. Actually our disk images at 2 mm agree with the previous ones at other wavelengths, as shown in § 4.2. It would be almost impossible that a point-like source happens to mimic the disk obtained previously owing to the radio seeing.

We have determined Rout, Σ(r), i, and PA of the disk by applying the two disk models to the disk image at 2 mm. These disk parameters are sensitive to the disk image. In the χ2 fitting, we assumed a geometrically thin disk for simplicity. Thus the flux density from the model disks at frequency ν, Fν,isgivenby
    2 Fν(r)= Pbeam+seeing(r − r )Bν(T (r ))(1 − exp(−τν(r )/ cos i)) cos idr /d , (1) where Pbeam+seeing is the effective beam pattern including the seeing, Bν is the Planck func- tion, and τν = κνΣ . The peak positions of the 2 mm continuum emission were considered as the stellar positions, i.e., the disk centers. As the first guess about the disk parameters −2 we adopted Σ0 =1gcm at r = 100 AU and p = 1. The initial values of Rout, i,andPA were the best-fit values in the Gaussian fitting (see Table 3).

By simultaneously applying the same models to the SED we have determined Rin, T (r), β, and Σ0. These parameters are well estimated from the energy spectrum from millimeter to near-infrared wavelengths. Although Σ0 can be determined in the above image fitting, we also treated it as a free parameter in the SED fitting. This is mainly because the disk models could not be well fitted to the SED particularly over the range from millimeter to submillimeter wavelengths for the fixed value of Σ0 determined in the image fitting. For the central star only the stellar radius was treated as a free parameter, for simplicity, with fixing the effective temperature, which is shown in Table 2: the stellar radius was determined –15– so as to reproduce the flux densities from near-infrared to ultraviolet wavelengths, without considering the excess at ultraviolet. These stellar parameters affect only the inner radius of the disk, which is not discussed in this paper. Detailed analysis of the stellar properties is beyond this work. The cycle of the image fitting followed by the SED fitting was repeated until all the disk parameters were converged within uncertainties. The convergence was achieved after a few repetitions, because the two groups of the disk parameters are not strongly coupled with each other. Although two different best-fit values for Σ0 were determined by both the image and SED fitting, we adopt the best-fit value by the image fitting in the following. There exists a serious problem of runaway increase of the disk outer radius in the above model fitting. The main reason is as follows: the outer part of the model disk comes to have surface brightness equal to or lower than the rms noise level when the model disk becomes more extended than the observed disk, and as a result, the contribution of the outer part of the model disk to χ2 becomes small, which makes the sensitivity of the radius to χ2 very weak. Such problems are usually associated with analysis of data having finite S/N (e.g., Mundy et al. 1996). In order to suppress the runaway increase we impose the following constraint on the model fitting: The total flux density of a model disk should be equal to the observed one, Fν(D or AB(+C)). Since Fν(AB+C) is only 70% of Fν(D) for the two sources IQ Tau and LkCa 15, we consider the two total flux densities of Fν(D) and Fν(AB(+C)) in the constraint. With the constraint we treat the disk outer radius as a dependent variable. To estimate all the disk parameters in a consistent way, higher-resolution images with higher S/N at many frequencies will be required.

5.2. Fitting results

The best-fit results for the images and the SEDs are shown in Figures 3 & 4, respectively. The corresponding best-fit parameters with errors are listed in Table 4 for the disks. These errors include the rms noise on the images and the uncertainties in our and previous flux measurements, but mutual coupling among the disk parameters is not considered. If the coupling is taken into account, the errors would become greater by a factor of ∼ 1.5. In the images, the disk models were fitted to pixels higher than the 1.5σ levels over the regions of −4 ≤ ∆α, ∆δ ≤ +4. Figure 3 shows that the model fitting seems fairly good and the residual intensities are comparable to the 1.5σ levels except for some non-axisymmetric components. Although there seem to exist non-axisymmetric components for Haro 6-5B, HL Tau, DM Tau, DO Tau, and GM Aur, as described in § 4.2, we equally treated all the components higher than the 1.5σ levels, and applied the axisymmetric models in our fitting. –16–

This is because we have no firm theoretical disk model including such non-axisymmetry. Therefore, it is likely that the estimated disk parameters are somewhat affected by the non-axisymmetry.

Our flux measurements strongly suggest that Rout of the disks around IQ Tau and LkCa 15 is larger than that expected from the disk images shown in Figure 2. Since Fν(AB+C) is only about 70% of Fν(D) for the two sources, as shown in Figure 1, the best-fit value of Rout with the constraint using Fν(D) becomes larger than that for Fν (AB+C) (see the positive errors of Rout in Table 4). Of course, calculated images for the models with Rout based on Fν(AB+C) agree well with the observed images in Figure 2. Considering S/N in Figure 2, it is very likely that the outer parts of the large disks around the two sources are embedded in the noise levels because of low brightness. The model fitting to the SEDs over the frequency range from 1011 to 1014 Hz seems good, as shown in Figure 4, as well as the image fitting. For the two embedded sources Haro 6-5B and HL Tau, the stellar parameters of the effective temperatures and radii were fixed in such a way that the star/disk systems reproduce the flux densities at near-infrared, but the excess at ultraviolet was ignored in the fitting. For the other sources, the best-fit values of the stellar radii were reasonable values of ∼ 0.01 AU. The stellar luminosities determined by the SED fitting, however, do not agree with those in literature (see Table 2), probably because our analysis to derive the stellar parameters is simplified. In the best-fit curve of CY Tau a shallow dip is seen at ∼ 1012 Hz because of the lower limit of 8 K for the disk temperature.

5.3. Evolution and diversity of the disks in their accretion stage

Diversity in the disk properties has been reported by previous surveys. The diversity in the temperature distribution and mass of the disks was revealed by Beckwith et al. (1990). Furthermore, recent HST imaging found several silhouette disks with various radii in the Orion region (McCaughrean & O’Dell 1996). On the other hand, in this study we will investigate the disk diversity in more detail by comparing the disk properties on the basis of the disk images taken under almost uniform conditions. In addition to the disk diversity, previous infrared and millimeter observations have accumulated much evidence for the overall evolution of the disks during the stellar evolution from protostars to WTTSs through CTTSs (e.g., Looney et al. 2000; Strom et al. 1989; Osterloh & Beckwith 1995; Duvert et al. 2000). If the disks really evolve, can we find out some evidence for the disk evolution in the CTTS stage? We will try to extract the disk –17– evolution from the diversity in the disk properties.

5.3.1. Clock for the disk evolution

First of all, we need to select good measures of the disk evolution, i.e., the clock. There are at least two good candidates for the clock: the age of the central star and the mass accretion rate (the accretion luminosity) of the disk. The two clocks become equivalent if the disk evolution completely synchronizes with the stellar evolution. The stellar age is the most fundamental clock for a star and is usually determined from the comparison between observations and theoretical calculations on HR diagrams (e.g., Kenyon & Hartmann 1995). We made new determinations of the stellar ages and masses from the effective temperatures and luminosities in literature (see Table 2). Here we did not use the stellar luminosities obtained from our SED fitting, because they depend on our adopted models. The disk activity tends to decrease with the stellar evolution from the CTTS stage of ∼ 106 yr to the WTTS stage of ∼ 107 yr (e.g., Hartmann et al. 1998; Calvet et al. 2000). Since the timescale of Keplerian rotation is likely to control the dynamical evolution of the disk, the stellar age normalized by the Kepler time must be also considered. The normalized age, however, is essentially the same as the stellar age itself, because the Kepler time is weakly dependent on the stellar mass, which almost falls into a narrow range of (0.5 - 1) M, as shown in Table 2. ˙ The clock of the disk mass accretion rate, Macc, or the disk accretion luminosity, Lacc,is theoretically thought to be an important parameter, which is closely related to the evolution of viscous accretion disks. In the standard model for viscous accretion disks, the timescale of ˙ the viscosity processes characterizes the disk evolution, and both Macc and Lacc decrease with the disk evolution. These two parameters, however, can not be derived directly from any observations and have been estimated from other observable quantities such as the infrared excess luminosity and the luminosities of the line and continuum emission from the boundary layer between the star and the disk (e.g., Valenti, Basri, & Johns 1993; Hartigan, Edwards, & Ghandour 1995; Gullbring et al. 1998; Hartmann et al. 1998). We will now examine the infrared excess luminosity and the Hα line luminosity as potential disk clocks.

The infrared excess luminosity, LIR, comes to be equal to Lacc after subtraction of the luminosity of reprocessed stellar radiation, Lrep, while the Hα line luminosity, LHα, is thought to represent the activity of T Tauri winds or the mass ejection rate. Since the mass ejection rate is very likely to be proportional to the mass accretion rate, a good correlation between

LHα and LIR is seen, as shown in Figure 5 (Cabrit et al. 1990). We obtained the best-fit curve given by

LHα ≈ 0.002LIR ∼ 0.002Lacc. (2) –18–

The luminosity LIR is roughly equal to Lacc for large LIR. In Figure 5 the scatter around the best-fit curve seems to increase with decreasing LIR, probably because both the contribution of Lrep to LIR and that of the luminosity of the chromospheric radiation to LHα increase.

There exist uncertainties in estimating LIR and LHα. The estimation of LIR from SEDs requires some disk models and the subtraction of Lrep is highly dependent on the models (e.g., Miyake & Nakagawa 1995; Kenyon & Hartmann 1987; Gullbring et al. 1998; Basri &

Batalha 1990). On the other hand, the estimation of LHα does not depend on any models, although the luminosity often shows time variability and is uncertain by a factor of ∼ 2

(Cabrit et al. 1990). We prefer LHα as the disk clock, because it can directly be observed. The luminosity is expected to decrease with the disk evolution. In the following, we use both the stellar age and LHα as the measures of the disk evolution. For the two embedded sources Haro 6-5B and HL Tau, however, there exists no definite estimate of either measure.

5.3.2. Disk radius

The disk outer radius seems to increase with decreasing LHα, as shown in Figure 6. This increasing trend is likely to be a signature of the disk evolution, because we can not find any distinct correlation between the disk radius and the stellar parameters of mass and luminosity. In the case of the LHα clock the trend seems clear except for the two protostars in both the models 1 & 2. Although the luminosity is thought to have uncertainty by a factor of ∼ 2, the disk radius likely increases with a scatter as the luminosity decreases, or the disk evolves. The trend, however, becomes unclear against the stellar age, and this situation is not improved even for the age normalized by the Kepler time. These results may suggest that the disk evolution does not synchronize well with the stellar evolution. Although the decrease in the disk activity has been observationally reported over a long time span of CTTS to WTTS, it is likely that the disk evolution over a short span comparable to the timescale of the disk accretion is somewhat embedded in the diversity of disk formation and evolution processes, such as different termination times of mass supply from envelopes and various timescales of the viscosity controlling the disk evolution. In fact, the distribution of WTTSs slightly overlaps with that of CTTSs on HR diagrams (e.g., Strom et al. 1989; Stahler & Walter 1993). The increase in the disk radius with the evolution can be interpreted as radial expansion of an accretion disk due to outward transport of angular momentum (Lynden-Bell & Pringle 1974; Hartmann et al. 1998). The viscosity transports angular momentum from the inner to outer parts of the disk, resulting in the accreting motion of the inner part and the expansion of the outer part. If we compare the disk expansion with the similarity solution in the standard –19– model for viscous accretion disks, we are able to obtain some insight into the viscosity. In the similarity solution the disk outer radius can be related to the Hα line luminosity, as shown by Equation (A11). The relation is applied to the data in Figure 6a. The best-fit −2/5 curve is given by Rout = 128(LHα/0.01L) AU for the model 2 under the condition that γ ≥ 0, suggesting γ ≈ 0. The disk expansion with γ ≈ 0 is supported by the fact that p ∼ 0 for the model 2, as shown in Figure 7: the index γ is equivalent to the power-law index p of the surface density distribution in the similarity solution. Figure 6 suggests that the disk expands radially from ∼ 100 AU to ∼ 500 AU at a time interval of ∼ 107 yr. If this is the case, then the viscosity parameter α is estimated to be ∼ 0.01 (see Equation (A7)), which is predicted in theoretical models for the MHD turbulence caused by the B-H mechanism (Balbus & Hawley 1998). The B-H mechanism is one of the most promising candidates to generate the viscosity. Recently, Tamura et al. (1999) have revealed from the detection of submillimeter polarization that toroidal components of the magnetic field are dominant in some disks, which would put a tight constraint on the viscosity models. The radius of ∼ 100 AU at the beginning of the expansion is roughly consistent with the centrifugal radii of four protostars (35 AU for B335, 40 AU for HL Tau, 82 AU for IRAS04169, and 113 AU for L1527), which are derived from the rotating motion of the infalling envelopes with specific angular momentum conserved (Ohashi et al. 1997). Here we exclude a protobinary system L1551 IRS 5 (Looney, Mundy, & Welch 1997), because the disk formation process in such a binary system with its separation comparable to the disk sizes can be significantly different from that in the case of single stars. On the other hand, the large radii of the expanded disks agree with the spatial extent of the silhouette disks in Orion, the dust debris disks found around Vega-like stars, and the hypothetical extended Kuiper belt as a possible source of short-period comets beyond the orbit of Neptune (e.g., McCaughrean & O’Dell 1996; Holland et al. 1998; Backman & Paresce 1993; Mumma, Weissman, & Stern 1993). Note, however, that recent observations (e.g., Chiang & Brown 1999; Gladman et al. 2001) suggest significant depletion of the Edgeworth-Kuiper Belt Objects beyond ∼ 50 AU, which might suggest disk truncation due to a stellar encounter (McCaughrean & O’Dell 1996; Ida, Larwood, & Burkert 2000; Kobayashi & Ida 2001) or inward movement of solid material (Stepinski & Valageas 1997; Kornet, Stepinski, & R´o˙zyczka 2001). A scatter of ∼ 100 AU remains in the disk radius even after subtracting the increasing trend in Figure 6. This scatter can be attributed to the disk formation process in an infalling envelope around a protostar (e.g., Nakamoto & Nakagawa 1994). Since the specific angular momentum of the envelope gas is shown to be conserved during infall (Momose et al. 1998; Ohashi et al. 1997), a scatter in the specific angular momentum probably generates various centrifugal radii, resulting in the scatter in the disk radius. This is supported by the fact that the centrifugal radii of the four protostars have a similar scatter of ∼ 50 AU. It is to –20– be noted that the scatter in the disk radius also increases from ∼ 50 AU to a few hundreds AU with the disk evolution, as shown in Figure 6a. The disk images around the two protostar candidates Haro 6-5B and HL Tau are ex- tended, and their disk outer radii seem to deviate from the increasing trend, as shown in Figure 6a. The large radii, however, are probably due to remaining dense envelopes. Actu- ally, the envelope around Haro 6-5B is clearly seen as a reflection nebula in the HST image by Krist et al. (1998), and Hayashi et al. (1993) found the infalling envelope around HL Tau with the NMA. The extraction of the disk components from the continuum images of Haro 6-5B and HL Tau should be done by observing gas kinematics in line emission. A similar large sub-Keplerian disk with 2000 AU radius was found around L1489 IRS by Hogerheijde (2001). Since the disk shows a sign of infall as well as rotation, he suggested that the disk is in a transient phase between a large disk growing in a collapsing envelope and a compact disk around a T Tauri star. The extended disks around Haro 6-5B and HL Tau in our sample are also likely to be in such a transient phase. On the other hand, Ohashi et al. (1997) found another large Keplerian disk with 580 AU radius around IRAS 04365: The disk shows no infalling motion, though the disk rotation is opposite to the core rotation. Therefore, we can not definitely exclude the possibility that the disks around LkCa 15 and IQ Tau in our sample were also initially large: a large scatter in the disk radius without any evolution in Figure 6a. Nevertheless, we stress that the various disk radii shown in Figure 6a can be well described by the increasing trend with a scatter, rather than by a large scatter without any trend. The maximum value of the Hα line luminosity at the beginning of the disk radial expansion is probably determined by the gravitational stability of the disks. In this survey we selected single T Tauri stars and thus the disks around them are expected to be stable against the gravitational instability. The disk stability is parameterized by the Q value

(Toomre 1964). Assuming that 0 ≤ γ ≤ 1, T0 = 300 K at 1 AU, M∗ =0.5M, and q =0.5, the stability condition that Q ≥ 1 leads to the relation of

M / M < . R / 1/4 t . ( disk(0) 1 ) ∼ 0 1( out(0) 1AU) at =0 (3)

0.5 3 0.5 Here Q = csΩK /(πGΣ), cs =(kT/µmH) , ΩK =(GM∗/r ) , Σ is given by Equation (A1), G is the gravitational constant, k is the Boltzmann constant, µ is the mean molecular weight, and mH is the mass of a hydrogen atom. Furthermore, if we assume that α =0.01, R∗ =0.01 AU, and Rdisk(0) = 50 AU at t = 0, we obtain the final relation of L ∼ . L < . L , Hα 0 002 acc ∼ 0 01  (4) or L ≤ L(max)T −(5−2γ)/(4−2γ) L(max) ∼ . L . Hα Hα g with Hα 0 01  (5) –21–

Here we adopt Equation (A10). This relation agrees with Figures 5 and 6. Consequently, the presence of the upper limit of the luminosity would mean a stable low-mass disk around a single star. < The disk inner radii shown in Table 4 are as small as ∼ 0.1 AU, indicating that the disks in our sample do not have developed gaps or large inner holes, which might be generated by forming giant planets. These small inner radii agree with the fact that most of the disks in our sample are in the active accretion stage preceding possible planet formation, though the relatively large inner radius of GM Aur might suggest the presence of a planet (e.g., Marsh & Mahoney 1992). Note that the inner radius of ∼ 1 AU for Haro 6-5B would be an artifact caused by our simplified disk models, where the self-shielding effect was not taken into account (see Yokogawa et al. 2001). After the disk accretion stage, the gaps or the inner holes would grow owing to the growth of protoplanets at ∼ 107 yr (e.g., Strom et al. 1989; Skrutskie et al. 1990; Calvet et al. 2002). Finally, the protoplanetary disks would evolve into mature planetary systems surrounded by dust rings in the main-sequence stage (e.g., Holland et al. 1998; Greaves et al. 1998; Schneider et al. 1999).

5.3.3. Disk surface density

The power-law index p of the surface density distribution has large scatters, as shown in Figure 7. The index p is mainly distributed in ranges of 0 - 1.5 and 0 - 0.5 for the disk models 1 and 2, respectively. The p values are probably linked to the angular momentum distributions in the envelopes because the disks form in the infalling envelopes. Mundy et al. (1996) tried to determine p, for the first time, in the disk around HL Tau, and showed 0 ≤ p ≤ 1, consistent with our results. Furthermore, the scatters in Figure 7 agree well with the theoretical prediction of 0 ≤ p ≤ 1 in the protostar stage by Nakamoto & Nakagawa (1994). In the model 2, the low values of p (≈ 0) for about a half of our sample are quite consistent with the disk radial expansion with γ ≈ 0 in Figure 6, as discussed in §5.3.2. This fact suggests that the low p values are not mainly due to low spatial resolutions in our survey. Understanding of the surface density distribution provides us some insight into the viscosity processes. The α model for the viscosity relates the power-law index p of the surface density distribution to the index q of the disk temperature (see Appendix). Since p becomes equal to γ in a steady state, or p ≡ γ in the similarity solution, the relation of γ = p =3/2 − q holds assuming α is constant. The range of 0.5 ≤ q ≤ 0.7 in Figure 12, however, leads to the range of 0.8 ≤ p ≤ 1.0, which is not consistent with the large scatters in Figure 7. The disagreement might suggest that α has radial dependence. On the other –22– hand, the large scatters might indicate some diversity in the viscosity processes of accretion disks. These observational results may put some constraints on more elaborate models for accretion disks (e.g., Hartmann et al. 1998), which must address the following questions: What is the origin of the viscosity? How is the viscosity well described? The disk surface density at 100 AU seems constant with decreasing Hα line luminosity, as shown in Figure 8. Here the surface density for the model 2 corresponds to the power-law r/r −p −4 < L < −2L part of Σ0( 0) . No distinct trend in Σ100AU over a wide range of 10 ∼ Hα ∼ 10  suggests various values of ν0 in the viscosity as well as the scatter in the index γ, stated above, because LHα ∝ Lacc ∝ νΣ (see Appendix). However, the disk evolution as an accretion disk suggested in Figure 6 predicts the decrease in the surface density expressed by Equation (A12). In the standard model for accretion disks, the viscosity is assumed to be independent of time and the surface density decreases with the evolution, resulting in the decrease in the accretion luminosity. The disagreement might be an artifact caused by F < our flux-limited sample, because our survey could not cover weak sources ( ν ∼ 20 mJy), 0 −2 −3 which are expected to be located at the area of Σ100AU < 10 gcm and LHα < 10 L in Figure 8a. In other words, our survey suggests that the disk surface density at 100 AU 1 −2 −1 is smaller than 10 gcm , and therefore a more sensitive survey covering Σ100AU ∼ 10 gcm−2 is required to reveal the decreasing trend in the surface density. There is another possible interpretation of the nearly constant surface density: the dynamical accretion of tenuous envelope components continues at the outer edges of the disks with small mass −8 −1 accretion rates of ∼ 10 M yr , which could compensate for the decrease in the surface density. The surface density at 100 AU agrees well with the extrapolated value at 100 AU in the Hayashi model (Hayashi et al. 1985). Since the surface density at large radii is closely related to the formation of the Edgeworth-Kuiper Belt Objects and short-period comets, all the disks in our survey have enough surface densities to make such small bodies (e.g., Yamamoto, Mizutani, & Kadota 1994). The surface density falls into a range of 0.1 − 10 gcm−2: its scatter is within two orders of magnitude as well as the scatter of the disk mass. On the other hand, the power-law index p has small values (see Figure 7): only three sources show values as large as 1.5 predicted in the Hayashi model, and the other sources have flatter distributions of the surface density compared with the Hayashi model. These flat distributions would seriously affect both the timescale of planet building (Lissauer 1987) and the formation of Jovian planets, if the low values of p in the outer parts of the disks hold even in the innermost parts, where planets will be formed. One possible solution to this problem is reshuffle of solids within the disks, i.e., inward movement of solids (Stepinski & Valageas 1997; Kornet et al. 2001). According to their models, it seems possible to change flat radial distributions of solids into steeper distributions consistent with the Hayashi model, because –23– the disk mass in our sample is comparable to the mass of the solar nebula. Anyway, it is very important to accurately determine p all over the disks by more sensitive and higher-resolution imaging in the near future.

5.3.4. Disk mass

The estimated disk mass is not proportional to the total flux density, as shown in Figure 9. This is because each disk has different distributions of the surface density and the temperature. Although the estimation of the disk mass can be highly dependent on the disk model adopted, our derived disk masses based on the two models agree with each other and range from 0.001 to 0.1 M, which is quite consistent with the previous results by Beckwith & Sargent (1996). This mass range strongly supports the Hayashi model with the minimum- mass nebula, and puts a tight constraint on any generalized model for planet formation. In addition, no distinct correlation in Figure 9 would indicate that our flux-limited sample uniformly covers the wide mass range in spite of a small sample number of 13. Figure 10 shows no distinct evolutionary trend in the disk mass except for DR Tau, RY Tau, and DO Tau with exceptionally small β, which is consistent with the previous results by Beckwith et al. (1990). However, no decrease in the disk mass with the evolution seems inconsistent with the standard model for accretion disks. As discussed in Appendix, we have M ∝ L1/(5−2γ) γ ≈ the relation of disk Hα . The power-law index of this relation is small for 0, and therefore the predicted trend is probably embedded in the scatter of the disk mass. On the other hand, no distinct increasing trend in the disk mass supports our interpretation of the disk radial expansion, because large disk radii are not simply due to large disk masses.

5.3.5. Disk temperature

We have confirmed the decrease in the disk temperature with the evolution found by Beckwith et al. (1990), as shown in Figure 11. The decrease is most likely due to the decrease in the disk accretion rate, and the scatter of the disk temperature around the evolutionary trend probably reflects different mass accretion rates. Our rough analysis shows that the disk temperature at 1 AU seems higher than those for flat and flaring passive disks, suggesting that the accretion processes are dominant in the CTTS stage. The temperature at 1 AU, however, seems lower than the equilibrium temperature in a transparent disk assumed in the Hayashi model. If we assume that the disk temperature at small radii is determined mainly by the mass accretion in the disk and that the stellar radiation is negligible, the disk –24– temperature at 1 AU, for example, is proportional to the Hα line luminosity as given by T ∝ L1/4 1AU Hα (see Equation (A15)). In fact, the decrease in the temperature in Figure 11a 1/4 can be described as T1AU = 300(LHα/0.01L) K, although the power-law index q of 3/4 for the temperature distribution in the viscous accretion disks is inconsistent with our results of q ≈ 0.5 in Figure 12. In addition, the decrease in the temperature for the LHα clock seems clearer than that for the stellar age, suggesting that the Hα line luminosity represents the disk evolution better than the stellar age. There seems no distinct evolutionary trend in the power-law index q of the temperature distribution. Figure 12 shows 0.45

5.3.6. Dust opacity law

The power-law index β of the frequency dependence of the dust opacity coefficient has been thought to be a good indicator of the growth of dust particles. Interstellar dust particles are well known to have β ∼ 2 (Hildebrand 1983). In contrast, Figure 13 shows that all the sources except DR Tau, RY Tau, and DO Tau have β ∼ 1, which is consistent with the previous results (Beckwith & Sargent 1991). Note that even the two protostars have β ∼ 1. These facts probably mean that dust particles grow in the innermost regions of protostellar envelopes, or the accretion disks (Moriarty-Schieven et al. 1994; Hogerheijde & Sandell 2000). If dust particles really grow in the accretion disks, their growth might produce an apparent flat radial profile of the disk surface density with p ∼ 0. This is because the depletion of small dust particles at small radial distances is expected to proceed more quickly than that –25– at large distances, and as a result, an intensity profile of dust continuum emission becomes flat in spite of a steep radial profile of the surface density, which leads to an apparent flat distribution of the surface density. For the three sources of DR Tau, RY Tau, and DO Tau with exceptionally small β, it is possible that the growth of dust particles is very rapid or the optical thickness of these disks is extremely large (Koerner et al. 1995; Chen, Zhao, & Ohashi 1995). We find no anomalous values of β (< 0) such as Moriarty-Schieven et al. (1994) obtained, mainly because we appropriately treated the contribution of optically thick inner regions of the disks. There seems no distinct evolutionary trend in β during the CTTS stage. Miyake & Nakagawa (1993) showed that β remains ∼ 1 even when dust particles grow. In Figure 13 one might see a slight increase with the evolution, but it would be an artifact caused by the exceptionally small values for DR Tau, RY Tau, and DO Tau.

5.4. Initial conditions for planet formation

A disk just at the end of the accretion stage provides the initial conditions for planet building. We propose that the disk around IQ Tau is a good candidate. Our imaging survey suggests that the source is about to terminate the radial expansion. Another candidate is the source V836 Tau, for which weak continuum emission from a possible disk was detected by Duvert et al. (2000). Since the intensity of V836 Tau is too weak to be imaged, one of the best candidates is IQ Tau. We compare the physical properties of the disk around IQ Tau with those of the solar nebula, i.e., the initial conditions adopted in the Hayashi model (Hayashi et al. 1985), one of the standard models for the origin of our solar system. The outer radius of the disk around IQ Tau is estimated to be about 300 AU, which is much larger than both the nebula radius of 36 AU in the Hayashi model and the radial extent of ∼ 50 AU of the Edgeworth-Kuiper Belt Objects. The large disk radius might suggest the presence of the extended Kuiper belt as a source of short-period comets, or require the redistribution of solids within the disk. The mass of the disk around IQ Tau is estimated to be 0.04 M, indicating that the disk is not self-gravitating. The disk mass is larger than the nebula mass in the Hayashi model, mainly because the disk is large. The surface density at 100 AU of the disk around IQ Tau is consistent with the extrapolated value at 100 AU in the Hayashi model. The power-law index p of the surface density distribution, however, is 0.8, much smaller than the value of 1.5 in the Hayashi model. If p =0.8 holds even in the inner part of the disk, the surface density at 1 AU becomes considerably small, making the formation of Jovian planets very difficult and the timescale of planet building longer. These serious problems can be solved by the –26– inward movement of solids within the disk, or the flat distribution might be an artifact due to the quick growth of dust grains in the disk inner region. Only higher-resolution imaging will reveal the disk inner region. The disk temperature at 1 AU for IQ Tau is derived to be 180 K, which is cooler than the temperature of 250 K for a transparent disk in radiative equilibrium with L∗ =0.65L. In the Hayashi model the nebula is assumed to be transparent at visible wavelength and the equilibrium temperature at 1 AU is 280 K for 1 L. The power-law index q of the temperature distribution for IQ Tau is 0.6, slightly larger than the value of 0.5 in the Hayashi model. These facts imply that the condensation region of icy materials is extended toward the central star IQ Tau, compared with our solar system: the condensation of H2O ice helps the growth of the cores of giant planets. The diversity found in the physical properties of the protoplanetary disks is likely to be one of the main causes that produce the diversity seen in the planetary systems including both our solar system and extra-solar systems being discovered (e.g., Marcy & Butler 1998). It has not been well understood as yet, however, which property of the protoplanetary disks produces the difference between the Hot Jupiters and our planets. Theoretically it is of great significance to generalize the planet formation processes so as to be applicable to any planetary system. The initial conditions, i.e., the starting point of the generalized models, should be provided by imaging observations with extremely high resolutions and sensitivities. Great advances in understanding of planet formation must be achieved by a next-generation array, Atacama Large Millimeter and submillimeter Array (ALMA) to be operated in ∼ 2010 in Chile.

6. Summary

We have conducted an imaging survey of the protoplanetary disks around 13 single T Tauri stars, two of which are embedded. The 2 mm dust continuum emission from the disks has been imaged with a maximum spatial resolution of one arcsecond by using the NMA. The disk images have been successfully obtained under almost uniform conditions for all the sources. We have derived the disk properties of the outer radius, the surface density distribution, the mass, the temperature distribution, and the β index of the dust opacity coefficient, by analyzing both our images and the SEDs on the basis of the two disk models: the usual power-law model and the standard model for viscous accretion disks. The correlations between the derived disk parameters and the disk clocks have been investigated. As a result, we have found possible evolution of the disks as accretion disks and some diversity in the disk properties. Our main conclusions are as follows: (1) The outer radius of the disks increases with decreasing the Hα line luminosity. This –27– trend can be interpreted as the radial expansion of accretion disks due to outward transport of the angular momentum of the disk matter, which has been predicted by the standard model for viscous accretion disks. In addition, the increasing rate of the radius suggests that γ ≈ 0 and α ∼ 0.01 in the similarity solution for the accretion disks. (2) One of the good clocks for the disk evolution during the disk accretion stage is the Hα line luminosity. (3) The disk radius has a scatter of a few hundreds AU after subtraction of the evolu- tionary trend. The scatter would be attributed to that of the specific angular momentum of infalling envelopes around protostars. (4) The surface density at 100 AU does not decrease with the evolution, which seems inconsistent with the standard model for the accretion disks. The disagreement might be an artifact caused by our insufficient sensitivity. In contrast, the surface density at 100 AU ranges from 0.1 to 10 g cm−2, which is consistent with the extrapolated value at 100 AU in the Hayashi model. This fact suggests that the outer parts of the disks have enough matter to form small bodies like the Edgeworth-Kuiper Belt Objects. (5) The power-law index p of the surface density distribution mainly falls into a range of 0 to 1, and is likely to be smaller than the index of 1.5 predicted in the Hayashi model. The low values of p, however, agree with the disk radial expansion with γ ≈ 0. Such flat distributions of the surface density might prohibit the formation of giant planets in the innermost regions of the disks, unless solids move inwards within the disks. On the other hand, the large scatters of p might indicate some diversity in the viscosity processes. (6) No distinct evolutionary trend in the disk mass is found by this study, as in the previous studies. Furthermore, our derived disk mass ranges from 0.001 to 0.1 M,as previously found. The agreement suggests that our sample uniformly covers the wide mass range in spite of its small sample number. (7) We have confirmed the decrease in the disk temperature with the evolution, which is previously suggested. The trend becomes clearer in the case of the Hα line luminosity, suggesting that the line luminosity is a good clock for the evolution. The decrease in the 1/4 temperature can be described by the standard model, where T ∝ Lacc . On the other hand, the power-law index q of the temperature distribution ranges from ∼ 0.5 to 0.7, and there is no evolutionary trend in q. (8) The index β of the dust opacity law in our sample is around 1, a typical value seen in T Tauri stars. Since even the two protostars have β ∼ 1, it seems probable that dust particles quickly grow in the accretion disks. –28–

(9) The physical properties of the disk around IQ Tau would provide one example of the initial conditions for planet formation, because our study suggests that the object is one of the most evolved sources in the disk accretion stage.

We are grateful to the staff of the Nobeyama Radio Observatory (NRO) for both operat- ing the Millimeter Array and helping us with data reduction. We also thank an anonymous referee for providing invaluable suggestions that improved this paper.

A. A similarity solution for viscous accretion disks

For a geometrically thin disk with viscous accretion, we have the following simple simi- larity solution assuming that the viscosity is independent of time and has a power-law form γ of ν(r)=ν0(r/r0) (e.g., Lynden-Bell & Pringle 1974; Hartmann et al. 1998). The surface density at radius r and time t, Σ(r, t), is given by       r −γ a r 2−γ r, t T −(5−2γ)/(4−2γ) − , Σ ( )=Σ0 g exp (A1) r0 Tg r0

T t/t t r2/ a − γ 2ν a where g =√ 0 +1,√ 0 = 0 3 (2 ) 0,and is a constant. The mass accretion rate, M˙ π r ∂ ν r acc(= 6 ∂r( Σ)), is given by      −   − a r 2 γ 2a(2 − γ) r 2 γ M˙ r, t πν T −(5−2γ)/(4−2γ) − − acc( )=3 0Σ0 g exp 1 (A2) Tg r0 Tg r0 =3πνΣ for r ≈ 0. (A3)

The total mass of the disk, Mdisk(t), is calculated by integrating the surface density over the entire disk area and we have     − 2πr2Σ a R 2 γ M t 0 0 T −1/(4−2γ) − in disk( )= g exp (A4) a(2 − γ) Tg r0 2πr2Σ = 0 0 T −1/(4−2γ) for R ≈ 0. (A5) a(2 − γ) g in

If we define the disk outer radius, Rout, as the radius within which 95% of the total mass are −(2−γ) contained, we have a/Tg =3(Rout/r0) for Rin ≈ 0 and the radial expansion of the disk is described by R t R T 1/(2−γ). out( )= out(0) g (A6) –29–

Furthermore, the timescale of the disk expansion is typically given by

5 −1 t0 ∼ 10 (α/0.01) yr (A7) for Rout(0) = 50 AU, γ =1,T0 = 280 K at 1 AU, and M∗ =0.5M. Here we adopt the α prescription for the viscosity (Shakura & Sunyaev 1973): the viscosity is given by ν = αcsH, 0.5 where α is a dimensionless parameter, cs(= (kT/µmH) ) is an isothermal sound velocity, 3 1/2 and H(= (2kTr /µmHGM∗) ) is a scale height of the disk. The similarity solution indicates that the accreting motion approaches a steady flow with a nearly constant mass accretion rate in the inner regions. In such a steady state, the disk temperature is determined from the local balance between the release of the gravitational energy of accreting matter and the radiative cooling on the disk surfaces, assuming an optically thick disk. The radial profile of the temperature is given by

˙ 1/4 −3/4 T (r)=(3GM∗Macc/(8πσ)) r for r R∗, (A8)

˙ ˙ where σ is the Stefan-Boltzmann constant and Macc is approximately given by Macc(Rin,t) with Equation (A3). If the temperature distribution is generally given by a power-law form −q of T (r)=T0(r/r0) , then the above α model leads us to the relation of γ =3/2 − q with constant α,andγ =3/4forq =3/4. The disk accretion luminosity, Lacc, is expressed by

Lacc = GM∗M˙ acc/2R∗ (A9) 3πν Σ GM∗ ≈ 0 0 T −(5−2γ)/(4−2γ). g (A10) 2R∗

−(5−2γ)/(4−2γ) Using the empirical relation of Equation (2), we have LHα ∝ Lacc ∝ Tg ,and as a result, the disk outer radius, the surface density at a fixed radius, the disk mass, and the temperature at a radius are described as a function of the Hα line luminosity. The disk outer radius Rout(t) is written in terms of LHα as follows:

R t /R L /L(max) −2/(5−2γ), out( ) out(0) = ( Hα Hα ) (A11) L(max) ∼ . L where Hα 0 01  (see Equation (4)). Similarly, the surface density at a small radius −(5−2γ)/(4−2γ) varies as Σ (t) ∝ Tg , where the exponential term is negligible, and we have

Σ(t) ∝ LHα. (A12)

−1/(4−2γ) Since Mdisk(t) ∝ Tg for Rin ≈ 0, we have the relation of

M t ∝ L1/(5−2γ). disk( ) Hα (A13) –30–

Finally, we have the following relation based on Equation (A8):

T ∝ L1/4. Hα (A14)

In particular, the temperature at 1 AU is given by

1/4 T1AU =240(LHα/0.01L) K (A15) for R∗ =0.01 AU.

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This preprint was prepared with the AAS LATEX macros v5.0. –36–

190

Fν(D) [mJy]

180 Fν(AB+C) [mJy]

Fν(AB) [mJy] 170

160

150

140

130

60

50

40 Total Flux Density [mJy]

30

20

10

0 Haro 6-5B HL Tau CY Tau RY Tau DL Tau DM Tau AA Tau DO Tau DR Tau GM Aur IQ Tau DN Tau LkCa 15 Source

Fig. 1.— Total flux densities of our target sources at 2 mm measured with the NMA. Error bars are also shown. Fν(D), Fν(AB+C), and Fν(AB) mean the total flux densities measured with spatial resolutions of 5,2,and1, respectively. –37–

4 4 4 4 Haro 6-5B HL Tau CY Tau RY Tau 3 3 3 3 2 2 2 2 1 1 1 1 0 0 0 0 -1 -1 -1 -1 -2 -2 -2 -2 -3 -3 -3 -3 -4 -4 -4 -4 -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4

4 4 4 4 DL Tau DM Tau AA Tau DO Tau 3 3 3 3 2 2 2 2 1 1 1 1 0 0 0 0 -1 -1 -1 -1 -2 -2 -2 -2 -3 -3 -3 -3 -4 -4 -4 -4 -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4

4 4 4 4 DR Tau GM Aur IQ Tau DN Tau 3 3 3 3 2 2 2 2 1 1 1 1 0 0 0 0 -1 -1 -1 -1 -2 -2 -2 -2 -3 -3 -3 -3 -4 -4 -4 -4 -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4

4 3 LkCa 15 2 1 0 -1

DECoffset [arcsec] -2 -3 -4 -4 -3 -2 -1 0 1 2 3 4 -RAoffset [arcsec]

Fig. 2.— Images of 2 mm dust continuum emission toward 13 T Tauri stars obtained with the NMA. The contour lines start at 1.5σ and −1.5σ levels with intervals of 1.5σ. The 1σ levels are 3.0, 2.8, 2.1, 2.7, 2.0, 1.9, 1.3, 1.8, 1.7, 2.0, 1.6, 2.0, and 2.3 mJy beam−1 for Haro 6-5B, HL Tau, CY Tau, RY Tau, DL Tau, DM Tau, AA Tau, DO Tau, DR Tau, GM Aur, IQ Tau, DN Tau, and LkCa 15, respectively. The negative levels are written by broken lines. The hatched ellipse at the bottom-left corner of each panel means the synthesized beam size in FWHM. Both the axes are measured from the peak positions listed in Table 1 in the unit of arcsecond. –38–

Haro 6-5B (obs) Haro 6-5B (cal) Haro 6-5B (obs-cal) HL Tau (obs) HL Tau (cal) HL Tau (obs-cal) 600 600 600 600 600 600 400 400 400 400 400 400

200 200 200 200 200 200

0 0 0 0 0 0

-200 -200 -200 -200 -200 -200

-400 -400 -400 -400 -400 -400 -600 -600 -600 -600 -600 -600 -600 -400 -200 0 200 400 600 -600 -400 -200 0 200 400 600 -600 -400 -200 0 200 400 600 -600 -400 -200 0 200 400 600 -600 -400 -200 0 200 400 600 -600 -400 -200 0 200 400 600

CY Tau (obs) CY Tau (cal) CY Tau (obs-cal) RY Tau (obs) RY Tau (cal) RY Tau (obs-cal) 600 600 600 600 600 600 400 400 400 400 400 400

200 200 200 200 200 200

0 0 0 0 0 0

-200 -200 -200 -200 -200 -200

-400 -400 -400 -400 -400 -400 -600 -600 -600 -600 -600 -600 -600 -400 -200 0 200 400 600 -600 -400 -200 0 200 400 600 -600 -400 -200 0 200 400 600 -600 -400 -200 0 200 400 600 -600 -400 -200 0 200 400 600 -600 -400 -200 0 200 400 600

DL Tau (obs) DL Tau (cal) DL Tau (obs-cal) DM Tau (obs) DM Tau (cal) DM Tau (obs-cal) 600 600 600 600 600 600 400 400 400 400 400 400

200 200 200 200 200 200

0 0 0 0 0 0

-200 -200 -200 -200 -200 -200

-400 -400 -400 -400 -400 -400 -600 -600 -600 -600 -600 -600 -600 -400 -200 0 200 400 600 -600 -400 -200 0 200 400 600 -600 -400 -200 0 200 400 600 -600 -400 -200 0 200 400 600 -600 -400 -200 0 200 400 600 -600 -400 -200 0 200 400 600

AA Tau (obs) AA Tau (cal) AA Tau (obs-cal) DO Tau (obs) DO Tau (cal) DO Tau (obs-cal) 600 600 600 600 600 600 400 400 400 400 400 400

200 200 200 200 200 200

0 0 0 0 0 0

-200 -200 -200 -200 -200 -200

-400 -400 -400 -400 -400 -400 -600 -600 -600 -600 -600 -600 -600 -400 -200 0 200 400 600 -600 -400 -200 0 200 400 600 -600 -400 -200 0 200 400 600 -600 -400 -200 0 200 400 600 -600 -400 -200 0 200 400 600 -600 -400 -200 0 200 400 600

DR Tau (obs) DR Tau (cal) DR Tau (obs-cal) GM Aur (obs) GM Aur (cal) GM Aur (obs-cal) 600 600 600 600 600 600 400 400 400 400 400 400

200 200 200 200 200 200

0 0 0 0 0 0

-200 -200 -200 -200 -200 -200

-400 -400 -400 -400 -400 -400 -600 -600 -600 -600 -600 -600 -600 -400 -200 0 200 400 600 -600 -400 -200 0 200 400 600 -600 -400 -200 0 200 400 600 -600 -400 -200 0 200 400 600 -600 -400 -200 0 200 400 600 -600 -400 -200 0 200 400 600

IQ Tau (obs) IQ Tau (cal) IQ Tau (obs-cal) DN Tau (obs) DN Tau (cal) DN Tau (obs-cal) 600 600 600 600 600 600 400 400 400 400 400 400

200 200 200 200 200 200

0 0 0 0 0 0

-200 -200 -200 -200 -200 -200

-400 -400 -400 -400 -400 -400 -600 -600 -600 -600 -600 -600 -600 -400 -200 0 200 400 600 -600 -400 -200 0 200 400 600 -600 -400 -200 0 200 400 600 -600 -400 -200 0 200 400 600 -600 -400 -200 0 200 400 600 -600 -400 -200 0 200 400 600

LkCa 15 (obs) LkCa 15 (cal) LkCa 15 (obs-cal) 600 600 600 400 400 400

200 200 200

0 0 0

-200 -200 -200 DECoffset [AU] DECoffset [AU] DECoffset [AU] -400 -400 -400 -600 -600 -600 -600 -400 -200 0 200 400 600 -600 -400 -200 0 200 400 600 -600 -400 -200 0 200 400 600 -RAoffset [AU] -RAoffset [AU] -RAoffset [AU]

Fig. 3.— Model fitting to the disk images. The adopted model is the model 1 (see text). Each set of three panels corresponds to each source. In each set, the left, middle, and right panels mean the observed image (obs), the model image (cal), and the residual image (obs- cal), respectively. The contour lines in all the panels are the same as in Figure 2. The x and y axes are in the unit of AU assuming the distance to Taurus is 140 pc. –39–

1035 1035 1035 1035 1034 Haro 6-5B 1034 HL Tau 1034 CY Tau 1034 RY Tau 1033 1033 1033 1033 1032 1032 1032 1032 1031 1031 1031 1031 1030 1030 1030 1030 1029 1029 1029 1029 1028 1028 1028 1028 1027 1027 1027 1027 1026 1026 1026 1026 1025 1025 1025 1025 1010 1011 1012 1013 1014 1015 1016 1010 1011 1012 1013 1014 1015 1016 1010 1011 1012 1013 1014 1015 1016 1010 1011 1012 1013 1014 1015 1016

1035 1035 1035 1035 1034 DL Tau 1034 DM Tau 1034 AA Tau 1034 DO Tau 1033 1033 1033 1033 1032 1032 1032 1032 1031 1031 1031 1031 1030 1030 1030 1030 1029 1029 1029 1029 1028 1028 1028 1028 1027 1027 1027 1027 1026 1026 1026 1026 1025 1025 1025 1025 1010 1011 1012 1013 1014 1015 1016 1010 1011 1012 1013 1014 1015 1016 1010 1011 1012 1013 1014 1015 1016 1010 1011 1012 1013 1014 1015 1016

1035 1035 1035 1035 1034 DR Tau 1034 GM Aur 1034 IQ Tau 1034 DN Tau 1033 1033 1033 1033 1032 1032 1032 1032 1031 1031 1031 1031 1030 1030 1030 1030 1029 1029 1029 1029 1028 1028 1028 1028 1027 1027 1027 1027 1026 1026 1026 1026 1025 1025 1025 1025 1010 1011 1012 1013 1014 1015 1016 1010 1011 1012 1013 1014 1015 1016 1010 1011 1012 1013 1014 1015 1016 1010 1011 1012 1013 1014 1015 1016

1035 1034 LkCa 15 1033

] 32 -1 10 1031

[erg s 30

ν 10 F

ν 29

2 10 d

π 1028 4 1027 1026 1025 1010 1011 1012 1013 1014 1015 1016 ν [Hz]

Fig. 4.— Model fitting to the SEDs of our sample. The adopted model is the same as in Figure 3. The gray filled circles with error bars indicate the SED data including ours. The other data are compiled from the previous papers (Adams, Emerson, & Fuller 1990; Beckwith & Sargent 1991; Beckwith et al. 1990; Bouvier et al. 1995; Cohen 1983, 1980; Dutrey et al. 1996, 1998; Duvert et al. 2000; Guilloteau & Dutrey 1998; Hayashi et al. 1993; Kenyon & Hartmann 1995; Koerner et al. 1993, 1995; Krist et al. 1998; Looney et al. 2000; Mundy et al. 1996; Myers et al. 1987; Ohashi et al. 1991; Osterloh & Beckwith 1995; Reipurth et al. 1993; Rydgren & Vrba 1983; Saito et al. 1995; Sargent & Beckwith 1991, 1987; Strom, Strom, & Vrba 1976; Strom et al. 1989, 1988; Vrba, Rydgren, & Zak 1985; Weintraub, Sandell, & Duncan 1989; Wilner et al. 1996). The best-fit curves are written by solid lines, and the stellar contribution is indicated by broken lines. –40–

100

10-1

] 10-2 [L α α α α -3 H 10 L

10-4

10-5 10-2 10-1 100 101 102 LIR [L ]

Fig. 5.— Hα line luminosity vs IR luminosity. The data are compiled from Cabrit et al. (1990). A distinct correlation is seen and the best-fit curve of (LHα/1 L)= 1.0 0.002(LIR/1 L) is shown by a broken line. –41–

(a) 1000

IRAS 04365+2535

LkCa 15

IQ Tau

Haro 6-5B DR Tau DM Tau CY Tau HL Tau DL Tau AA Tau

[AU] GM Aur DN Tau L1527 100 disk DO Tau IRAS 04169+2702

R RY Tau

HL Tau B335

10 10-2 10-3 10-4

LHα [L ]

(b) 1000

IRAS 04365+2535

LkCa 15 IQ Tau

Haro 6-5B CY Tau AA Tau DM Tau HL Tau DR Tau GM Aur [AU] DN Tau DL Tau L1527 100 disk IRAS 04169+2702 DO Tau RY Tau R

HL Tau B335

10 106 107 Age [yr]

Fig. 6.— (a) Disk outer radius vs Hα line luminosity. The filled and open circles with error bars correspond to the models 1 & 2, respectively (see text). The best-fit curve, shown by −2/5 a broken line, is Rout = 128(LHα/0.01L) AU for the model 2 under the condition that γ ≥ 0. The left panel shows the results for the two embedded sources without the abscissa and shows the centrifugal radii derived from Ohashi et al. (1997). (b) Disk radius vs stellar age. –42–

2.5 (a)

2.0 CY Tau

DN Tau 1.5 Hayashi model DR Tau

HL Tau

p 1.0

IQ Tau LkCa 15 0.5 AA Tau DL Tau DM Tau Haro 6-5B

0.0 GM Aur RY Tau DO Tau

-0.5 10-2 10-3 10-4

LHα [L ]

2.5 (b)

2.0

CY Tau

1.5 DN Tau Hayashi model DR Tau

HL Tau p 1.0 IQ Tau

LkCa 15 AA Tau 0.5 DM Tau

Haro 6-5B DL Tau

0.0 GM Aur DO Tau RY Tau

-0.5 106 107 Age [yr]

Fig. 7.— (a) Power-law index of the disk surface density distribution vs Hα line luminosity. The filled and open circles with error bars correspond to the models 1 & 2, respectively. The horizontal broken line shows p =1.5 adopted in the Hayashi model. The left panel is for the embedded sources. (b) Power-law index vs stellar age. –43–

102 (a)

] 101 -2 DL Tau GM Aur CY Tau [g cm RY Tau AA Tau Hayashi HL Tau DN Tau model DO Tau LkCa 15 100AU 100 IQ Tau Σ Σ Σ Σ Haro 6-5B DM Tau

DR Tau

10-1 10-2 10-3 10-4

LHα [L ]

102 (b)

] 101 -2 DL Tau CY Tau GM Aur

[g cm RY Tau AA Tau LkCa 15 Hayashi HL Tau model DN Tau IQ Tau 100AU 100 DO Tau Σ Σ Σ Σ DM Tau Haro 6-5B

DR Tau

10-1 106 107 Age [yr]

Fig. 8.— (a) Disk surface density at 100 AU vs Hα line luminosity. The filled and open circles with error bars correspond to the models 1 & 2, respectively. Note that the surface density for the model 2 represents the power-law part excluding the exponential term. The horizontal broken line shows the surface density extrapolated at 100 AU in the Hayashi model. The left panel is for the embedded sources. (b) Surface density vs stellar age. –44–

100

CY Tau 10-1

LkCa 15 DL Tau HL Tau IQ Tau GM Aur DN Tau Haro 6-5B [M ] AA Tau disk DM Tau M 10-2 DR Tau RY Tau DO Tau

10-3 101 102 Total Flux Density [mJy]

Fig. 9.— Disk mass vs total flux density at 2 mm (Fν(AB+C) for AA Tau, IQ Tau, and LkCa 15, and Fν(AB) for the other sources). The filled and open circles correspond to the models 1 & 2, respectively. –45–

100 (a)

CY Tau 10-1

] LkCa 15 HL Tau DL Tau GM Aur IQ Tau

[M Haro 6-5B DN Tau AA Tau disk Hayashi model M 10-2 DM Tau

DR Tau RY Tau DO Tau

10-3 10-2 10-3 10-4

LHα [L ]

100 (b)

CY Tau 10-1 ] LkCa 15 HL Tau DL Tau DN Tau GM Aur IQ Tau [M Haro 6-5B AA Tau disk Hayashi model M 10-2 DM Tau DR Tau RY Tau DO Tau

10-3 106 107 Age [yr]

Fig. 10.— (a) Disk mass vs Hα line luminosity. The filled and open circles with error bars correspond to the models 1 & 2, respectively. The horizontal broken line shows the nebula mass predicted by the Hayashi model. The left panel is for the embedded sources. (b) Disk mass vs stellar age. –46–

500 (a)

400 RY Tau

HL Tau Haro 6-5B 300 DR Tau Hayashi model [K]

DO Tau

1AU DN Tau

T 200 GM Aur IQ Tau DM Tau DL Tau LkCa 15 AA Tau

100 CY Tau

0 10-2 10-3 10-4

LHα [L ]

500 (b)

400 RY Tau

HL Tau Haro 6-5B 300 DR Tau Hayashi model [K]

DO Tau 1AU

T 200 IQ Tau DM Tau GM Aur DN Tau DL Tau AA Tau LkCa 15

100 CY Tau

0 106 107 Age [yr]

Fig. 11.— (a) Disk temperature at 1 AU vs Hα line luminosity. The filled and open circles with error bars correspond to the models 1 & 2, respectively. The thick broken line shows 1/4 the best-fit curve of T1AU = 300(LHα/0.01L) K, where the power-law index of 1/4 in the standard model is fixed. The horizontal broken line shows the temperature at 1 AU in the Hayashi model. The left panel is for the embedded sources. (b) Disk temperature vs stellar age. –47–

1.0 (a) 0.9 CY Tau 0.8

0.7 RY Tau DR Tau IQ Tau 0.6 Haro 6-5B DL Tau DN Tau GM Aur AA Tau DO Tau DM Tau

q 0.5 Hayashi model HL Tau LkCa 15 0.4

0.3

0.2

0.1

0.0 10-2 10-3 10-4

LHα [L ]

1.0 (b) 0.9 CY Tau 0.8

0.7

RY Tau IQ Tau 0.6 Haro 6-5B DR Tau DN Tau DL Tau AA Tau GM Aur DO Tau DM Tau

q 0.5 Hayashi HL Tau LkCa 15 model 0.4

0.3

0.2

0.1

0.0 106 107 Age [yr]

Fig. 12.— (a) Power-law index of the disk temperature distribution vs Hα line luminosity. The filled and open circles with error bars correspond to the models 1 & 2, respectively. The horizontal broken line shows q =0.5 adopted in the Hayashi model. The left panel is for the embedded sources. (b) Power-law index vs stellar age. –48–

2.0 (a)

1.5 LkCa 15

GM Aur DN Tau AA Tau Haro 6-5B IQ Tau HL Tau β β β β 1.0 DL Tau CY Tau

DM Tau

0.5

RY Tau DR Tau DO Tau

0.0 10-2 10-3 10-4

LHα [L ]

2.0 (b)

1.5 LkCa 15

GM Aur DN Tau Haro 6-5B AA Tau HL Tau IQ Tau β β β β 1.0 DL Tau CY Tau

DM Tau

0.5

DR Tau RY Tau DO Tau

0.0 106 107 Age [yr]

Fig. 13.— (a) β index in the dust opacity law vs Hα line luminosity. The filled and open circles with error bars correspond to the models 1 & 2, respectively. The left panel is for the embedded sources. (b) β index vs stellar age. –49–

Table 1. Source list

Sourcea RA(1950)b Dec(1950)b Fν (D)c[mJy] Fν (AB)d[mJy] Fν (AB + C)e[mJy]

Haro 6-5B 04h18m56.20s +26d50m31.6s 35 ± 436± 8 − HL Tau 04h28m44.39s +18d07m35.0s 161 ± 17 153 ± 15 −

CY Tau 04h14m27.68s +28d13m27.6s 32 ± 432± 3 − RY Tau 04h18m50.92s +28d19m33.7s 52 ± 655± 6 − DL Tau 04h30m36.06s +25d14m23.3s 52 ± 648± 5 − DM Tau 04h30m54.70s +18d03m56.1s 25 ± 325± 4 − AA Tau 04h31m53.47s +24d22m43.4s 22 ± 313± 220± 2 DO Tau 04h35m24.20s +26d04m54.4s 37 ± 537± 4 − DR Tau 04h44m13.19s +16d53m23.5s 51 ± 644± 5 − GM Aur 04h51m59.78s +30d17m13.4s 37 ± 431± 4 −

IQ Tau 04h26m47.73s +26d00m14.6s 29 ± 411± 221± 2 DN Tau 04h32m25.70s +24d08m51.5s 27 ± 325± 3 − LkCa 15 04h36m18.39s +22d15m11.6s 28 ± 515± 520± 2

aHaro 6-5B and HL Tau are thought to be borderline sources between protostars and classical T Tauri stars. IQ Tau, DN Tau, and LkCa 15 are presumably borderline sources between classical and weak-line T Tauri stars. The other sources are typical classical T Tauri stars. bPeak positions in the 2 mm continuum images obtained with the NMA in the AB or AB+C configura- tions. cTotal flux density measured with NMA 5beam.

dTotal flux density with 1beam. eTotal flux density with 2beam. –50– ]  L [ g α H L [mag] f c R 02, except Haro 6-5B and HL . ˚ A] [ e =0 α Z [mag] WH d V A [AU] c = 23 and 24 mag just toward the embedded sources of Haro ∗ V R A ]  L [ c (fit) ∗ L ]  L [ b ∗ Table 2. Stellar properties L [K] b eff T [yr] a ] Age . The stellar radii are determined from the SED fitting based on the disk model 1. For Haro 6-5B and HL Tau, the radii  4 eff M [ a σT ∗ 2 ∗ M πR line luminosities are calculated following the work by Cabrit et al. (1990). α (fit)= 4 ∗ Kenyon & Hartmann (1995)except Haro 6-5B and HL Tau. The temperatures of the two protostars areOsterloh determined & so Beckwith as (1995)for to Haro reproduce 6-5B. the Beckwith et al. (1990)for HL Tau, DL Tau, and DR Tau. Kenyon & Hartmann (1995)for the The H The stellar masses and ages are calculated by the internet server (Siess, Dufour, & Forestini 2000)with L Cabrit et al. (1990)for RY Tau, DL Tau, AA Tau, DN Tau, DO Tau, DR Tau, and GM Aur. Strom et al. (1989)for the other sources. Strom et al. (1989). IQ Tau 0.52 1.50E+06 3785 0.65 0.54 0.008 1.25 8 12.43 2.79E-04 Source HL TauRY Tau (0.55)-DL Tau 2.37 (4060)- 0.76 - 2.11E+06 2.53E+06 (1.13)(0.010)(3.50) 5080 4060 (13.47)- 7.60 0.77 4.66 0.91 0.013 0.009 1.84 1.35 22 138 10.28 12.11 8.78E-03 7.13E-03 CY Tau 0.48 1.91E+06 3720 0.47 0.29 0.006 0.10 70 12.38 1.13E-03 AA TauDR Tau 0.77 0.74 2.66E+06 4060 7.20E+05 4060 0.74 2.50 0.41 1.62 0.006 0.012 0.49 0.95 21 77 12.05 10.91 6.10E-04 8.96E-03 DN Tau 0.56 1.17E+06 3850 0.91 0.74 0.009 0.49 15 11.40 7.93E-04 DO Tau 0.56 8.84E+05 3850 1.20 1.31 0.012 2.64 101 13.11 5.35E-03 DM Tau 0.47GM Aur 4.32E+06 3720 1.17 1.29E+07 0.25 4730 0.20 0.83 0.005 0.52 0.00 0.005 140 0.14 12.99 71 1.20E-03 11.22 3.43E-03 LkCa 15 1.10 6.40E+06 4350 0.74 0.73 0.007 0.62 13 11.31 8.03E-04 a b c d e f g Haro 6-5B (0.10)- (4000)- - (1.06)(0.010)(3.80) (19.50)- 6-5B and HL Tau, respectively, to take into account the attenuation of the stellar radiation due to the disks. SEDs in the range from near IR to UV. The stellar luminosities of DL Tau and DR Tauother are sources. from Beckwith In et calculating al. the (1990). stellar contribution to the SEDs, we assume are assumed to be 0.01 AU. Tau. For the masses of Haro 6-5B and HL Tau, we refer to Osterloh & Beckwith (1995)and Beckwith et al. (1990),respectively. –51–

Table 3. Source sizes based on Gaussian fitting

a a a b b b c Source Smajor [arcsec] Sminor [arcsec] PA [deg] Bmajor [arcsec] Bminor [arcsec] PA [deg] Seeing [arcsec]

Haro 6-5B 3.40 ± 0.51 1.91 ± 0.49 155 ± 8 1.2 1.1 134 0.38 HL Tau 1.04 ± 0.03 0.60 ± 0.04 144 ± 2 1.2 1.1 7 0.37 CY Tau 0.91 ± 0.10 0.50 ± 0.12 68 ± 8 1.3 1.1 62 0.28 RY Tau 0.72 ± 0.06 0.37 ± 0.09 27 ± 7 1.0 1.0 33 0.21 DL Tau 1.15 ± 0.06 0.78 ± 0.08 52 ± 6 1.0 1.0 47 0.26 DM Tau 2.46 ± 0.18 0.92 ± 0.17 134 ± 4 1.2 1.1 55 0.32 AA Tau 1.34 ± 0.10 0.61 ± 0.13 86 ± 5 1.4 1.3 51 0.26 DO Tau 1.01 ± 0.05 0.80 ± 0.07 67 ± 9 1.0 1.0 71 0.18 DR Tau 0.60 ± 0.05 0.49 ± 0.07 156 ± 17 1.1 1.0 4 0.25 GM Aur 1.47 ± 0.10 0.84 ± 0.11 57 ± 6 1.0 1.0 33 0.33 IQ Tau 1.54 ± 0.11 0.81 ± 0.15 41 ± 6 1.6 1.5 154 0.53 DN Tau 0.50 ± 0.11 0.19 ± 0.28 109 ± 19 1.0 1.0 37 0.21 LkCa 15 2.09 ± 0.18 0.60 ± 0.37 79 ± 5 2.1 1.6 146 0.48

aBeam-deconvolved sizes in the 2 mm images.

bSizes of the synthesized beams in FWHM. cSizes of the seeing disks in FWHM. –52– 2 1 4 2 1 5 2 2 5 2 aint 4 6 5 4 6 8 2 2 4 5 4 7 5 16 12 11 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± tion disk 2152 1144 385 531 4155 448 588 3128 443 464 775 329 1144 678 335 588 361 385 4131 532 3128 461 565 3128 14 120 12 120 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± [deg] PA [deg] 1703 69 22 53 18 52 44 16 59 11 43 26 66 5610 66 45 13 28 29 25 28 71 03 53 29 28 3030 79 26 79 65 10 42 21 51 21 80 29 77 18 43 61 67 10 42 13 25 21 80 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 04 96 87 37 02 23 51 29 21 92 10 07 95 08 18 08 48 29 96 69 12 39 30 29 90 82 ...... 0500 1 07 0 02 0 0 04 1 04 1 04 1 0404 0 1 02 0 02 1 04 1 00 0 02 1 0402 1 02 1 1 01 0 06 0 03 0 02 1 02 0 04 0 01 0 02 0 03 0 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± qβi 58 51 87 63 57 54 47 62 55 59 55 63 51 55 63 59 47 53 81 52 59 64 62 54 59 54 ...... 20 30 50 60 20 60 80 50 70 30 70 50 30 30 70 44 0 70 13 0 33 0 17 0 18 0 2217 0 0 60 13 0 22 0 [K] ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 1AU 02 364 12 337 13 297 0935 87 406 10 162 09 153 2309 184 183 11 152 02 365 14 155 1616 301 11 186 69 211 18 197 151 58 231 14 182 46 190 23 93 31 231 65 405 14 304 09 162 12 181 ...... 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± pT 21 05 00 79 01 24 57 01 81 62 78 02 72 00 65 81 15 00 47 54 65 00 00 41 00 11 ...... ] 2 − 10 0 40 0 23 1 10 0 21 0 4131 1 0 8826 0 0 0719 0 22 0 0 41 0 5523 0 1727 0 02 0 34 0 0 0 84 0 11 0 60 1 71 0 31 0 06 1 15 0 ...... 2 1 0 0 0 0 4 0 0 1 0 0 0 0 0 1 0 4 0 0 0 1 0 3 0 0 (1)disk model 1 (2)disk model 2 ± ± [g cm ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 32 44 23 91 61 93 41 33 99 21 55 82 05 75 15 66 38 88 61 32 80 87 95 74 35 29 ...... 100AU Σ ] 015 2 013 0 022 1 031021 3 2 018011 6 1 018011 5 020 1 1 012 4 017004 3 028025 1 146 6 050 2 6 2 006 1 068 15 004 0 097 1 005 1 014022 3 10 003 0 014 1  ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 M [ ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± disk 050 031 031 129 006 053 029 042 038 059 048 028 006 040 044 025 058 004 078 012 033 004 006 051 006 023 ...... M Table 4. Physical properties of the protoplanetary disks 003 0 540 0 403 0 012032 0 0 009 0 012 0 254020 0 0 031 0 003 0 013 0 078260 0 026 0 021 0 890 0 0 012 0 011 0 022 0 019 0 012 0 031009 0 0 080 0 023 0 ...... 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 [AU] ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± (AB or AB+C). in ν F R 063 164 851 038 143 023 016 618 035 012 063 016 130 638 045 032 013 031 032 017 030 032 141 023 133 023 ...... 0 0 0 0 0 0 0 0 0 0 0 0 65 1 85 0 28 0 29 0 47 0 41 0 8428 0 0 66 0 (D)and 27 51 29 37 38 111 0 52 0 469 0 141 0 70 86 86 62 83 114 134 128 0 ν [AU] +28 − +63 − +63 − +91 − +47 − ± ± ± ± ± ± ± ± ± +403 − +332 − +191 − +123 − +522 − +437 − +443 − a F ± ± ± ± ± out R (model)= ν F If positive and negative errors are not the same, the positive errors are defined by the differences between the two disk radii derived with the flux constr IQ Tau 329 IQ Tau 598 Source HL Tau 322 HL Tau 459 RY TauDL Tau 81 152 RY TauDL Tau 116 208 CY Tau 211 CY Tau 229 AA Tau 214 DN Tau 147 DN Tau 174 DR Tau 193 DR Tau 221 AA Tau 260 DO Tau 98 DO Tau 143 GM Aur 233 DM Tau 220 LkCa 15 483 GM Aur 151 LkCa 15 383 DM Tau 422 Note. — In the disk model 1, a disk is assumed to have the usual power-law forms of the surface density and temperature variations. In the disk model 2, the a Haro 6-5B 318 Haro 6-5B 412 variation of the diskmodel, surface while density the has radial a profile power-law of form the with disk an temperature exponential is tail, similar the to same that as in in the one model of 1 the (see similarity text solutions for of further the details). standard accre using