Herschel Observations of Protoplanetary Disks in Lynds 1641^*

L1641 is a relatively well-studied region, and the works of Fang et al. (2009, 2013), Hsu et al. (2012), Caratti o Garatti et al. (2012), and Kim et al. (2013, 2016) provide stellar properties for a subset of its YSO population. Gâlfalk & Olofsson (2008) derive an age of ∼1 Myr for region. Fang et al. (2013) found similar median ages, with 1.5 Myr for the "clustered" YSOs and 1.6 Myr for the "isolated" YSOs. We adopt 1.5 Myr for our analysis.⁸

From the 169 Class II objects discussed in Section 2, we remove objects that meet any of the following criteria: (1) their Herschel maps show nearby close sources or nebulosity; 49 are flagged for this reason. (2) They have visual extinctions, A_V, equal to or greater than 15; this applies to 18 systems. (3) They have colors uncharacteristic of classical T Tauri stars (CTTSs); this applies to one object. Including the targets with nearby sources or nebulosity would make the 70 μm fluxes for those objects more uncertain due to contamination, thus making our analysis using the 70 μm data unreliable. Objects with A_V ≥ 15 will have higher uncertainties and we remove them to avoid introducing biases. The one source that does not have colors characteristic of a CTTS is an A star and should not be compared directly to the rest of the sample. We present the Herschel photometry and stellar properties for these flagged objects in the Appendix, but these objects are not included in the rest of the analysis. The 104 protoplanetary systems analyzed in this work are marked in the L1641 column density map shown in Figure 1 (Stutz & Kainulainen 2015; Stutz & Gould 2016).

For our sample of 104 systems, we obtained literature values of spectral types for 75 objects, A_V values for 98 objects, luminosities (L) for 77 objects, and mass accretion rates ($\dot{{\text{}}M}$) for 61 objects. These values and their references are listed in Table 2. More than 75% of the spectral types in this sample come from Hsu et al. (2012) using the Hernández et al. (2004) spectral-typing process, with typical uncertainties of less than one subclass. The median spectral type for the sample is M1 (Figure 2). Additionally, for the flagged sample we have spectral types for 31, A_V values for 59, luminosities for 40, and mass accretion rates for 24 objects. These values are available in Table 8 in the Appendix.

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Table 2.  Stellar Properties

M12 Num	SpT	SpT References	AV	AV References	L	L References	log $\dot{{\text{}}M}$	$\dot{{\text{}}M}$ References
					(L⊙)		(${\text{}}{M}_{\odot }\,{\mathrm{yr}}^{-1}$)
198	M2.0	a	7.71	b	2.17	b	<−8.29	b
217	⋯	⋯	10.82	CTTS J–H	⋯	⋯	⋯	⋯
223	M2.5	a	0.91	b	0.25	b	−8.2	c
225	K4.0	a	5.46	b	4.04	b	<−8.5	c
227	K0.0	a	7.2	c	2.501	c	⋯	⋯
228	⋯	⋯	6.21	CTTS J–H	⋯	⋯	⋯	⋯
231	M1.5	a	4.7	b	0.95	b	−8.6	c
232	⋯	⋯	6.21	CTTS J–H	⋯	⋯	⋯	⋯
233	⋯	⋯	6.72	b	3.54	b	⋯	⋯
250	K1.0	a	5.43	b	11.71	b	−8.2	b
263	M2.5	a	5.3	c	0.527	c	<−9.4	c
269	M3.0	b	6.92	b	0.47	b	−8.02	b
278	K6.0	a	4	c	0.896	c	<−8.6	c
282	M2.5	a	2.92	b	0.35	b	−9.2	b
284	⋯	⋯	10.82	CTTS J–H	⋯	⋯	⋯	⋯
291	⋯	⋯	8.25	CTTS J–H	⋯	⋯	⋯	⋯
294	M3.0	a	3.4	b	0.32	b	−9.15	c
296	M2.0	a	2.14	b	0.35	b	−9.05	b
307	K7.0	a	4.1	c	0.575	c	−7.75	c
313	M3.0	a	5.3	CTTS J–H	⋯	⋯	⋯	⋯
342	M2.0	a	1.41	b	0.95	b	<−8.52	b
378	K7.0	d	6.88	b	1.95	b	−6.74	d
383	F	d	6.6	b	59.31	b	⋯	⋯
387	M0	d	7.9	d	1.126	d	⋯	⋯
399	K5.0	d	0.03	b	5.83	b	−6.69	d
400	M0.0	a	4.5	d	0.047	d	⋯	⋯
402	M1.5	d	8	b	1.05	b	<−9.09	b
403	K3.5	d	5.07	b	1.9	b	−8.4	d
411	M7.5	a	0.59	b	0.4	b	−8.22	b
428	⋯	⋯	8.46	b	1.05	b	⋯	⋯
429	M3.0	a	8.2	d	1.723	d	−7.15	d
434	K7.0	a	0.9	b	0.55	b	−8.05	c
463	K7.0	a	4.26	b	1.03	b	<−8.96	b
466	M4.0	a	0	c	0.014	c	−10.7	c
468	⋯	⋯	8.55	CTTS J–H	⋯	⋯	⋯	⋯
471	⋯	⋯	9.07	CTTS J–H	⋯	⋯	⋯	⋯
476	M3.0	a	4.2	d	0.187	d	−8.92	d
477	M1.0	a	4.6	d	0.566	d	−8.26	d
483	G1.0	d	10.6	d	8.105	d	−6.54	d
485	M0.0	a	6.33	b	0.52	b	<−9.55	b
487	M0.5	a	2.1	c	1.134	c	<−9.15	c
488	⋯	⋯	10.77	CTTS J–H	⋯	⋯	⋯	⋯
491	⋯	⋯	12.26	CTTS J–H	⋯	⋯	⋯	⋯
494	⋯	⋯	9.85	b	0.36	b	⋯	⋯
512	M5.5	a	4.1	CTTS J–H	⋯	⋯	⋯	⋯
525	M1.0	a	4.26	b	1.13	b	−6.8	c
530	⋯	⋯	9.94	CTTS J–H	⋯	⋯	⋯	⋯
556	M1.5	a	0.87	b	0.33	b	−8.67	b
561	⋯	⋯	13.1	b	2	b	⋯	⋯
574	M3.0	a	5.7	c	0.509	c	−8.3	c
579	⋯	⋯	14.6	b	4.17	b	⋯	⋯
582	K3.0	a	0.39	b	0.14	b	−9.92	b
597	⋯	⋯	3.38	CTTS J–H	⋯	⋯	⋯	⋯
598	⋯	⋯	11.95	CTTS J–H	⋯	⋯	⋯	⋯
619	⋯	⋯	12.54	CTTS J–H	⋯	⋯	⋯	⋯
626	M4.0	a	4.2	c	0.8	c	−7.75	c
633	M3.0	a	4.2	b	1.07	b	<−8.47	b
637	K8.0	a	1.5	c	0.978	c	−8.8	c
641	K4.5	d	8.89	b	2	b	⋯	⋯
644	M1.0	d	10.67	CTTS J–H	⋯	⋯	⋯	⋯
653	K2.0	a	3.4	c	2.045	c	−8.4	c
654	M0.5	a	4.3	d	0.243	d	−8.94	d
663	⋯	⋯	11.26	CTTS J–H	⋯	⋯	⋯	⋯
666	⋯	⋯	11.34	CTTS J–H	⋯	⋯	⋯	⋯
673	M5.0	a	2.7	d	0.136	d	⋯	⋯
677	⋯	⋯	10.7	CTTS J–H	⋯	⋯	⋯	⋯
680	M2.0	a	2.6	d	0.183	d	⋯	⋯
689	M3.0	a	3.5	c	0.339	c	−8.8	c
729	K7.0	a	0.9	d	0.939	d	⋯	⋯
734	M2.4	a	1.4	d	1.324	d	−7.29	d
751	M3.5	a	1.41	b	0.31	b	−9.31	b
755	M1.0	a	0.66	b	0.45	b	−7.9	c
761	M0.0	b	8.25	b	1.04	b	⋯	⋯
762	M2.5	a	2.31	b	0.85	b	−7.79	d
792	⋯	⋯	0.0	CTTS J–H	⋯	⋯	⋯	⋯
798	M1.5	a	8.01	b	1.41	b	−8.09	d
811	M5.0	a	0.4	c	0.197	c	−9.85	c
818	M3.0	a	0	c	0.228	c	−10.1	c
832	G8.0	e	0.42	CTTS J–H	⋯	⋯	⋯	⋯
847	G3.0	a	2.61	b	10.44	b	−7.89	b
848	K7.5	a	3.94	b	2.05	b	−6.35	b
853	M0.0	a	0.24	b	0.15	b	<−10.6	c
874	M4.0	a	0	d	0.187	d	−8.9	d
887	G0.5	c	9.8	c	13.259	c	−6.9	c
914	M5.0	a	0.4	d	0.232	d	−9.83	d
920	M2.5	a	0.21	b	0.3	b	<−8.82	b
926	K6.5	a	0.9	d	0.45	d	⋯	⋯
930	M0.0	a	1	d	1.839	d	−7.93	d
971	G8.0	a	2.8	c	4.469	c	<−8.15	c
980	⋯	⋯	0.35	b	8.82	b	−7.85	b
994	M1.0	a	2.3	d	0.569	d	−8.92	d
1001	K8.0	a	2.6	c	0.723	c	−8.35	c
1006	M0.5	a	0.41	b	0.45	b	<−9.62	b
1007	M1.5	a	0.2	d	0.355	d	−8.18	d
1011	K4.0	a	0.9	c	1.736	c	−7.65	c
1020	K2.0	a	12.3	b	1.6	b	⋯	⋯
1039	M0.0	a	1.78	b	1.5	b	<−8.77	b
1086	M0.0	a	0.08	b	0.66	b	−8.51	b

Note. For sources with an A_V reference of CTTS J–H, visual extinctions were measured in this work as described in Section 3.1.

References. (a) Hsu et al. (2012), (b) Kim et al. (2016), (c) Fang et al. (2013), (d) Fang et al. (2009), (e) Hsu et al. (2013).

A machine-readable version of the table is available.

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Of the 98 A_V determinations, 77 were obtained from the literature (see the Appendix for the flagged sample). The rest of the A_V determinations were calculated in this work using the observed J, H, and K_S photometry bands from the Two Micron All-Sky Survey (2MASS, Skrutskie et al. 2006), the Mathis (1990) reddening law (with an R_V = 3.1; the extinction curve in the JHK-bands is similar for the McClure (2009) law and we adopt A_H/A_J = 0.624 and A_K/A_J = 0.382), and by calculating the intrinsic color from the CTTS locus from Meyer et al. (1997),

$$\begin{eqnarray}&&{(J\mbox{--}H)}_{\mathrm{CTTS}}=0.58\pm 0.11{(H\mbox{--}{K}_{S})}_{\mathrm{CTTS}}+0.52\pm 0.06,\end{eqnarray} \tag{ 2 }$$

after converting the locus to the 2MASS photometric system using conversions presented by Carpenter (2001). For objects with Megeath et al. (2012) ID numbers 612 and 792, the A_V calculated in this manner leads to slightly negative values that are nonphysical, and thus here we adopt A_V values of 0.0 (Table 2). Figure 3 shows the J–H versus H–K_S color–color diagram for the sample with the CTTS locus shown as a solid red line. The dwarf branch from Bessell & Brett (1988) and the CTTS locus shown in Figure 3 have been converted to the 2MASS photometric system as described above. The distribution of the visual extinction values gathered from the literature and calculated in this work from the CTTS locus for the sample is shown in Figure 4. The A_V distribution indicates that the literature values tend to be lower than those calculated from the CTTS locus, likely because the literature values (taken mostly from Fang et al. 2009, 2013 and Kim et al. 2016) were calculated using photospheric colors based on available spectral types, which biases the distribution toward low extinction values where spectral typing is more accurate.

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The values for L and $\dot{{\text{}}M}$ are collected from the literature (Fang et al. 2009, 2013; Kim et al. 2013, 2016). Fang et al. (2009, 2013) use the Hα, Hβ, and He i line luminosities, as well as the full width of Hα at 10% to derive $\dot{{\text{}}M}$. Accretion rates from Fang et al. (2009, 2013) agree within 50%–250%. Kim et al. (2013, 2016) use the Paγ, Paβ, and Brγ line luminosities. Accretion rates from Kim et al. (2016) range from being ∼3% to ∼7000% of the values in Fang et al. (2013) for the same object. These differences in mass accretion rates could be due to variability and/or differences in observational proxies (i.e., the lines used to calculate the mass accretion rates). We adopt the most recently published values.

We note that using Infrared Telescope Facility/SpeX (Rayner et al. 2003) observations Kim et al. (2016) determined objects with Megeath et al. (2012) ID numbers 250 and 980 to be binaries. Kounkel et al. (2016), using the Hubble Space Telescope and the NICMOS and WFC3 cameras, searched for binaries with separations between 100 and 1000 au; 25 out of 169 of the objects in our Class II sample are included in this survey. Of these 25, 8 are found to be binaries; these are objects 315, 369, 421, 523, 526, 561, 579, and 950. Of the known binaries, only two (561 and 579) remain in the sample after object flagging. With roughly 15% of the sample studied for binaries (and mostly at large separations), we cannot rule out that other objects in our sample are binaries.

We constructed the SEDs of the sources using data from 2MASS at the J-, H-, and K_S- bands, Spitzer IRAC at bands 3.6, 4.5, 5.8, and 8 μm, and Spitzer MIPS at 24 μm, along with the Herschel PACS 70 μm photometry presented in this work. Additionally, 40 of the objects have V- and I-band photometry from Hsu et al. (2012) and 64 have Spitzer Infrared Spectrograph (IRS; Houck et al. 2004) spectra from Kim et al. (2013, 2016) with an additional 18 available in the Combined Atlas of Sources with Spitzer IRS Spectra archive (Lebouteiller et al. 2011). In Figure 5, we present the SEDs of the 98 sources that have A_V determinations. In Figure 13, we show the SEDs of the 58 sources that have A_V determinations, but have been flagged as described in Section 3.1. Objects without A_V values lack 2MASS data, and thus we cannot calculate A_V from the CTTS locus. The values of A_V that are listed in Table 2 were used to correct the photometry, and spectroscopy if available, for extinction. Objects with A_Ks < 0.3 are dereddened using the Mathis (1990) extinction law with R_V = 3.1, and those with A_Ks ≥ 0.3 are dereddened using the McClure (2009) extinction curves. Each object is listed with its corresponding identification number from Megeath et al. (2012); spectral type, if available; and A_V.

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View extended figure (13 panels)

The dereddened SEDs are plotted with the photospheric fluxes corresponding to the adopted spectral type (Kenyon & Hartmann 1995), scaled at the J-band where the peak of the photospheric emission of K- and M-type stars occurs, although excesses from accretion, hot gas, and hot dust may still contribute at this wavelength (Fischer et al. 2011; McClure et al. 2013a). For objects with spectral types later than M6.0 (i.e., M7.0 or M7.5 objects), M6.0 photospheres are shown. For objects with spectral type K8.0, K7.0 photospheres are shown. This is due to the lack of M7.0 and K8.0 spectral types in Kenyon & Hartmann (1995). The SEDs are also plotted with the median dereddened SED of Taurus in the K5.0–M2.0 spectral type range (Furlan et al. 2006). We use the Taurus median as a proxy for a typical protoplanetary disk. The Taurus median is scaled at the H-band, where the flux is still dominated by stellar emission in T Tauri stars. Here we have extended the median out to the 70 μm PACS wavelength using data from the Herschel Open Time Key Project's Gas in Protoplanetary Systems, presented in Howard et al. (2013). Additionally, we plot the Taurus quartiles (i.e., the range within which 50% of the objects fall) for reference.

To characterize the properties of the disks in our sample, we first determine if our objects are FDs or TDs. FDs have optically thick material extending from the dust sublimation radius near the star to the outer disk. TDs contain holes or gaps in their dust distribution that can be inferred from a dip in the NIR and MIR flux in the SED (see Espaillat et al. 2014 for a review).

A considerable number of ways to identify TDs are outlined in the literature. These include the use of photometric colors and spectral indices, essentially slopes between photometry or spectral points (e.g., Furlan et al. 2006, 2009, 2011; Gutermuth et al. 2008; Fang et al. 2009, 2013; Cieza et al. 2010; Merín et al. 2010; Muzerolle et al. 2010; Koenig et al. 2012; Kim et al. 2013, 2016; Ribas et al. 2013). The photometric bands for the colors and indices should be chosen such that they effectively separate the TDs from the FD population, while minimizing biases from emission features, such as silicate features. We implement some of the color and index criteria used in the literature on our sample; we did not use classification systems that relied on IRS spectra as we do not have these data for our entire sample. Here we list the criteria implemented on our sample:

• 1.  
  Fang et al. (2009, 2013) use K_S–[5.8] versus [8]–[24] with boundaries for TDs defined by [8]–[24] ≥ 2.5 and K_S–[5.8] ≤ 0.56 + ([8]–[24]) × 0.15.
• 2.  
  Merín et al. (2010) use [3.6]–[8] versus [8]–[24] with two regions separating systems with only photospheric fluxes and systems with some excess flux above the photosphere in the IRAC bands: Region A: 0.0 < [3.6]–[8] < 1.1 and 3.2 < [8]–[24] < 5.3; Region B: 1.1 < [3.6]–[8] < 1.8 and 3.2 < [8]–[24] < 5.3. Kim et al. (2013), using their Orion A sample, finds that Region A corresponds to classical TDs, defined as disks with Inner Disk Excess Fractions (IDEF) < 0.25 (see Equation (8) of Kim et al. 2013) and Region B to weak-excess TDs, 0.25 ≤ IDEF < 0.5, and pre-TDs, IDEF ≥ 0.5.
• 3.  
  Cieza et al. (2010) use [3.6]–[24] > 1.5 and [3.6]–[4.5] < 0.25 as the broad criteria for TDs in their Ophiuchus sample.
• 4.  
  Muzerolle et al. (2010) use slopes α_{[3.6]–[5.8]} < −1.8 and −1.5 ≤ α_[8]–[24] ≤ 0.0 for weak-excess sources (sources with some MIR excess that is above the photosphere but below what we would expect for optically thick material; see Muzerolle et al. 2010 for full discussion) and α_[8]–[24] > 0.0 for classical TDs.

The criteria discussed here are shown in Figure 6. To classify disks as transitional in our sample, we require that they meet three out of the four criteria listed above (we consider objects to be candidate TDs if they meet one or two, but not three, of the criteria). We identify 24 objects in our sample as TDs and an additional 25 as candidates; the TDs and candidate TDs are listed in Table 3. This leads to ∼23% TDs among the sample with available A_V, eight of which are newly identified. The eight newly identified TDs have Megeath et al. (2012) ID numbers of 227, 250, 278, 387, 403, 619, 689, and 811. A TD fraction of ∼23% is higher than the ∼10% at 1.5 Myr inferred by Muzerolle et al. (2010) using Spitzer photometry. The discrepancy could be due to broader selection criteria than Muzerolle et al. (2010), a bias that the TDs tend to be bright at PACS 70 μm because of illuminated edge of the disk (discussed more in Section 4.2), or to uncertainties in the age of L1641. We note that a TD identification criterion using Wide-field Infrared Survey Explorer (WISE; Wright et al. 2010) 12 μm photometry and Herschel PACS 70 μm photometry has been used by Ribas et al. (2013), Bustamante et al. (2015), and Rebollido et al. (2015). These groups used the spectral index α_[12]–[70] > 0.0 to identify TDs. A similar method can be used for this sample, using the IRAC 8 μm data instead of the WISE photometry. Using α_[8]–[70] > 0.0 as an identifier for this sample leads to 21 TDs. Compared to the identifications reported in this work, 16 overlap with the TD sample of 24 (67% overlap), 3 with the candidate TDs, and 2 with the FDs.

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We present the median SEDs for the entire L1641 sample (Table 4), as well as the FD median (Table 5) and TD median (Table 6), along with their quartiles. In Figure 7, we show the median of the FD and TD samples in L1641 compared to the Taurus median. We use photometry from the V-band to 70 μm to construct the median. We determine the dereddened fluxes using the A_V values listed in Table 2, as described in Section 3.2. In order to reduce spread in the SEDs due to differences in stellar parameters and unknown distances, we normalize each object's SED to the median H-band flux for the sample, as is done for the Taurus median computed by Furlan et al. (2006), before computing the median at each photometry point. Flagged objects are not included in the median SED.

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The median SED for the whole sample (FDs and TDs) is dominated by the FD flux as they are more numerous. The L1641 FD median follows the Taurus median quite well. When the TDs are separated from the FDs, a clear distinction arises between the two, namely, that the TD median is much lower than the FD median at NIR and MIR wavelengths, while it is slightly higher at 70 μm. A similar result was found for TDs with Herschel data in Chamaeleon by Ribas et al. (2013).

We use the D'Alessio irradiated accretion disk (DIAD) models (D'Alessio et al. 1998, 1999, 2001, 2006) to study to our protoplanetary disk sample, allowing us to identify trends in disk properties. These models calculate the structure and emission of accretion disks irradiated by the central star. The models include the effects of dust settling, by having two grain populations, each with a size distribution n(a) ∝ a^−3.5 between a_min and a_max. The population in the upper layers has a_max = 0.25 μm, while the disk midplane population has a_max = 1 mm (D'Alessio et al. 2006). For both populations, a_min = 0.005 μm. We assume a dust mixture of silicates and graphite with silicate mass abundances relative to gas of 0.004 and graphite fractional abundances of 0.0025, with opacities calculated with Mie theory using optical constants from Dorschner et al. (1995) for olivine silicates and from Draine & Lee (1984) for graphite. The input model parameters that we keep fixed are the disk radius, the height of the directly irradiated inner disk edge or the "wall," the viscosity parameter (α), which is parameterized using the methods of Shakura & Sunyaev (1973), and the stellar parameters for the central star (mass, radius, and effective temperature). We vary the following input parameters: the mass accretion rate ($\dot{{\text{}}M}$), the inclination angle of the disk to the line of sight (i), and the dust-to-gas mass ratio in the upper layers of the disk in units of the total dust-to-gas mass ratio (, with a value of 1 corresponding to an unsettled disk, while a low value of 0.0001 corresponds to a factor 10⁴ depletion of small grains in the upper layers of the disk atmosphere). We discuss the input values in Section 4.2.

We calculate theoretical SEDs from our grid of models, convolving the theoretical fluxes with the filter response at each band to create synthetic colors that we use to calculate the indices for each model. We select the ${n}_{{K}_{S}-[70]}$ index (see Equation (1)) as an indicator of the total flux in the PACS 70 μm band relative to the star. We adopted K_S-band magnitudes as our anchoring point instead of a shorter wavelength filter because the K_S-band is less affected by reddening and a large fraction of our sample is heavily reddened (Figure 4). We performed a two-sample Kolmogorov–Smirnov (K–S) test between the ${n}_{{K}_{S}-[70]}$ and n_J−[70] distributions for the total samples, the FDs, and the TDs separately and find that there is no significant difference between using J and K_S as our anchoring point.

In this work, we focus on the effects of varying mass accretion rates, inclinations, and dust settling and how these parameters affect our index comparison with observations in L1641. The mass of the disk depends on the surface density and radius. The surface density depends on $\dot{{\text{}}M}$/α. Here, when we fix α and the radius and vary $\dot{{\text{}}M}$, we are essentially varying the mass of the disk (D'Alessio et al. 1998).

Inclination can have significant effects on model SEDs (D'Alessio et al. 2006), especially in the edge-on case. When the disk is edge-on, the star is extinguished by the disk. We note that we do not expect any of the disks in this sample to be edge-on, as the extinction of the central star in the optical and near-infrared wavelengths would have prevented these disks from being identified as Class II objects. Besides the edge-on case, the closer to edge-on the disk is, the more of the hot inner wall that will be observed, making the NIR flux higher than a disk that is closer to face-on. The DIAD models assume a completely vertical inner wall at the dust sublimation radius for FDs; in the future, changes may be made to include curved walls (McClure et al. 2013b).

Dust settling is a marker of disk evolution. As small dust grains settle from the upper layers of the disk atmosphere to the disk midplane, the flux from MIR to FIR wavelengths decreases (D'Alessio et al. 2006). The NIR is dominated by the wall, which is unaffected by disk settling. Thus, NIR fluxes anchor our indices such that we can probe the degree of dust settling with the FIR.

We adopt the median spectral type of the sample, M1, to describe the stellar properties that we use as input to the model. Although the sample contains a wide range of spectral types, the use of a spectral index anchored at a stellar-dominated wavelength mitigates the effects caused by the choice of one spectral type to represent the sample. We adopt an age of 1 Myr and a mass of ∼0.4 M_⊙ and a radius of ∼2 R_⊙ for our M1 star, consistent with the Siess et al. (2000) isochrones. We use a ${T}_{\mathrm{eff}}$ of ∼3600 K based on stellar temperatures given in Kenyon & Hartmann (1995). We vary (1) the mass accretion rate as log₁₀($\dot{{\text{}}M}$) from −10 to −7.5 ${\text{}}{M}_{\odot }\,{\mathrm{yr}}^{-1}$, sampled in steps of 0.5, (2) the inclination of the disk parameterized by μ = cos(i) from 0.1 to 0.9, in steps of 0.1, and at 0.99 (nearly edge-on at μ = 0.1, ∼84°, and very nearly face-on at μ = 0.99, ∼8°), and (3) the dust settling, log₁₀(), from −4 to 0, in steps of 1.0. This results in a grid of 300 models. We use a constant value of α set to 0.01 and an outer radius for the disk at 300 au. In Figure 8, we show that the synthetic model index distribution covers the observed distribution.

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We do not include forward modeling analysis of the TDs in this work. Depending on the size of the gap in the disk, the 70 μm flux could be dominated by the wall emission, as opposed to the disk emission. Figure 9 shows an example TD model with a wall temperature of 150 K and a gap size of ∼13 au showing that for only the most unsettled disks does the disk emission dominate over the wall emission at 70 μm. As we aim to derive information about the disk properties, we cannot use 70 μm fluxes unless the gap size is known. Future work will be done to use the DIAD models to model the TDs in detail.

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We use forward modeling to reproduce the observed FD L1641 ${n}_{{K}_{S}-[70]}$ distribution using the DIAD models. We carried out 100 realizations (above this, there is minimal change), and in each, we randomly sampled the cumulative distributions of the grid of log₁₀($\dot{{\text{}}M}$), cos(i), and log₁₀() and calculated the ${n}_{{K}_{S}-[70]}$ index for that combination of model parameters. We repeated this until we found the distributions of these parameters that reproduced the observed distribution of ${n}_{{K}_{S}-[70]}$. We adopted a uniform distribution for parameters μ (assuming that disks are randomly oriented) and , i.e., we sample uniformly from an even distribution. To determine a cumulative distribution for the mass accretion rate, we used the $\dot{{\text{}}M}$ values from the literature for the sample, as listed in Table 2. We only have mass accretion rates for a subset of the sample and these are likely biased toward higher values because these are more easily measured. Therefore, we do not expect to reproduce the observed distribution. We restrict the analysis sample to have mass accretion rates between 10^−7.5 and 10⁻¹⁰ ${\text{}}{M}_{\odot }\,{\mathrm{yr}}^{-1}$. The highest mass accretion rates (>10^−7.5 ${\text{}}{M}_{\odot }\,{\mathrm{yr}}^{-1}$) are largely associated with high visual extinction values and we remove these to avoid uncertain measurements. We also remove low mass accretion rates (<10⁻¹⁰ ${\text{}}{M}_{\odot }\,{\mathrm{yr}}^{-1}$) as these may have chromospheric contamination (Ingleby et al. 2011). The adopted distribution of mass accretion rates is shown in the second column of Figure 10. We note that we use the 63 FD objects that have ${n}_{{K}_{S}}-[70]$ values (i.e., are not missing K_S-band measurements or extinctions), have mass accretion rates between 10^−7.5 and 10⁻¹⁰ ${\text{}}{M}_{\odot }\,{\mathrm{yr}}^{-1}$, and have not been flagged (for contamination by close sources and/or nebulosity, high A_V, or colors not representative of CTTS objects), as discussed in Section 2.

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The resulting mean of the distributions that reproduce the observed ${n}_{{K}_{S}-[70]}$ distribution are shown in Figure 10, where the error bars indicate the standard deviation between the results of each realization. The results in Figure 10 indicate that the observed $\dot{{\text{}}M}$ distribution is biased toward high values and that we may be missing lower accretion rate values for the sample, as expected. The distribution for μ is consistent with a uniform distribution, except at very high inclinations. This is to be expected, as it is difficult to identify edge-on disks. The distribution is relatively uniform except at the extreme unsettled and settled ends of the parameter space. We discuss this result and put it into context with other regions in Section 5.

We also performed this analysis on a Taurus sample in order to compare the two regions. The Taurus ${n}_{{K}_{S}-[70]}$ indices are calculated for 23 Class II objects with K_S-band magnitudes from Kenyon & Hartmann (1995) and PACS 70 μm fluxes from Howard et al. (2013). We only use objects that were not classified as TDs in Howard et al. (2013) and have A_V values in Kenyon & Hartmann (1995). The objects are dereddened in the same way as described for the L1641 sample in Section 3.2. We compare the FD ${n}_{{K}_{S}-[70]}$ distributions for the two samples in Figure 11, finding that L1641 peaks at a slightly higher index. Mass accretion rates for 51 systems in Taurus are taken from Hartmann et al. (1998) and White & Ghez (2001) to produce the cumulative distribution that is sampled to reproduce the distribution of ${n}_{{K}_{S}-[70]}$. The sample for which we have mass accretion rates for and for which we have ${n}_{{K}_{S}-[70]}$ overlaps partially. We restrict the objects in the Taurus sample to have $-10.0\leqslant {\mathrm{log}}_{10}(\dot{M})\leqslant -7.5$ as we did for L1641. We use the same model grid and parameter space for both L1641 and Taurus, as the index distributions cover the same range. These results are shown in Figure 12 and are discussed in Section 5.

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