MUSTANG 3.3 mm CONTINUUM OBSERVATIONS OF CLASS 0 PROTOSTARS

We characterize the emission between two wavelengths by assuming that the SED follows a power law (S_ν ∝ ν^α) and calculating the spectral index

$$\begin{equation} \alpha _{\lambda _1/\lambda _2} = \frac{\log (S_{\lambda _1}/S_{\lambda _2})}{\log (\lambda _2/\lambda _1)} \;\;. \end{equation} \tag{ 1 }$$

The spectral index is an empirical property of the observed SED that can be related to the underlying properties of the emission.

We have tabulated the observed spectral indices from 0.86 mm to 3.3 mm in Table 1. The spectral index between 1.25 mm and 3.3 mm is calculated from the peak flux density (mJy beam⁻¹) in both the MAMBO 1.25 mm observations of Motte & André (2001) and our MUSTANG observations. A direct comparison is feasible since the solid angles of the IRAM 30 m at 1.25 mm and the GBT 100 m at 3.3 mm are both approximately equivalent to the solid angle of a Gaussian beam with FWHM of 11''. Unfortunately, we were not able to make the same peak flux density comparison with SCUBA observations at 0.862 mm since the effective beamsize of the JCMT is 16''. Therefore, we calculate the spectral index between 0.86 mm and 3.3 mm using matched 20'' diameter aperture photometry. The typical millimeter spectral index varies between α_1.25/3.3 = 2.6 and 3.8 with an average of 〈α_1.25/3.3〉 = 3.2 for this sample. Due to L1448NW being at the edge of the map, we were unable to calculate the flux in a 20'' aperture at 3.3 mm for this source. Excluding L1448NW, the average 〈α_0.86/3.3〉 = 3.0 is very similar to the spectral index calculated at 1.25 mm for the same sources.

If we assume the dust opacity (κ_ν cm² g⁻¹) follows a single-power law at (sub)millimeter wavelengths (κ_ν ∝ ν^β), then the dust opacity index, β, may be found from the ratio of fluxes at the two wavelengths

$$\begin{equation} \frac{S_{\lambda _1}}{S_{\lambda _2}} = \left(\frac{\lambda _2}{\lambda _1} \right)^{(3+\beta)} \frac{{\rm exp}(h\nu _2/kT_{d}) - 1}{{\rm exp}(h\nu _1/kT_{d}) - 1} \;\;\;. \end{equation} \tag{ 2 }$$

Equation (2) assumes that a single dust temperature, T_d, characterizes the emission at both wavelengths. The derived β can be sensitive to the choice of the dust temperature. For instance, at submillimeter wavelengths of 442 and 862 μm, β varies by a factor of 0.5 for assumed dust temperatures that range from 10 to 20 K (Shirley et al. 2000). The variation is less severe when two wavelengths longer than 1 mm are compared; nevertheless, a suitable single dust temperature must be found.

In reality, there are strong gradients in the density and temperature increasing toward the center of the core. Since the dust continuum emission of the envelopes of four of the sources in this survey has been modeled using radiative transfer, we may use the calculated temperature profiles T(s) and constrained density profiles n(s) along each line-of-sight distance, s, to estimate the appropriate characteristic dust temperature within an aperture. We define the isothermal envelope temperature, T^env_iso, as the single dust temperature that characterizes the observed emission from density and temperature gradients within a central aperture. For a telescope with normalized beam pattern, P_n(θ, ϕ), the isothermal envelope temperature in a central aperture is derived from the equation for specific intensity of optically thin dust emission convolved with the telescope beam pattern,

$$\begin{eqnarray} && B_{\nu }\big(T_{{\rm iso}}^{\rm{env}}\big) \int _{\Omega } \int _{s} P_n(\theta,\phi) n(s) ds d\Omega \nonumber\\ &&\quad = \int _{\Omega } \int _{s} P_n(\theta,\phi) B_{\nu }[T(s)] n(s) ds d\Omega, \end{eqnarray} \tag{ 3 }$$

where B_ν is the Planck function. Solving for T^env_iso gives

$$\begin{equation} T_{{\rm iso}}^{\rm{env}}\, {=}\, (h\nu /k) \left[ \ln \left(1\, {+}\, \frac{\int _{\Omega } P_n(\theta,\phi) N(\theta,\phi) d\Omega }{\int _{\Omega } P_n(\theta,\phi) \int _{s} \frac{n(s) ds}{\mbox{exp}(h\nu /kT(s)) - 1} d\Omega }\right) \right]^{-1}, \end{equation} \tag{ 4 }$$

where N(θ, ϕ) is the column density at an impact parameter θ away from the protostar. The line-of-sight distance is related to the impact parameter θ geometrically by s² + θ² = r² where r is the radial distance from the protostar (see Adams 1991; Shirley et al. 2003).

The isothermal envelope temperatures for the best-fit one-dimensional dust continuum models of Class 0 sources in this survey are shown in Figure 2. T^env_iso depends on the frequency of the observations. The T^env_iso curves in Figure 2 flatten at millimeter wavelengths. As a result, it is a good assumption to assume a single characteristic dust temperature at both wavelengths in Equation (2) as long as both of those wavelengths are greater than 0.6 mm (ΔT^env_iso < 0.5 K). The single temperature assumption starts to break down at submillimeter wavelengths, although the variation in T^env_iso is not strong. For instance, the typical difference in T^env_iso between Herschel Space Observatory SPIRE wavelengths (250–500 μm) is slightly less than 2 K. If the emission within the central aperture is dominated by envelope emission, then the curves in Figure 2 constrain the appropriate dust temperature to use in calculating β. In Section 3.2, we explore the effects of the contribution of the disk.

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We constrain the dust opacity index using Equation (2) from a plot of β versus the characteristic dust temperature (Figure 3). The β curves for the 0.86–3.3 mm flux ratio (blue curves) and 1.25–3.3 mm flux ratio (red curves) are shown as solid lines in Figure 3. The dashed lines represent the ±1σ statistical uncertainty in the flux at each wavelength. At the characteristic T^env_iso ∼ 16 K for this sample of Class 0 protostars, β does not have a strong dependence on the dust temperature. In general, the derived opacity index agrees within the statistical calibration uncertainty between β_0.86/3.3 and β_1.25/3.3. Typical values range from β_mm = 0.8 to 2.2 with an average value of 〈β_mm〉 = 1.5 ± 0.4. In the next section, we interpret the derived β_mm by accounting for the contribution from disk and envelope emission.

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In all cases except for L1527, the β_0.86/3.3 curve agrees within the statistical error bars with the β_1.25/3.3 curve. For L1527, the offset may be accounted for by a systematic calibration error of 20% at one or more of the three wavelengths (e.g., with SCUBA, MAMBO, and or MUSTANG calibration). The dominant source of uncertainty in determining β_mm is the uncertainty in the flux ratio at two wavelengths. The uncertainty is lower if the two wavelengths are more widely spaced (e.g., lower for the ratio 0.86/3.3 versus 1.25/3.3). An uncertainty in the fluxes of 20% results in a typical uncertainty of ±0.4 in β_1.25/3.3 and ±0.3 in β_0.86/3.3. An accurate flux calibration at both wavelengths and deep photometry is required to minimize this uncertainty. A second possibility for this discrepancy is that the emission between 0.86 mm and 3.3 mm is mixing different fractions of grain populations from the envelope and disk. This possibility is explored in the next section.

The total flux observed at 3.3 mm is the sum of emission from the protostellar envelope, disk, and wind (or jet)

$$\begin{equation} S_{\nu }^{\rm{dust}} = S_{\nu }^{\rm{env}} + S_{\nu }^{\rm{disk}} + S_{\nu }^{ff} \;\;\;. \end{equation} \tag{ 5 }$$

At submillimeter wavelengths, the dust continuum emission from Class 0 sources is expected to be dominated by their massive envelopes. At millimeter wavelengths, this assumption may no longer be valid and the disk emission may be a significant fraction of the total emission in the central beam. At centimeter wavelengths, thermal radio continuum emission (free–free emission) from the protostellar jet or wind becomes the dominant emission mechanism (Anglada 1995). The centimeter wavelength free–free emission can be variable from Class 0 protostars (see Shirley et al. 2007). Unfortunately, we do not have simultaneous centimeter continuum observations at the same epoch as the MUSTANG observations; however, extrapolation from published VLA fluxes and spectral indices indicates that the expected contribution from free–free emission at 3.3 mm is expected to be small (<10%). Therefore, we ignore the free–free contribution to the 3.3 mm fluxes in the following analysis.

While it may be advantageous to study dust properties at millimeter wavelengths close to the Rayleigh–Jeans limit, the interpretation of single-dish observations becomes more difficult since derived β_mm is an amalgamation of disk plus envelope opacities. This is a particular issue for calculating β since dust grains may undergo coagulation in the dense environments of Class 0 disks, and therefore the resulting β_mm is expected to be lower than for dust in the protostellar envelope (e.g., Henning & Stognienko 1996; Dominik & Tielens 1997; Poppe et al. 2000; Draine 2006; Birnstiel et al. 2010). We can identify the disk contribution at (sub)millimeter wavelengths from the visibility amplitudes of interferometric observations. Since disks are typically small (R < 100 AU corresponding to θ < 1''), they appear as unresolved structures on baselines shorter than the characteristic size. The disk flux may be determined from the flux level of a flattening in the visibility amplitudes and a spectral index from interferometric observations at two wavelengths. In this section, we estimate the fraction of disk emission at 3.3 mm and the impact on our interpretation of β_mm for four of the sources (L483, L1527, B335, and L1448C) with published multi-wavelength interferometric observations.

L483 is perhaps the easiest example to analyze as the emission from a disk is thought to be very weak for this source, even at wavelengths as long as 3 mm (Jørgensen 2004, 2007, 2009). This source has been observed with the SMA and OVRO at wavelengths ranging from 0.8 mm to 3.0 mm and no evidence for a compact component is seen in the visibility amplitudes. Jørgensen et al. (2009) estimate a negligible disk mass compared to the envelope mass. The measured spectral index for L483 observed in this paper is consistent with the spectral index observed by Jørgensen et al. (2007) with the SMA between 0.8 and 1.25 mm (α_0.8/1.25 = 3.7 on baselines >40 kλ). Therefore, our observations at 3.3 mm are probing the envelope structure and are not significantly contaminated by emission from a disk. The observed range of β_mm = 1.57–2.00 is consistent with the opacities typically assumed for radiative transfer models of Class 0 envelopes. For instance, the widely used OH5 opacities have a β_OH5 = 1.85 for coagulated dust grains at a density of 10⁶ cm⁻³ for 10⁵ years with thin ice mantles (Table 2, Column 5 of Ossenkopf & Henning 1994; also see Table 2 of Shirley et al. 2005 for a summary of β for various dust models).

In contrast, the emission from L1527 at 3.3 mm within a central MUSTANG aperture appears to be dominated by the disk emission. L1527 has also been observed by Jørgensen et al. (2007) with the SMA at 0.8 and 1.3 mm where a very flat spectral index of α_0.8/1.3 = 1.9 is observed on baselines >40 kλ. Observations of this source have also been made using BIMA with the combination of four array configurations at 2.7 mm where a distinct flattening in visibility amplitudes is observed on baselines >10 kλ with a flux of 42 mJy (Y. Shirley, unpublished observations). The observed spectral index between 1.3 and 2.7 mm is consistent with SMA results (α_1.3/2.7 = 1.8). All of the observed emission in the MUSTANG central beam may be accounted for by extrapolating this disk flux and interferometric spectral indices to 3.3 mm. The spectral indices reported in Table 1 are larger than the interferometric spectral indices because a significant fraction (⩾50%) of the flux in the single dish apertures at 0.86 and 1.25 mm is still coming from the envelope. If the envelope dust has a steeper opacity index than the disk dust opacity index, the resulting β_mm = 0.82–1.42 is then an overestimate of the true β in the disk. Despite this uncertainty in the true disk β, the low value is consistent with evidence of evolution of the dust properties in the L1527 disk and indicates that different dust opacities that are used to model the envelope are needed (cf. Shirley et al. 2002).

B335 is a popular target with extensive interferometric observations; however, not all of these observations agree on the disk contribution. B335 was studied extensively at 1.2 and 3.0 mm with the Plateau de Bure Interferometer (PdBI) by Harvey et al. (2003). They found a slight flattening in the visibility amplitudes for baselines greater than 60 kλ. The observed flux on these long baselines was 21 ± 3 mJy at 1.2 mm and ≈2 mJy at 3.0 mm. The resulting spectral index is 2.6, slightly less than the spectral index of 3.1 observed in this paper. Extrapolation of the 3.0 mm disk flux to 3.3 mm results in a small contribution to the MUSTANG flux (1.7 mJy or 10% of the 3.3 mm flux). The PdBI observations do not agree with the flux estimates from extrapolation of the Jørgensen et al. (2007) SMA observations on baselines greater than 40 kλ which predict a 3.3 mm disk flux of 7 mJy or approximately 50% of the MUSTANG flux. The real contribution of the disk flux for B335 is probably somewhere in between these estimates. It is likely that the Jørgensen et al. (2007) fluxes include a contribution from the envelope since emission has been seen on baselines longer than 40 kλ from the envelope toward B335 (Harvey et al. 2003).

With the caveat that our estimates of β_mm toward B335 may have some contribution (10%–50%) from a disk component, the derived opacity index β_mm = 1.18–1.45 is significantly lower than has been observed in the outer envelope (Shirley et al. 2010). A comparison of the opacity ratios at 442/2.2 μm and 862/2.2 μm from near-infrared extinction observations and submillimeter continuum images yields a β_submm = 2.1–2.5 for lines of sight greater than 15'' from the protostar (lines of sight where background stars were detected in NICMOS observations by Harvey et al. 2001). This may be direct evidence of a change in the opacity in the inner envelope and disk of B335. Our derived β_mm is consistent with the slope of OH2 dust (β_OH2 = 1.35; Ossenkopf & Henning 1994; Shirley et al. 2005) for coagulated dust grains with no ice mantles as might be expected in the warm inner envelope of a Class 0 protostar. However, contribution from disk emission may be responsible, in part, for this perceived lowering of β_mm. If we subtract the maximal disk flux from 1.25 mm and 3.3 mm photometry, then β_mm = 1.6–2.4 which is consistent with the Shirley et al. (2010) outer envelope determination. Unfortunately, until observations are performed with an interferometer at two wavelengths that match the wavelengths of single dish continuum observations (e.g., 0.86 mm and 3.3 mm with ALMA), then this level of uncertainty in determining the envelope β_mm in the central aperture toward B335 will persist.

Observations of L1448C were also made by Jørgensen et al. (2007) with the SMA. Spitzer Space Telescope imaging of this region (Jørgensen et al. 2006) has discovered that this source is actually two Class 0 sources in 8'' proximity (L1448C(N) and L1448C(S)). Because the southern source is significantly weaker than the northern source (only 7% of the flux of the northern source in the 1.25 mm SMA observations; Jørgensen et al. 2007), it is likely that the MUSTANG fluxes are predominantly from the northern source. The observed spectral index between 0.8 and 1.25 mm on baselines greater than 40 kλ is significantly shallower than the spectral indices observed with single dish telescopes. Extrapolating the SMA results to 3.3 mm indicates that as much as 50% of the MUSTANG flux may be due to the emission from the disk. Again, without multi-wavelength interferometric observations that match the wavelength of single-dish observations, we cannot accurately determine the envelope β_mm. The range of β_mm = 1.10–1.52 observed toward L1448C is very similar to the range observed toward B335. Accounting for the maximal disk contribution can also increase the range of β_mm in the envelope by 0.5.

Unfortunately, there are no published u, v amplitude curves for L1448N and L1448NW; therefore, we are unable to assess the disk contribution to the MUSTANG fluxes in these two cases. Interferometric observations have been made at 1.3 and 2.7 mm toward the L1448 IRS3 region which includes these two sources (Looney et al. 2000; Kwon et al. 2006). L1448N is comprised of two bright sources separated by 10'' (labeled L1448 IRS3 A and B in Looney et al. 2000) which makes separating their visibility amplitudes very difficult. We note that L1448NW has the highest range in β_mm = 1.94–2.23. Given such high values and the expectation that the opacity index is lower in disks, it seems unlikely that L1448NW has a significant disk contribution. However, the only way to confidently constrain the envelope and disk opacities is to analyze the visibility amplitudes at multiple wavelengths with an interferometer.