On the Beam Filling Factors of Molecular Clouds
DDraft version February 16, 2021
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On the Beam Filling Factors of Molecular Clouds
Qing-Zeng Yan, Ji Yang, Yang Su , Yan Sun, and Chen Wang Purple Mountain Observatory and Key Laboratory of Radio Astronomy,Chinese Academy of Sciences, 10 Yuanhua Road, Qixia District, Nanjing 210033, People’s Republic of China (Dated: February 16, 2021)
Submitted to ApJABSTRACTImaging surveys of CO and other molecular transition lines are fundamental to measuring the large-scale distribution of molecular gas in the Milky Way. Due to finite angular resolution and sensitivity,however, observational effects are inevitable in the surveys, but few studies are available on the extentof uncertainties involved. The purpose of this work is to investigate the dependence of observationson angular resolution (beam sizes), sensitivity (noise levels), distances, and molecular tracers. To thisend, we use high-quality CO images of a large-scale region (25 . ◦ < l < . ◦ | b | < ◦ ) mappedby the Milky Way Imaging Scroll Painting (MWISP) survey as a benchmark to simulate observationswith larger beam sizes and higher noise levels, deriving corresponding beam filling and sensitivity clipfactors. The sensitivity clip factor is defined to be the completeness of observed flux. Taking the entireimage as a whole object, we found that CO has the largest beam filling and sensitivity clip factorsand C O has the lowest. For molecular cloud samples extracted from images, the beam filling factorcan be described by a characteristic size, l / = 0 .
762 (in beam size), at which the beam filling factoris approximately 1/4. The sensitivity clip factor shows a similar relationship but is more correlatedwith the mean voxel signal-to-noise ratio of molecular clouds. This result may serve as a practicalreference on beam filling and sensitivity clip factors in further analyses of the MWISP data and otherobservations.
Keywords:
Molecular clouds (1072); Interstellar clouds (834); Interstellar molecules (849) ; Extra-galactic astronomy(506); Astronomy data modeling(1859) INTRODUCTIONMolecular clouds are a kind of neutral interstellar medium (ISM) (Heyer & Dame 2015), characterized with lowtemperatures (Mathis et al. 1983) and relatively high densities (Dame et al. 2001). In terms of morphology, molecularclouds are clumpy (Norman & Silk 1980) with fractal boundaries (Falgarone et al. 1991; Stutzki et al. 1998; Stanimirovicet al. 1999), and many show filamentary structures at large scale (Bally et al. 1987; Molinari et al. 2010; Andr´eet al. 2010). Having this particular kind of structure, the surface brightness temperature of molecular clouds isinhomogeneous (Burton et al. 1990), causing non-unity beam filling factors subjected to observations with finite beamsizes and sensitivities. However, beam filling factors are usually assumed to be unity in the calculation of physicalproperties, such as the excitation temperature, the optical depth, and the mass, which is inaccurate and may introducesystematic errors.Observationally, the beam filling factor appears to be an item of diminishing the brightness temperature (e.g.,Mangum & Shirley 2015). In the low temperature approximation, the specific form of the radiative equation is T mb = η ( T ex − T bg ) (1 − exp ( − τ )) , (1) Corresponding author: Ji [email protected],[email protected] a r X i v : . [ a s t r o - ph . GA ] F e b Yan et al. where T mb is the observed brightness temperature, η is the beam filling factor, T ex is the excitation temperature, T bg is the background temperature, and τ is the optical depth. However, in practical observations, T mb is clipped dueto the limited sensitivity, i.e., the brightness temperature below sensitivity is clipped to be zero in the observationaldata. We refer to this sensitivity effect (flux completeness) as the sensitivity clip factor, ξ . Obviously, the observedbrightness temperature of molecular clouds is an interplay of sensitivity and angular resolution. ξ , the sensitivity clip factor, is defined as the ratio of the observed flux and the total flux. The total flux correspondsto the flux at perfect sensitivity, and can be estimated by extrapolation from the observed flux at different sensitivitylevels. The observed flux, however, corresponds to the flux at finite sensitivity.Geometrically, the beam filling factor can also be defined by η ≈ Ω s Ω A , (2)where Ω s is solid angle of objects within the antenna beam and Ω A is the beam solid angle, respectively. Apparently,for an object with a uniform brightness temperature across Ω A , Ω s equals Ω A , i.e., η = 1, otherwise η < ∼
100 pc; Motte et al. 2018), the beam filling factors of extragalactic GMCs under Atacama Large MillimeterArray (ALMA) observations may be significantly less than unity. They concluded that the beam filling factor maycause the virial parameter ( α vir ) to be overestimated, due to the underestimation of molecular cloud mass. Dassa-Terrier et al. (2019) derived a surface beam filling factor of ∼ CO cloud clumps toward M31 (with 11-pcresolution), and in the central region, the beam filling factor of dense gas is less than 0.02 with a resolution of about100 pc (Melchior & Combes 2016). For some studies, such as metallicity gradients (Acharyya et al. 2020) and radiativetransfer analyses subjected to sub-beam structures (Leroy et al. 2017), the beam size effect is pivotal.Observations of molecular clouds in the Milky Way are also not free of beam dilution and sensitivity clip effects,particularly for molecular clouds with small angular sizes. Due to the inhomogeneity, beam smoothing diminishes thepeak and edge brightness temperatures, and when observed with finite sensitivities and spatial resolutions, both theflux and the brightness temperature are underestimated. For instance, Yan et al. (2020) found a completeness of 80%for the observed flux of local molecular clouds in the first Galactic quadrant. This completeness is expected to belower for molecular clouds in distant spiral arms. In addition to the flux completeness, Gong et al. (2018) found withnumerical simulations that the X CO factor would increase by a factor of 2 if the beam size increases by a factor of100 (from 1 to 100 pc). In addition to molecular clouds, the beam filling factor of H i gas is also less than unity. Forinstance, Heiles & Troland (2003) derived a value of ∼ i gas. The beam filling factor can be thelargest error source for analyses of small objects, for example, in the study of molecular outflows (Flower et al. 2010).Conventionally, η can be estimated in two ways. The first method uses Equation 1 and an optically thick spectral linewith an assumed excitation temperature, but this is inaccurate due to unknown optical depths and guessed excitationtemperatures. The other approach is based on the assumption of Gaussian source distribution (Pineda et al. 2008).Under this assumption, the beam filling factor is estimated to be Θ / (cid:0) Θ + Θ (cid:1) , where Θ b and Θ s are the beam sizeand full width at half maximum (FWHM) of the source, respectively. This is a good approximation for stars anddense cores, but for molecular clouds, whose surface brightness temperature distributions are usually none-Gaussian,the beam filling factor may not follow this convolution approach.In this paper, we use images of three CO isotopologue lines in the first Galactic quadrant (25 . ◦ < l < . ◦ | b | < ◦ ,and − < V LSR ¡139 km s − ) to study observational effects on spectroscopic survey data of molecular clouds, includingthe beam filling and sensitivity clip factors. This region has been mapped by the Milky Way Imaging Scroll Painting(MWISP) CO survey (Su et al. 2019) with high sensitivity ( ∼ CO) and medium angular resolution (about50 (cid:48)(cid:48) ). The high dynamical range of the MWISP survey in scale makes this region a superb data set for studying thebeam filling factor. The entire data set is roughly divided into four spiral arms based on their radial velocities, andexaminations of beam filling factors are subsequently performed on those arm segments.This paper is organized as follows. The next section (Section 2) describes the CO data, cloud identification methods,and the beam filling and sensitivity clip factor models. Section 3 presents results of beam filling and sensitivity clip eam filling factors Table 1.
Observation parameters of three CO isotopologue lines of the MWISP survey.Tracer Rest frequency Effective critical density a HPBW T sys δv rms noise(GHz) (10 cm − ) ( (cid:48)(cid:48) ) (K) (km s − ) (K) CO ( J = 1 →
0) 115.27 ∼ ∼ CO ( J = 1 →
0) 110.20 ∼ ∼ O ( J = 1 →
0) 109.78 ∼
18 52 140-190 0.167 ∼ a The effective critical density takes account of radiative line trapping (Yan et al. 2019). factors, including collective and individual molecular clouds. Discussions are presented in Section 4, and we summarizethe conclusions in Section 5. DATA AND METHODS2.1.
CO data
We select a region in the first Galactic quadrant (25 . ◦ < l < . ◦ | b | < ◦ , and − < V LSR <
139 km s − ) to studythe beam filling and sensitivity clip factors, and this region has been uniformly mapped by the MWISP CO survey (Suet al. 2019). Observations were performed with the Purple Mountain Observatory (PMO) 13.7-m millimeter telescope,containing three CO isotopologue line maps, CO ( J = 1 → CO ( J = 1 → O ( J = 1 → (cid:48)(cid:48) , 52 (cid:48)(cid:48) , and 52 (cid:48)(cid:48) , and the velocity resolutions are0.158, 0.166, and 0.167 km s − , respectively. The pixel size of the regridded map is 30 (cid:48)(cid:48) . The rms noise of CO isabout 0.49 K, ∼ CO and C O. See Table 1 for a summary of the observation parameters.In order to investigate the beam filling and sensitivity clip effects at different distance layers, we roughly split eachof the three isotopologue data cubes into four arm segments (Reid et al. 2016) along the V LSR axis: (1) the Localarm (-6 to 30 km s − ); (2) the Sagittarius arm (30 to 70 km s − ); (3) the Scutum arm (70 to 139 km s − ); (4) theOuter and Outer Scutum-Centaurus arm ( −
79 to − − ). Based on kinematic distances (A5 model in Reidet al. 2014), distances of the four spiral arms are approximately 1, 3, 6, and 15 kpc, respectively. The Perseus arm islargely overlapped with the Local arm in V LSR space, so we ignored the Perseus arm. Consequently, three CO linescollectively yield 12 data cubes. 2.2.
Molecular cloud samples
In order to investigate the beam filling and sensitivity clip factors of different molecular cloud species, we use theDBSCAN algorithm to draw samples from the position-position-velocity (PPV) cubes (Yan et al. 2020). DBSCANignores internal structures of molecular clouds and identifies independent structures in PPV space, sufficing for thebeam filling factor studies.In PPV space, DBSCAN has two parameters, MinPts and the connectivity. The connectivity (three types in PPVspace) defines the neighborhood of each voxel, i.e., whether two voxels are connected. For a given voxel, if the number ofits neighboring voxels (including itself) is ≥ MinPts, it is a core point, and connecting core points and their neighborsdefine a molecular cloud. As discussed in Yan et al. (2020), for small MinPts values, the three connectivity typesprovide similar cloud samples, so we simply use connectivity 1 and MinPts 4. The minimum cutoff of the data cubeis 2 σ ( ∼ CO and ∼ CO and C O), and in practice, the rms noise calculation is accurate to eachspectrum.We applied the post selection criteria to remove small DBSCAN clusters that are likely to be noise (Yan et al. 2020).The post selection criteria contain four conditions: (1) the voxel number is ≥
16; (2) the peak brightness temperatureis ≥ σ ; (3) the projection area contains a beam (a compact 2 × (cid:48)(cid:48) × (cid:48)(cid:48) ); (4) the velocitychannel number is ≥ https://scikit-learn.org/stable/modules/generated/sklearn.cluster.DBSCAN.html Yan et al.
Beam filling and sensitivity clip factors
The beam filling and sensitivity clip factors have different applications. The beam filling factor is used to correct T mb to obtain accurate excitation temperatures and optical depths, while the sensitivity clip factor is used to correctthe observed flux, which is more related to, e.g., the mass of molecular clouds. η strongly depends on the beam size,and we refer to η as the value at the angular resolution of the data. For a single pixel, ξ is either unity or zero, but foran image or a molecular cloud, the observed flux above cutoffs is the inverse cumulative distribution function of CObrightness temperatures, and ξ describes the observed fraction of the flux. η and ξ can be estimated in two approaches: (1) based on the entire image and (2) based on molecular cloudsamples. In the first image-based case, the whole data cube is taken as a single molecular cloud, while in the secondsample-based case, the estimation is performed for each molecular cloud sample identified with the method describedin Section 2.2.The variation of η and ξ is modeled with extrapolation functions. Without a physically motivated theory at hand,we use an empirical function. However, the extrapolation function should be simple and versatile, applicable to bothimage-based and sample-based cases.In order to model η and ξ , we produce two data sets based on the MWISP CO data. The first data set simulatesa series of observations at different beam sizes and is used to estimate η . For the convenience of calculation, we keepthe pixel size constant in smoothing. The second data set, however, resembles observations with the same beam sizebut different sensitivity clips and is used to estimate ξ .2.3.1. Beam filling factors
For an isolated Gaussian source, the variation of its peak brightness temperature is proportional to the beam fillingfactor, but for molecular clouds, which are irregular and have non-uniform brightness distribution, we can take eachvoxel in a data cube as a Gaussian peak. In this case, the beam filling factor can be examined voxel by voxel based onintensity, but the signal-to-noise ratio (SNR) of a single voxel is low, causing large errors in curve fitting, particularlyfor extended weak components of molecular clouds. In order to obtain high SNRs, we take each molecular cloud as anobject and use the mean T mb to derive an average beam filling factor.For specific observation data, the rms noise and the angular resolution are coupled. Smoothing operations decreaseboth T mb and the rms noise, and in the estimation of η , voxels are need to be above the sensitivity level in allsimulated observations. In other words, voxels that are below the sensitivity level in a smoothing case are discarded.The sensitivity level we used is 2 σ , consistent with DBSCAN parameters. However, σ is different between smoothingcases.The procedure of obtaining η contains three main steps: (1) identifying molecular clouds, (2) smoothing data cubes,and (3) modeling η . The first step applies the procedure of producing molecular cloud samples (see Section 2.2) onraw CO data.In the second step, data cubes are smoothed to simulate observations with larger beam sizes. The smoothingoperation is performed with the spectral-cube package in Python language. The beam size varies by factors from1.5 to 10 with an interval of 0.5, giving 18 smoothing cases in total. For the convenience of comparing, we keep thevoxel size unchanged. The rms noise is calculated with the Outer arm cubes, which contain the largest amount ofnoise voxels, and we use the rms of negative values in the spectra as a proxy of the rms noise.In the third step, we estimate η based on the variation of mean T mb with respect to the beam size. The mean T mb is obtained by averaging brightness temperatures over voxels that are above the sensitivity clip levels in all smoothingcases. η is obtained through extrapolation, and taking the mean T mb at the zero-beam point as observations withinfinite angular resolutions ( η = 1), the fraction of T mb at a specific beam size is the corresponding beam filling factor.For MWISP molecular clouds, we use the fraction at the MWISP beam size as their beam filling factors.We found that the mean T mb roughly contains two components, a linear part and an exponential part, which canbe well described by a four-parameter function: T (Θ) = a exp ( − b Θ) − c Θ + d, (3)where, Θ represents the beam size, y is the corresponding observed flux, and a , b , c , and d are four parameters to bedetermined. Equation 3 is approximately linear when b is small and is also able to fit flux variations that decrease https://spectral-cube.readthedocs.io/en/latest/index.html eam filling factors b values). The superiority of this function over polynomials is that the meaning of Equation 3 is moreclear, and Equation 3 is a monotonic function, which satisfies the intuition that flux decreases with larger beam sizes.We use Equation 3 to extrapolate the value of the mean T mb to zero. The variation of the mean T mb is fitted with curve fit in the Python package
SciPy with flux errors considered. The error of the mean T mb is estimated with (cid:112)(cid:80) i σ i /N , where N is the voxel number and σ i is the rms noise of each voxel. σ i decreases with beam sizes but N is constant. Specifically, the beam filling factor is estimated with Equation 3 using η = T (Θ MWISP ) T (0) , (4)where T (Θ MWISP ) and T (0) is the mean T mb at the MWISP beam size and at zero-beam size, respectively. Errorsof T (Θ MWISP ) and T (0) are obtained with first derivatives of T at Θ MWISP and 0, respectively, together with thecovariance of a , b , and c , and the error of η is subsequently estimated with propagation of errors.2.3.2. Sensitivity clip factors
In this section, we present the method of deriving ξ . We use the cutoff as a proxy of the sensitivity clip levels,simulating observations at different sensitivities but with the same angular resolution. In this context, the flux abovethe cutoff is the inverse of the cumulative distribution function of the brightness temperature.The procedure of modeling ξ is similar to that of η . By definition, ξ approaches unity as the sensitivity goes infinity,and ξ corresponds to the completeness of the flux at a specific sensitivity level. The cutoffs range from 2 σ to 20 σ withan interval of 0.2 σ . ξ is the fraction of observed flux at 2 σ with respect to the zeroth flux obtained with extrapolation.Equation 3 cannot model the flux variation with respect to the sensitivity (the brightness temperature cutoff).Instead, we found that the quadratic equation suggested by Rosolowsky & Leroy (2006) is more appropriate. However,toward high cutoffs, the observed flux of molecular clouds usually decreases rapidly to zero, so we use a sigmoid termthat contains the Gaussian CDF to model this zero tail.specifically, the observed flux ( f ) above the cutoff ( x ) is approximately erf( z ) = √ π (cid:82) z exp( − t ) dt,F ( x ) = (cid:16) (cid:16) x − µ √ δ (cid:17)(cid:17) ,f ( x ) = (cid:0) a ( x − b ) + c (cid:1) (1 − F ( x )) , (5)where F ( x ) is the cumulative distribution function (CDF) of Gaussian distribution N ( µ, δ ), x is in units of rms noise( σ ), and f ( x ) is the observed flux above x . Due to the sigmoid item that contains the Gaussian CDF, f ( x ) is forcedto approximate 0 for large x values. In total, Equation 5 contains 5 parameters: a , b , c , µ , and δ , and for a normalfitting, a , b , and δ should be positive. ξ is estimated subsequently with ξ = f ( σ cut ) f (0) , (6)where σ cut is the T mb cutoff (in the unit of rms noise) of molecular clouds. RESULTS3.1.
Image-based beam filling factors
In this section, we demonstrate the results of imaged-based beam filling factors. Image-based beam filling factorsare calculated by taking the whole data cube as a single molecular cloud. The observed mean T mb toward four spiralarm segments is listed in Table 2, including all smoothing cases. No C O emission is detected toward the Outer arm,i.e., the beam filling factor of C O in the Outer arm is approximately zero.As examples, we display variations of the mean T mb and the image-base η of local molecular clouds in Figure 1. Therelative errors are small, about 1 × − . The patterns of the mean T mb variations are similar for three CO lines, andEquation 3 fits the variation of the mean T mb well, except a slight deviation for the mean T mb of the raw data. Thissystematic shift of the mean T mb between raw and smoothing data is discussed in Section 4.1. As expected, CO hasthe highest η , while C O has the lowest.Table 3 summarizes η of three CO lines in four spiral arm segments. η of CO and CO in the Local, Sagittarius,and Scutum arm are approximately unity, while η of C O are significantly lower. In the Outer arm, however, both CO and CO have low beam filling factors.
Yan et al.
Table 2.
Variation of the mean T mb with respect to beam sizes for molecular clouds in four Galactic arm segments. Beam sizesArm Line 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10(K) CO 2.55 2.46 2.42 2.39 2.37 2.35 2.33 2.31 2.29 2.28 2.26 2.25 2.23 2.22 2.21 2.19 2.18 2.17 2.16Local CO 1.22 1.15 1.12 1.10 1.08 1.06 1.05 1.03 1.02 1.01 0.996 0.985 0.975 0.965 0.955 0.946 0.937 0.928 0.920C O 0.950 0.836 0.792 0.762 0.736 0.713 0.692 0.672 0.654 0.637 0.622 0.607 0.593 0.580 0.567 0.556 0.544 0.534 0.524 CO 2.72 2.61 2.56 2.52 2.48 2.45 2.42 2.40 2.37 2.35 2.32 2.30 2.28 2.26 2.24 2.23 2.21 2.19 2.18Sagittarius CO 1.22 1.12 1.08 1.04 1.01 0.986 0.961 0.938 0.917 0.897 0.878 0.861 0.844 0.829 0.814 0.800 0.787 0.774 0.762C O 0.867 0.724 0.657 0.608 0.566 0.529 0.497 0.468 0.442 0.419 0.398 0.378 0.361 0.344 0.329 0.316 0.303 0.291 0.280 CO 3.06 2.96 2.92 2.88 2.85 2.82 2.80 2.77 2.75 2.73 2.71 2.69 2.67 2.65 2.63 2.62 2.60 2.59 2.57Scutum CO 1.29 1.20 1.16 1.12 1.09 1.06 1.04 1.02 0.994 0.974 0.955 0.938 0.921 0.906 0.892 0.878 0.865 0.852 0.840C O 0.914 0.765 0.694 0.642 0.598 0.560 0.527 0.497 0.471 0.448 0.426 0.407 0.390 0.374 0.359 0.346 0.333 0.322 0.311 CO 1.93 1.68 1.53 1.41 1.30 1.21 1.13 1.06 0.998 0.942 0.891 0.844 0.802 0.764 0.729 0.696 0.666 0.638 0.613Outer CO 0.924 0.753 0.660 0.590 0.533 0.486 0.446 0.412 0.382 0.356 0.332 0.312 0.293 0.276 0.261 0.247 0.234 0.223 0.212C O 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
Note —The beam size is in units of the MWISP beam, 49 (cid:48)(cid:48) for CO and 52 (cid:48)(cid:48) for CO and C O. Table 3.
Image-based beam filling and sensitivity clipfactors of the MWISP survey.
Arm Line V LSR η ξ (1) (2) (3) (4) (5) CO 0.982 ± ± CO [ 0, 30] 0.959 ± ± O 0.877 ± ± CO 0.980 ± ± CO [ 30, 70] 0.941 ± ± O 0.722 ± ± CO 0.987 ± ± CO [ 70,139] 0.956 ± ± O 0.713 ± ± CO 0.777 ± ± CO [-79, -6] 0.613 ± ± O – – –
Sample-based beam filling factors
The procedure of deriving beam filling factors for individual molecular clouds is similar to that of image-based beamfilling factors, and the only difference is that the mean T mb for each molecular cloud is calculated over its own region.This region is determined with raw (unsmoothed) MWISP data using DBSCAN (down to 2 σ ), and voxels involved inthe estimate of the mean T mb are required to be above 2 σ level in all smoothing cases.The beam filling factor of each molecular cloud is estimated with Equation 3 based on the variation of mean T mb with respect to the beam size. As examples, we show η of four CO molecular clouds in the Local arm in Figure2. Usually, molecular clouds whose mean T mb decreases approximately linearly have high beam filling factors, whilemolecular clouds with exponentially decreasing mean T mb have low beam filling factors.We found that η is correlated with the angular size l of molecular clouds. The angular size is defined as an equivalentdiameter derived with l = (cid:114) Aπ − Θ , (7) eam filling factors M ean T m b ( K ) T = 0.25exp( − − CO in the Local arm ( η = 0.943 ± (a) M ean T m b ( K ) T = 0.20exp( − − CO in the Local arm ( η = 0.914 ± (b) M ean T m b ( K ) T = 0.31exp( − − O in the Local arm ( η = 0.850 ± (c) Figure 1.
Variation of the mean T mb against the beam size. Only local molecular clouds are displayed: (a) CO, (b) CO,and (c) C O. Red points are the observed flux for each smoothing case with different beam sizes, and the error bar is smallerthan the marker size. The blue lines are fitted with red points using Equation 3. where A is the angular area and Θ MWISP is the beam size of the MWISP survey. Figure 3 demonstrates the η variationof CO clouds against their angular sizes. Evidently, compared with the radial velocity (the color code), which isusually used as a distance indicator, η is more related to the angular size. η is approximately unity for molecularclouds with large angular sizes ( ≥ (cid:48) ), but decrease sharply for small ones.Given the large dispersion of η , we only look for a first-order approximation for the relationship between η and l .The function we used is η = η max l (cid:0) l + l / (cid:1) , (8)where l / is the angular size corresponding to η = 0 . η max . With this model, the range of η is from 0 ( l →
0) to η max ( l → ∞ ), and compared with other models (see Section 4.2), Equation 8 yields a smaller rms residual and has aclear physical meaning. Theoretically, η max should equal one, but η max is slightly less than one in practice, possiblydue to the error of simulated data. The right side of Equation 8 is a ratio of the angular area of molecular cloudsto the observed angular area enlarged by the beam, consistent with the beam filling factor definition. Consequently,we use Equation 8 to model the relationship between η and l . η max and l / is solved with curve fit of the Python package
SciPy , considering the error of η .Results of CO molecular clouds are described in Figure 3, and values of l / for three CO lines and four spiralarms are summarized in Table 4. Remarkably, values of η max and l / is approximately equal for molecular clouds inall four arms, suggesting that molecular clouds with close angular sizes have approximately equal beam filling factorsdespite being at different distances.Given the similarity of η max and l / in four arm segments, we fit overall values with all molecular clouds. Asdemonstrated in Figure 4, the overall fitting gives values of l / = 0 . ± .
001 and η max = 0 . ± .
000 , i.e., η isapproximately η = 0 . l ( l + 0 . , (9)where l is the angular size of molecular clouds in units of the beam size.3.3. Image-based sensitivity clip factors
Similar to the beam filling factor, the sensitivity clip factor can also be estimated by taking all emission as a singleobject. Figure 5 displays results of ξ for local molecular clouds. As can be seen, Equation 5 fits the flux variation well,but show slight deviations around 2 σ cutoff. This is because at 2 σ , the observed flux may not be complete due to theinsufficient SNR.Table 3 lists results of all four arm segments. Clearly, among three CO lines, CO has the highest ξ , while C Ohas the smallest ξ . As to arm segments, the Scutum and Outer arm has the highest and lowest ξ , respectively, whilethe rest two arms have medium ξ . Yan et al. M ean T m b ( K ) G048.6+04.0 (6.7 km s − ) T = 0.36 exp ( − − η = 0.888 ± (a) M ean T m b ( K ) G039.6+03.4 (7.9 km s − ) T = 1.84 exp ( − − η = 0.672 ± (b) M ean T m b ( K ) G037.9+02.9 (6.8 km s − ) T = 1.95 exp ( − − η = 0.557 ± (c) M ean T m b ( K ) G042.9 − − ) T = 3.72 exp ( − − η = 0.273 ± (d) Figure 2.
Same as Figure 1 but for four typical CO clouds in the Local arm: (a) G048.6+04.0 at 6.7 km s − , (b) G039.6+03.4at 7.9 km s − , (c) G037.9+02.9 at 6.8 km s − , and (d) G042.9 − − . Sample-based sensitivity clip factors
To make ξ consistent with η , the minimum cutoff of brightness temperature for individual molecular clouds is 2 σ .In Figure 6, fitting results show that Equation 5 describes the flux variation well for individual molecular clouds.We found that ξ are correlated with the mean voxel SNR of molecular clouds. This relationship is insensitive tomolecular cloud tracers and distances, and can be described with Equation 8 but with a slight adjustment of zeropoints: ξ = ( x − x ) (cid:0) x − x + x / (cid:1) , (10)where x is the zero point and x / is the mean voxel SNR (with respect to x ) at which ξ = 1 / a , b , and δ are positive and 0 < ξ < DISCUSSION eam filling factors (a) (b)(c) (d) Figure 3. η of individual CO clouds against their angular sizes in four spiral arm segments: (a) the Local arm, (b) theSagittarius arm, (c) the Scutum arm, and (d) the Outer arm. The color code represents V LSR , and see Equation 8 for the formof black solid lines.
Simulated data
In this work, we used simulated data instead of practical observations, which may cause systematic errors. The rawData is clipped at a certain sensitivity level, and all smoothing cases are based on clipped images. Consequently, T mb in simulated data is possibly systematically smaller (than practical observations) due to the clip effect of the raw data,particularly for voxels near the edge of molecular clouds.This systematic shift of simulated T mb is demonstrated in Figure 1. The mean T mb of the raw data is slightly largerthan the fitted value. This discrepancy can be examined with practical observations, which do not have this issue.4.2. Beam filling factors and the angular size
Although we use Equation 8 to describe the relationship between the beam filling factor and the angular size, thechoice of functions is not unique. We compared two additional function forms, and found that judging by the rmsresidual, Equation 8 outperforms the other two models. One of the two models uses the function η = l ( l + a ) , (11)0 Yan et al.
Figure 4. η - l relationship with all molecular cloud samples in four spiral arm segments, including CO, CO, and C Osamples. See Equation 9 for the form of the black solid line.
Table 4.
Beam filling and sensitivity clip factor relationships in the firstGalactic quadrant based on the MWISP survey. η = η max l (cid:16) l + l / (cid:17) ξ = ( x − x (cid:16) x − x x / (cid:17) Arm Line η max l / x x / (1) (2) (3) (4) (5) (6) CO 0.930 ± ± ± ± CO 0.913 ± ± ± ± O 0.896 ± ± ± ± CO 0.939 ± ± ± ± CO 0.911 ± ± ± ± O 0.889 ± ± ± ± CO 0.927 ± ± ± ± CO 0.916 ± ± ± ± O 0.864 ± ± ± ± CO 0.898 ± ± ± ± CO 0.864 ± ± ± ± O – – – –
Note — l / is in arcmin, while x and x / are in rms noise. eam filling factors σ )0.00.20.40.60.8 F l u x ( K k m s − a r c m i n ) × CO clouds in the Local arm ξ = 0.766 ± (a) σ )0.00.20.40.60.81.0 F l u x ( K k m s − a r c m i n ) × CO clouds in the Local arm ξ = 0.675 ± (b) σ )0123 F l u x ( K k m s − a r c m i n ) × C O clouds in the Local arm ξ = 0.450 ± (c) Figure 5.
Fitting of image-based flux variation with respect to cutoffs in the Local arm: (a) CO, (b) CO, and (c) C O.Flux variations (blue lines) are modeled with Equation 5, and the corresponding ξ (see Equation 6 for the definition) is derivedwith the ratio of modeled flux values at 2 σ to that at zero. The error bar is smaller than the marker size. σ )0200400600 F l u x ( K k m s − a r c m i n ) G033.5 − − ) ξ = 0.733 ± (a) σ )050100150200250 F l u x ( K k m s − a r c m i n ) G032.5+00.3 (23.5 km s − ) ξ = 0.718 ± (b) σ )0100200300 F l u x ( K k m s − a r c m i n ) G040.5+00.5 (26.5 km s − ) ξ = 0.389 ± (c) Figure 6.
Same as Figure 5 but for three CO clouds in the Local arm: (a) G033.5 − − , (b) G032.5+00.3at 23.5 km s − , and (c) G040.5+00.5 at 26.5 km s − . where a is a parameter. Equation 11 resembles the convolution of Gaussian distributions (Pineda et al. 2008), whilethe third model has a form of η = a (1 − exp( − bl )) , (12)where a and b are two parameters and l is the angular size of molecular clouds. In this case, η → l →
0, while f → a as l → ∞ , i.e., a is the maximum beam filling factor.To test which model performs best, we split CO molecular cloud samples in the Local arm into two categories: (1)the training set and (2) the validation data set. The training set is used to fit the model, while the validation data isused to verify the model. We examined two cases, having 20% and 30% validation data ratios, respectively, and theweighted rms residual (chi-square) of the validation data is used as an indicator of modeling qualities. As shown inFigure 8, Equation 8 possess the best performance.4.3.
Beam filling factors of molecular clouds
Beam filling factors of small molecular clouds are largely uncertain, while beam filling factors of large molecularclouds are well modeled. According to the relationship between the beam filling factor and the angular size ofmolecular clouds, beam filling factors are approximately unity for relatively large molecular clouds, and decrease fasttoward small molecular clouds. Beam filling factors are less than 0.5 for molecular clouds with angular size less than ∼ Yan et al.
Figure 7.
The relationship between the sample-based ξ and the mean voxel SNR. The mean voxel SNR is the mean SNR ofall voxels in a molecular cloud. In the relationship fitting, all MWISP molecular clouds samples are used, including four armsegments and three CO isotopologue lines. For clarity purposes, only molecular clouds with relative errors less than 20% aredisplayed. The mean voxel SNR is in units of rms noise, and see Equation 5 for the black solid line. Molecular cloud samples in this work is built with the DBSCAN detection scheme, but an alternative algorithmwould yield different molecular cloud samples. The variation of beam filling and sensitivity clip factors with respectto molecular cloud samples is possibly significant and will be investigated in the future.Due to the uncertainty of beam filling factors, estimations of excitation temperatures and optical depths for smallmolecular clouds are subject to large errors. This is usually the case for extragalactic observations, in which mostmolecular clouds are unresolved. At least a factor of 2 should be used to calibrate the brightness temperature in theapplication of radiative transfer equations.We estimate beam filling factors of observations toward Giant molecular clouds (GMCs) in size of ∼ ∼ η is about 0.5. For those observations, the α vir would be overestimated by a factor of2. For GMCs at a medium distance of ∼ η of ∼ η of GMCs would be approximately 0.9. Seen by PMO13.7-m, dense cores in a close high-mass star forming region, Orion ( ∼
400 pc) have an angular size of 1-2 beam sizes,and their η would be about 0.4. Consequently, observations with relatively low angular resolutions would significantlyunderestimate the brightness temperature.4.4. Sensitivity clip factors of molecular clouds eam filling factors (a) (b) Figure 8.
A comparison of three functions to model the relationship of η and the angular size ( l ). The molecular cloud samplesused are CO clouds in the Local arm. Panel (a) uses 80% of the samples to fit the model and the rest 20% for testing, whilethe validation data ratio in panel (b) is 30%. The right side of each panel plots the variation of the weighted rms residual of thevalidation samples with respect to a maximum relative error threshold for each model. See Equation 8 (blue), 11 (green), and12 (red) for the form of three models. As examples, on the left side of each panel, we display the training (gray) and validation(black) samples (maximum relative errors less than 0.2), together with the fitting of three models. η GMCs in local group galaxies with ALMAGMCs at 5 kpc with CfA 1.2-mGMCs at 5 kpc with PMO 13.7-mDense cores in Orion with PMO 13.7-m
Figure 9.
Beam filling factors of molecular clouds in different observational cases. See Equation 9 for the form of the blacksolid line. We list four typical situations with respect to GMCs (two Galactic cases and an extragalactic one) and dense cores(in Orion).
Results of the sensitivity clip factors reveal that measured flux is incomplete. According to the relationship in Figure7, the sensitivity clip factor is about 0.5 for a cloud with a mean voxel SNR of 3.3. This suggests that a large fractionof flux is missed for barely detected molecular clouds. Apparently, physical properties of C O ( J = 1 →
0) are largelyuncertain due to their low beam filling and sensitivity clip factors.4
Yan et al.
Table 5.
Observation parameters of four surveys.
Survey Name Spectral line Beam size l b V
LSR δv rms noise Cloud number([ ◦ , ◦ ]) ([ ◦ , ◦ ]) (km s − ) (km s − ) (K)CfA 1.2-m a CO ( J = 1 →
0) 8.5 (cid:48) [17, 75] [-3, 4.9] [-87, 140] 1.3 ∼ b CO ( J = 1 →
0) 46 (cid:48)(cid:48) [25.8, 49.7] [-1, 1] [-5, 70] 0.2 ∼ c CO ( J = 1 →
0) 46 (cid:48)(cid:48) [102.5, 141.5] [-3, 5.4] [-100, 20] 0.8 ∼ d CO ( J = 3 →
2) 16 (cid:48)(cid:48) [24.75, 48.75] [-0.275, 0.275] [-5, 70] 1.0 ∼ a ) Dame et al. (2001). ( b ) Jackson et al. (2006). ( c ) Heyer et al. (1998). ( d ) Dempsey et al. (2013). (a) (b) Figure 10.
Beam filling and sensitivity clip factors of molecular clouds in four CO surveys (see Table 5). For clarity purposes,molecular clouds with beam filling factor relative errors larger than 20% were removed. As a comparison, we display resultsderived with the MWISP survey in black solid lines: (a) the η and the angular size relationship (see Equation 9) and (b) the ξ and the mean voxel SNR relationship (see Figure 7). The sensitivity clip factor is remarkably consistent between molecular clouds. This universality suggests that thebrightness temperature distribution of most molecular clouds is similar. As shown in Figure 7, the dispersion of ξ issmall, meaning that molecular clouds with the same mean voxel SNR miss a similar fraction of flux, this only happenswhen their distributions of brightness temperatures are the same.4.5. Comparison with other surveys
In order to see the variation of beam filling sensitivity clip factors under other observations with different spectral linetracers, beam sizes, and sensitivities, we compare four CO surveys with the MWISP survey. Table 5 lists observationalparameters and PPV ranges of four examined surveys, and for the large-scale CO survey (Dame et al. 2001) conductedwith CfA 1.2-m, we choose a uniformly sampled region that has a large Galactic latitude coverage in the first Galacticquadrant.With the same smoothing and cloud identification procedure, we calculated η and ξ of molecular clouds for the fourCO surveys. As demonstrated in Figure 10, their beam filling and sensitivity clip factors are remarkably consistentwith that derived with the MWISP survey. This indicates that despite of having different sensitivities, beam sizes,and even spectral lines, beam filling and sensitivity clip factors of molecular clouds show similar relationships withmolecular cloud sizes and mean voxel SNRs.Our analysis methodology is applicable to other phases of the ISM. Berkhuijsen (1999) studied the volume fillingfactor of multiple phases of the ISM, including H ii , H i , molecular, and dust clouds, and the results suggest similar eam filling factors SUMMARYWe studied beam filling and sensitivity clip factors of molecular clouds by simulating observations with large beamsizes and low sensitivities using the MWISP CO survey in the first Galactic quadrant. The beam filling factor is usedto calibrate the brightness temperature, and the sensitivity clip factor is used to estimate the completeness of the flux.The beam filling factor is modeled with a two-component function according to the variation of the mean T mb withrespect to beam sizes, while the sensitivity clip factor is modeled using a quadratic function with a fast decreasingtail. In order to examine the collective and individual properties, we derived beam filling and sensitivity clip factorsbased on both the entire images and molecular cloud samples.The main results can be summarized as follows:1. Beam filling factors of CO and CO are approximately unity in the Local ( ∼ ∼ ∼ ∼ ∼ ∼
15 kpc), respectively. C Ohowever, decreases significantly with distance, and is approximately zero in the Outer arm. The sensitivity clipfactor shows similar variations with the beam filling factors, but is systematically lower by ∼ l in the beam size unit and can be approximatedwith 0 . l / ( l + 0 . . The average beam filling factors of molecular clouds identified with DBSCAN can bederived using this correlation.3. We derived a relationship between the observed flux and the mean voxel SNR ( x ), and the ratio of the observedflux to the total flux is approximately ( x − . / ( x − .
224 + 0 . . This relationship can be used toestimate the total flux.4. The η -size and ξ -sensitivity relationships seem to be universal suggested by the comparison with other existingCO surveys. ACKNOWLEDGMENTSWe would like to show our gratitude to support members of the MWISP group, Xin Zhou, Zhiwei Chen, ShaoboZhang, Min Wang, Jixian Sun, and Dengrong Lu, and observation assistants at PMO Qinghai station for their long-term observation efforts. We are also immensely grateful to other member of the MWISP group, Ye Xu, HongchiWang, Zhibo Jiang, Xuepeng Chen, Yiping Ao for their useful discussions. This work was sponsored by the Ministryof Science and Technology (MOST) Grant No. 2017YFA0402701, Key Research Program of Frontier Sciences (CAS)Grant No. QYZDJ-SSW-SLH047, National Natural Science Foundation of China Grant No. 11773077, U1831136, and12003071, and the Youth Innovation Promotion Association, CAS (2018355). Facilities:
PMO 13.7-m
Software: astropy (Astropy Collaboration et al. 2013), SciPyREFERENCES