Person:

Goodman, Alyssa

We present a hierarchical Bayesian method for fitting infrared spectral energy distributions (SEDs) of dust emission to observed fluxes. Under the standard assumption of optically thin single temperature (T) sources, the dust SED as represented by a power-law-modified blackbody is subject to a strong degeneracy between T and the spectral index (\beta). The traditional non-hierarchical approaches, typically based on (\chi^2) minimization, are severely limited by this degeneracy, as it produces an artificial anti-correlation between T and (\beta) even with modest levels of observational noise. The hierarchical Bayesian method rigorously and self-consistently treats measurement uncertainties, including calibration and noise, resulting in more precise SED fits. As a result, the Bayesian fits do not produce any spurious anti-correlations between the SED parameters due to measurement uncertainty. We demonstrate that the Bayesian method is substantially more accurate than the (\chi^2) fit in recovering the SED parameters, as well as the correlations between them. As an illustration, we apply our method to Herschel and submillimeter ground-based observations of the star-forming Bok globule CB244. This source is a small, nearby molecular cloud containing a single low-mass protostar and a starless core. We find that T and (\beta) are weakly positively correlated—in contradiction with the (\chi^2) fits, which indicate a T-(\beta) anti-correlation from the same data set. Additionally, in comparison to the (\chi^2) fits the Bayesian SED parameter estimates exhibit a reduced range in values.

The physical properties of molecular clouds are often measured using spectral-line observations, which provide the only probes of the clouds' velocity structure. It is hard, though, to assess whether and to what extent intensity features in position-position-velocity (PPV) space correspond to "real" density structures in position-position-position (PPP) space. In this paper, we create synthetic molecular cloud spectral-line maps of simulated molecular clouds, and present a new technique for measuring the reality of individual PPV structures. Using a dendrogram algorithm, we identify hierarchical structures in both PPP and PPV space. Our procedure projects density structures identified in PPP space into corresponding intensity structures in PPV space and then measures the geometric overlap of the projected structures with structures identified from the synthetic observation. The fractional overlap between a PPP and PPV structure quantifies how well the synthetic observation recovers information about the three-dimensional structure. Applying this machinery to a set of synthetic observations of CO isotopes, we measure how well spectral-line measurements recover mass, size, velocity dispersion, and virial parameter for a simulated star-forming region. By disabling various steps of our analysis, we investigate how much opacity, chemistry, and gravity affect measurements of physical properties extracted from PPV cubes. For the simulations used here, which offer a decent, but not perfect, match to the properties of a star-forming region like Perseus, our results suggest that superposition induces a ~40% uncertainty in masses, sizes, and velocity dispersions derived from(^{13})CO (J = 1-0). As would be expected, superposition and confusion is worst in regions where the filling factor of emitting material is large. The virial parameter is most affected by superposition, such that estimates of the virial parameter derived from PPV and PPP information typically disagree by a factor of ~2. This uncertainty makes it particularly difficult to judge whether gravitational or kinetic energy dominate a given region, since the majority of virial parameter measurements fall within a factor of two of the equipartition level α ~ 2.

We investigate how uncertainties in flux measurements affect the results from modified blackbody spectral energy distribution (SED) fits. We show that an inverse correlation between the dust temperature T and spectral index β naturally arises from least-squares fits due to the uncertainties, even for sources with a single T and (\beta). Fitting SEDs to noisy fluxes solely in the Rayleigh–Jeans regime produces unreliable T and (\beta) estimates. Thus, for long wavelength observations ((\lambda > \sim 200 \mu m)), or for warm sources ((T >\sim 60 K)), it becomes difficult to distinguish sources with different temperatures. We assess the role of noise in recent observational results that indicate an inverse and continuously varying (T –\beta) relation. Though an inverse and continuous (T –\beta) correlation may be a physical property of dust in the interstellar medium, we find that the observed inverse correlation may be primarily due to noise.

We use a new contour-based map analysis technique to measure the mass and size of molecular cloud fragments continuously over a wide range of spatial scales ((0.05 \leq r/pc \leq 10)), i.e., from the scale of dense cores to those of entire clouds. The present paper presents the method via a detailed exploration of the Perseus molecular cloud. Dust extinction and emission data are combined to yield reliable scale-dependent measurements of mass. This scale-independent analysis approach is useful for several reasons. First, it provides a more comprehensive characterization of a map (i.e., not biased toward a particular spatial scale). Such a lack of bias is extremely useful for the joint analysis of many data sets taken with different spatial resolution. This includes comparisons between different cloud complexes. Second, the multi-scale mass-size data constitute a unique resource to derive slopes of mass-size laws (via power-law fits). Such slopes provide singular constraints on large-scale density gradients in clouds.

In this paper, we present a dust emission map of the starless core TMC-1C taken at (2100 \mu m). Along with maps at 160, 450, 850, and 1200 μm, we study the dust emissivity spectral index from the (sub)millimeter spectral energy distribution, and find that it is close to the typically assumed value of (\beta = 2). We also map the dust temperature and column density in TMC-1C, and find that at the position of the dust peak ((A_{V} \sim 50)) the line-of-sight-averaged temperature is (\sim 7) K. Employing simple Monte Carlo modeling, we show that the data are consistent with a constant value for the emissivity spectral index over the whole map of TMC-1C.

Power-law mass-size and linewidth-size correlations, two of "Larson's laws," are often studied to assess the dynamical state of clumps within molecular clouds. Using the result of a hydrodynamic simulation of a molecular cloud, we investigate how geometric projection may affect the derived Larson relationships. We find that large-scale structures in the column density map have similar masses and sizes to those in the three-dimensional simulation (position-position-position, PPP). Smaller scale clumps in the column density map are measured to be more massive than the PPP clumps, due to the projection of all emitting gas along lines of sight. Further, due to projection effects, structures in a synthetic spectral observation (position-position-velocity, PPV) may not necessarily correlate with physical structures in the simulation. In considering the turbulent velocities only, the linewidth-size relationship in the PPV cube is appreciably different from that measured from the simulation. Including thermal pressure in the simulated line widths imposes a minimum line width, which results in a better agreement in the slopes of the linewidth-size relationships, though there are still discrepancies in the offsets, as well as considerable scatter. Employing commonly used assumptions in a virial analysis, we find similarities in the computed virial parameters of the structures in the PPV and PPP cubes. However, due to the discrepancies in the linewidth-size and mass-size relationships in the PPP and PPV cubes, we caution that applying a virial analysis to observed clouds may be misleading due to geometric projection effects. We speculate that consideration of physical processes beyond kinetic and gravitational pressure would be required for accurately assessing whether complex clouds, such as those with highly filamentary structure, are bound.

We measure the mass and size of cloud fragments in several molecular clouds continuously over a wide range of spatial scales ((0.05 <\sim r/pc <\sim 3)). Based on the recently developed "dendrogram-technique," this characterizes dense cores as well as the enveloping clouds. "Larson's Third Law" of constant column density, (m(r) \alpha r^2), is not well suited to describe the derived mass-size data. Solar neighborhood clouds not forming massive stars ((< \sim 10 M \odot); Pipe Nebula, Taurus, Perseus, and Ophiuchus) obey (m(r) \leq 870 M \odot (r/pc)^{1.33}). In contrast to this, clouds forming massive stars (Orion A, G10.15 – 0.34, G11.11 – 0.12) do exceed the aforementioned relation. Thus, this limiting mass-size relation may approximate a threshold for the formation of massive stars. Across all clouds, cluster-forming cloud fragments are found to be—at given radius—more massive than fragments devoid of clusters. The cluster-bearing fragments are found to roughly obey a mass-size law (m \ \alpha \ r^{1.27}) (where the exponent is highly uncertain in any given cloud, but is certainly smaller than 1.5).

We investigate the effect of line-of-sight temperature variations and noise on two commonly used methods to determine dust properties from dust-continuum observations of dense cores. One method employs a direct fit to a modified blackbody spectral energy distribution (SED); the other involves a comparison of flux ratios to an analytical prediction. Fitting fluxes near the SED peak produces inaccurate temperature and dust spectral index estimates due to the line-of-sight temperature (and density) variations. Longer wavelength fluxes in the Rayleigh-Jeans part of the spectrum ((>{\sim} 600 \mu m) for typical cores) may more accurately recover the spectral index, but both methods are very sensitive to noise. The temperature estimate approaches the density-weighted temperature, or "column temperature," of the source as short wavelength fluxes are excluded. An inverse temperature-spectral index correlation naturally results from SED fitting, due to the inaccurate isothermal assumption, as well as noise uncertainties. We show that above some "threshold" temperature, the temperatures estimated through the flux ratio method can be highly inaccurate. In general, observations with widely separated wavelengths, and including shorter wavelengths, result in higher threshold temperatures; such observations thus allow for more accurate temperature estimates of sources with temperatures less than the threshold temperature. When only three fluxes are available, a constrained fit, where the spectral index is fixed, produces less scatter in the temperature estimate when compared to the estimate from the flux ratio method.