A New Technique for Mapping Distances Across the Perseus Molecular Cloud Using CO Observations and Stellar Photometry

A New Technique for Mapping Distances Across the Perseus Molecular Cloud Using CO Observations and Stellar Photometry

Catherine Zucker Harvard Astronomy, Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA 02138, USA Edward F. Schlafly Lawrence Berkeley National Laboratory, One Cyclotron Road, Berkeley, CA 94720, USA Gregory M. Green Kavli Institute for Particle Astrophysics and Cosmology, Physics and Astrophysics Building, 452 Lomita Mall, Stanford, CA 94305, USA Joshua S. Speagle Harvard Astronomy, Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA 02138, USA Stephen K. N. Portillo Harvard Astronomy, Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA 02138, USA Douglas P. Finkbeiner Harvard Astronomy, Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA 02138, USA Alyssa A. Goodman Harvard Astronomy, Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA 02138, USA
Abstract

We present a new technique to determine distances to major star-forming regions across the Perseus Molecular Cloud, using a combination of stellar photometry and spectral-line data. We start by inferring the distance and reddening to thousands of stars across the complex from their Pan-STARRS1 and 2MASS photometry, using a technique presented in Green et al. (2014, 2015) and implemented in their 3D “Bayestar” dust map of three-quarters of the sky. We then refine their technique by using the velocity slices of a CO spectral cube as dust templates and modeling the cumulative distribution of dust along the line of sight towards these stars as a linear combination of the emission in the slices. Using a nested sampling algorithm, we fit these per-star distance-reddening measurements to find the distances to the CO velocity slices towards each star-forming region. We determine distances to the B5, IC348, B1, NGC1333, L1448, and L1451 star-forming regions and find that individual clouds are located between pc, with a per-cloud statistical uncertainty of 7 to 15 pc, or a fractional distance error between 2% and 5%. We find that on average the velocity gradient across Perseus corresponds to a distance gradient, with the eastern portion of the cloud (B5, IC348) about 30 pc farther away than the western portion (NGC1333, L1448). The method we present is not limited to the Perseus Complex, but may be applied anywhere on the sky with adequate CO data in the pursuit of more accurate 3D maps of molecular clouds in the solar neighborhood and beyond.

1 Introduction

As the most active site of star formation within the solar neighborhood ( 300 pc, Bally et al., 2008), the Perseus Molecular Cloud complex has been the subject of a wealth of continuum and spectral-line observations in recent years. These studies have targeted both the global properties of the molecular cloud and the properties of discrete, high-density pockets of gas and dust in which small groups or clusters of stars are forming (e.g. L1451, L1448, NGC1333, B1, IC348, and B5; see Figure 1). Distance estimates to these well-known star-forming regions show a wide degree of dispersion—varying anywhere between 200-350 pc—and are based on a variety of techniques. Nevertheless, accurate distance measurements are critically important for constraining properties like clump mass or star formation efficiency of the gas in this local, and subsequently high-resolution, environment.

Černis (1990, 1993) use interstellar extinction to find distances to both the eastern (IC348 at 260 pc) and western portions (NGC1333 at 220 pc) of Perseus. Because optical light is extinguished by molecular clouds, Černis (1990, 1993) determine photometric distances to unextinguished foreground and extinguished background stars, thereby constraining the distance of the jump in extinction. More recently, Hirota et al. (2008, 2011) determine distances to the western portion of Perseus, by obtaining trigonometric parallax measurements of water masers associated with young stellar objects embedded in the NGC1333 and L1448 regions—finding a distance of 235 pc () for the former and 232 pc () for the latter. Adopting a slightly different technique, Lombardi et al. (2010) calculates a distance to Perseus by comparing the density of low-extinction foreground stars (determined by the NICEST color excess method; Lombardi, 2009) with the prediction for foreground stellar density from the Robin et al. (2003) Galactic model, finding a distance of 260 pc to B1 and B5 and 212 pc to L1448.

Schlafly et al. (2014) is the first to systematically map distances across the entire Perseus Complex. Schlafly et al. (2014) first obtains distances and reddenings to batches of stars in over a dozen sightlines throughout Perseus using optical photometry (as outlined in Green et al., 2014). Then, Schlafly et al. (2014) adopts a model whereby the stellar distances and reddenings are caused by a single dust screen, which they find the distance to using an MCMC analysis. The Schlafly et al. (2014) results suggest a distance gradient, but often with large uncertainties on individual lines of sight () and with further overall distances than suggested in Hirota et al. (2008, 2011), typically lying around 260-315 pc.

One possible explanation for the distance discrepancies is that the Perseus Complex consists of clouds at several distances along the line of sight, a scenario bolstered by the large velocity gradient () observed across the cloud in CO (Ridge et al., 2006). However, unlike for clouds outside the solar neighborhood, this velocity gradient cannot be mapped to a distance gradient via a Galactic rotation curve (by the so-called “kinematic distance” method, see Roman-Duval et al., 2009) because the distance resolution of the kinematic method is coarser than the typical peculiar motions for objects so close to the Sun.

In this analysis, we build upon the work of Schlafly et al. (2014) and present a new technique to map velocities to distances across the Perseus Molecular Cloud, by combining information on the spatial distribution of CO emission with distance and reddening estimates towards thousands of stars from Green et al. (2018) (hereafter G18). By using the velocity slices of a CO spectral cube as dust templates and modeling the cumulative distribution of dust along the line of sight as a linear combination of optical depth in CO velocity slices, we perform a Monte Carlo analysis to determine which distance configuration for the velocity slices is most consistent with the distance and reddening estimates to sets of stars in regions across Perseus.

In §2, we introduce the key data sets in this analysis, including the photometry used to derive the stellar distance and reddening estimates and the CO spectral-line data used as a dust tracer. In §3, we briefly explain how the stellar distance and reddening estimates are derived from the photometric catalogs discussed in §2. In §4, we discuss our target selection and the batch of stars we consider towards each region of interest. In §5, we present our Bayesian model for the line-of-sight distribution of dust as a function of the CO velocity slices. In §6, we discuss the nested sampling algorithm we use to perform the parameter estimation. In §7, we present our distance estimates to the CO slices for major star-forming regions across the Perseus Molecular Cloud and compare our results with distance estimates from the literature. We discuss the implications of our results in §8 and conclude in §9.

2 Data

Determining distances to molecular clouds based on the distribution of their CO emission is a two-step process. In the first step, we obtain the per-star posterior probability density function (PDF) of distance and reddening for stars towards the Perseus Molecular Cloud, based solely on their optical (Pan-STARRS1, hereafter PS1) and near-infrared (2MASS) photometry. In the second step, we use the slices of a CO spectral cube as dust templates—by multiplying each slice by a gas-to-dust conversion coefficient—and sample for the probable range of distances to each velocity slice given the distance and reddening estimates to stars towards the same region. In this section, we briefly describe the key data sets necessary for this analysis.

2.1 Pan-STARRS1 Photometry

The Pan-STARRS1 survey is a deep optical survey of the three quarters of the sky north of (Chambers et al., 2016). The PS1 observations are obtained using a 1.8-meter telescope situated on Mount Haleakala in Hawaii, which has been equipped with a Gigapixel camera with a field of view and pixel scale of 0.258. The survey observes in five broadband filters, the bands, spanning nm (Chambers et al., 2016). The images are processed by the PS1 Image Processing Pipeline, which automatically performs photometric and astrometric measurements on the reduced data (Magnier et al., 2016). The stellar posteriors on distances and reddening we utilize in this work (from G18; see §3) are based on catalog coadds of single-epoch photometry derived from the PS1 DR1 steradian survey, which reaches typical single-exposure depths of 22.0, 21.8, 21.5, 20.9, and 19.7 magnitudes (AB) in the , , , , and bands, respectively (Chambers et al., 2016).

2.2 2MASS Photometry

The Two Micron All Sky Survey (2MASS) is a near infrared survey of the full sky targeting the , , and bandpasses at (Skrutskie et al., 2006). The 2MASS observations are obtained via two 1.3-meter telescopes located on Mount Hopkins, Arizona and Cerro Tololo, Chile, both of which are equipped with a three-array survey camera with an field of view and a pixel scale of 2 (Skrutskie et al., 2006). Like PS1, the image processing pipeline automatically performs photometric and astrometric measurements on the reduced images. The resulting catalogs achieve typical point-source depths of 15.8, 15.1, and 14.3 magnitudes (Vega) for the , , and bands, with bright sources possessing photometric uncertainties on the order of mag. To derive the stellar posteriors on distance and reddening from G18 we specifically use the 2MASS “high-reliability” catalog (see §3.3 in G18 for more details).

2.3 CO COMPLETE Data

We employ maps of the (1-0) transition in Perseus, taken from the COMPLETE Survey of Star Forming Regions111All the COMPLETE data are publicly available and can be downloaded from the Harvard Dataverse here. The corresponding DOI for the COMPLETE cube of Perseus is doi:10904/10072 (Ridge et al., 2006). We have chosen the COMPLETE data due to its high-angular resolution (half-power beamwidth ) and its relatively low noise (mean rms per channel 0.35 K). Its spectral resolution in the (1-0) line is . The total areal coverage of the COMPLETE survey towards Perseus is 6.25 x 3 deg.

We have chosen the (1-0) line primarily because it is more abundant and has more extended emission, which allows us to target larger areas of the sky (and therefore more stars), enabling better distance estimates. In theory, we could employ maps targeting the rarer CO isotopologues (e.g. ) in lieu of or in addition to the data. There are several arguments to be made for using the (1-0) line in particular, most notably that it is optically thinner and tends to be more linear with extinction than the (1-0) line (Pineda et al., 2008). According to Pineda et al. (2008), the line typically becomes self-absorbed in Perseus (and thus non-linear with reddening) around an mag, while the line typically saturates around an mag. Thus, in theory, allows us to probe one magnitude deeper, and thereby spatially closer to the densest star-forming cores. However, gaining an extra magnitude does not translate to an appreciable increase in the number of stars available for analysis, since few stars are visible behind such a high dust column density. Moreover, because of the higher critical density of (it forms at mag, vs. mag for ; see Tables 4 & 6 in Pineda et al., 2008), it is a comparatively poor dust template in the more diffuse, extended envelopes of the clouds, from which we draw the bulk of our stars.222For the clouds we target in §4 (see Figure 1), we find that the inclusion of regions with total line-of-sight visual extinction mag typically only increases our star count by approximately a few dozen to a few hundred, enlarging our sample size by only about 5-15%. As discussed in §4, we delineate cloud boundaries using integrated intensity contours. However, since is less extended (both spatially and kinematically) we would need to adopt a comparatively higher CO integrated intensity threshold to produce reliable dust templates. In cases like L1448, the adoption of even a generous low-level integrated intensity contour (roughly coincident with the mag formation threshold for ) reduces our star count by a factor of three, because we target less area towards the outer envelopes of the clouds (better traced by ), from which most of our stars are drawn.

Thus, all the results presented in this work utilize the line, towards sightlines where CO does not self-absorb. We defer the potential inclusion of and/or to future studies, where we can implement a more complex model (e.g. one which takes into account the critical density of various CO lines and probes a larger density regime within each cloud).

3 Obtaining Stellar Distance and Reddening Estimates

Based solely on the PS1 (§2.1) and 2MASS (§2.2) photometry, we obtain stellar distance and reddening estimates for stars across Perseus from the work of G18. The methodology used to derive the distance and reddening posteriors is given in Green et al. (2014, 2015). In brief, G18 infers a distance modulus , reddening , and stellar type for a star, where is parameterized by the star’s absolute PS1 -band magnitude, , and its metallicity, . By adopting a set of stellar templates that map the star’s intrinsic stellar type to its absolute magnitude in different photometric bands, G18 obtains the following relation for the modeled apparent magnitudes for each star:

(1)

where the vector notation indicates that , , and are considered over a range of passbands (e.g. ]). Then, fixing the extinction curve and assuming (Schlafly et al., 2016)333The Schlafly et al. (2016) work does not directly measure because their observations are insensitive to the gray component of the extinction vector. Rather, Schlafly et al. (2016) builds a proxy for using the quantity , where and are the Pan-STARRS1 and band magnitudes and is the WISE band two magnitude. Thus, the value we quote is actually the proxy Schlafly et al. (2016) calculates for their mean extinction vector. See §5.3 in Schlafly et al. (2016) for more details., the likelihood of observing a star with apparent PS1 and 2MASS magnitudes , assuming independent Gaussian photometric uncertainties , is a multivariate normal with mean and standard deviation , evaluated at . Prior knowledge of the number density and metallicity of stars across the Galactic disk and halo (Jurić et al., 2008; Ivezić et al., 2008), as well as on the stellar luminosity function (Bressan et al., 2012), is also incorporated into the model. Combining the likelihood function and priors, G18 draws from the posterior distribution function of , , and for individual stars using an affine-invariant ensemble Markov Chain Monte Carlo (MCMC) sampler (Goodman & Weare, 2010). After marginalizing over the parameters and taking a kernel density estimate of the samples, G18 produces a two-dimensional gridded stellar posterior describing the probable range of the distance and reddening to each star over the domain mag and mag. These are the stellar posteriors we implement in our model (see §5).

4 Cloud Selection

We target every major star-forming region inside the boundaries of the CO COMPLETE survey (see §2.3) in this analysis. This includes B5, IC348, B1, NGC1333, L1448, and L1451. We show an extinction map of the Perseus complex in Figure 1a (Pineda et al., 2008) and box these regions with green rectangles, which are apparent as pockets of very high optical depth. We also show a integrated intensity map of the Perseus complex in Figure 1b (Ridge et al., 2006). To define boundaries around each cloud we apply integrated intensity contours to the cloud’s corresponding emission (, depending on the region).444The lowest integrated intensity contour is applied to B5 at a level of , while the highest is applied to NGC1333 at a level of In cases where we cannot obtain a closed contour, we find the intersection between a reasonable semi-closed integrated intensity contour and the “classical” regions of extinction as established in previous studies of Perseus (green rectangles in Figure 1a; see, for instance Rosolowsky et al., 2008; Bally et al., 2008; Kirk et al., 2007; Sadavoy et al., 2014). These boundaries are ad hoc, but they are inclusive of previous definitions of these regions, and we find that our results are robust to modest changes in boundary definition. The final boundaries we use for each region are shown as green polygons in Figure 1b. The right ascension and declination values corresponding to the geometric centroid of each polygon are given in Table 1.

While all the stars we consider lie inside the green polygonal boundaries shown in Figure 1b, we make two additional cuts for stars lying inside these regions. First, we discard stars whose MCMC chains are unconverged, which is determined using a Gelman-Rubin convergence diagnostic in G18 (Gelman & Rubin, 1992).555In more detail, G18 produces four separate MCMC chains for each star, and if the Gelman-Rubin diagnostic computed using these four chains is , the star is considered converged. Towards Perseus, the fraction of unconverged stars is very low, and accounts for of stars towards each region we consider Second, we discard stars based on their total line-of-sight extinction. Specifically, at higher reddening, we know that the line becomes optically thick and begins to self-absorb, such that the total integrated CO emission we observe flattens above some extinction threshold (see Figure 6 in Pineda et al., 2008). Above this threshold, CO can no longer be used as a reliable tracer of dust.

Pineda et al. (2008) quantifies the extinction threshold above which becomes self-absorbed for Perseus. In more detail, Pineda et al. (2008) combines the COMPLETE data with a dust extinction map, produced by applying the NICER (Near-Infrared Color Excess method Revised; Lombardi & Alves, 2001) algorithm to 2MASS photometry, to estimate the total line-of-sight extinction towards Perseus. This NICER extinction map from Pineda et al. (2008) is the same shown in Figure 1a. Pineda et al. (2008) finds that on average, emission in Perseus becomes optically thick around mag. Consequently, we mask out all regions inside the green polygons (Figure 1b) with mag. Specifically, we regrid the NICER extinction map to the same pixel scale as the COMPLETE cube and exclude all stars that fall inside CO pixels with corresponding NICER mag. The relative area we mask inside the green polygonal boundaries in Figure 1b is quite high, typically on the order of 25-75%, so we are, in reality, only probing the outer, lower density envelopes of each region. We mark the positions of the final stars we use in our analysis with gold scatter points in Figure 1c. For each region, between 2,000 and 3,500 stars are used in the parameter estimation described in §6.

Figure 1: Panel a: The Perseus Molecular Cloud, as seen in visual extinction (Pineda et al., 2008). We frame each region of interest (B5, IC348, B1, NGC1333, L1448, and L1451) via the green rectangles. Panel b: The Perseus Molecular Cloud, as seen in integrated emission (Ridge et al., 2006). To delineate boundaries around the cloud, we apply integrated intensity contours to this map (at , depending on the region) and find the intersection between these contours and the classical regions of extinction, as shown in panel (a). The final boundaries are shown via the green polygons in panel (b). Panel c: The final positions of the stars (yellow scatterpoints) used in our analysis for each region, after masking out lines of sight with high extinction ( mag).
\colnumbers
Table 1:
Cloud Name RA Dec Velocity Range Downsampled Slice Count Slice Velocities Slice Width
L1451 51.0 30.5 -1.1 – 7.4 4 [0.3, 2.3, 4.2, 5.8] 2.1
L1448 51.2 30.9 -2.2 – 7.9 5 [-0.7, 0.8, 2.9, 4.8, 6.6] 2.0
NGC1333 52.2 31.2 -1.3 – 10.1 6 [-0.1, 1.7, 3.5, 5.5, 7.3, 8.8] 1.9
B1 53.4 31.1 -0.2 – 9.6 5 [1.1, 2.8, 4.8, 6.7, 8.2] 2.0
IC348 55.8 31.8 4.2 – 11.8 4 [5.4, 7.4, 9.0, 10.4] 1.9
B5 56.9 32.9 6.7 – 12.4 3 [8.3, 9.7, 11.0] 1.9

Note. – The velocity of the downsampled CO slices that we model as dust screens in our analysis. In column (1) we list the name of the star-forming region. In column (2) we list the right ascension for the cloud, computed using the geometric centroid of each green polygonal boundary shown in Figure 1b. In column (3) we list the declination for the cloud, computed using the same geometric centroid as in (2). In column (4) we show velocity range towards each region in which CO emission is not saturated and also above the noise level. Given the native spectral resolution of the CO data (; Ridge et al., 2006) there are typically dozens of velocity slices in this velocity range towards each cloud. As a result, we have downsampled the cube along the spectral axis, and in column (5) we list the number of downsampled velocity slices towards each region. In column (6) we summarize the mean CO-weighted velocity of each downsampled CO slice, computed using only pixels along sightlines where the CO does not self-absorb (see §4 and §5.1). In column (7) we indicate the spectral width of the downsampled slices towards each region; to maintain uniformity across the complex, the spectral width is chosen to be close to while covering the desired velocity range for each cloud with an integer number of slices

5 Model

Our model for the cumulative reddening in E() along the line of sight out to distance modulus —hereafter called the “reddening profile”—is denoted by:

(2)

where is some set of parameters describing the reddening profile. As in G18, the posterior probability density of our parameters is determined by the product of the integral over through the individual stellar posterior density functions (derived from the set of PS1+2MASS photometry {}) following :

(3)

where ) constitutes our priors and the product of the integral through the individual stellar posterior arrays is our likelihood function. This is simply Bayes’ rule, modulo the normalizing constant in the denominator and assuming independence among stars.

G18 parameterizes the reddening profile as a piecewise linear function in distance modulus, so is given by a set of parameters denoting the increase in reddening, , in fixed distance bins equally spaced in distance modulus between mag (63 pc) and mag (63 kpc). We, however, do not sample the increase in reddening directly or in fixed distance bins along the entire line of sight. Rather, for an individual star lying in CO pixel we parameterize Perseus’ contribution to the line-of-sight reddening profile towards this pixel as the sum of the CO emission in a set of velocity slices } that lie at distances } and whose corresponding CO emission intensities } can be converted to reddening by multiplying by a set of gas-to-dust coefficients }. Since the CO emission along the line of sight differs for stars in different pixels, the reddening profile will vary from star to star.

We also assume that there is an angularly uniform foreground dust cloud, whose reddening is not accounted for in the total CO emission given by the velocity slices. We parameterize the foreground dust cloud as lying at a distance , and with a reddening contribution in given by . For a star in CO pixel , the reddening profile is parameterized as follows:

(4)

where our free parameters are , , , and . The profile takes the form of a step function. Under the assumption that our velocity slices are ordered by distance, the total line-of-sight reddening out to a distance for a star coincident with CO pixel is then:

In Figure 2, we show a cartoon line-of-sight reddening profile for a very simple one-slice model (lying at velocity ), where the CO emission towards our region of interest is described by a single distance component . In such a model, we can parameterize the total line-of-sight reddening profile as having a jump in reddening at distance , plus a second jump in reddening at distance , the magnitude of which is given by the CO emission in slice coincident with the star, multiplied by some gas-to-dust coefficient . The free parameters defining our reddening profile are , , , and , which are fixed across all stars. The only variation in the reddening profile from star to star stems from the magnitude of the reddening jump at distance , which is dependent on the CO pixel coincident with each star on the plane of the sky. This single-template model is similar to the one implemented in Schlafly et al. (2014), with the exception that they use the angular distribution of the Planck emission (Planck Collaboration et al., 2011) instead of CO velocity slices as their dust templates and fix to 0 pc. In comparison to Planck, our more complex multi-slice CO velocity model is better able to trace the underlying molecular and probe the structure of the cloud along the line of sight.

We discuss the CO slice distance parameters () in more detail in §5.1, the gas-to-dust coefficients () in §5.2, the foreground dust cloud parameters in §5.3 and our method for handling outliers in §5.4.

5.1 Cloud Distance Parameters

For each cloud of interest (B5, IC345, B1, NGC1333, L1448, and L1451), we have a set of parameters that describe the distance moduli to velocities slices containing the CO emission for that cloud. The typical velocity range of the CO emission observed towards each cloud spans , meaning that, in general, there are several dozen to hundreds of velocity slices along the line of sight. However, because these velocity channels are highly correlated with one another, we choose to downsample the cube along the spectral axis. Because we want to preserve the total CO emission along the line of sight, we downsample by summing to produce downsampled cubes with a velocity slice channel width of . Specifically, towards each region (green polygonal boundaries in Figure 1b), we determine the minimum and maximum velocity slice in which CO emission appears above the noise level and the line is not self-absorbed.666 Recall that we determine where becomes self-absorbed using the analysis of Pineda et al. (2008), which finds that saturates and becomes non-linear with reddening around an mag. Thus, we only consider the CO emission over the same area we select our stars (i.e. sightlines inside the green polygonal boundaries in Figure 1b with corresponding NICER mag). These areas are shown overlaid with yellow points (constituting our target stars) in Figure 1c. See discussion in §4 for more details. We then group the velocity slices across this velocity range into batches (where is chosen so that the velocity range of each batch is ), sum the CO emission in each batch of slices, and assign the downsampled slice the CO-weighted mean velocity of that batch. The CO-weighted mean slice velocities are computed using only the pixels along sightlines where CO never self-absorbs.6 This choice of downsampled spectral resolution () yields between 3-6 velocity slices towards our target regions, meaning that each cloud is composed of 3-6 distance components (). This slice range allows for the freedom to place the velocity slices at multiple distances along the line of sight, while preventing our parameter space from containing overly redundant parameters. The velocity range of each cloud, along with the number of downsampled velocity slices and the CO-weighted mean velocity of each slice, can be found in Table 1.

We place a uniform, flat prior on the distance moduli to the velocity slices {, , …} in the range , corresponding to 200 pc d 400 pc. This is consistent with the range of potential distances to the Perseus Molecular Cloud from the literature (see Hirota et al., 2008, 2011; Černis, 1990, 1993; Schlafly et al., 2014; de Zeeuw et al., 1999). None of our distance parameters, particularly those associated with slices with higher amounts of CO emission, are strongly prior-constrained, so we anticipate our adoption of a flat prior in distance modulus (rather than distance) to have a negligible effect on our results.

5.2 Gas-to-Dust Conversion Factor Parameters

For each of our velocity slices in each cloud, we have a set of parameters that describes how gas relates to dust at each slice distance . Specifically, it converts the amount of integrated CO emission in in each downsampled velocity slice to the amount of reddening, E(), in units of magnitudes. In theory, we could fix this parameter in our model. However, both parameters that constitute this conversion coefficient (i.e. the factor and the -to-reddening factor, as discussed below) are not well-constrained, varying in column density and likely physical density of the cloud. We estimate a central value for the prior on the CO-to-reddening coefficient and allow it to vary about this value in our fits.

To derive an average literature value for this coefficient, we adopt a -factor of from Dame et al. (2001), where the -factor converts integrated intensity to column density. Next, assuming that all the hydrogen traced by the reddening is in molecular form—the same assumption made in Pineda et al. (2008)—we adopt an ratio of from Bohlin et al. (1978), which converts column density to reddening. Combining the two, we get an average factor equal to . In practice, we integrate over the velocity channels (defined in §5.1) by summing the CO emission in Kelvin and multiplying by the native channel width () of the CO COMPLETE survey. This gives the CO brightness in each velocity slice in , which we can then multiply by the factor sampled at each iteration to produce the total E() in magnitudes.

We place a normal Gaussian prior on our factors, {, , …}. For simplicity, we treat the {, , …} parameters as multiplicative factors in front of the average gas-to-dust coefficient we adopt from the literature (, as computed above). A value of would yield a gas-to-dust conversion coefficient of , while a value of would yield a gas-to-dust conversion coefficient of . We set the mean of the Gaussian prior equal to the average value from the literature () and set the standard deviation to . The larger value of constitutes a relatively loose prior on the conversion coefficients, and is consistent with the underlying uncertainty in this factor from the literature.777It is well known that the formation of CO (and thus the -factor) is dependent on the metallicity of the gas and the strength of the background UV radiation field, which can cause CO molecules to disassociate (Shetty et al., 2011; Pineda et al., 2008). Pineda et al. (2008) finds that the factor within Perseus can vary by at least a factor of two, depending on which star-forming region is used in the fit (e.g. NGC1333 vs. B5; see their Table 4) or whether it is determined in saturated or unsaturated regimes. As discussed further in §7, we tend to infer coefficients in the range , which is lower than the mean we adopt for our Gaussian prior—indicating that this prior is being “tugged” on to produce a smaller value for the gas-to-dust coefficient.

5.3 Foreground Dust Cloud Parameters

Our model includes two foreground dust cloud parameters, whose reddening contribution is completely independent from the CO emission associated with the Perseus Molecular Cloud. Specifically, we parameterize the foreground dust cloud as lying at a distance modulus , and with a reddening contribution in given by . We assume there are no background clouds beyond the distance to Perseus, which is consistent with the bulk of the reddening profiles given by the G18 method for sightlines towards our regions of interest. Introducing the set of foreground dust cloud parameters {, } serves two purposes.

First, towards every region, we observe a set of stars that lie at distances prior to Perseus ( pc) and whose reddening lies between zero and the bulk of the reddening associated with Perseus’ CO features. Thus, there is a physical basis for including these parameters. The second reason is that the foreground dust cloud accommodates the fact that does not form in regions with mag. The parameter then accounts for the reddening in a regime where CO and reddening are non-linear, and corrects for the “missing” reddening below mag which appears in the stellar posteriors but not in the total line-of-sight CO emission. As a result, this parameter is somewhat correlated with the CO formation threshold. Since we do not take into account the critical density of CO, the model will not be a perfect description of the data. However, we are most interested in the distances to the velocity slices (§5.1), which are only very weakly covariant with our nuisance parameters (i.e. the and parameters described in this section, plus the gas-to-dust coefficients described in §5.2).

Like the cloud distances, we place a flat prior on the distance modulus to the foreground reddening cloud , where the lower bound is the minimum distance modulus represented in our gridded stellar posteriors () and the upper bound is the minimum distance modulus to the velocity slices corresponding to Perseus, {, , …}. We additionally place a flat prior on the reddening of the foreground dust cloud, , restricting it to mag. Our adopted upper bound of mag is the largest plausible reddening jump given that we explicitly mask out any sightlines with mag. In practice, the parameters never approach this value, typically leveling off at around mag.

5.4 Accounting for Outliers

We include one final free parameter in our model—denoted —to reduce the influence of outlier stars on our Equation 3. This is critical, given that our gridded stellar posteriors can sometimes be wildly incorrect, due to errors in the stellar photometry or the presence of stars not well-accounted for in our models (e.g. white dwarfs, young stellar objects, quasars, variable stars). To this purpose, a final free parameter is implemented as part of the Gaussian mixture model described in Hogg et al. (2010). quantifies the probability that any individual star is “bad” and unlikely to be drawn from our model. Following Eqn. (17) in Hogg et al. (2010), the total likelihood for an individual star in the context of our mixture model is given by:

(5)

is the original likelihood for the star (computed by taking an integral following over through the star’s distance and reddening posterior) and is the same integral taken through a stellar posterior array that is completely flat (i.e. every cell has an identical probability). This is similar to adding a small constant to the likelihood of every star.

We place a flat prior on our term. We set the lower bound on to 0.0 and the upper bound to 0.1, given that of stars lie on the main sequence in the solar neighborhood (Bahcall & Soneira, 1980) and should be accurately captured by the stellar models used to derive our per-star posteriors on distance and reddening from G18.

Figure 2: Cartoon line-of-sight reddening profile assuming a model with a single velocity slice (where the cloud can be described by single distance component ). Such a reddening profile is defined by the free parameters (foreground cloud distance), (foreground cloud reddening), (distance modulus to single velocity slice) and (gas-to-dust conversion coefficient in single velocity slice). The quantity constitutes the CO emission for star lying in pixel in the single velocity slice . The reddening profile is overlaid on an idealized joint posterior on distance and reddening (grayscale ellipsoid) for individual star lying in CO pixel . The star’s likelihood is simply the integral over following the reddening profile (solid red line); the dashed red lines are excluded from the integration. In §6 the likelihood from this star would be multiplied together with the likelihoods of thousands of other stars to get the total likelihood contribution (see Equation 3).

6 Parameter Estimation using Nested Sampling

For each of our target regions (see Figure 1) we sample a set of model parameters (, , {, , …}, {, ,…}, where is the number of velocity slices) using the nested sampling code dynesty888https://dynesty.readthedocs.io (Speagle et al. 2018, in prep). Nested sampling (Skilling, 2006) is similar to traditional MCMC algorithms in that it generates samples that can be used to estimate the posterior PDF given in Equation 3. The nested sampling algorithm relies on iteratively drawing samples (or “live” points) from the constrained prior distribution, where the likelihood value of a new sample must be greater than the lowest likelihood value of existing samples. In this way, the live point with the lowest likelihood is replaced by a new live point of higher likelihood at every iteration. As this process progresses, the live points sample a smaller and smaller region of the prior “volume”. One continues sampling until some stopping criterion is reached and the remaining live points occupy the region of highest likelihood.

There are several reasons to use dynesty over more commonly used affine-invariant ensemble MCMC samplers. Ensemble MCMC sampling codes like emcee (Foreman-Mackey et al., 2013) have been shown to have “undesirable properties” in higher-dimensional parameter spaces (). When the number of model parameters is high, ensemble samplers may not only fail to converge to the target distribution, but also visually appear converged without have done so (Huijser et al., 2015). Nested sampling has been shown to perform well in higher-dimensional parameter spaces and is also efficient at exploring multi-modal distributions (Feroz et al., 2009; Handley et al., 2015). Additional benefits of dynesty include a user-defined stopping criterion that can act as a convergence metric, so the number of generated samples does not have to be pre-defined beforehand. Finally, dynesty can quantify the statistical uncertainties in a single run using a combined reweighting/bootstrapping procedure (Higson et al., 2017a, b).

Since dynesty can be run in multiple modes, we provide the exact setup used to derive our results in §11.1 of the Appendix. While less efficient at producing independent samples, using a more traditional sampler from emcee produces results that are consistent with the dynamic nested sampler when comparing runs for the cloud with the largest parameter space (NGC1333).999In more detail, in order to test whether our results are robust to changes in sampler, we repeat the analysis for NGC1333 using the affine-invariant MCMC ensemble sampler from emcee (Foreman-Mackey et al., 2013). Specifically, we run with 100 walkers, where each walker takes 20,000 steps. We set so only every tenth sample is saved. Prior to this, we run 1000 burn-in steps. Using the stacked chain (flattened along the walker axis), we compute the median of the samples from the chain for each parameter. We find that the median of the samples agrees with our dynesty results within the uncertainties we report in Table 2. Since we report the 16th and 84th percentiles of the dynesty samples via the upper and lower bounds, this means that the 50th percentile of the samples derived from our emcee run falls within the 16th and 84th percentiles of the samples from our dynesty run.

To get a sense of how the sampling operates, in Figure 3 we show a video which illustrates the progression of our dynesty samples over the course of a run. We select nine stars used in our B5 fit. For each star, we use the current sample (summarized in the table at top) along with the CO emission coincident with each star to build up the star’s reddening profile (red lines), which we then overlay on its distance-reddening posterior from G18 (grayscale background in each panel). Integrating over along each reddening profile, we show the individual log-likelihood for each star in the upper right hand corner of every panel. Adding all these individual log-likelihoods together, we get the total-log likelihood for this batch of nine stars (), which we list atop the panels. The samples are drawn sequentially from the dynesty chain as we converge towards the region of highest likelihood. However, the total log-likelihood for these nine stars does not necessarily increase as the video progresses, since the samples are determined by the best fit to the full sample of 2,898 stars rather than just these nine.

Finally, we note that when calculating the line-of-sight integral for the individual log-likelihoods, we interpolate between cells in the two-dimensional stellar posterior array. This mitigates the negative effects of binning on our results.

\includemedia

[ activate=onclick, flashvars= modestbranding=1 &autohide=1 &showinfo=0 &rel=0 ]https://youtube.com/v/7mkdW04oqLo

Figure 3: If this video is not supported in your document viewer, click here. Video showing samples (summarized in the table at top) from our dynesty run towards the B5 region. These are not fair samples, but are intended to illustrate how variations in the reddening profile affect the log-likelihood. Nine stars are selected out of the 2,898 total towards the region. Each star lies in a different CO pixel, so the reddening profile varies from star to star and is dependent on the CO emission in each velocity slice corresponding to that pixel. The reddening profile is built using the sample summarized in the table at top. Individual stellar log-likelihoods (listed in the upper right corner of each panel) are calculated by integrating the reddening profile (red lines) over through each star’s distance-reddening posterior (background grayscale of each panel) and taking the logarithm. In each stellar posterior, the region of highest probability is marked with a lime green scatter point. The total log-likelihood for all nine stars () is listed above the panels.

7 Results

In Table 2, we summarize the results for the six star-forming regions targeted in this study—L1451, L1448, NGC1333, B1, IC348, and B5. Specifically, we report the 50th percentile of the samples for each parameter: , , {, , …}, {, ,…}, where again, is determined by the number of downsampled velocity slices towards each cloud (see Table 1). In order to properly estimate the posterior, we weight the samples by their posterior mass, as discussed in the dynesty documentation. For the {, ,…} parameters, recall that we report the value of a multiplicative factor in front of the nominal gas-to-dust coefficient we adopt from the literature (see §5.2). In addition to reporting the median value, we provide the 16th and 84th percentiles via the upper and lower bounds, equivalent to the range for a Gaussian distribution. We also include corner plots—showing different projections of our dynesty samples in the -dimensional parameter space—in the Appendix.

Our coefficients, while not usually poorly constrained, are likely afflicted by systematics. The most pertinent systematic is the fact that a better model for the CO-to-reddening conversion factor takes into account the critical density of CO. We adopt a simplistic model to account for this, by letting our coefficients float and also coupling the CO formation threshold to the foreground dust cloud parameter. We build in these parameters to provide our dust templates (the velocity slices) more freedom to arrive at the correct distances in light of the non-linearity between CO emission and reddening. However, the fact remains that a better model for the critical density could more robustly predict the regional variations in the amount of reddening per . We generally favor a smaller CO-to-reddening conversion factor than the nominal factor we derive from the literature (§5.2), possibly suggesting a smaller factor than given in Dame et al. (2001) or a larger -to-reddening factor than given in Bohlin et al. (1978). We caution, however, that our coefficients should not be considered accurate representations of how CO emission translates to reddening along the line of sight.

We discuss the properties of individual clouds in comparison with the literature values in more detail in the next section. We also provide an average distance and distance uncertainty to each cloud (Table 3). The average distances are computed using the distances to the set of velocity slices towards each cloud, and we weight each slice according to its total contribution to the line-of-sight reddening. Specifically, to compute the average distance to each cloud we perform a Monte Carlo-based averaging procedure where samples are drawn at random (again, weighted by their posterior mass) from “noisy” realizations of our original dynesty chain. For each Monte Carlo draw we first construct a “simulated” set of samples using dynesty’s simulate_run() function. This allows us to account for statistical uncertainties in the average cloud distances we report in Table 3. Then, for each realization, after drawing from our “simulated” set of samples, we compute a reddening-weighted average distance to the cloud using the following formula:

(6)

where is the mean CO emission in the velocity slice, is the gas-to-dust coefficient in the slice for each realization, and is the distance to the slice for each realization. We repeat this process 500 times, producing 500 realizations of the average reddening-weighted distance to the cloud. The final average distance and uncertainty to each cloud that we report in Table 3 are the mean and standard deviation of the set of 500 average reddening-weighted distances. We show histograms of the Monte Carlo realizations of the average reddening-weighted distances towards each region in Figure 4, which are discussed further in §8.

In Table 3 we also report the average reddening-weighted velocity for each cloud. Since the CO-weighted mean velocity of each slice is fixed (see Table 1 and §5.1 for details on how this is calculated) and our coefficients are well-constrained, we do not perform a Monte Carlo averaging procedure, but simply weight each velocity slice by the mean CO in each slice times the median coefficient for that slice (summarized in Table 2). We also report the peak reddening velocity for each cloud; this corresponds to the velocity of the slice with the highest reddening, and which dominates the total dust column density along the line of sight (see Table 1 for the velocities of the slices we consider towards each region)

Finally, we emphasize that we do not take into account systematic uncertainty in our distance estimates stemming from the reliability of our stellar models (used to derive the per-star distance and reddening posteriors) which could be as high as (see Schlafly et al., 2014). However, any systematic uncertainty should affect all the cloud distances in the same way, so while the absolute distance of the clouds can be affected by up to , the relative distance between the clouds should remain relatively unchanged.

Figure 4: Monte Carlo realizations of the average reddening-weighted distance to each cloud, as described at the beginning of §7. Each “count” is a reddening-weighted average distance computed using Equation 6. From top to bottom, the panels are sorted in order of increasing right ascension. We report the mean of the distribution in each panel (vertical blue lines) as the final overall average distance to the cloud in Table 3. Likewise, the uncertainty we report in Table 3 is the standard deviation of each distribution.
{turnpage}
\colnumbers
Table 2:
\colnumbers
[-.3in] Cloud
mag mag mag mag mag mag mag mag
pc pc pc pc pc pc
B5
IC348
B1
NGC1333
L1448
L1451
**footnotetext: The upper and lower bounds on these parameters are uncertain to and have been rounded off to two significant figures

Note. – The results of our parameter estimation for major star-forming regions across the Perseus Molecular Cloud. For each parameter, we report the 50th percentile of the samples. The 16th and 84th percentiles are given via the upper and lower bounds. The columns are summarized as follows: (1) Name of the star-forming region. (2) the distance modulus of the foreground reddening cloud. Directly under it we list the corresponding distance in parsecs. (3) the reddening in of the foreground cloud. (4) the fraction of “bad” stars, implemented as part of a Gaussian mixture model in an attempt to mitigate outliers. (5)-(10) the distance moduli of the CO velocity slices corresponding to each region (c.f. Table 1). Directly under them we list the corresponding distances in parsecs. (11)-(16) multiplicative factors in front of the nominal gas-to-dust coefficient adopted in this work (), which are built into the model to account for the uncertainty in how we translate CO emission to reddening.

\colnumbers
Table 3:
Cloud Name Average Distance Modulus Average Distance Average Velocity Peak Reddening Velocity
mag pc
L1451 3.8 4.2
L1448 4.1 4.8
NGC1333 5.7 7.3
B1 6.2 6.7
IC348 8.4 9.0
B5 9.7 9.7

Note. – Average region-by-region distances and velocities for clouds across the Perseus complex. In column (1), we list the name of the star-forming region. In column (2), we show the average reddening-weighted distance to the downsampled velocity slices towards each region (see the beginning of §7 for details on how this is calculated). The uncertainty provided only accounts for the statistical uncertainty and does not include any systematic uncertainty due to the reliability of the stellar models implemented in our per-star distance and reddening posterior inference; the systematic uncertainty is unknown, but could be as high as 10% (Schlafly et al., 2014). In column (3), we convert the average distance modulus from column (2) into its corresponding distance in parsecs. In column (4), we show the reddening-weighted average velocity for each cloud (see §7). In column (5) we list the velocity slice which contains the highest amount of reddening, dominating the dust column density along the line of sight (see column (5) in Table 1 for the velocity slices we consider towards each region).

7.1 B5

We consider three velocity slices towards B5 (with CO-weighted velocities of 8.3, 9.7, and ) and determine of , , and mag, respectively. This corresponds to distances of , , and pc. We find that the different velocity components of B5 correspond to nearly identical distances, and all the slices are consistent with being at the same distance given the uncertainty on each slice. We find an average distance to B5 of mag ( pc). B5 is farthest away in distance, in comparison to IC348, B1, NGC1333, L1448, and L1451.

The line-of-sight reddening profile for B5 determined using the median of the samples from our dynesty run is shown in red in Figure 5. The line-of-sight reddening profiles determined by drawing random weighted samples from the same run are shown in blue, which gives a sense of the underlying uncertainty in the parameters. While the CO intensity (and thus the magnitude of the reddening jump corresponding to each slice) changes from pixel to pixel, for illustrative purposes we take the average CO value of each velocity slice to draw the profiles. In the background grayscale, we show the stacked stellar posteriors on distance and reddening for all the stars used in the analysis; it has been normalized so that each distance column has the same amount of “ink.” Finally, we overlay the most probable distance and reddening for each star in lime green, obtained by extracting the cell in each gridded stellar posterior array with the maximum probability.

We place B5 about 50 pc further away than Černis (1993). That study performs optical photometry on dozens of stars in the vicinity of IC348 (including B5). From this multi-band optical photometry Černis (1993) infers a spectral type and intrinsic color for each star by assuming some reddening law and determining where the stars’ unreddened colors intersect the stellar locus of main sequence stars in various color-color projections. The reddening then follows from the difference between the star’s observed and intrinsic colors, which can subsequently be used to determine the distance to the stars through the adoption of a fixed value. Černis (1993) is able to roughly bracket B5 between lower extinction foreground stars and higher extinction background stars, constraining the distance to B5 to , which is significantly lower than our distances of pc. However, considering that Černis (1993) states their uncertainties can be as high as 25%, our two distances can still be reconciled.

Our distances to B5 agree well with the distances to the same region from Schlafly et al. (2014), which are derived using a technique similar to the one presented in this work. Recall that Schlafly et al. (2014) finds distances to sightlines distributed systematically across the Perseus Molecular Cloud, by modeling the line-of-sight reddening distribution as caused by a single dust cloud at some distance , with an angular distribution given by Planck (Planck Collaboration et al., 2011). Schlafly et al. (2014) then samples the most probable distance to the cloud, by determining which reddening profile is most consistent with distance and reddening posteriors (Green et al., 2014, 2015) for stars along the same line of sight. Considering sightlines in the immediate vicinity of B5 ( from our region of interest shown in Figure 1), with uncertainties 20%, Schlafly et al. (2014) determines distances of pc, pc, and pc so they also favor higher distances on average than Černis (1993).

Figure 5: Line-of-sight reddening profile for the B5 star-forming region. The full profile is integrated out to mag (see Figures 2 and 3), but for illustrative purposes we only show the reddening profile in the range mag. The red line indicates the reddening profile parameterized by the median of the samples for each parameter (summarized in the table above the figure), derived from our dynesty run. Since the magnitude of the reddening jump depends on the amount of CO emission, we have taken the average of the CO emission in each slice to obtain an estimate of the reddening to B5. The blue lines are random samples from the same chain used to derive the median. In the background grayscale, we show 2,898 distance-reddening stellar posteriors used in the parameter estimation problem (see G18), stacked on top of one another; the stacked stellar posterior has been normalized so that each distance modulus bin has the same amount of “ink”. We plot the most probable distance-reddening position for each star as a lime green scatter point, obtained by extracting the cell with the maximum probability in each of the 2,898 gridded stellar posteriors. These lime green points are shown only for reference and are not used in the fit, which always integrates through the full posterior for each star.

7.2 Ic348

We consider four velocity slices towards IC348 (with CO-weighted velocities of 5.4, 7.4, 9.0, and ) and determine of , , , and mag, respectively. This corresponds to distances of , , , and pc. Like B5, we find that the different velocity components of IC348 correspond to nearly identical distances, and all the slices are consistent with being at the same distance given the uncertainty on each slice. We find an average distance to IC348 of mag ( pc). The line-of-sight reddening profile for IC348 determined using the median of the samples from our dynesty run is shown in red in Figure 6.

It is important to emphasize that IC348 has by far the largest foreground reddening cloud ( mag). We are hesitant to say this is physical. Rather, this could be due an abnormally high CO formation threshold in IC348. Pineda et al. (2008) approximates the CO formation threshold on a region-by-region basis in Perseus by linearly fitting vs. in unsaturated lines of sight ( mag). Pineda et al. (2008) then determines values for the parameter denoted“(see Eqn. 18 in Pineda et al., 2008). “” is very roughly the minimum extinction below which there is no emission. Pineda et al. (2008) find that IC348 has the highest parameter of all the regions they consider, equivalent to 1.6 mag (see Table 4 in Pineda et al., 2008). It is, for example, a factor of three higher than that found for NGC1333 and 50% higher than that found for B5. Since our foreground reddening parameter is weakly coupled to the CO formation threshold, we believe this is increasing the value of . is intended to be a nuisance parameter in our model, and is built in solely to provide the velocity slices enough freedom to arrive at the correct distances. Thus, we would not discount our distances due to the large foreground reddening parameter, but would caution that these distances are the most uncertain in comparison to our other regions, where the foreground reddening parameter is more physical (see Table 2). This larger uncertainty is clearly reflected in the large dispersion in the average distance to IC348 from our Monte Carlo averaging procedure (described in the beginning of §7), as seen in Figure 4.

The distance to IC348 has been much debated in the literature since the 1950’s. Almost all distances are photometric and rely on directly or indirectly determining the color excess in , which can then be used to calculate the distance similarly to the Černis (1990, 1993) method. Like B5, Černis (1990) places IC348 at a distance of . However, similar photometric-based studies by Trullols & Jordi (1997) and Harris et al. (1954) determine distances to IC348 of 240 pc and 316 pc, respectively, so there is little agreement in the literature, particularly in the absence of more modern studies. We note that our distances to IC348 are in agreement with the distance of pc that Schlafly et al. (2014) calculates for a sightline intersecting IC348 (again, derived using a very similar technique as presented in this work).

Figure 6: Line-of-sight reddening profile for the IC348 star-forming region in the range mag. The meaning of points, lines, and the background grayscale are the same as in Figure 5, except a total of 2,061 stars are used in the analysis.

7.3 B1

We consider five velocity slices towards B1 (with CO-weighted velocities of 1.1, 2.8, 4.8, 6.7, and 8.2 ) and determine of , , , , and mag, respectively. This corresponds to distances of , , , , and pc. The velocity slices towards B1 show a larger distance dispersion than B5 and IC348, with most of the reddening at mag, then some around mag, and then very little around mag, the latter of which is very poorly constrained. However, this is only mild evidence for multiple distance components, and could amount to 10 pc given the uncertainties on individual slices. We find an average distance to B1 of ( pc). The line-of-sight reddening profile for B1 determined using the median of the samples from our dynesty run is shown in red in Figure 7.

Again, using the same technique as implemented in Černis (1993) for B5 and IC348, Černis & Straižys (2003) determines a distance of pc to B1 by calculating the extinction and distance to 98 stars towards the B1 sightline. Like B5 and IC348, we find that our distances to B1 are systematically further away, this time on the order of pc. Regardless of its absolute distance, it is likely that B1 lies somewhere in between the eastern (B5, IC348) and western (NGC1333, L1448) portions of the cloud along the line of sight.

Figure 7: Line-of-sight reddening profile for the B1 star-forming region in the range . The meaning of points, lines, and the background grayscale are the same as in Figure 5, except a total of 2,255 stars are used in the analysis.

7.4 Ngc1333

We consider six velocity slices towards NGC1333 (with CO-weighted velocities of -0.1, 1.7, 3.5, 5.5, 7.3, and 8.8 ) and determine of , , , , and mag, respectively. This corresponds to distances of , , , , , and pc. We find an average distance to NGC1333 of ( pc). While four of the slices are well-constrained and at a similar distance of ( 280 pc), we find that the distance to the first and second slices, at the lowest velocities of -0.1 and 1.7 km/s are significantly closer in distance (at 246 and 259 pc). Looking at a 3D volume rendering of the CO emission in NGC1333 (from e.g. the CO COMPLETE survey page) we see that there is actually a lower velocity wisp-like structure in front of the bulk of the higher velocity CO emission constituting this cloud. We find that the velocity of our first and second slices, and , are spatially and kinematically coincident with this wispy structure in front of NGC1333, suggesting our algorithm is able to place (albeit with larger uncertainties) this structure at a different distance than the bulk of the cloud. The larger uncertainties are due to the fact that these slices are contributing only a small amount of CO emission to the total along the line of sight. Thus, they make weaker dust templates, leading to more poorly constrained distances. Nevertheless, it is promising that our method is able to separate this wisp structure from the rest of the cloud. The line-of-sight reddening profile for NGC1333 determined using the median of the samples from our dynesty run is shown in red in Figure 8. A small jump in reddening is apparent before the main reddening jump, which corresponds to the wisp’s average reddening contribution.

In comparison to the literature, we place NGC1333 on average 45 pc further away than Hirota et al. (2008), which determines a distance of to NGC1333 using astrometry of masers associated with a young stellar object (YSO) embedded inside the cloud. Considering just the slices coincident with the wisp, the difference is only about 20 pc; however, since the YSO could not have formed inside the wisp, this would only be a plausible scenario to account for the discrepancy if the YSO had migrated towards the wisp since birth. We note that the Hirota et al. (2008) result is in agreement with the photometric distance to NGC1333 from Černis (1990) (220 pc), which uses the same technique as implemented in Černis (1993) for B5. Hirota et al. (2008) estimates their uncertainties to be on the order of 8%, in comparison to the 25% uncertainty quoted by Černis (1990). We find the typical uncertainty on our average distance to be about 3%. However, we reiterate that this only accounts for statistical uncertainty and neglects any systematic uncertainty, which is unknown but could be as high as 10%. When added in quadrature, this produces total combined uncertainties on the order of . We note that Gaia data (from the Tycho-Gaia Astrometric Solution; Gaia Collaboration et al., 2016) supports a farther distance to NGC1333 (beyond 275 pc) which we discuss further in §8. Finally, as in the case of B5 and IC348, the Schlafly et al. (2014) distance to a sightline near NGC1333 (d= pc) is in agreement with our CO-based distances.

Figure 8: Line-of-sight reddening profile for the NGC1333 star-forming region in the range mag. The meaning of points, lines, and the background grayscale are the same as in Figure 5, except a total of 3,412 stars are used in the analysis.

7.5 L1448

We consider five velocity slices towards L1448 (with CO-weighted velocities of -0.7, 0.8, 2.9, 4.8, and 6.6 ) and determine of , , , , and mag, respectively. This corresponds to distances of , , , , and pc. All of the velocity slices are consistent with being at the average distance to the cloud, which we find to be ( pc). The line-of-sight reddening profile for L1448 determined using the median of the samples from our dynesty run is shown in red in Figure 9.

As is also the case with NGC1333, we place L1448 approximately 50 pc further away than the parallax distance derived from maser parallax measurements. Using a similar technique to Hirota et al. (2008), Hirota et al. (2011) monitor maser activity associated with a YSO embedded in the L1448 star-forming region, and determine a parallax distance of . Finally, we note that Schlafly et al. (2014) also prefer a farther distance to L1448, placing the cloud at a distance of pc, which is 20 pc from the average distance we find for the cloud.

7.6 L1451

We consider four velocity slices towards L1451 (with CO-weighted velocities of 0.3, 2.3, 4.2, and 5.8 ) and determine of , , , and mag, respectively. This corresponds to distances of , , , and pc, respectively. L1451 is the only region which shows evidence of the velocity slices lying at large distance separation, with the two major reddening components (at mag and mag) lying over 50 pc apart. Approximately 85% of the average reddening attributed to L1451 lies at the far distance. Though less meaningful given the evidence of there being multiple distance components, we find an average distance to L1451 of ( pc). The line-of-sight reddening profile for L1451 determined using the median of the samples from our dynesty run is shown in red in Figure 10.

In comparison to the other regions discussed, there have been fewer attempts to determine a distance to L1451. Most studies which target L1451 either assume an average distance to the cloud based on a compilation of previous distance estimates for the entire Perseus complex ( 250 pc, Pineda et al., 2011) or else assign the same distance determined via trigonometric parallax measurements for the neighboring L1448 and NGC1333 regions ( 230 pc, Storm et al., 2016; Maureira et al., 2017). The most targeted distance estimate to the cloud comes from Schlafly et al. (2014), which considers a sightline about half a degree away from the L1451-mm dense core (see Pineda et al., 2008). For that sightline, Schlafly et al. (2014) determines a distance of pc, which is in between the two distances where we place most of the cloud.

Figure 9: Line-of-sight reddening profile for the L1448 star-forming region in the range mag. The meaning of points, lines, and the background grayscale are the same as in Figure 5, except a total of 2,061 stars are used in the analysis.
Figure 10: Line-of-sight reddening profile for the L1451 star-forming region in the range mag. The meaning of points, lines, and the background grayscale are the same as in Figure 5, except a total of 2,170 stars are used in the analysis.

8 Discussion

Across the entire Perseus Complex, the distances to our velocity slices typically vary between pc. The slices constituting the bulk of the reddening for each cloud typically lie within of each other, while the overall dispersion across the entire cloud is on the order of mag. From east to west we find the average distance to B5, IC348, B1, NGC1333, L1448, and L1451 to be 310 pc, 296 pc, 292 pc, 279 pc, 278 pc, and 294 pc, respectively. We summarize our results in Figure 11, which includes plots showing how the declination (panel a), velocity (panel b) and average reddening-weighted distance (panel c) for each region varies as a function of right ascension, as well as how the cloud declination (panel d) and average reddening-weighted distance (panel e) varies as a function of cloud velocity. The background colorscale in panels (a), (c), and (d) are different projections of a combined volume rendering of the (red) and (blue) Perseus COMPLETE cubes. A movie showing the full 3D volume rendering of this cube can be found on the Dataverse.

In more detail, we discuss how velocity maps to distance in Perseus in §8.1 and how our CO-reddening-based distance estimates compare to those derived from maser parallax observations in §8.2.

8.1 Mapping Velocities to Distances

Because we sample for the distances to the velocity slices of CO spectral cubes, we can explicitly tie our distance measurements to the velocity structure of the molecular gas. Thus, for the first time, we can show that the velocity gradient from east to west roughly maps to a corresponding distance gradient. This distance gradient is apparent in Figure 11e, which shows average distance as a function of both peak-reddening velocity (dark colored points) and average reddening-weighted velocity (light colored points). We note that a potential distance gradient has been proposed before (for instance, in Ridge et al., 2006) but could not be confirmed with any confidence due to the fact that our traditional method of translating molecular clouds’ line-of-sight velocities to distances (via a Galactic rotation curve e.g. Roman-Duval et al., 2009) cannot be applied locally because peculiar motions dominate over the motions due to Galactic rotation at such close separation.

In more detail, Figure 11e shows that the the average velocity of the slices towards IC348 and B5 clouds lies at , while the slices towards NGC1333, and L1448 typically lie between . As a result, the typical velocity gradient of across the cloud translates to a distance gradient of 30 pc. We note that L1451 appears to be the outlier in our distribution, lying at an average distance of mag (294 pc), which is similar to the distance to IC348 and B5 on the eastern side of the cloud. However, it has velocities more consistent with NGC1333 and L1448 on the western side. L1451 is also the only cloud which shows evidence of the different velocity components lying at large distance separation ( 50 pc). If we neglect L1451, however, we find that the distances to L1448, NGC1333, B1, IC348, and B5 monotonically increase with average-reddening weighted velocities, with higher average velocities producing higher average distances. If we only consider the weaker distance component for L1451 (containing only 15% of the total non-foreground line-of-sight reddening), our monotonic trend would hold throughout the cloud. However, this weaker distance component is also likely an extension of the same wisp like structure we observe in front of NGC1333.

To put this distance gradient in context, we emphasize that projected length of Perseus spans on the plane of the sky, which corresponds to a length of pc assuming our average distance to the complex ( pc). This means that the line-of-sight extent of Perseus could be equal to or as high as its projected length, giving it an aspect ratio between and .

While a single cloud distance for all of Perseus—usually around 250 pc—is often adopted (e.g. Curtis & Richer, 2010; Arce et al., 2010; Enoch et al., 2006; Rebull et al., 2007; Campbell et al., 2016) our results suggest that this may be inappropriate. Again, this potential distance gradient has been discussed extensively in the literature (including in many of the cloud-wide studies that adopt a single cloud distance); however, the distance gradient has been difficult to implement due to the diversity of distance mapping techniques and the high uncertainties associated with certain methods (e.g. the photometric distance method from Černis, 1990, 1993; Černis & Straižys, 2003). Nevertheless, since properties like clump mass are dependent on the distance squared, distance discrepancies of just a few tens of parsecs could produce variations in these properties as high as 50%. Our CO-reddening based distances are systematically calculated across the cloud and have typical combined statistical, sampling, and systematic uncertainties , so they can be used in lieu of traditional photometric methods to constrain the distances to star-forming regions in the Perseus Molecular Cloud that lie at different velocities.

8.2 Comparison to Parallax-Based Distances

Our distances to the NGC1333 and L1448 star-forming regions are about 50 pc farther away than those derived via trigonometric parallax measurements of water masers from Hirota et al. (2008, 2011). We determine distances to NGC1333 and L1448 of , with typical statistical uncertainties on the order of 3%, though there is additional systematic uncertainty which is unknown but could be as high as 10% (Schlafly et al., 2014); this is due to the reliability of the stellar models used in the creation of our per-star distance and reddening posteriors. Hirota et al. (2008, 2011) determine a maser-based distance of pc to the NGC1333 and L1448 star-forming regions, and they estimate their uncertainties to be . Our measurements are only about away from the Hirota et al. (2008, 2011) results, and a distance of pc could be consistent with both sets of observations.

However, we also emphasize that Gaia stellar parallax measurements towards stars along the NGC1333 sightline (given by the Tycho-Gaia Astrometric Solution or TGAS; Gaia Collaboration et al., 2016) favor the farther, CO-reddening-based distance we present in this work. This is illustrated in Figure 12. There are two TGAS stars embedded near the dense star-forming clump of NGC1333. Because the NICER method (employed throughout this work to mask out regions where the CO is self-absorbed) underestimates the total column density in the very densest regions (e.g. the core of NGC1333; Pineda et al., 2008), we obtain total line-of-sight extinction measurements towards these stars from a combination of Herschel and Planck observations (Zari et al., 2016). Converting the Zari et al. (2016) total line-of-sight column densities towards these two stars101010The corresponding TGAS Source IDs for these stars are 121392062300963584 (BD+30547) and 121406905707934464 (BD+30549) to total line-of-sight visual extinction, we obtain values of mag and mag.111111We perform the conversion from column density to using the relation given in Bohlin et al. (1978) TGAS parallax measurements place both these stars at a distance of pc (). Given that these stars are observed in the optical (G band), it is unlikely that both would be visible behind such a high dust column density, which would suggest that the bulk of the reddening must be beyond the TGAS parallax distance of 275 pc in order for the stars to be observed. The star with the highest total line-of-sight extinction (BD+30547, at mag) is also widely considered to be foreground to NGC1333 in the literature (e.g. Černis, 1990; Preibisch, 2003). In this vein, Preibisch (2003) derives the extinction towards the star BD+30547 via fits to its X-ray spectrum, determining an ( mag), which further supports the idea that BD+30547 lies in front of NGC1333. Our CO-reddening-based distance of d280 pc to NGC1333 (just beyond the TGAS distance to BD+30547) is thus more consistent with Gaia. We perform the same exercise using the TGAS data towards L1448. The results are inconclusive but are not inconsistent with either our distance or the Hirota et al. (2011) maser distance.

Soon, Gaia DR2 and later releases will provide even richer datasets for comparison with our CO-reddening-based distances. In addition to providing a point of comparison for the maser parallax distances towards Perseus, they can also be incorporated into Bayestar—the G18 program used to derive our per-star distance and reddening posteriors. By cross-matching the Gaia stars with the existing PS1 and 2MASS photometry, Gaia parallax measurements could provide better distance localization, thereby shrinking the probable range of distances to the star in our distance-reddening stellar posteriors. This in turn can better constrain the distances to the CO velocity slices in our model.

Figure 11: Panel a: Combined (red) and (blue) integrated intensity map of Perseus (Ridge et al., 2006). The boundaries we define for each region (same as in Figure 1b) are shown in black. The centroid of each polygon is marked with a different colored point. Panel b: an RA-velocity diagram of Perseus, with the same colorscale as panel (a). The colored points show the peak-reddening velocity (dark points) and average-reddening weighted velocity (light points) as a function of right ascension for each cloud. The errorbars in right ascension show the horizontal extents of the polygons overlaid in panel (a). Panel c: Average reddening-weighted distance to each region as a function of right ascension. Panel d: a velocity-DEC diagram of Perseus, with the same colorscale as panel (a). The colored points show the declination as a function of peak reddening velocity (dark points) and average-reddening weighted velocity (light points). The errorbars in declination show the vertical extents of the polygons overlaid in panel (a). Panel e: The average reddening-weighted distance to each region as a function of its peak reddening velocity (dark points) and average reddening-weighted velocity (light points). With the exception of L1451 (at a slightly higher distance), we find that the cloud distances increase monotonically with both average reddening-weighted velocity and right ascension. In total, the velocity gradient of maps to a distance gradient of about 30 pc. The uncertainty provided on distance only accounts for the statistical uncertainty and does not include any systematic uncertainty due to the reliability of the stellar models implemented in our per-star distance and reddening posterior estimation; the systematic uncertainty is unknown but could be as high as 10% (Schlafly et al., 2014). The systematic uncertainty, however, should only affect the absolute distance, and not the relative distance between clouds.
Figure 12: TGAS stars (Tycho-Gaia Astrometric Solution; Gaia Collaboration et al., 2016) towards the NGC1333 star-forming region (green polygon). In the left panel, we plot the location of the TGAS stars (round points, color-coded according to distance modulus) on top of a Herschel & Planck-based column density map from Zari et al. (2016). We also overlay the location of the maser (black star) used to derive the trigonometric parallax distance to NGC1333 from Hirota et al. (2008). In the right panel, we plot total line-of-sight visual extinction (from Zari et al., 2016) as a function of distance modulus for the same TGAS stars. We mark our average CO-reddening-based distance and the trigonometric parallax distance to the Hirota et al. (2008) maser via the black vertical lines. The uncertainties are highlighted via the light gray rectangles. We note that the uncertainty on our distance only accounts for the statistical uncertainty and does not include any systematic uncertainty due to the reliability of the stellar models implemented in our per-star distance and reddening posterior estimation; the systematic uncertainty is unknown but could be as high as 10% (Schlafly et al., 2014). Note how there are two TGAS stars at with very high line-of-sight extinction that are in close spatial proximity to the maser. Given that the TGAS stars are observed in Gaia’s G-band, if NGC1333 was located at the maser distance, it would be unphysical for both of these stars to be observed behind such a high dust column density. This suggests that the bulk of the cloud might be farther than the maser parallax distance (beyond , or 275 pc), which is consistent with the NGC1333 distance derived in this work.

9 Conclusion

We present a catalog of distances to major star-forming regions in the Perseus Molecular Cloud in the velocity range to . We produce the catalog using a two-step process. First, we infer the distance-reddening posteriors for batches of stars across Perseus based on the technique presented in G18. Then, we model the reddening along the line of sight towards these stars as a linear combination of the optical depth of CO velocity slices. The result is a set of distances tied to the velocity slices defining the structure of the molecular gas towards these clouds, which we then sample using a Monte Carlo method. We target the B5, IC348, B1, NGC1333, L1448, and L1451 star-forming regions, and find typical cloud distances of 310 pc, 296 pc, 292 pc, 279 pc, 278 pc, and 294 pc, respectively. On average, the velocity gradient maps to a corresponding distance gradient of pc, with the eastern half of Perseus systematically farther away than the western half. The exception is L1451, which appears to show evidence of multiple distance components, with the bulk of its reddening lying at a farther distance than its neighbors L1448 and NGC1333. Excluding L1451, we find that the average velocities of L1448, NGC1333, B1, IC348, and B5 increase monotonically with velocity, with higher average velocities producing higher average distances.

Typical uncertainties on our distances are on the order of , with of that due to statistical uncertainties, and 10% (Schlafly et al., 2014) due to the reliability of the stellar models used to derive our per-star distance and reddening posteriors121212We add the statistical and systematic uncertainties in quadrature to estimate the combined 11% uncertainty on our distances.. We note that we place the western portion of the Perseus Molecular Cloud (NGC1333 and L1448) approximately 50 pc () further away than the distances derived from maser parallax measurements towards the same regions (Hirota et al., 2008, 2011). We find support for our farther distances from the TGAS stars (Tycho-Gaia Astrometric Solution; Gaia Collaboration et al., 2016) towards NGC1333, but our distances and the maser distances can still be reconciled if NGC1333 and L1448 lie in the distance range pc. The arrival of the Gaia DR2 data will likely be able to confirm or deny our prediction that the clouds lie closer to 270-280 pc than 230 pc, as it will include many fainter stars than currently available in TGAS.

We have only scratched the surface of what is possible through the combination of stellar photometry, CO observations, and Bayesian statistics. The accuracy of our cloud distances are directly linked to the reliability of our per-star distance and reddening posteriors, which are derived from near-infrared and optical photometry (see G18). The addition of deeper near-infrared surveys (e.g. UKIDSS; Warren et al., 2007) as well as the release of upcoming all-sky surveys like Gaia and LSST will only better constrain stellar distances and reddenings, and enable us to probe deeper into dust-enshrouded regions like Perseus. Moreover, our technique is flexible enough to incorporate models with multiple clouds along the line of sight, which would prove useful towards the inner galaxy. Thus, the same technique presented here for the Perseus Molecular Cloud could be applied to local molecular clouds across the full sky, paving the way for accurate cloud distances (tied to the distribution of CO molecular gas) in the solar neighborhood and beyond.

10 Acknowledgements

We would like to thank Mark Reid for his expertise regarding the uncertainty surrounding distance measurements derived from maser parallax observations, Thomas Dame for his expertise on the structure and properties of CO molecular gas in the galaxy, and Charlie Conroy for insight regarding the stellar models we implemented to derive the per-star distance and reddening posteriors. All three also provided valuable comments on an early draft of this work.

11 Appendix

11.1 Dynesty Setup

We use the following dynesty code to produce the chains used for parameter estimation in §7.

    sampler = dynesty.NestedSampler(log-likelihood, prior-transform, ndim, bound=‘multi’, sample=‘rwalk’, update_interval=6., nlive=300, walks=50)
    sampler.run_nested(dlogz=0.1)

where “log-likelihood” is our log-likelihood function (described in §5) and “prior-transform” is a function that transforms our priors (described in §5) from an ‘ndim’-dimensional unit cube to the parameter space of interest. We set our convergence threshold, “dlogz”, equal to 0.1.

Figure 13: Corner plot derived from our dynesty run towards the B5 region.
Figure 14: Corner plot derived from our dynesty run towards the IC348 region.
Figure 15: Corner plot derived from our dynesty run towards the B1 region.
Figure 16: Corner plot derived from our dynesty run towards the NGC1333 region.
Figure 17: Corner plot derived from our dynesty run towards the L1448 region.
Figure 18: Corner plot derived from our dynesty run towards the L1451 region.

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