The Distances to Molecular Clouds at High Galactic Latitudes based on GAIA DR2

TL;DR

This paper determines distances to high-latitude molecular clouds using Gaia DR2 parallax and extinction data, employing Bayesian analysis to identify extinction breakpoints, and validates the method with known star-forming regions.

Contribution

It introduces a reliable method for measuring molecular cloud distances using Gaia DR2 data, providing first-time distances for 13 clouds and confirming previous measurements.

Findings

Distances for 52 molecular clouds are determined with about 5% systematic error.

Most clouds are at high Galactic latitudes, with 13 measured for the first time.

The method's reliability is validated against well-known star-forming regions.

Abstract

We report the distances of molecular clouds at high Galactic latitudes (|b|>10$^\circ$) derived from parallax and G band extinction (A$_{\rm G}$) measurements in the second Gaia data release, Gaia DR2. Aided by Bayesian analyses, we determined distances by identifying the breakpoint in the extinction A$_{\rm G}$ towards molecular clouds and using the extinction A$_{\rm G}$ of Gaia stars around molecular clouds to confirm the breakpoint. We use nearby star-forming regions, such as Orion, Taurus, Cepheus, and Perseus, whose distances are well-known to examine the reliability of our method. By comparing with previous results, we found that the molecular cloud distances derived from this method are reliable. The systematic error in the distances is approximately 5%. In total, 52 molecular clouds have their distances well determined, most of which are at high Galactic latitudes, and we provide reliable distances for 13 molecular clouds for the first time.

Tables

Table 1. Table 1: Comparing with VLBI distance measurements.

Cloud	Gaia DR2	VLBI	References
	(pc)	(pc)	(pc)
L1641	${408}_{-4}^{+4}$	428$\pm$10	1
NGC 2068	${412}_{-4}^{+4}$	388$\pm$10	1
Mon R2	${862}_{-10}^{+10}$	${893}_{-40}^{+44}$	2
Ophiuchus	${131}_{-2}^{+2}$	137.3$\pm$1.2	3
Perseus	${310}_{-4}^{+4}$	293$\pm$22 or 321$\pm$10	4

Equations

$$\Delta A_{\mathrm{G}}=\frac{1}{2}\left(84th\_percentile-16th\_percentile\right).$$

$$\int_{-\infty}^{x}p\left(I|\upmu,\upsigma,\rm SP,\uplambda\right)\mathrm{d}I=\left\{\begin{array}[]{l}\int_{-\infty}^{x}\frac{1}{\sqrt{2\uppi}\upsigma}\exp\left({-\frac{(I-\upmu)^{2}}{2\upsigma^{2}}}\right)\mathrm{d}I,x\leq\mathrm{SP},\\ \\ \int_{-\infty}^{\mathrm{SP}}\frac{1}{\sqrt{2\uppi}\upsigma}\exp\left({-\frac{(I-\upmu)^{2}}{2\upsigma^{2}}}\right)\mathrm{d}I\\ +\int_{\mathrm{SP}}^{x}\uplambda\exp\left(-\uplambda\left(I-\rm SP\right)\right)\mathrm{d}I,x>\mathrm{SP},\end{array}\right.$$

$$\text{upphi}\left(x\right)=\frac{1}{\sqrt{2\text{uppi}}}{\int }_{\infty }^x{e}^{-t^2/2}\mathrm{d}t$$

$$f_i=\text{upphi}\left(\frac{\mathrm{D}-{\mathrm{d}}_{\mathrm{i}}}{\Delta {\mathrm{d}}_i}\right),$$

$$p\left(A_{\mathrm{G}i}|\upmu,\upsigma\right)=\left\{\begin{array}[]{ll}\frac{\frac{1}{\upsigma\sqrt{2\uppi}}\exp\left(-\frac{1}{2}\left(\frac{A_{\mathrm{G}i}-\upmu}{\upsigma}\right)^{2}\right)}{\frac{1}{2}\left(\mathrm{erf}\left(\frac{A_{\mathrm{G}}^{\mathrm{max}}-\upmu}{\sqrt{2}\upsigma}\right)+\mathrm{erf}\left(\frac{\upmu-A_{\mathrm{G}}^{\mathrm{min}}}{\sqrt{2}\upsigma}\right)\right)},&A_{\mathrm{G}}^{\mathrm{min}}\leq A_{\mathrm{G}i}\leq A_{\mathrm{G}}^{\mathrm{max}},\\ &\\ 0,&\mathrm{otherwise},\end{array}\right.$$

$$\mathrm{erf}\left(z\right)=\frac{2}{\sqrt{\text{uppi}}}{\int }_0^z{e}^{-t^2}\mathrm{d}t.$$

$$P{F}_i=p\left(A_{\mathrm{G}i}|{\text{upmu}}_1,\sqrt{{\text{upsigma}}_1^2+\Delta A_{\mathrm{G}i}^2}\right).$$

$$P{B}_i=p\left(A_{\mathrm{G}i}|{\text{upmu}}_2,\sqrt{{\text{upsigma}}_2^2+\Delta A_{\mathrm{G}i}^2}\right).$$

$$p\left(A_{\mathrm{G}i}|{\text{upmu}}_1,{\text{upsigma}}_1,{\text{upmu}}_2,{\text{upsigma}}_2,\mathrm{D}\right)=f_{i}P{F}_i+\left(1-f_i\right)P{B}_i,$$

$$\left\{\begin{array}[]{rcl}\rm D&\sim&\mathcal{U}\left(\rm D_{\rm min},\rm D_{\rm max}-50\ pc\right),\\ &&\\ \rm I\upsigma_{1}&\sim&\mathcal{E}\left(2\right),\\ &&\\ \upmu_{2}&\sim&\mathcal{E}\left(\upmu_{50}\right),\\ &&\\ \rm I\upsigma_{2}&\sim&\mathcal{E}\left(\rm I\upsigma_{50}\right),\end{array}\right.$$

Full text

1 Introduction

Determining the distances of molecular clouds, the birthplaces of stars, is usually difficult. Many distance measurement methods, such as the photometric parallax method and the period-luminosity relation, are not applicable to molecular clouds. However, the distance relates directly to many important properties of molecular clouds, especially the mass and size, making their physical states and relationships with Galactic spiral arms largely uncertain.

There are a few approaches to derive molecular cloud distances, but they usually either yield large uncertainties or are not applicable to diffuse and translucent molecular clouds at high Galactic latitudes. For instance, maser astrometry with multiple-epoch Very Long Baseline Interferometry (VLBI) measurements (Xu et al. 2009; Zhang et al. 2013; Ortiz-León et al. 2017a), cannot be performed towards molecular clouds that lack masers. Furthermore, kinematic distance estimates based on modeled rotation curves (Roman-Duval et al. 2009; Rice et al. 2016) suffer from large uncertainties and ambiguities.

Optical extinction provides another approach to determine the cloud distances. However, the optical extinction derived from counting stars towards molecular clouds  (Bok 1937; Magnani & de Vries 1986) involves large uncertainties. A more straightforward but sophisticated way is to examine the variation of optical extinction with respect to distance along the line of sight (see Knude & Hog 1998). For example, using this method, Schlafly et al. (2014) provides a large distance catalog of 18 well-known star-forming regions and 108 high-Galactic-latitude molecular clouds cataloged by Magnani et al. (1985, hereafter denoted as MBM). For each star, they derived its distance and extinction simultaneously from Pan-STARRS1 (Kaiser et al. 2010; Green et al. 2014) photometry and subsequently estimated the molecular cloud distances according to the breakpoint of the extinction. However, as Schlafly et al. (2014) have pointed out, their distances may have a $\sim$10% systematic uncertainty.

The release of the second Gaia data (Gaia Collaboration et al. 2016, 2018), Gaia DR2, providing parallaxes for about 1.3 billion stars, a large proportion of which have G band extinction ($A_{\mathrm{G}}$) measurements, has advanced this approach substantially. The correspondence between the extinction $A_{\mathrm{G}}$ map (Arenou et al. 2018) and the distribution of molecular clouds (Dame et al. 2001) suggests that the extinction $A_{\mathrm{G}}$ is capable of detecting molecular clouds. Although, as warned by Gaia Collaboration et al. (2018), the extinction $A_{\mathrm{G}}$ has large uncertainties, it is found to be reliable on average (see Andrae et al. 2018).

Gaia DR2 permits the analysis of the extinction caused by molecular clouds more precisely than previously possible and offers an independent means of examining distance estimates of Schlafly et al. (2014). The rest of this paper is organized as follows. In §2 and §3, we introduce the methodology of reducing Gaia data and the process of deriving molecular cloud distances. We present the distance catalog in §4 and compare the results with previous studies in §5. The conclusions are summarized in §6.

2 The Gaia DR2 and Planck 857-GHz  Data

We primarily investigate diffuse molecular clouds at $|b|>10$∘, which are less likely contaminated by other molecular clouds either adjacent or overlapping along the line of sight. Despite being located at high Galactic latitudes, they are usually less than 1 kpc from the Sun and are still located in the Galactic plane (Magnani et al. 1996; Schlafly et al. 2014), They tend to occupy large areas on the sky, making them able to optically extinct a large number of stars. Consequently, the extinction $A_{\mathrm{G}}$ imposed on stars behind molecular clouds is able to be investigated statistically.

We selected Gaia DR2 stars according to their parallaxes and extinction $A_{\mathrm{G}}$. First, we removed those stars with parallaxes $<$ 0.5 mas, corresponding to an upper distance threshold of 2 kpc. Molecular clouds at high Galactic latitudes are usually near the Sun ($<$ 1 kpc), and a 2-kpc cutoff suffices for our study. If the relative errors of stellar parallaxes exceed 20%, the corresponding distances would differ from the reciprocal of their parallaxes by a large amount (Bailer-Jones 2015), and consequently, we only kept those stars whose relative parallax errors are less than 20%. As to the extinction $A_{\mathrm{G}}$, we require $A_{\mathrm{G}}$  $>$ 0, which rejects 13 stars. In total, we have 30,259,242 Gaia stars meet the criteria.

The extinction $A_{\mathrm{G}}$ error of a single star, $\Delta\mathrm{A}_{G}$, is estimated with

[TABLE]

where $\rm 84th\_percentile$ and $\rm 16th\_percentile$ are provided in Gaia DR2. To derive the distance and its standard deviation, we drew 10000 samples from the Gaussian distribution $\mathcal{N}(\varpi,\Delta\varpi)$, where $\varpi$ and $\Delta\varpi$ are the parallax and its error in Gaia DR2, and the mean and standard deviation of the distance correspond to the mean and standard deviation of the reciprocal of the 10000 samples, respectively.

We used Planck 857-GHz images (Planck Collaboration et al. 2014) to trace the molecular clouds rather than IRAS 100-$\mathrm{\upmu}$m and CO spectral lines. Although the IRAS 100-$\mathrm{\upmu}$m images (Schlegel et al. 1998; Miville-Deschênes & Lagache 2005) have a comparable spatial resolution ($5$′) with the Planck 857-GHz survey, the sensitivity of the Planck 857-GHz survey is higher, and consequently, the Planck 857-GHz survey classifies Gaia stars more accurately and produces slightly better results, details of which are discussed in §4.1. Planck 857-GHz  data is more complete than CO survey data (Dame et al. 2001) at high Galactic latitudes, and although Planck 857-GHz emission cannot distinguish molecular cloud components that overlap each other along the line of sight, this situation is not severe at high Galactic latitudes, and usually only one nearby molecular cloud component is present towards one direction. The Planck 857-GHz images are only used to classify Gaia DR2 stars, not to build dust models.

3 METHOD

In this section we describe the method of deriving molecular cloud distances with the Gaia DR2 and Planck 857-GHz data. Principally, molecular clouds increase the extinction $A_{\mathrm{G}}$ of all stars behind them, thus producing breakpoints in the extinction $A_{\mathrm{G}}$ along the line of sight. Consequently, identifying the breakpoints in the extinction $A_{\mathrm{G}}$ is the essential point in our method.

First, we collect two classes of Gaia stars, on- and off-cloud stars, based on Planck 857-GHz emission. On-cloud stars, showing breakpoints in the extinction $A_{\mathrm{G}}$, are those Gaia stars towards molecular clouds. We define as “off-cloud stars” those Gaia stars around molecular clouds, and because they are not affected by molecular clouds, off-cloud stars have no breakpoints in the extinction $A_{\mathrm{G}}$. Practically, the breakpoints are determined with only on-cloud stars, while off-cloud stars are used to confirm the breakpoints by eye. The schematic diagram of Figure 1 depicts the extinction $A_{\mathrm{G}}$ feature of on- (green) and off-cloud (blue) stars.

In the second step, the distances are determined using Bayesian inference with on-cloud stars and are subsequently confirmed with off-cloud stars by eye. We use two molecular clouds, Taurus (Pineda et al. 2010; Ward-Thompson et al. 2016) and Gemini (Li et al. 2015), to illustrate the process of determining distances.

3.1 On- and off-cloud stars

The on- and off-cloud stars are classified according to Planck 857-GHz intensity thresholds. The on-cloud stars are those whose Planck 857-GHz emission is stronger than an intensity threshold (the signal level), while towards off-cloud stars, the Planck 857-GHz emission is fainter than a lower threshold (the noise level). We determined the two thresholds through fitting a mixed distribution combining two functions, Gaussian + exponential, as described below.

First, we manually drew a box region for each molecular cloud, and only those Gaia stars in this box region were considered. This region contains at least part of the molecular cloud and incorporates an extra nearby region. This extra region, where the Planck 857-GHz emission is significantly lower than the molecular cloud region, contains off-cloud stars. Noteworthy is that these box regions are not necessarily the same with traditional boundaries of molecular clouds, i.e., they may be larger or smaller than the entire molecular clouds as long as those regions contain sufficient Gaia stars.

Secondly, we assigned each Gaia star a Planck 857-GHz emission value according to the Planck 857-GHz data. As the histograms demonstrate in Figure 2, the Planck 857-GHz emission roughly contains two components: (1) a background noise that has approximately Gaussian distribution in intensity; (2) the molecular cloud emission that resembles an exponential distribution.

We use four parameters to model this mixed distribution: the mean ($\upmu$) and the standard deviation ($\upsigma$) of the Gaussian part, a switch point ($\rm SP$), and the rate ($\uplambda$) of the exponential distribution. The cumulative distribution function (CDF) of the mixed distribution (the likelihood) is

[TABLE]

where $I$ is the observed Planck 857-GHz intensity.

The four parameters are estimated by maximising this likelihood, and the noise and signal level are subsequently determined according to the two distributions. As demonstrated in Figure 2, we used a noise level of Gaussian mean $\upmu$, and a signal level of exponential median $\left(\rm SP+\ln\left(2\right)/\uplambda\right)$.

There is a trade-off in the choice of the signal level. Lower signal levels keep more stars, which are good for statistical analysis, but involve many stars having low extinction $A_{\mathrm{G}}$, smearing the breakpoints. Higher signal levels make the breakpoints evident but fewer stars remain. The signal level, the median value of the exponential component, is a compromised choice. However, in §4.2, we show that as long as the breakpoints are detected, derived distances are insensitive to the choice of signal levels.

The signal and noise levels classify on- and off-cloud stars, respectively. Although only on-cloud stars are employed to calculate distances, the off-cloud stars are useful as a reference to confirm the breakpoints.

3.2 Estimating the distance

With the extinction $A_{\mathrm{G}}$ and their corresponding distances of on- and off-cloud stars, we are now in the position to calculate the distance of molecular clouds. We built a Bayesian model to estimate the distance with on-cloud stars and solved for the parameters in the model with MCMC sampling. We emphasize that off-cloud stars are not involved in the model but only used to confirm the breakpoint by eye.

The extinction $A_{\mathrm{G}}$ of on-cloud stars includes three components: (1) the foreground $A_{\mathrm{G}}$, (2) the background $A_{\mathrm{G}}$, and (3) the transition values. When passing through a molecular cloud along the line of sight, the extinction $A_{\mathrm{G}}$ gradually increases from the foreground $A_{\mathrm{G}}$ (which is usually 0 mag) level to the background $A_{\mathrm{G}}$ level. Here, we ignored the third component and focused on the average distance in our model. Because the minimum ($A_{\mathrm{G}}^{\mathrm{min}}$) and maximum ($A_{\mathrm{G}}^{\mathrm{max}}$) values of the extinction $A_{\mathrm{G}}$ is 0 and 3.609 mag, respectively, we follow the procedure of Andrae et al. (2018) using a truncated Gaussian distribution to model the on-cloud star extinction $A_{\mathrm{G}}$.

Our model involves four parameters, the cloud distance ($\rm D$), the extinction $A_{\mathrm{G}}$ dispersion ($\upsigma_{1}$) of foreground stars, and the extinction $A_{\mathrm{G}}$ ($\upmu_{2}$) and its dispersion ($\upsigma_{2}$) of background stars. We found the extinction $A_{\mathrm{G}}$  of foreground stars ($\upmu_{1}$) is precisely zero but for only four molecular clouds, so we use $\upmu_{1}=0$ in most cases. These four molecular clouds have additional small molecular cloud components in front of them, so they will be treated particularly. We used a lower distance cutoff to remove the foreground stars of the small molecular cloud components, and their $\upmu_{1}$ were also modeled, i.e., five parameters in total for them.

With those four parameters and a list of distances $\mathrm{d}_{i}$ with standard deviations $\Delta\mathrm{d}_{i}$, which were calculated with the reciprocal of 10000 parallax samples (see §2), and extinction $A_{\mathrm{G}i}$ (with standard deviations $\Delta A_{\mathrm{G}i}$), we derive an appropriate likelihood below. This approach requires no binning, and the binned extinction $A_{\mathrm{G}}$ and distances are only used for eye confirmation.

We calculated the likelihood of each star on the condition of being in front of or behind the molecular cloud. Denoting the CDF of the standard normal distribution as

[TABLE]

and given $\rm D$, the probability for a star to be in front of the molecular cloud is

[TABLE]

while the probability to be located behind the molecular cloud is $1-f_{i}$.

The form of a truncated Gaussian distribution (Andrae et al. 2018), with mean $\upmu$ and standard deviation $\upsigma$, is

[TABLE]

where

[TABLE]

In order to consider the measure error of the extinction $A_{\mathrm{G}}$, we convolved the standard deviation of the extinction $A_{\mathrm{G}}$, $\Delta A_{\mathrm{G}i}$, with $\upsigma_{1}$ and $\upsigma_{2}$. Consequently, the likelihood of foreground stars is

[TABLE]

Similarly, the likelihood of background stars is

[TABLE]

Consequently, the likelihood of a star is

[TABLE]

The total likelihood is the product of the likelihoods over all the stars, and we solve this model with MCMC sampling. In order to obtain a high sampling rate, we may need to set smart priors. The prior distribution of $\rm D$ is assumed to be uniform, so that the model can uniformly search the switch point, i.e., the breakpoint of the extinction $A_{\mathrm{G}}$, along the line of sight. Denoting the minimum and maximum of $\mathrm{d}_{i}$ as $\rm D_{\rm min}$ and $\rm D_{\rm max}$, the prior of $\rm D$ is $\mathcal{U}\left(\rm D_{\rm min},\rm D_{\rm max}-50\ pc\right)$, where $\mathcal{U}$ represents the uniform distribution and the 50 pc is set to avoid touching the edge. In practice, instead of sampling $\upsigma_{1}$ and $\upsigma_{2}$, we sampled their reciprocal, denoted as $\rm I\upsigma_{1}$ and $\rm I\upsigma_{2}$. The prior distributions of $\rm I\upsigma_{1}$, $\upmu_{2}$, and $\rm I\upsigma_{2}$ are assumed to be exponential, and here, we use a form of $\mathcal{E}\left(\beta\right)$ for the exponential distribution, where $\beta$ is the mean. We summarize the priors as

[TABLE]

where $\mathcal{U}$ and $\mathcal{E}$ represent the uniform and exponential distributions, respectively, $\upmu_{50}$ and $\rm I\upsigma_{50}$ are the mean and reciprocal standard deviation of extinction $A_{\mathrm{G}}$  of stars with distances >$$\left(\rm D_{\rm max}-50\ pc\right), and the initial guess of 2 mag*-1* for $\rm I\upsigma_{1}$ is derived from the reciprocal of 0.45 mag, which is the typical extinction $A_{\mathrm{G}}$  standard deviation of clustering stars derived by Andrae et al. (2018).

In order to decrease the autocorrelation time of the MCMC samples, we thinned the samples. Because the acceptance rate is about 25%, we thinned the samples by a factor of 15, considering only every 15th step in each chain, and the autocorrelation time is small (about 4) after thinning. We calculated eight independent chains, and each chain has 1000 thinned samples (with extra 50 burn-in, which means the first 50 thinned samples were removed), i.e., 8000 thinned posterior samples for each parameter. In order to increase the sampling speed, we used the Gibbs sampler (Geman & Geman 1984), i.e., we changed one parameter at a time and used Metropolis–Hastings algorithm (Metropolis et al. 1953; Hastings 1970) for each time until all four parameters have obtained the next values. The state transition function of the parameters are all Gaussian, whose standard deviations are 50 pc for $\rm D$, 0.3 mag for $\upmu_{2}$, and 0.3 mag*-1* for $\rm I\upsigma_{1}$ and $\rm I\upsigma_{2}$.

We investigated the convergence of each chain with the Gelman–Rubin diagnostic (Gelman & Rubin 1992). The potential scale reduction factor (PSRF), $\hat{R}$, is less than 1.01 for all molecular clouds but Taurus (whose $\hat{R}$ is about 6), which has two close components. Consequently, the MCMC chains have converged but for Taurus.

In order to verify the MCMC results, we compared the Gibbs sampler with the affine-invariant ensemble sampler (Goodman & Weare 2010) implemented by the Python MCMC module EMCEE (Foreman-Mackey et al. 2013). The EMCEE package usually produces many outliers, but when it occasionally generates less outliers, it gave the same results with the Gibbs sampler.

In order to balance the star numbers of the two truncated Gaussian components in Equation 5 and to avoid the interference of farther molecular clouds along the line of sight, we set a distance cutoff for each molecular cloud. Because we need to know the distance before setting the distance cutoff, this is a recursive process. Usually, a distance cutoff of 1000 pc is sufficient, and we adjust this cutoff for far or near molecular clouds if necessary. Close molecular clouds usually have too few foreground stars compared to background stars, and small fluctuation of $A_{\mathrm{G}}$ in the background stars would make their distances wrongly recognized.

A lower cutoff for foreground stars is unnecessary but for four molecular clouds, which are the far component of Cepheus, Mon R2, Polaris, and Rosette. In oder to remove the effect of foreground components, we removed on-cloud stars that are nearer than 400, 500, 325, and 1000 pc for them and the corresponding $\upmu_{1}$ are estimated to be 0.683, 0.499, 0.295, and 0.839 mag, respectively. Because of having 5 parameters to model, we calculated 10 chains for these four molecular clouds, i.e., 10000 thinned samples for each of them.

3.3 Testing the model

We tested this model of calculating molecular cloud distances with simulated data before applying it to Gaia DR2 stars. The simulated a molecular cloud is at a distance of $\rm D=300$ pc, and its foreground star extinction is 0 mag, i.e., $\upmu_{1}=0$. The dispersions of the foreground and background stars $A_{\mathrm{G}}$ are 0.3 and 0.6 mag, respectively, i.e., $\upsigma_{1}=0.3$ mag and $\upsigma_{2}=0.6$ mag. We added scatter $A_{\mathrm{G}}$ errors of 20$\sim$50% on the extinction $A_{\mathrm{G}}$, and stars whose extinction $A_{\mathrm{G}}$ $<$0 or $>$ 3.609 mag were removed.

We changed four parameters, the background star extinction $\upmu_{2}$, number of stars per pc, the relative parallax errors, and the distance cutoff, to see the variation on the resulting distances. As demonstrated in Figure 3, our model detected the distances successfully in all cases. Unsurprisingly, smaller jumps of extinction $A_{\mathrm{G}}$, fewer Gaia DR2 star samples, and larger parallax errors would cause larger distance errors, while the distance cutoff has no effect in the distance determination. Remarkably, although the distance errors became larger, the distance was still consistently recognized.

In Gaia DR2, the errors on distances are usually better than 10% for stars nearer than 1 kpc. Therefore, the two most important factors that may affect our results are the number of on-cloud stars and the magnitude of $A_{\mathrm{G}}$ caused by molecular clouds.

3.4 The distance of the Taurus and Gemini molecular clouds

We applied our model to the Taurus and Gemini molecular clouds. The corner plots of the four parameters, $\rm D$, $\upsigma_{1}$, $\upmu_{2}$, and $\upsigma_{2}$, are displayed in Figure 4b, together with their means and 95% Highest Posterior Densities (HPDs). On the bottom right, the dashed blue boxes of (a) and (b) are the manually chosen cloud regions (see §3.1), which contain molecular clouds and extra noise regions. The blue and green contours represent the noise and signal thresholds, respectively, which were determined in §3.1 (see Figure 2). On the top right, we display the distances and on- and off-clouds stars with green and blue points, respectively. The derived distances to the Taurus and Gemini molecular clouds are $145_{-16}^{+12}$ and $725_{-18}^{+16}$ pc, respectively.

Interestingly, the Taurus molecular cloud seems to have two components. Farther than 130 pc, there are still many stars having low extinction $A_{\mathrm{G}}$  until about 150 pc. The Taurus distance given by Torres et al. (2009) is 161.2$\pm$0.9 pc, which may be corresponding to this component.

4 Molecular clouds distances

4.1 Planck 857-GHz versus IRAS 100-$\mu$m data

We demonstrate that Planck 857-GHz is a better dust tracer than IRAS 100-$\mathrm{\upmu}$m; or at least is more well-suited to our method. As an example, we show the distances of the molecular cloud HSVMT 5 produced with Planck 857-GHz and IRAS 100-$\mathrm{\upmu}$m in Figure  5.

The distance estimated with Planck 857-GHz  is $352_{-3}^{+4}$ pc, and $349_{-6}^{+4}$ pc for IRAS 100-$\mathrm{\upmu}$m. As indicated in Figure 5, the average extinction of on-cloud stars classified with Planck 857-GHz is higher than that with IRAS 100-$\mathrm{\upmu}$m, suggesting that the on-cloud stars identified with Planck 857-GHz is less contaminated by other Gaia stars that have low extinction $A_{\mathrm{G}}$. Consequently, the extinction values, $A_{\mathrm{G}}$, of the background stars are better separated from the foreground stars with Planck 857-GHz emission.

4.2 The impact of the parameter choice

In this subsection, we examine the reliability of our method and its dependence on parameters. In our method, we primarily involve two subjective parameters: the manually chosen molecular cloud region and the signal level cutoff in the Planck 857-GHz intensity. In total, we found 52 molecular clouds have their distances well determined with the chosen molecular cloud regions and signal levels.

In order to see the influence of the two parameters, we shifted them one at a time and compared the distance variations. In Figure 6, we display the distance discrepancies and uncertainties after altering parameters. To reveal the effect of the molecular cloud region box, we shifted the molecular cloud region box along $l$ to the left and right by 10% of the box size in $l$, while the signal levels were altered by $\pm$20%. The distances show slight fluctuations in the molecular clouds, while their uncertainties widely agree with the distances calculated with the chosen parameters. Clearly, large distance discrepancies usually yield large uncertainties and no evident systematic deviations are present and only Gemini shows large distance deviations (with magnitudes $>$40 pc) in case (a), confirming our claim that correct recolonization of breakpoints guarantees comparable distances.

Consequently, our distance estimation is robust and only weakly dependent on the choice of region boxes and signal levels in a reasonable range. In addition, in order to make sure the breakpoints are genuine, it is necessary to confirm the breakpoints with off-cloud stars.

4.3 The Distance catalog

With the method described in §3, we calculated distances for many high-Galactic-latitude molecular clouds, most of which have been cataloged by Magnani et al. (1985), Magnani et al. (1996), and Schlafly et al. (2014). In these three catalogs, many molecular clouds belong to the same molecular could complexes or filaments, and we only give one distance for each complex or filament, because of the large dispersion in the extinction $A_{\mathrm{G}}$ and inadequate Gaia stars when dealing with small subregions.

We have reproduced the distances of many star-forming regions, whose distances are well determined with VLBI measurements, such as Orion (Kounkel et al. 2017), Mon R2 (Dzib et al. 2016), Ophiuchus (Ortiz-León et al. 2017b), and Perseus (Ortiz-León et al. 2018). In Table 1, we compare our results with VLBI-measured distances. Considering the distance errors and dispersions, the molecular cloud distance given by Gaia DR2 agrees quite well with VLBI measurements. We suggest that VLBI and Gaia DR2 may see different components of the molecular clouds, and the slight discrepancies may be due to the structure of the molecular clouds.

We summarize the distances in Table LABEL:tab:distance, listing 52 molecular clouds whose distances are well determined, i.e., the breakpoints are evident. Many molecular clouds cannot have their distance determined because (1) they are not well defined in Planck 857-GHz, (2) their optical depths are too low, or (3) their covering areas are too small. In the second last column, we mark those molecular clouds whose distances are not provided by Schlafly et al. (2014) or measured using VLBI with “Y”, 13 in total.

The distance figures of all molecular clouds have been provided on Harvard Dataverse111https://doi.org/10.7910/DVN/C6YO4T.

4.4 The systematic error

In this section, we estimate the systematic error in the distances.

First, as suggested by the testing results (see Figure 3), the distances show a systematic error of 1%-10%. In the testing data, we simulated parallax errors of 10%-20%, and within 1 kpc from the Sun, however the uncertainties of Gaia DR2 parallaxes are less than 10%. Therefore, the true systematic error is likely to be smaller.

Secondly, as shown in Figure 6, the root-mean-square (RMS) of the relative distance deviations in the four cases are 3% (a), 2% (b), 1% (c), and 2% (d), respectively, suggesting a systematic error of about 3% caused by subjective choices.

Thirdly, assuming the VLBI results (see Table 1) are the true distances, our distances show a systematic error of $\leq$ 6%. However, VLBI usually measures the parallax of one single star, which may deviate from the main part of the molecular clouds.

Fourthly, Lindegren et al. (2018) mentioned that the Gaia DR2 parallaxes have a systematic error of approximately 0.1 mas. Considering a molecular cloud at a distance of 500 pc (2 mas), this parallax systematic error causes a systematic error of about 5% in the distance.

Consequently, we estimate that the systematic distance error is about 5%, possibly slightly larger for distant molecular clouds.

5 Discussion

5.1 Comparison with previous results

In Figure 7, we draw the distances derived from Gaia DR2 against that provided by Schlafly et al. (2014). We averaged the distances weighted by the square inverse of their errors when multiple distances (corresponding to the same molecular cloud) are provided by Schlafly et al. (2014). The distances provided by Schlafly et al. (2014) display a slight systematic shift, $\sim$24 pc, towards the farther distances. In terms of relative errors, the systematic shift is, after removing one outliers ($>$70%), about 13%.

This systematic shift was predicted by Schlafly et al. (2014). They attribute the systematic errors to stellar models, dust models, and the reddening law. However, in addition to these, we found the molecular regions used by Schlafly et al. (2014) are much smaller than that in our work, which means their results may only represent the distances of molecular clouds in those small regions, and are less affected by the thickness of the molecular clouds along the line of sight. Alternatively, nearer clouds are brighter and easily to be chosen, which could cause the distances given by Schlafly et al. (2014) systematically smaller.

However, the Gaia DR2 data may also contribute to this systematic error. The $\sim$0.1 mas systematic parallax error in Gaia DR2 (Lindegren et al. 2018) may be one of the causes of the systematic distance discrepancy. At a typical distance of 500 pc, 0.1 mas error corresponds to $\sim$25 pc shift, which can explain this systematic shift. Furthermore, the extinction $A_{\mathrm{G}}$, which has large uncertainties, may contain unknown systematic errors.

One means of examining the discrepancy is to compare the distance modulus in Schlafly et al. (2014) with the Gaia DR2 parallaxes. However, this is beyond the scope of this work, and alternatively, in Figure 8, we display the distances of MBM11 and MBM7, which show large distance discrepancies. As shown in Figure 8, the distances of MBM11 and MBM7 are $279_{-3}^{+3}$ and $253_{-5}^{+6}$ pc, respectively. However, in the distance catalog of Schlafly et al. (2014), the distance of MBM11 molecular cloud is approximately 206$\pm$23 pc, which is the average distance of MBM11, 12, 13, and 14, while the distance of MBM7 is $148_{-11}^{+13}$ pc. Based on Gaia DR2, the distances of these two molecular clouds, particularly MBM11, are well determined.

5.2 Individual molecular cloud distances

In this subsection, we discuss the distances of several individual molecular clouds. Schlafly et al. (2014) obtained a distance of about 350 pc for the Ursa Major molecular clouds, which is much farther than previous studies, $\sim$110 pc (Penprase 1993). Using Gaia DR2 data, we have derived a distance of $412_{-12}^{+20}$ for the Ursa Major molecular clouds, which is close to that given by Schlafly et al. (2014).

Li et al. (2015) performed a survey of two CO isotopologue lines towards the same Gemini region as shown in Figure 4b, and they used a distance of 400 pc, but Gaia DR2 clearly shows that its distance is about 725 pc. Consequently, the masses and sizes of Gemini molecular cores calculated by Li et al. (2015) should be revised accordingly.

The Cepheus (Grenier et al. 1989; Yonekura et al. 1997; Kirk et al. 2009) is an interesting region. As shown in Figure 9, obviously, there are two components along the line of sight. The distance of the nearer one is $346_{-3}^{+3}$ pc, while the distance of the farther one is $1043_{-7}^{+6}$ pc. According to the CO observations of Grenier et al. (1989), the radial velocity range of the far component is about -12 km s*-1*, while the nearer component has a velocity of about 0 km s*-1*. The distances of the two components derived by Grenier et al. (1989) are $\sim$300 and 800 pc, respectively, which are consistent with our results.

6 Summary

Using the parallaxes and extinction $A_{\mathrm{G}}$ provided by Gaia DR2, we derived the distances for 52 molecular clouds, most of which are at high Galactic latitudes, i.e., $|b|>10$∘. The systematic error of the distances is about 5%, and 13 molecular clouds have reliable distances determined for the first time, In addition, we have confirmed the distances of many star-forming regions, such as Orion, Taurus, Cepheus, and Mon R2.

We used Planck 857-GHz data rather than CO data to trace molecular clouds because CO observations are incomplete at high Galactic latitudes. However, at low Galactic latitudes ($|b|\leq 10^{\circ}$), multi-wavelength observations with high spatial resolutions would classify on- and off-cloud stars more accurately.

Gaia DR2 has enabled us to determine the distances of many nearby molecular clouds efficiently. Although the large errors of extinction $A_{\mathrm{G}}$ in Gaia DR2 prevent us from examining distant ($>$ 2 kpc) molecular clouds, with the improved qualities of distances and extinction $A_{\mathrm{G}}$ in future Gaia data releases, we would be able to obtain the distances of many more molecular clouds.

Acknowledgements.

We thank Mario G. Lattanzi for his valuable comments and careful proofreading. We are grateful to an anonymous referee for his/her constructive comments, particularly on the distance likelihood and the MCMC method. This work was sponsored by the Ministry of Science and Technology (MOST) Grant No. 2017YFA0402701, the 100 Talents Project of the Chinese Academy of Sciences (CAS), the National Science Foundation of China under Grand NO. 11673051 and 11873019, the CAS Grand No. QYZDJ-SSW-SLH047, and the Key Laboratory for Radio Astronomy, CAS.