RESOLVED MEASUREMENTS OF X_CO IN NGC 6946

To investigate GMC evolution in galactic disks and resolve the physics which controls the star formation rate within GMCs, we will present resolved observations of GMCs across the disks of nearby galaxies in a CO Survey of Nearby Galaxies based on observations using the Combined Array for Research in Millimeter Astronomy (CARMA) interferometer and the Nobeyama Radio Observatory 45 m (NRO45) single dish telescopes. To date, 13 galactic disks have been observed with both CARMA and NRO45 as part of the survey, and 4 additional galaxies have been observed with the single dish. The design of the survey will be discussed in J. Koda et al. (2012, in preparation). In our survey, we will resolve individual GMCs in a significant sample of nearby spiral galaxies with a variety of morphologies in order to study the evolution of molecular clouds and star formation.

In this paper, we combine observations from CARMA and NRO45 in order to achieve high resolution, as well as extremely high image fidelity, and resolve individual extragalactic GMCs in NGC 6946 to highlight the results made possible by our survey. In Section 2 we describe the observations, and we present the results in Section 3 as well as a discussion of the boundaries of individual GMCs. We utilize the well-known CLUMPFIND algorithm (Williams et al. 1994) to derive sizes and velocity dispersions for individual GMCs. In Section 4, we discuss the properties of our detected GMC sample and determine the conversion factor X_CO within individual clouds. We summarize our conclusions in Section 5.

We observe NGC 6946 in the ¹²CO (1–0) transition with the Nobeyama 45 m single dish telescope,¹² using the Beam Array Receiver System (BEARS) instrument. Observations were performed during the early months of 2008, 2009, and 2010 (throughout our three-year observing program at NRO) as part of our CO Survey of Nearby Galaxies. BEARS is a multi-beam receiver with 25 beams, which are aligned in a 5 × 5 orientation. The FWHM of the 45 m dish is 15'' at 115 GHz (197 after regridding), and we observe with channel increments of 500 kHz and Hanning smooth for a velocity resolution of 2.54 km s⁻¹. After dropping the edge channels, we use a bandwidth of 265 MHz (690 km s⁻¹). For most of the scans, the system temperature in the double sideband (DSB) of BEARS ranges from 300 to 400 K; scans with T_sys much higher than 400 K (due to being observed during daylight hours) are heavily down-weighted and as a result do not contribute much to the final images. The ratio of the upper-to-lower sideband (known as the scale factor) was confirmed each year to be within a few percent by observing Galactic CO sources (e.g., asymptotic giant branch stars) with BEARS and the single sideband (SSB) S100 receiver and making subsequent corrections. We use the S40 receiver and Galactic masers for pointing. The pointing was checked roughly every two hours during the observations and was accurate to 2''–3''. We convert T_A* to T_mb assuming that the main beam efficiency of the telescope is 0.4 (i.e., T_mb = $T_{A^*}$/0.4).

Using on-the-fly (OTF) mapping, NGC 6946 was scanned in the R.A. and decl. directions, and positions external to the galaxy (OFF positions) were observed between scans. Extrapolating between OFF scans on opposite sides of the galaxy greatly reduced nonlinearities in the spectral baselines. The duration of each scan (ON + OFF) was ∼1 minute, and the entire galaxy was mapped in ∼40 minutes; a total of 36 usable maps were taken for a total of 24 hr of observation time (including ON, OFF, and slew time). The scans were separated by 5'', resulting in oversampling by a factor of three compared to the 15'' FWHM of the beam, which is necessary in order to achieve Nyquist sampling (596) of λ_CO/D = 1192 (where λ_CO is the observed wavelength, 2.6 mm, and D is the antenna diameter, 45 m). The data reduction and sky subtraction, performed by interpolating between the OFF scans for each OTF (ON) scan, were completed using the NOSTAR package developed at the NRO. Spatial baseline-subtracted maps were made separately from the scans in the R.A. and decl. directions in order to minimize systematic errors in the scan directions, and these were subsequently co-added. The rms noise of these single dish observations is 0.13 K (0.57 Jy beam⁻¹).

To complement the single dish observations, NGC 6946 was also observed in the ¹²CO (1–0) transition in 2009 April with the C and D configurations of CARMA. CARMA is a 15-element interferometer which combines six 10 m antennae (originally the Owens Valley Radio Observatory, or OVRO) with nine 6 m antennae (formerly the Berkeley–Illinois–Maryland Association, or BIMA) to achieve superior uv-coverage compared to either of its predecessors. The observations of NGC 6946 were performed using three dual sidebands, each with 63 channels, for a total bandwidth of ∼100 MHz (after removing six edge channels per sideband) and a channel width of 2.54 km s⁻¹. After a total of ∼21 hr on source (including calibrators), we achieve an rms of 0.73 K.

In order to combine the data from the single dish and interferometer, the NRO45 image is converted to visibilities, combined with the CARMA visibilities in the uv-plane, and the new uv data set is imaged together. We follow the procedure thoroughly described in Koda et al. (2011) for the imaging of M51, and we refer the reader to that paper for the details of the NRO45 deconvolution, combination in uv-space, and imaging process. In that paper, the relative uv-coverage of the single dish and interferometer observations enabled the single dish visibilities to be flagged beyond 4 kλ, as CARMA visibilities existed down to this value. We keep the NRO45 visibilities out to 10 kλ in order to ensure sufficient overlap between the two sets of uv-coverage. The rms of the combined cube is 0.11 Jy beam⁻¹ (1.9 K) using the combined synthesized beam, discussed below. We maintain the instrumental velocity resolution of 2.54 km s⁻¹ to optimize our ability to resolve, not only to detect, GMCs. The observing parameters for both the single dish and interferometric data sets are summarized in Table 1.

2.3.1. Combined Synthesized Beam

Combining interferometric and single dish information is non-trivial, as each is imprinted with the intrinsic beam size with which the observation is made. Typically, when deconvolving interferometric data, the process (e.g., CLEAN) removes the synthesized beam pattern at the location of each emission peak detected in the dirty map above some threshold and replaces it with a convolution beam, which is typically Gaussian in shape. The convolution beam in the case of a combined (interferometer + single dish) data set requires a beam comprised of weighted components from both telescopes and is effectively a superposition of two Gaussians with quite different FWHM sizes. Cleaning algorithms, such as invert within the interferometric data reduction package miriad, yield an output dirty beam whose point-spread function can be fitted with one Gaussian to achieve the beam size, but we find that this method does not lead to flux conservation since the two-component beam does not have a simple Gaussian shape.

In order to ensure that we conserve the flux in our combined NRO45+CARMA cube, we assume that the single dish observations accurately measure the total flux of NGC 6946. We calculate the flux in the single dish dirty map and compare it to the flux measured in the combined cleaned map; if the flux is conserved, these should be equal. We choose the beam size which leads to flux conservation by setting equal the single dish flux per single dish beam (with units Jy beam⁻¹_SD), both of which we know, and the flux in the combined dirty map per combined beam (with units Jy beam⁻¹_comb), where we know the flux but not the beam size:

$$\begin{equation} \frac{{\rm flux}_{{\rm SD}}\; ({\rm Jy}) }{{\rm beam}_{{\rm SD}}\; (^{\prime \prime })} = \frac{{\rm flux}_{{\rm combined}}\; ({\rm Jy}) }{{\rm beam}_{{\rm combined}}\; (^{\prime \prime })}. \end{equation} \tag{ 1 }$$

In this way, we calculate the intrinsic size of the combined beam (23 × 23) and use this value to clean the combined map. We refer the reader to Koda et al. (2011) for a complete discussion of this step in the data combination procedure.

NGC 6946 is a well-known nearby spiral galaxy. Its general properties are listed in Table 2, and our CO imaging is presented in Figures 1 and 2. The combined uv data sets are imaged with purely uniform weighting (robust = −5), where a weighting scheme is applied to the visibilities which is inversely proportional to the sampling density (in effect, weighting all cells the same at the cost of higher rms noise). This method takes advantage of the highest possible resolution of the array and yields a beam size of 23 (calculated as described in Section 2.3.1).

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Table 2. Properties of NGC 6946

R.A.J2000	20h34m523
Decl.J2000	60°09'14''
Morphologya	SABcd
Distanceb	6.08 Mpc
Optical velocitya	40 km s−1
Major diametera	115

Notes. References: ^a NASA/IPAC Extragalactic Database (NED); ^b Herrmann et al. (2008).

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We overlay the high-resolution CO imaging on a Hubble Space Telescope (HST) Hα image of the center of NGC 6946 in Figure 3. The CO emission traces the spiral arm and bar structures at the center of the galaxy. To the north and southeast of the nucleus, the CO emission and dust lanes apparent in the Hα image appear to be well matched. As the spiral arm structure consists of adjacent stellar light and dust features (inferred from the Hα absorption), where the CO does not match the absorption precisely, its peaks tend to be found on the edge of the absorption feature nearer to the stellar component.

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NGC 6946 is known to host a molecular bar (Ball et al. 1985), and we show the velocity field of NGC 6946 in Figure 4 to accentuate the regions where narrow spatial regions span large ranges in velocity. These are the offset ridges of the galactic bar, which largely coincide with the edges of the dust lanes seen in the Hα image.

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Given the distance of NGC 6946 and our spatial resolution of 68 pc, we expect to be sensitive to the largest GMCs and giant molecular associations (GMAs). The typical size of a Galactic GMC is 40 pc (Scoville & Sanders 1987), but measurements range from less than 20 pc to over 100 pc (Solomon et al. 1987; Scoville et al. 1987). We utilize CLUMPFIND (Williams et al. 1994), a well-known algorithm designed to search image cubes in three-dimensional (x–y–v) space for coherent emission, in order to decompose our detected CO emission into individual molecular clouds (e.g., Koda et al. 2009; Egusa et al. 2011).

CLUMPFIND assigns emission to individual clouds by contouring the emission in a data cube at multiples of the rms noise, identifying peaks, and following the peaks to lower intensities. We find CLUMPFIND to be the simplest algorithm for cloud decomposition, as it requires only two user-specified inputs: the contour increment and the minimum contour level (i.e., where to stop), and it assumes no physical priors in the clump decomposition process. We find that the most believable output is returned when using twice the rms noise of the data cube (0.11 Jy beam⁻¹) for both the increment and minimum contour level; Williams et al. (1994) also strongly recommend these values for the two parameters. We show an example of the clouds extracted by the algorithm in several subsequent velocity channels in Figure 5.

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All clumps returned by the algorithm have integrated flux measurements greater than 3σ (0.33 Jy beam⁻¹). Additionally, the velocity profile of each clump is visually inspected using the clplot package within CLUMPFIND, and any clumps with profiles which do not appear to be single entities (i.e., apparently blended profiles) are removed from the sample.

The algorithm returns the FWHM extents of each clump in the x-, y-, and v-directions, calculates the total two-dimensional areas of each clump from the pixels assigned and the total detected flux within the three-dimensional clump volume, and derives clump radii assuming the calculated areas are circular. We account for the beam size (the spatial resolution element) by subtracting in quadrature the beam size of a point source with the same peak temperature as the clump, measured out to the radius at which the increment equals the minimum contour level (where T = ΔT), from the radii calculated using the area method (as described in Appendix A of Williams et al. 1994). We account for the velocity resolution, or the (non-Gaussian) channel width, in our velocity dispersions by approximating the instrumental dispersion to be half of the channel width and subtracting this value in quadrature from the measured velocity dispersions. The equations are

$$\begin{equation} R = \sqrt{ R_{{\rm meas}}^{2} - \bigg (\frac{b}{2.355} \sqrt{2 \ln \left(\frac{T_{{\rm peak}}}{\Delta T} \right) } \bigg) ^{2} }, \end{equation} \tag{ 2 }$$

and

$$\begin{equation} \sigma _{v} = \sqrt{ \sigma _{v, {\rm meas}}^{2} - \left(\frac{\Delta v_{{\rm chan}}}{2} \right) ^{2} }. \end{equation} \tag{ 3 }$$

Rosolowsky & Leroy (2006) define more complex treatments of these quantities which require the extrapolations of clouds out to 0 K. We find good agreement between the values for the velocity dispersion (to within 4%) calculated using both methods, but the size measurements diverge for clouds smaller than our beam size. Since our measurements of size and velocity dispersion are measured out to a contour equal to twice the rms, not extrapolated to 0 K, we opt to use the Williams method to estimate the sizes and velocity dispersions of the clouds.

Following, e.g., Wilson & Scoville (1990), we take the uncertainties in our measurements to be 25% of the spatial beam size (17 pc) and half of our velocity channel width (1.3 km s⁻¹). These values are appropriate for the largest clouds in our sample, but the uncertainties in the clouds with sizes and velocity dispersions near our resolution limits may be larger. Running CLUMPFIND to a lower minimum contour level (1.5σ) creates more detections of faint clouds, but the resulting change in X_CO within the individual GMCs presented in this paper is well within these quoted measurement errors.

3.2.1. Derived Properties

We use the cloud measurements to calculate cloud virial masses in the following manner. Using the definition of potential energy from McKee & Zweibel (1992) and the kinetic energy in terms of the one-dimensional velocity dispersion, Williams et al. (1994) express the virial mass as

$$\begin{equation} M_{{\rm vir}} = \frac{ 5 \Delta R \sigma _{v}^{2} }{ \alpha _{{\rm vir}} G }, \end{equation} \tag{ 4 }$$

where ΔR is the circular radius of a clump, σ_v is the one-dimensional velocity dispersion, G is the gravitational constant, and α_vir is the "virial parameter" which describes a non-uniform density profile. If we parameterize the density profile as

$$\begin{equation} \rho (r) = \rho _{1} \bigg (\frac{R_1}{r} \bigg)^{\beta }, \end{equation} \tag{ 5 }$$

we find that integrating over $\it {dM = \rho (r) y dy dx}$ yields

$$\begin{equation} M_{{\rm vir}} = \frac{ 4 \pi \rho _{1} R_{1}^{3} }{ 3 - \beta }. \end{equation} \tag{ 6 }$$

Substituting the expressions in Equations (5) and (6) into the gravitational energy for a spherical body (Equation (2.32), Binney & Tremaine 2008) yields the relation

$$\begin{equation} \frac{3}{5} \alpha _{{\rm vir}} = \frac{ 3 - \beta }{ 5 - 2\beta }. \end{equation} \tag{ 7 }$$

It follows that for β = 0, α_vir = 1; for β = 1, α_vir = 10/9; and for β = 2, α_vir = 5/3 (as also mentioned by Williams et al. 1994).

If we assume that β = 1, as many authors do (e.g., Solomon et al. 1987; Rosolowsky & Leroy 2006; Bolatto et al. 2008), and G = 1/232 (using units of km s⁻¹, parsecs, and solar mass), substituting into Equation (4) yields

$$\begin{equation} M_{{\rm vir}} = 1040 R \sigma _{v}^{2}, \end{equation} \tag{ 8 }$$

as shown by Solomon et al. (1987) and similarly by Wilson & Scoville (1990). We assume that the GMCs are virialized, an assumption consistent with previous studies, and derive virial masses using the measurements of R and σ_v (corrected for their respective resolution elements) via Equation (8).

Solomon et al. (1987) describe the virial mass as

$$\begin{equation} M_{{\rm vir}} = \frac{ 3 f_{p} S \sigma _{v}^{2} }{ G }, \end{equation} \tag{ 9 }$$

where f_p is called a projection factor and the size of the cloud (S) is related to the effective radius (derived from the circular area of the cloud on the sky) such that R_eff = 1.91 S. However, this projection factor, indicated to equal 2.9, is not explicitly derived and as a result does not appear in many recent papers on this topic. Using the formulae shown above, it is trivial to derive that f_p = 2.9 when α_vir = 10/9.

To calculate X_CO, in effect the mass-to-light ratio for molecular clouds, we compare the virial masses of each cloud to the luminosities derived from each cloud's integrated CO flux. We calculate cloud luminosities via

$$\begin{equation} L_{{\rm CO}} = \frac{ 13.6 \lambda _{{\rm mm^{2}}} F_{{\rm CO}} }{ \theta _a \theta _b }, \end{equation} \tag{ 10 }$$

where λ is the observed wavelength (in mm), F_CO is the flux density in Jy beam⁻¹ km s⁻¹ arcsec², and θ_a and θ_b are the beam axes (arcsec). Finally, with M_vir in units of solar masses and L_CO in units of K km s⁻¹ pc², and including a factor of 1.36 to account for helium, we compute X_CO (the CO-to-H₂ conversion factor) as follows:

$$\begin{equation} X_{{\rm CO}} [{\rm cm}^{-2} {\rm (K \,km \,s}^{-1})^{-1}] = 4.60 \times 10^{19} \frac{ M_{{\rm vir}} }{ L_{{\rm CO}} }. \end{equation} \tag{ 11 }$$

From Figure 1, it is clear that most of the bright CO emission which is included in the resolved GMCs is coincident with spiral arms, but significant emission is also found in the inter-arm regions. The GMCs identified by the CLUMPFIND algorithm tend to be the largest, brightest clouds: those in the spiral arms. The inter-arm emission appears to be more extended, which is consistent with it being less likely to display an apparently self-gravitating velocity profile.

We resolve a total of 134 clouds. After discarding potential blends, as described in Section 3.2, 120 clouds remain in our GMC sample. Using the equations in Section 3.2 to account for the instrumental resolution elements, 64 clouds have a real velocity dispersion. We require that clouds have a velocity dispersion of at least 2.54 km s⁻¹ (twice our estimated instrumental dispersion) to be included in the sample, leaving 30 resolved clouds. The entire sample of 134 identified clouds is useful to examine the overall distribution of molecular gas which exists in GMCs, but the 30 fully resolved clouds will be the ones for which we derive virial masses, luminosities, and X_CO. These clouds are listed in Table 3.

Table 3. Properties of GMCs in NGC 6946

Number	R.A.J2000	Decl.J2000	R	σv	LCO	Mvir	XCO
			(pc)	(km s−1)	(105 K km s−1 pc2)	(105 M☉)	(1020 [cm−2 (K km s−1)−1])
1	20h 34m 5273	60° 9' 12''	117	11.5	118	161	0.630
2	20h 34m 5266	60° 9' 14''	155	14.7	152	348	1.05
3	20h 34m 5273	60° 9' 13''	154	7.89	88.5	99.7	0.518
4	20h 34m 5360	60° 9' 19''	189	9.04	117	161	0.633
5	20h 34m 5179	60° 9' 18''	123	8.86	55.8	101	0.829
6	20h 34m 5273	60° 9'	89.7	10.5	31.7	104	1.50
7	20h 34m 5280	60° 9' 30''	146	9.91	46.2	149	1.49
8	20h 34m 4805	60° 7' 53''	145	5.84	28.9	51.6	0.822
9	20h 34m 5166	60° 9' 23''	113	14.2	50.1	236	2.17
10	20h 34m 4148	60° 8' 47''	43.8	3.82	3.09	6.64	0.989
11	20h 34m 5220	60° 9' 27''	94.8	12.0	28.7	143	2.29
12	20h 34m 5193	60° 9' 19''	95.8	17.6	42.0	309	3.38
13	20h 34m 4577	60° 8'	138	5.52	24.4	43.7	0.825
14	20h 34m 4121	60° 8' 49''	40.4	4.89	4.59	10.0	1.01
15	20h 34m 4295	60° 9' 65	116	8.01	23.0	77.1	1.54
16	20h 34m 5072	60° 8' 25''	98.0	5.87	13.5	35.1	1.20
17	20h 34m 5508	60° 9' 51''	45.2	4.90	4.66	11.3	1.11
18	20h 34m 4322	60° 8' 35''	76.0	2.76	6.49	6.04	0.428
19	20h 34m 5976	60° 8' 18''	89.5	5.02	8.55	23.4	1.26
20	20h 35m 2913	60° 9' 13''	57.8	2.63	3.33	4.17	0.576
21	20h 34m 5588	60° 8' 25''	60.3	2.76	3.22	4.79	0.683
22	20h 34m 5601	60° 8' 15''	82.0	3.84	5.87	12.6	0.986
23	20h 34m 5809	60° 8' 37''	37.4	2.56	2.06	2.55	0.569
24	20h 34m 5380	60° 8' 10''	49.3	2.85	2.21	4.15	0.863
25	20h 34m 5534	60° 8' 53''	59.3	4.59	4.06	13.0	1.48
26	20h 34m 5467	60° 8' 50	56.9	5.02	3.75	14.9	1.83
27	20h 34m 5092	60° 9' 58''	90.3	4.19	6.66	16.5	1.14
28	20h 34m 5313	60° 10'	65.7	3.51	3.80	8.42	1.02
29	20h 34m 5032	60° 8' 27''	60.9	3.15	3.43	6.29	0.844
30	20h 34m 5307	60° 8' 59''	39.2	5.40	1.89	11.9	2.90

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The radii and velocity dispersions of the 30 fully resolved GMCs are shown in Figure 6, as are the measurements of resolved Milky Way disk clouds detected by Solomon et al. (1987) for comparison. Solomon et al. (1987) performed the seminal study of GMCs in the Galactic disk and showed that the relationship between the sizes and velocity dispersions of GMCs are not related by an exponent of $\frac{1}{3}$, as predicted by Kolmogorov turbulence (Larson 1981), but instead by an index of 0.5, as is consistent with clouds in virial equilibrium.

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However, more recently, comparisons of GMCs at the Galactic center using various tracers have been made to those throughout the disk, indicating that departures from the traditional relation do occur for clouds at the Galactic center (Miyazaki & Tsuboi 2000; Oka et al. 2001). The observed tendency is for clouds to exhibit higher velocity dispersions for a given size. Although our map covers the central region of NGC 6946, the measurements of its GMCs are largely consistent with the Galactic disk relation with some scatter, particularly for the largest clouds. The measurements of radius and velocity dispersion may even be consistent with a slope larger than 0.5, the value measured by several authors for Milky Way clouds within the disk and at the Galactic center (Solomon et al. 1987; Miyazaki & Tsuboi 2000; Oka et al. 2001).

For the largest clouds, the observed scatter goes to higher σ_v for a given size, and in fact, the six clouds which fall well above the Solomon relation are all located within ∼400 pc of the center of NGC 6946, consistent with the trend observed to be true for the center of our Galaxy. The smallest clouds which deviate from the Solomon relation are very close to our velocity resolution limit. While we are not sensitive to the full range of GMCs detected in the Galaxy by Solomon et al. (1987), our detected GMCs are largely consistent with the biggest clouds in the Galactic sample. While the Milky Way clouds included in the Solomon et al. (1987) study were measured using a contouring method which is physically distinct from the decomposition algorithm employed in this paper, the consistent trends present in both samples indicate similar underlying physics at work in both the center of NGC 6946 and the disk of our Galaxy.

The GMC luminosities are shown in Figure 7. The Galactic points and their linear fit (Solomon et al. 1987) are also shown for comparison. Again, our GMC sample is largely consistent with the measurements of the brightest Galactic clouds in the Solomon sample, though a fit to the NGC 6946 clouds would be shallower than the relation measured by Solomon et al. (1987). The brightest GMCs—the same clouds which fall above the Solomon radius–σ_v relation—appear to be underluminous for their velocity dispersion compared to the Galactic relation. Though blended clouds could artificially produce this relationship, since the total luminosity of multiple optically thick ¹²CO clouds along the same line of sight may not increase linearly with the amount of CO-emitting gas (depending on the geometry), the velocity dispersion would betray the presence of more than one cloud via multiple peaks. We individually inspect each cloud to select them on the basis of their apparently unblended velocity profiles, so blended clouds (at least at our velocity resolution limit) cannot explain this finding.

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The virial masses and CO luminosities of our NGC 6946 GMC sample are shown in Figure 8. Once again, we compare our measurements to the Galactic sample; the overlaid linear fit is derived in Solomon et al. (1987). The median mass of our GMC sample is comparable to the most massive Galactic disk GMCs, but we find excellent consistency between the trends. The histogram of our GMC sample is shown in Figure 9.

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Including a factor of 1.36 to account for heavier elements, the average value of X_CO for the GMC sample is 1.2 × 10²⁰ cm⁻² (K km s⁻¹)⁻¹; this value can vary by as much as a factor of two with our assumed uncertainties. This value is roughly a factor of two below the average value of the conversion factor which we calculate from the Solomon et al. (1987) Galactic disk clouds of 2.6 × 10²⁰ cm⁻² (K km s⁻¹)⁻¹ (derived using the same factor for heavier elements). If we calculate X_CO from the Solomon et al. (1987) clouds using only GMCs more massive than 5 × 10⁵ M_☉—approximately our completeness limit—we find a value of 1.9 × 10²⁰ cm⁻² (K km s⁻¹)⁻¹, which is in agreement with the value we derive for NGC 6946 within our uncertainty. It is also in agreement with the value derived by Dame et al. (2001) of 1.8 (±0.3) × 10²⁰ cm⁻² (K km s⁻¹)⁻¹ for CO emission in the solar neighborhood.

In addition to CO measurements, the conversion factor in the Galaxy has also been derived using γ-ray measurements (Strong & Mattox 1996), which yield a conversion factor of 1.9 × 10²⁰ cm⁻² (K km s⁻¹)⁻¹, in good agreement with the values listed above. This agreement between values of X_CO determined using both dynamical and non-dynamical methods has been claimed as evidence that Galactic GMCs are gravitationally bound and in equilibrium. Heyer et al. (2009) correct the Solomon et al. (1987) measurements for a more recently derived Galactic center distance and a factor of 1.36 to account for heavier elements and derive a mean surface density of 206 M_☉ pc⁻² for the entire Galactic disk GMC sample; the mean surface density that we calculate within our clouds is 204 M_☉ pc⁻².

The scatter in measurements of the conversion factor among studies of nearby galaxies has tended historically to be large, but more recently, studies of X_CO have exhibited relative agreement with the Galactic value. In Bolatto et al. (2008), ¹²CO (J = 1–0) and (J = 2–1) measurements of GMCs within Local Group dwarf galaxies and spiral galaxies are compiled and compared to the molecular clouds detected in the Milky Way. The sample includes dwarf galaxies as well as the Large Magellanic Cloud (LMC) and SMC to investigate the effect played by the environment on determinations of X_CO. These authors find that GMCs in Local Group dwarfs obey similar relationships to Milky Way GMCs and are consistent, to within a factor of two, with an X_CO of 2 × 10²⁰ cm⁻² (K km s⁻¹)⁻¹ with no detectable dependence on metallicity. A comparison of the GMC masses and luminosities of the NGC 6946 sample and the extragalactic GMC sample from Bolatto et al. (2008) is presented in Figure 10. Blitz et al. (2007) also study GMCs within nearby galaxies and find that X_CO is within a factor of two of 4 × 10²⁰ cm⁻² (K km s⁻¹)⁻¹. Even GMCs in the outer disk of M33 exhibit characteristics similar to those of Galactic GMCs (Bigiel et al. 2010). In both the studies by Bolatto et al. (2008) and by Blitz et al. (2007), however, the low-metallicity SMC is an outlier; in Blitz et al. (2007), it requires a conversion factor three times higher than the rest of the GMCs in the sample.

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As it is common for disk galaxies to exhibit a metallicity gradient, we investigate whether a radial dependence of the distributions of sizes, masses, and X_CO values of individual GMCs exists. Various observational studies which correlate the conversion factor with metallicity do so by studying them as a function of radius (or even averaged over entire disks) to investigate how X_CO may vary with metallicity in the presence of a metallicity gradient, especially in diffuse and low-metallicity gas (Arimoto et al. 1996; Leroy et al. 2011; Genzel et al. 2011). For instance, Nakai & Kuno (1995) show that M51 exhibits a radial gradient in the conversion factor in M51, with X_CO increasing with radius as the metallicity decreases. The same has also been shown to be true in the Galaxy (Sodroski 1991).

These properties are displayed in Figure 11. In this figure, the full detected cloud sample of 134 clouds is shown as dots. The 120 clouds with velocity profiles which point unambiguously to self-gravitation are circled in a color which indicates the mass (left) or value of X_CO (right) while the size of the circles indicates the radius of each cloud. The 30 clouds used in our analysis are indicated by asterisks. The most massive clouds (>10⁷ M_☉) are found at the very center of NGC 6946, as described in Section 4.2, and clouds with masses >5 × 10⁵ M_☉ are found where the density of clouds is the highest (i.e., on the spiral arms). No radial trend is apparent in X_CO, though the most central clouds tend to have values higher than 0.5 × 10²⁰ cm⁻² (K km s⁻¹)⁻¹, and all but one of the clouds with X_CO below this value are smaller than 70 pc.

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4.3.1. Metallicity

The fraction of emission in our CO cube which is identified as emission from the 30 GMCs with approximately Gaussian velocity profiles (suggesting self-gravitation) greater than our velocity resolution element is 19%; the total mass in this component is 2.2 × 10⁸ M_☉. If we include all of the GMCs identified by CLUMPFIND, which includes clouds which are spatially resolved but exhibit blended velocity profiles, the fraction increases to 31%. However, since the algorithm discards emission contours below 2σ, fainter emission around GMCs is not included. While this does not affect X_CO, as described in Section 3.2, it does artificially elevate the "diffuse" fraction that we find. For instance, the fraction of emission included in all GMCs in the 1.5σ run is 47% (compared to 31%).

The critical density for collisional excitation of CO (J = 1–0) emission is 300 cm⁻³ (Scoville & Sanders 1987), which is roughly the average gas density within a GMC (Solomon et al. 1987). The CO emission that is not identified as belonging to GMCs in our analysis is unlikely to be entirely diffuse (i.e., below the critical density). In addition, our velocity resolution of 2.54 km s⁻¹—optimal for resolving more massive GMCs—is not optimal for the detection of less massive GMCs.

We expect to be sensitive to clouds down to a limit of ∼5 × 10⁵ M_☉, but certainly clouds less massive than this limit will contribute as well. In their Galactic GMC sample, Scoville & Sanders (1987) find that ∼50% of the total H₂ mass resides in clouds less massive than 4 × 10⁵ M_☉, and the other half is found in clouds more massive than this value. Taking our estimate—that roughly half of the CO emission that we detect (in our 1.5σ run of CLUMPFIND) is assigned to clouds more massive than ∼5 × 10⁵ M_☉—in conjunction with the Scoville & Sanders (1987) result for the Galaxy indicates that the fraction of truly diffuse CO that we detect is likely small.