---
abstract: |
    The molecular components of three giant [[Hii]{}]{} regions (NGC 5461, NGC 5462, NGC 5471) in the galaxy M101 are investigated with new observations from single dish telescopes (James Clerk Maxwell Telescope and the NRAO 12-meter) and from the Owens Valley millimeter array.  Of the three [[Hii]{}]{} regions, only NGC 5461 had previously been detected in CO emission. 

    We calculate preliminary values for the molecular mass of the GMCs in NGC 5461 by assuming a CO-to-H$_2$ factor ($X$ factor) and then compare these values with the virial masses.  We find that the appropriate $X$ factor is 5 times smaller than the $X$ factor in the Milky Way despite the lower metallicity of M101.  We conclude that the data in this paper demonstrate for the first time that the value of $X$ may decrease in regions with intense star formation. 

    The molecular mass for the *association* of clouds in NGC 5461 is approximately $3 \times 10^7$  and is accompanied by 1–2 times as much atomic mass.  The observed CO emission in NGC 5461 is an order of magnitude stronger than in NGC 5462, while it was not possible to detect molecular gas toward NGC 5471 with the JCMT.  An even larger ratio of atomic to molecular gas in NGC 5471 was observed, which might be attributed to efficient conversion of molecular to atomic gas. 

    The masses of the *individual* clouds in NGC 5461, which are gravitationally bound, cover a range of ($2 \mbox{ -- } 8) \times 10^5$ , comparable with the masses of Galactic giant molecular clouds (GMCs).  Higher star forming efficiencies, and not massive clouds, appear to be the prerequisite for the formation of the large number of stars whose radiation is required to produce the giant [[Hii]{}]{} regions in M101. 
author:
- 'Jean Giannakopoulou-Creighton, Michel Fich and Christine D. Wilson'
title: Star Formation in the Giant H Regions of M101
---

internalcite

citex\[\#1\]\#2[@fileswauxout citeacite[forciteb:=\#2]{}[\#1]{}]{}

cite\#1\#2[(\#1@tempswa , \#2)]{} biblabel\#1

Introduction
============

Giant [[Hii]{}]{} regions are the most spectacular star-forming regions in normal galaxies and have been the object of many studies because of their brightness [@1].  Examples of giant [[Hii]{}]{} regions are found in galaxies of different brightness and morphology (e.g., Pellet 550 in M31, NGC 604 in M33, 30 Doradus in the Large Magellanic Cloud, and NGC 5461 in M101).  Especially impressive are the [[Hii]{}]{} regions observed in M101, which is a relatively nearby Sc spiral galaxy at $7.4 \pm 0.6$ Mpc [@126].  NGC 5471, one of the giant [[Hii]{}]{} regions in M101, is two orders of magnitude larger and brighter than W49, the largest [[Hii]{}]{} region in the Milky Way.  What is special about M101 that it produces such bright [[Hii]{}]{} regions?  One hypothesis is that these regions result from the unusual properties of the molecular gas from which the stars that ionize the gas originate [@4].  In this paper, we present data on the physical properties of the molecular gas in the giant [[Hii]{}]{} regions of M101 and discuss the implications for the formation of giant [[Hii]{}]{} regions. 

To explain the presence of massive star formation in the clouds associated with the giant [[Hii]{}]{} regions, Kenney et al. [ -@10] proposed that either the initial mass function is enhanced in massive stars, or the gas is consumed more efficiently in these regions.  The first idea has been investigated by Rosa and Benvenuti [-@41] in a study of four giant [[Hii]{}]{} regions in M101 with the Faint-Object Spectrometer on the Hubble Space Telescope.  They concluded that the initial mass function for stars with masses larger than 2  is similar to that in the Solar neighborhood.  This result is consistent with more recent studies of OB associations in inner part of M101 [@112].  On the other hand, there was an early indication that the star formation efficiency of massive stars in NGC 5461 is higher than the typical *total* star formation efficiencies observed in our Galaxy [@3]; however, the calculations of the molecular masses were rather crude.  In §\[s:star\], we find from new observations that the star formation efficiency is, indeed, *larger* than the efficiencies in star-forming regions of the Galaxy.  The conclusion in this paper is that higher star-formation efficiency is the key to the formation of giant [[Hii]{}]{} regions in M101. 

We present new observations of three giant [[Hii]{}]{} regions (NGC 5461, NGC 5471, and NGC 5462) from the James Clerk Maxwell Telescope[^1] (JCMT) in §\[s:jcmt\], the NRAO 12-meter Telescope in §\[s:nrao\], and the Owens Valley millimeter array in §\[s:ovro\].  These three regions are the brightest regions in M101 [@11].  The characteristics of the *individual* giant molecular clouds (GMCs) in NGC 5461 are discussed in §\[s:properties\] to deduce the molecular mass of the gas in the *associations* of GMCs.  Both LTE (local thermodynamic equilibrium) and LVG (large velocity gradient) methods, discussed in §\[lte\] and §\[lvg\], are used to find the physical properties of the associations of GMCs.  In §\[s:massestemp\], we compare the gas masses and temperatures of the three giant [[Hii]{}]{} regions.  We discuss the effect of star formation on the interstellar medium in §\[s:star\] and present our conclusions in §\[s:conclusions\]. 

Observations {#s:observations}
============

JCMT Data and Analysis {#s:jcmt}
----------------------

Receiver A2, used to observe the [$^{12}$CO $J=2\to1$]{} line, has a full-width at half-maximum (FWHM) of $20''$ and an efficiency $\eta_{fss}$ of 0.80, which is a correction for the radiation lost due to forward scattering and spillover.  Receivers B3i and B3 were both used to observe the [$^{12}$CO $J=3\to2$]{} line; their beam sizes are $14''$ and $13''$ respectively, while their $\eta_{fss}$ efficiencies are 0.70 and 0.75.  All the observations at the JCMT were obtained between 1995 January and 1997 February in position-switching mode.  The typical single sideband system temperatures were $300 \mbox{ -- } 400$ K for A2, $800 \mbox{ -- } 1000$ K for B3i, and $400 \mbox { -- } 600$ K for B3.  To reduce the data, we used SPECX, which is a spectral line data reduction code written by Rachael Padman (Cavendish Laboratory, Cambridge, U.K.) for JCMT data.  Linear baselines were removed after all the scans of the same position were averaged.  The data were binned to a frequency resolution of 5 MHz (which corresponds to 6.50 km s$^{-1}$ at 230 GHz and to 4.35 km s$^{-1}$ at 345 GHz) to achieve satisfactory noise levels. 

Tables \[tab:NGC5461\] and \[tab:NGC5462\] present the observed integrated antenna temperature, $\int{T_A^* \; dv}$, for various positions around each [[Hii]{}]{} region, while Figure \[fig:jcmt\] shows the three spectra at the peak positions in NGC 5461 and NGC 5462.  In addition, Tables \[tab:NGC5461\] and \[tab:NGC5462\] list the parameters obtained from a Gaussian fit to each of the spectra, where $V_{peak}$ is the velocity for the maximum antenna temperature, $T_{peak}$, and $\Delta V$ is the velocity width at half-maximum antenna temperature.  These velocity widths are large as noted by Skillman and Balick [-@100].  The uncertainties for each measurement of the line strength have been included in Tables \[tab:NGC5461\] and \[tab:NGC5462\]; the central velocity is not uncertain to more than a few percent, while the uncertainty in the velocity width is approximately 2–3 km s$^{-1}$. In the situations where the signal-to-noise ratio is low, we estimated an upper limit for the integrated intensity by choosing a velocity interval around the central velocity that maximizes the value of the integrated intensity. 

The uncertainties for the peak temperature, $T_{peak}$, quoted in Tables \[tab:NGC5461\] and \[tab:NGC5462\] are calculated for the smoothed data.  The uncertainties range between $7 \mbox{ -- } 24$ mK for $^{12}$CO and $ 2.5 \mbox{ -- }4 $ mK for $^{13}$CO.  In addition to the random uncertainties, it is possible that there are systematic uncertainties.  To check for systematic differences among observing sessions, we compared the data from different runs for positions where there are spectra with a high signal-to-noise ratio, for example, a calibrator or NGC 5461.  The variations in flux among observing sessions are typically 10%, so there could be a systematic error of 10% that is introduced from the flux calibration. 

$^{12}$CO $J=2\to1$ emission was detected at the $7 \, \sigma$ level or better toward all eleven points around NGC 5461.  The peak antenna temperatures ranged between 68 and 228 mK. Interestingly, the $^{12}$CO $J=3\to2$ (345 GHz) emission from three ($a,b,c$) out of the five central points observed in NGC 5461 was between $65 \mbox{ -- }
100$% as strong as the emission seen in the $^{12}$CO $J=2\to1$ spectra (Table \[tab:NGC5461\]). 

The strongest detections in NGC 5462 in both [$^{12}$CO $J=2\to1$]{} and $J=3\to2$ were for position $g$ and not for $a$, which coincides with the center of the region.  Special care was taken so that four positions around NGC 5462g were observed in [$^{12}$CO $J=3\to2$]{} in order to be convolved together so that they could be compared with the lower resolution [$^{12}$CO $J=2\to1$]{} data.  Finally, it was possible to get a significant detection of [$^{13}$CO $J=2\to1$]{} toward NGC 5461a (Table \[tab:NGC5461\]), while the detection toward NGC 5462g is more questionable (Table \[tab:NGC5462\]). 

To our surprise, there was no detection of $^{12}$CO $J=2\to1$ toward NGC 5471, one of the brightest [[Hii]{}]{} regions in M101.  To calculate an upper limit for the integrated intensity from the five positions observed, we added the positive integrated intensity only.  The upper limit derived in this way for NGC 5471 is 0.70 K km s$^{-1}$ integrated over 40 km s$^{-1}$.

National Radio Observatory (NRAO) 12-meter {#s:nrao}
------------------------------------------

On 1996 November 6, we used the NRAO 12-meter telescope to observe NGC 5461, NGC 5462, and NGC 5471 in the rotational transition of $^{12}$CO $J=1\to0$ (Table \[tab:nrao\]).  The beam is large enough ($55''$) to encompass each of the three giant [[Hii]{}]{} regions; the integration times were 60 minutes, 120 minutes, and 78 minutes for NGC 5461, NGC 5462, and NGC 5471, respectively.  Each individual scan was 6 minutes.  Typical system temperatures at 115 GHz were $350\mbox{ -- }400$ K. The 256 channel 1 MHz (2.6 km s$^{-1}$) dual polarization filterbank was configured in series mode to gain a factor of $\sqrt{2}$ in the noise level.  The telescope software presents the data in units of corrected radiation temperature, $T_R^*$, as opposed to the corrected antenna temperature, $T_A^*$, that the JCMT system delivers.  The telescope scale was checked by observing K3-50 (observed peak $T_R^\ast= 24
\pm 0.1$ K); the uncertainty in the $T_R^\ast$ calibration was estimated to be about 15%. 

The data were reduced with UniPOPS.  Linear baselines were removed from two of the spectra ([NGC ]{}5461 and [NGC ]{}5471) while a polynomial baseline was removed from the spectrum of [NGC ]{}5462.  The data were smoothed to a resolution of 10 km s$^{-1}$ (or 4 MHz at 115 GHz) to improve the signal-to-noise ratio.  The [$^{12}$CO $J=1\to0$]{} spectra are shown in Figure \[fig:nrao\], and the integrated intensities are given in Table \[tab:nrao\]. 

Owens Valley Millimeter-Wave Interferometer Data {#s:ovro}
------------------------------------------------

NGC 5461 and NGC 5462 were observed with the Owens Valley (OVRO) Millimeter-Wave Interferometer during 1996 February and April and during 1997 February and April.  This millimeter array has 6 antennae, which have diameters of 10.4 m. Two configurations of the array (A and C) were combined so that the synthesized beam is $2\farcs62 \times 2\farcs03$ for NGC 5461 and $3\farcs14 \times
2\farcs73$ for NGC 5462.  The total track length was 16 hours for each region in each configuration.  Typical single-sideband system temperatures at the zenith were $600 \mbox{ -- }
1000$ K. All four independent correlator modules of the digital spectrometer system were used to observe the [$^{12}$CO $J=1\to0$]{} line with an effective bandwidth of 126 MHz (328 km s$^{-1}$) and a resolution of 1 MHz (2.6 km s$^{-1}$). 

To reduce the data from the OVRO interferometer, we used the `mma` software package, which is written and maintained by the Caltech millimeter interferometry group [@173].  For the two tracks of NGC 5461, Neptune and the quasar 3C273 were used for the flux calibration measurements, which agreed to within $20\%$. For the two tracks of NGC 5462, one was calibrated with Uranus observations, and the other with the quasar 3C345.  The flux measurements from these two tracks agreed to within $10\%$.

The gain calibrator was the quasar 1418+546 \[$\alpha(1950)=14{\mbox{${}^{\rm{\scriptsize{h}}}$}}18{\mbox{${}^{\rm{\scriptsize{m}}}$}}06\fs200$ and $\delta(1950)=+54\arcdeg36\arcmin57\farcs80$\], which is unfortunately fairly weak.  The average measured flux during the NGC 5461 observations was 0.80 Jy, while the flux during the NGC 5462 observations a year later was 0.55 Jy.  Since it is possible that the intrinsic brightness of the gain calibrator changed during the course of a year, we adopt these two separate values for the remaining analysis. 

The data sets were edited to remove poor data with the main criterion being that the coherence on the gain calibrator was higher than 50%.  We determined passband calibration for the NGC 5461 tracks using the quasars 3C273 and 3C454.3.  For the NGC 5462 tracks, the quasars 3C454.3 and 3C345 were used for passband calibration. 

After the basic reduction was completed, we used Miriad [@178] to map and clean the data.  Because the signal-to-noise ratio was relatively small, we used natural weighting.  All channel maps were cleaned to the $1.5 \, \sigma$ level with fewer than 1000 iterations.  The rms noise and the maximum signal for the maps integrated over 52 km s$^{-1}$ were 0.025 Jy beam$^{-1}$ and 0.12 Jy beam$^{-1}$ for NGC 5461, and 0.022 Jy beam$^{-1}$ and 0.09 Jy beam$^{-1}$ for NGC 5462.  The rms noise and the maximum signal for the 1 MHz channel maps were 0.10 Jy beam$^{-1}$ and 0.36 Jy beam$^{-1}$ for NGC 5461, and 0.10 Jy beam$^{-1}$ and 0.33 Jy beam$^{-1}$ for NGC 5462. 

Once the cleaning process was completed, we identified candidate GMCs in the maps.  Three different types of plots were used.  The *integrated map* is a plot that has been integrated over a wide velocity range (52 km s$^{-1}$) to include all the emission.  The *channel maps* are a series of maps where the region is plotted by integrating over one channel only (2.6 km s$^{-1}$).  Finally, the *optimum map* for each cloud is integrated only over the velocity range in which a given feature is visible at the $3 \, \sigma$ level or better. 

It must be stated in no uncertain terms that the process of identifying GMCs is fairly subjective.  We have chosen three relatively conservative criteria so that the results inspire some confidence.  The first criterion used to identify the GMCs is that the feature should be at least $2 \, \sigma$ in the integrated map.  The second criterion required features to persist at the $3 \, \sigma$ level over two consecutive channels; this method, however, could result in underestimating the number of clouds, especially the ones with a narrow velocity width.  The best velocity range for each GMC candidate was determined from the channel maps, and for each feature, the optimum map integrated for the appropriate velocity range was generated.  The third criterion required the feature to have signal of at least $3 \, \sigma$ in the optimum map.  It is possible for the second criterion to be satisfied but not the third if the features drift slightly from channel-to-channel so that the ‘optimum’ integrated flux is less than 3$ \, \sigma$ above the noise. 

The integrated map of NGC 5461 (Figure \[fig:61int\]) has thirty-three features with peak fluxes of at least $2 \, \sigma$. Of the thirty-three features that appear in the integrated map, nine can be seen at the $3 \, \sigma$ level in two consecutive channels of the channel maps.  In addition, we have also considered two other features (4 and 6 in Figure \[fig:61int\]) that appear on the integrated map and in three consecutive channels at the $2 \, \sigma$ level.  Of the eleven GMC candidates, one is eliminated by the third criterion, i.e., the total flux in the optimum map is not three times higher than the noise of the optimum map.  The rms noise determined in the optimum maps varies between $0.05 \mbox{ -- } 0.069$ Jy beam$^{-1}$ depending on the number of channels over which the signal has been integrated. 

The characteristics of the ten GMCs are given in Table \[tab:ovro\_gma\].  The positions were determined from the optimum maps.  The sizes are not deconvolved from the synthesized beam; no cloud appears significantly larger than the beam in both dimensions, and thus we can only place upper limits on the true size of the clouds.  The cloud positions are given for the center of the peak of the GMC in terms of offsets from the field center with an estimated uncertainty of $0\farcs5$. Two features, clouds 7 and 10, have similar coordinates, but they are separated in velocity space.  Table \[tab:ovro\_gma\] also includes the integrated flux measured from the optimum map.  We calculated the equivalent brightness temperature, $T_B$, by multiplying the peak flux in a one channel map by the conversion factor from Janskys to Kelvin (17.34 K/Jy).  These brightness temperatures (4.5 – 8.1 K) are larger than those found in the GMCs toward M33 [@34] despite the fact that the GMCs in M33 are resolved and much closer.  These large brightness temperatures suggest that the GMCs in M101 are close to being resolved. 

The emission from NGC 5462 is much weaker than that from NGC 5461; in fact, the emission is so much weaker that there are no features that appear in consecutive channels at the $3 \, \sigma$ level.  If all the positive features are added, then the upper limit to the total flux from the region is $<16$ Jy km s$^{-1}$.

Molecular mass from the Empirical Method {#s:properties}
========================================

Mass of individual GMCs in NGC 5461 {#empirical}
-----------------------------------

The column density can be calculated via an empirical relation based on data that suggest that the column density of hydrogen, $N_{H_2}$, is linearly proportional to the observed radiation temperature integrated over the emission line of $^{12}$CO $J=1\to0$ [@71; @139].  The constant of proportionality, $X$, is defined by [@153] $$X = {N_{H_2} \over \int T_R^* \, dv }. 
    \label{a_twentyfive}$$ The range of $X_{Gal}$ found in the literature is 1 – 12 $\times 10^{20}$ cm$^{-2} \,$ (K km s$^{-1}$)$^{-1}$ [@139], and we assume a value of $X_{Gal}= 3 \times 10^{20}$ cm$^{-2} \,$ (K km s$^{-1}$)$^{-1}$. However, the $X$ factor probably depends upon the metallicity of the gas.  It has been suggested that galaxies with lower metallicity than that of the Milky Way have higher values of $X$ [@37] with $$\begin{aligned}
\log(X/X_{Gal})&=&(5.95 \pm 0.86)\nonumber\\
  &&\quad-(0.67 \pm 0.10) [12+ \log({\rm O}/{\rm H})]. 
\label{a_twentyfivea}\end{aligned}$$ The giant [[Hii]{}]{} regions of M101 have low metallicities, so the value of $X$ should be adjusted: the value of \[12+ log(O/H)\] for NGC 5461 is $8.39 \pm 0.08$, while for NGC 5471 it is $8.05 \pm 0.05$ [@14]. 

The uncertainties introduced by the value of $X$ may be severe, especially in a galaxy with intense star formation [@159].  However, the use of the $X$ factor is a standard way to estimate molecular cloud masses, so we adopt an initial value of $X$ correcting for the known metallicity of these regions and compare the resulting molecular gas mass with another, usually more reliable, virial mass to determine an appropriate value of $X$ for the [[Hii]{}]{} regions in M101.  As a starting point, we use the value of $X_{NGC~5461}=6 \times 10^{20}$ cm$^{-2} \,$ (K km s$^{-1}$)$^{-1}$ for NGC 5461.  The hydrogen column density can be obtained from equation (\[a\_twentyfive\]), and the molecular mass in terms of flux density, $S_{\nu}$, can be written as [@34] $$\begin{aligned}
    M_{mol} =&& 1.61 \times 10^4 \, {\mbox{$M_{\sun}$}}\
\left(    
{X \over {3 \times  10^{20} {\rm cm}^{-2} \, (\rm{K} \, \rm{km} \, 
\rm{s}^{-1})^{-1}}}
\right) \nonumber \\
&&\quad\times 
\left({d \over {\rm Mpc}}\right)^2 
 \int {S_{\nu} \over {\rm Jy}} \, {dv \over {\rm km} \,  {\rm s}^{-1}}. 
\label{a_twentysevena}\end{aligned}$$ We use equation (\[a\_twentysevena\]) to calculate the masses of the individual GMCs (Table \[tab:sizeline\]); we have corrected the fluxes (and masses) for the primary beam falloff.  With this equation, the individual features in NGC 5461 have masses of $(2\mbox{ -- }9) \times 10^6$  and appear to be much more massive than the GMCs in the Milky Way, which have masses typically around $10^5$ . 

As a check on these masses, we compare the molecular masses to the virial masses [@179].  If the virial mass is less than or equal to the molecular mass, then the clouds are considered gravitationally bound.  The virial mass is adapted \[by converting $\sigma_{1D}$ to $\Delta V_{\rm FWHM}$\] from Rand [-@33]: $$M_{vir} = 95 \, {\mbox{$M_{\sun}$}}\ 
\left({\Delta V_{{\rm FWHM}} \over {\rm km \, s^{-1}}}\right)^2
\left({D \over {\rm pc}}\right). 
\label{d_two}$$ We define the diameter, $D$, to be $1.4 \, D_{FWHM}$ to include all the emission from the GMC and for consistency with previous studies [@34].  In our calculations, we have assumed an upper limit for $D_{FWHM}$ of 80 pc because none of the GMCs are resolved from the beam in both dimensions; therefore, the virial masses estimated are upper limits based on a diameter of 110 pc.  These masses have been tabulated in Table \[tab:sizeline\] along with the masses found by the empirical method and the expected diameter, $D_{exp}$, given the observed velocity dispersion, $\Delta V_{obs}$ and the size:line-width relation [@118; @160]: $${\Delta V \over {\rm km \, s^{-1}}}= 1.2 \biggl({D \over {\rm pc}}\biggr)^{0.5}. 
\label{d_one}$$

We find that the virial mass of each cloud is generally an order of magnitude smaller than the molecular mass calculated from the metallicity corrected $X$ factor and is similar to the masses of the GMCs in the Milky Way.  We have more confidence in the universality of the virial theorem and, thus, this large discrepancy suggests that a smaller value of $X$ for the giant [[Hii]{}]{} regions in M101 is more appropriate.  The physical justification for adopting a different value of $X$ is that $X$ depends on temperature, and the observed high brightness temperature indicates that the gas is hot.  If these clouds obey the Galactic size:line-width relation, they have rather low filling factors within the OVRO beam and their true brightness temperatures would be even higher than what is observed.  It is possible that the high temperature of the gas lowers the value of $X$ in a star forming region [@159].  The average value of the ratio $M_{mol}$/$M_{vir}$ for the GMCs in NGC 5461 is $10 \pm 4$, which suggests that the appropriate value of $X_{NGC~5461}$ is $6  \times 10^{19}$ cm$^{-2} \,$ (K km s$^{-1}$)$^{-1}$ or approximately five times smaller than the canonical Galactic value (and ten times smaller than the metallicity corrected value).  We adopt the virial mass as the most correct estimate of the mass of the GMCs.  These observations are the first to demonstrate a clear decrease in the value of $X$ due to heating by intense star formation.  We adopt the same value for $X_{NGC~5462}$ because the two giant [[Hii]{}]{} regions have similar metallicities.  \[Xranges\] For NGC 5471, we use the value $X_{NGC~5471}$ of $12 \times 10^{19}$ cm$^{-2} \,$ (K km s$^{-1}$)$^{-1}$ because of its lower metallicity. 

Mass of the Associations of GMCs
--------------------------------

After examining the masses of the individual GMCs, we use equation (\[a\_twentysevena\]) to calculate the mass of the entire association of GMCs for each region using the appropriate value of $X$. With observations from the NRAO 12-meter telescope (Table \[tab:nrao\]), we obtain the integrated intensity by multiplying the integrated $T_R^*$ with the factor 34 Jy/K. From the integrated intensity, we calculate the total molecular mass from equation (\[a\_twentysevena\]) (Table \[tab:ansummary\]). 

The integrated flux from the OVRO map was measured from the integrated map over a velocity range of 52 km s$^{-1}$ of [$^{12}$CO $J=1\to0$]{} emission.  The value for the integrated flux density is $33
\pm 13$ Jy km s$^{-1}$, which corresponds to $(60 \pm 23) \times 10^5$ .  This result from the OVRO millimeter array is smaller than the mass obtained from the single dish data (12-meter telescope).  To explain the difference, one might evoke the presence of a constant, broadly distributed contribution to the intensity, which would have been undetected by the interferometer [@34].  The same explanation might be appropriate for the measurements of NGC 5462 (Table \[tab:ansummary\]). 

The mass of molecular gas associated with NGC 5461 is greater than the masses associated with the other two [[Hii]{}]{} regions.  Although the calculations indicate that NGC 5462 has a slightly smaller molecular mass than NGC 5471, their allowed values are the same to within their uncertainties.  The uncertainties in the total molecular mass are derived from the uncertainty in the integrated radiation temperature; the (potentially more significant) systematic uncertainty in the value of $X$ has been ignored. 

LTE Analysis {#lte}
============

In this section, we calculate the optical depth and the column density based on the assumption that the gas is in *local thermodynamic equilibrium* (LTE).  For the LTE analysis, we used $^{12}$CO and $^{13}$CO $J=2\to1$ data obtained at the JCMT.  The LTE method is frequently used to calculate the physical properties of molecular gas especially when isotopomers of CO or of other molecules such as CS and NH$_3$ have been observed \[e.g.,  Lada and Fich [-@161]; Giannakopoulou et al. [ -@162]\].  The $^{12}$CO $J=2 \to 1$ and $^{13}$CO $J=2 \to 1$ transitions have very similar frequencies, so the ratio of the transparency of the gas in the two transitions is approximately given by $${1-e^{- {}^{12}\tau_{\nu}} \over   1-e^{- {}^{13}\tau_{\nu}}} =
{  k \, ^{12} T_R / h \,\nu_{12} -e^{-h \, \nu_{12} / k \, T_{bg}}  \over 
   k \, ^{13} T_R / h\,  \nu_{13} -e^{-h \, \nu_{13} / k \, T_{bg}}    }, 
\label{a_nine}$$ where $\nu$ is the frequency of the transition, $\tau_{\nu}$ is the optical depth, $T_R$ is the radiation temperature, and $T_{bg}=2.73$ K is the temperature of the microwave background. 

We can calculate $^{13}\tau_{\nu}$ if we assume a value for the abundance ratio $\psi = [^{12}$CO\]/\[$^{13}$CO\].  This ratio may increase with the galactocentric distance [@79]: $\psi$ has been measured to be as low as 24 within 4 kpc and as high as 79 at 12 kpc from the center of our Galaxy.  For the distance of the two giant [[Hii]{}]{} regions from the center of M101 (between 4 and 6 kpc), we expect the most appropriate ratio to be at the lower end of these values.  \[psivalues\]

Since we have assumed that the gas is in LTE, the kinetic temperature $T_k$ is equal to the excitation temperature and is given by $$T_k = \frac{h\nu/k}{\ln\{ (1-e^{-\tau}) [kT_R/h\nu -
 (e^{h\nu/kT_{bg}}-1)^{-1}]^{-1} +1\}}.   
    \label{a_eleven}$$ Appendix A contains a detailed discussion of how we obtained the radiation temperature.  The values of the kinetic temperature are between 8 and 25 K, which are fairly low for a star forming region.  Even lower values of 4–6 K are found for NGC 5462.  However, these are average temperatures over large volumes of gas.  Near young stars, the temperature of the gas will likely be much higher.  The smaller value of $X$ found for the GMCs in NGC 5461 is possible evidence of higher temperatures, as are the large brightness temperatures of 4 – 8 K observed in the [$^{12}$CO $J=1\to0$]{} line. 

From references such as Mitchell et al. [ -@138] and Giannakopoulou et al. [ -@162], we find that the total integrated $^{13}$CO column density, $N_{^{13}CO}$, is given by $$\begin{aligned}
    N_{^{13}CO} =&&
{3k \over 8  B_r  \pi^3  \mu^2}  \int  
{e^{h  B_r  l(l+1)/k  T_k} \over (l+1)} \nonumber \\
&&\qquad\times
{T_k + {h  B_r / 3 k} \over 1-e^{-h  \nu /k  T_k}}\, {}^{13}\tau_v  dv,
    \label{a_twentyone}\end{aligned}$$ where the rotational moment of CO is $B_r=5.764 \times 10^{10}$ Hz and the electric dipole moment of CO is $\mu=1.12 \times 10^{-19}$ esu cm [@165].  For the error analysis of the column density, we use a robust bootstrap method [@168].  \[bootstrap\] We estimated the probability distribution of the column density, via equation (\[a\_twentyone\]), by a Monte Carlo method in which Gaussian random numbers representing $^{12}T_A^*$ and $^{13}T_A^*$ are generated with population means equal to the measured temperatures and with population standard deviations equal to the measured uncertainties in the temperatures.  From the characteristics of these distributions, one can obtain the uncertainty that is associated with the total column density. 

To obtain the column density $N_{^{13}CO}$ for each region, we summed the column densities per channel over all channels.  For a channel to be included, we required that its peak $^{12}$CO signal is at least $3\sigma$, and its peak $^{13}$CO is at least $2\sigma$. The median column density for NGC 5461 is $6.5 \times 10^{15}$ cm$^{-2}$. These criteria result in a lower limit of $1.2 \times 10^{15}$ cm$^{-2}$ for the value of the column density (and consequently of the mass) of NGC 5462, which only has one channel that meets the restrictions. 

The column density is used to obtain the total molecular mass for the three regions.  Since the main constituent of the interstellar medium is molecular hydrogen, the $^{13}$CO column density obtained above needs to be extrapolated to that of $^{12}$CO, $ N_{^{12}CO}$, which in turn will be converted to the H$_2$ column density, $N_{H_2}$. The values of the isotopomer ratio, $\psi$, and of the ratio of the column density of molecular hydrogen to the column density of CO are uncertain.  We adopt a value of $\psi=20$ because the giant [[Hii]{}]{}regions are fairly near the center of M101 (also §\[lvg\]).  In addition, we need to determine a suitable value for the proportionality constant \[$N_{H_2}$\]/\[$N_{^{12}CO}$\].  The canonical value of this parameter is $10^4$ for the Galaxy [@152].  However, this ratio depends on metallicity.  Since NGC 5461 and NGC 5462 are 4 times more metal poor, and NGC 5471 is 9 times more metal poor than the Milky Way [@14], we adopt values of $4 \times 10^4$ and $10^5$, respectively, for these [[Hii]{}]{} regions. 

The mass of the H$_2$ gas \[hydrogenmass\]is found by multiplying the total H$_2$ column density, $N_{H_2}$, by the beam area, $A$, and the mass of a hydrogen molecule $m_{H_2}$. Finally, the fractional helium abundance (10% by number) is taken into account in order to obtain the total mass within a beam.  Thus, the total molecular gas mass for NGC 5461 and NGC 5462 is $$    M _{mol}= 6.3 \times 10^{-9} \, {\mbox{$M_{\sun}$}}\,
  \left( {N_{^{13}CO} \over {\rm cm^{-2}}}\right)
  \left( {A \over {\rm pc^2}} \right). 
    \label{a_twentyfour}$$ The masses of the molecular gas are given in Table \[tab:ansummary\].  These masses are much larger than Galactic GMCs, which have typical values of $10^5$  [@56].  Large masses are expected from regions as big as those observed (700 pc or $20''$), which would contain an association of GMCs. 

LVG analysis {#lvg}
============

There are two extreme non-LTE approaches in dealing with the radiative transfer problem.  One approach is to assume that there is no global motion: all the emission is a result of small-scale thermal motions and turbulence.  The second approach is to assume that the gas cloud has a large velocity gradient (LVG), which means that the CO emission from one part of the cloud will be Doppler-shifted to a frequency that will not be re-absorbed by gas in other parts of the cloud [@60].  Under these conditions, the emission from a molecule will only be absorbed by neighboring molecules, and the radiative transfer can be solved locally [@58].  The first approach is called the microturbulence method, and the second approach is called the LVG method.  The two non-LTE approaches typically produce results that agree within the errors of the methods, which are a factor of three due to uncertainties in geometry [@59].  We have used just one non-LTE method, the LVG method, to compare with our LTE analysis [@59; @23]. 

To estimate the physical conditions in the GMCs of M101, we have used the LVG code written by Jessica Arlett and Lorne Avery for a spherical, uniform cloud.  This program computes the radiation temperatures of various CO transitions for ranges of kinetic temperature, density of H$_2$, and an abundance parameter, which is defined as $(N_{CO}/N_{H_2})/(dV/dr)$. These radiation temperatures were fitted to the observed values (Table \[tab:ratios\]) by minimizing a $\chi^2$ statistic to give the best fit for kinetic temperature, density, and abundance.  In addition to these parameters, the LVG results depend on the isotopomer ratio, $\psi = $\[$^{12}$CO\]/\[$^{13}$CO\].  We found that the best fits occur for $\psi=20$, which is in agreement with the observed value in the Milky Way [@79]. 

For NGC 5461, we found reasonable fits both for low temperatures ($T_k=30 \mbox{ -- }90$ K) and for high temperatures ($T_k=230 \mbox{ -- }250$ K); the best fit was for $T_K=60$ K. However, the best fit for density, $n_{H_2} = 3 \times 10^3$ cm$^{-3}$, was the same for both ranges of temperature.  A GMC with diameter 112 pc and this density has a mass of $1.3 \times 10^8$ , which is two orders of magnitude larger than those obtained with the empirical method (Table \[tab:ansummary\]).  The clouds are not resolved, and we do not expect the cloud to be completely uniform: the volume-averaged density is usually substantially smaller than $10^3$ cm$^{-3}$ [@34].  By comparing the results from the LVG analysis and the empirical method, we conclude that the filling factor is on the order of 1% for the clumpy material.  This low filling-factor indicates that the gas emitting the CO has formed dense clumps, which are surrounded by a lower density envelope.  A similar situation is observed in the molecular gas of 30 Doradus [@179]: the intense radiation field is considered the reason for the dissociation or ionization of the gas in the envelope. 

The LVG analysis cannot be done for NGC 5462 because there are only two available ratios (Table \[tab:ratios\]).  The LVG analysis is not conclusive with only two ratios [@54], but one can use the available ratios to make some general statements about the physical properties of the gas.  Because the $^{12}$CO ($J=3 \to 2$)/($J=2 \to 1$) ratio for NGC 5462 is larger than that of NGC 5461, the molecular gas in NGC 5462 may be warmer than the molecular gas around NGC 5461.  \[a:tempratio\] One expects from the Boltzmann equation that, for usual temperatures of quiescent molecular gas (10 K), the second level is more populated than the third level, which means that for cold gas the ratio $^{12}$CO ($J=3 \to 2$)/($J=2 \to 1$) is less than unity; however, in the case of NGC 5462, the ratio is larger than unity, which implies that the gas is probably considerably hotter than typical quiescent gas. 

Masses and Temperatures {#s:massestemp}
=======================

NGC 5461 is the brightest [[Hii]{}]{} region in CO emission; therefore, the data set for NGC 5461 is the most complete of all the giant [[Hii]{}]{} regions in M101.  There are two estimates (from the LTE and from the empirical estimate using the NRAO data) for the mass of the association of GMCs in NGC 5461, which range in value from ($15 \mbox{ -- } 40) \times 10^6$ .  Although the discrepancy of a factor of 3 is not surprising when we consider the uncertainties in the methods involved, the difference could be partially due to an underestimate of $\eta_c$ (see Appendix).  It is likely that there are GMCs in NGC 5461 that were not detected and would have contributed in increasing the coupling efficiency. 

Both the LTE method and the empirical method suggest that NGC 5462 has roughly an order of magnitude less molecular mass than NGC 5461.  This conclusion is consistent with the fact that the molecular gas in NGC 5462 was not detected with the Owens Valley array.  NGC 5462 might not be as massive as NGC 5461, but it may be hotter than NGC 5461 since the $^{12}$CO ($J=3 \to 2$)/($J=2 \to 1$) ratio in NGC 5462 is even higher than the extraordinary cloud NGC 604-2 in M33.  Unfortunately, the uncertainty of this ratio for NGC 5462 is fairly high; within its uncertainty, it agrees with normal clouds in M33 and IC10 [@156]. 

NGC 5471 \[s:def71\] is significantly weaker in CO emission than NGC 5462 and NGC 5461, so the only observations that were made of NGC 5471 were in [$^{12}$CO $J=1\to0$]{}and $J=2 \to1$ emission.  Unlike NGC 5462, NGC 5471 is several kpc away from the center of M101 and, therefore, has considerably lower metallicity than regions closer to the center of M101.  With the higher adopted value for $X$ for this region, the upper limit to the molecular mass in NGC 5471 is comparable to that of NGC 5462.  If the $X$ factor is in fact even larger, then the mass of the molecular gas in NGC 5471 could even be comparable to that of NGC 5461.  However, a more probable explanation for the deficiency of CO emission in NGC 5471 is that there is genuinely less molecular gas present.  A clue to this problem is the presence of large amounts of atomic gas near the giant [[Hii]{}]{} region [@kamphuis:93].  Perhaps the atomic gas was converted to molecular gas more slowly in NGC 5471 than in the other two giant [[Hii]{}]{} regions, or perhaps more of the molecular gas has been rapidly dissociated to atomic gas.  There is more discussion on this topic in §\[s:star\]. 

One possible result of star formation is that the molecular gas becomes hotter.  One of the diagnostics of high temperatures is the $^{12}$CO ($J=3 \to 2$)/($J=2 \to 1$) ratio.  NGC 5461 has a similar $^{12}$CO ($J=3 \to 2$)/($J=2 \to 1$) ratio to the clouds in M33 that are associated with [[Hii]{}]{} regions [@55].  However, the ratio for NGC 5461 does not approach the value for the GMC NGC 604-2, which is, indeed, very high compared to other ratios seen in M33 and other galaxies.  An explanation for the lower value of the $^{12}$CO ($J=3 \to 2$)/($J=2 \to 1$) ratio in NGC 5461 compared to that found in other giant [[Hii]{}]{} regions is that the clouds in NGC 5461 may have a mixture of warm and hot gas.  The dominant warm component could be similar in temperature and density to the GMCs in M33 associated with [[Hii]{}]{} regions [@55]. 

Star Formation {#s:star}
==============

The main motivation of this paper is to understand the connection between the molecular gas in M101 and the existence of the bright stars that produce the giant [[Hii]{}]{} regions.  In the Milky Way, the masses of GMCs that are affiliated with OB associations are (1 – 7)$\times 10^5$  [@65].  From this observational result, Williams and McKee [-@65] suggest that the probability a GMC will contain a very bright massive star (O9.5) increases as the mass of the cloud increases.  However, the GMCs in NGC 5461 have masses that are much smaller than those required to form many bright O stars according to Fig.  7 of Williams and McKee [-@65].  Perhaps the star-forming efficiency, and not the mass of the clouds, is the key issue for regions with intense star formation. 

Intense star formation influences and is influenced by the surrounding ambient ionized and atomic gas.  It is instructive to consider the correlation of molecular gas with ionized and atomic gas.  One might expect that molecular and ionized hydrogen, which is observed via the H$\alpha$ transition, should be associated with each other.  In Figure \[fig:halpha\], the CO peaks are compared with a recent H$\alpha$ map that was kindly provided by Robert C. Kennicutt.  The CO clouds are denoted by stars to distinguish them from the contours of the H$\alpha$ map. 

The CO peaks are typically displaced from the peak of the H$\alpha$ image by $2\mbox{ -- }5''$ or, at the distance of M101, $70\mbox{ -- }180$ pc.  A possible reason, which was used to explain the same type of anticorrelation in M51 [@33], is that the presence of gas with high surface density increases the extinction from the associated dust.  The radiation from the ionized gas is obstructed, and therefore the H$\alpha$ emission is anticorrelated with the molecular gas. 

Figure \[fig:hi\] presents the CO clouds in NGC 5461 superimposed on a high resolution ($9''$) [[Hi]{}]{} contour map kindly provided by Robert Braun.  The GMCs are denoted with stars.  The CO peaks are displaced compared to the peak of the [[Hi]{}]{} image by $3
\mbox{ -- }10''$ or, at the distance of M101, $100\mbox{ -- }360$ pc.  A possible explanation for this displacement is that the radiation from the young massive stars in the [[Hii]{}]{} regions may have dissociated the molecular gas to atomic gas; this explanation was used for M51 where a similar displacement was observed [@164].  Unfortunately, the uncertainty of the positions of the [[Hi]{}]{} map, $4''$, is fairly high; it is possible that the observed offset is not significant. 

A large scale study of the atomic gas in M101 was conducted with $6''$ resolution [@63].  Two of the several positions observed coincide with the giant [[Hii]{}]{} regions NGC 5461 and NGC 5471.  The atomic masses for the two regions are $5 \times 10^7 $  and $ 4
\times 10^7$ , respectively, which are a few times higher than the corresponding molecular mass observed in similar beams.  This result is consistent with the observed ratios of atomic to molecular gas in a sample of 27 Sc galaxies [@57]. 

Despite the fact that the masses of the atomic gas for NGC 5461 and NGC 5471 are similar, their empirical *molecular* masses are different by a factor of 3.  Why does NGC 5461 have so much more molecular gas than NGC 5471?  One possibility is that NGC 5461 has a more efficient mechanism to convert atomic to molecular gas than NGC 5471 does.  The efficiency of the conversion might be enhanced by large-scale gravitational effects that increase gas interactions and remove loosely bound atomic gas [@57].  Alternatively, it is possible that the star-formation mechanism in NGC 5471 has had more time to dissociate the molecular gas (to form atomic gas) by the intense radiation from the young stars [@164; @38].  Unfortunately, the available data are not sufficient to find the relative ages of the giant [[Hii]{}]{} regions in order to distinguish between the two possibilities. 

The sum of the masses of the molecular, ionized, and the atomic gas for both regions (Table \[tab:efficiencies\]) is in fair agreement with the total gas mass calculated from the dust and the assumed gas-to-dust ratio.  We calculate the dust mass from the 60 $\mu$m-wavelength fluxes obtained from the Infrared Astronomical Satellite (IRAS) Point Source Catalog with the equation [@66]: $$M_{dust} = {4 a  \rho \over 3 Q_{em}} \, {S_{\nu} d^2 \over
B_{\nu}(T)}, 
\label{d_three}$$ where $a$ is the radius of the dust grain, $\rho$ is the density of the dust grain, $Q_{em}$ is the dust grain emission efficiency, $S_{\nu}$ is the measured flux density, $d$ is the distance to the dust (7.4 Mpc for the giant [[Hii]{}]{} regions in M101), and $B_{\nu}(T)$ is the Planck function.  The flux at 60 $\mu$m for NGC 5461 is 9.65 Jy, and for NGC 5471 is 1.81 Jy (NGC 5462 is not detected as a point source).  For the calculation, it has been assumed that $\rho=3$ g cm$^{-3}$, $Q_{em}/a
= 340$ cm$^{-1}$, and $T=30$ K [@66], which are canonical values.  With these assumptions, the dust mass for NGC 5461 is $5 \times 10^5$  while for NGC 5471 the dust mass is $9 \times 10^4$ .  If the gas-to-dust ratio is assumed to be a typical value of 600 for Sc galaxies \[Young and Scoville [-@57] and references therein\], then the total gas mass in NGC 5461 is $3 \times 10^8$ , and the total gas mass in NGC 5471 is $5 \times 10^7$ , which is in agreement with the sum of all gas components in these regions (Table \[tab:efficiencies\]). 

The star-formation efficiency is the ratio of the mass of stars formed in the region to the sum of the stellar and molecular masses of the region [@176].  Recently, the mass of stars ($M >  2$ ) in a small portion of NGC 5461 has been estimated from UV fluxes [@41]; however, to compare this result with our data, we must scale the mass of stars to include all stars down to 0.1 , and we must scale the flux to correct for the small aperture used ($1''$).  We estimate that the first scale factor is 2.6 from Table 9 in Miller and Scalo [-@177], and the second scale factor is 4 for both NGC 5461 and NGC 5471 from Figure 1 in Rosa and Benvenuti [-@41].  The masses of the stars and the star formation efficiencies in NGC 5461 and NGC 5471 calculated in this manner are given in Table \[tab:efficiencies\].  The star forming efficiencies are slightly larger than the efficiencies of a few percent (1% $\mbox{ -- }$ 4%) observed in several clouds in the Milky Way such as the Taurus-Auriga, Orion A, and Orion B clouds [@175].  Thus, the GMCs in these giant [[Hii]{}]{} regions are somewhat more efficient in creating stars than molecular clouds in our Galaxy.  Perhaps the mass of the atomic gas should be included in the calculation of the star efficiency if the molecular gas has already been dissociated to [[Hi]{}]{}.  However, the [[Hi]{}]{}  has been found at great distances from the molecular cloud especially in the case of NGC 5471.  It is not obvious that the [[Hi]{}]{} is directly connected with the star forming regions. 

Conclusions {#s:conclusions}
===========

In this paper, the molecular components of three giant [[Hii]{}]{} regions in the spiral galaxy M101 have been investigated with new observations from two single dish telescopes (JCMT and NRAO 12-meter) and from the OVRO millimeter array. 

\(1) NGC 5461 is the only [[Hii]{}]{} region with strong enough emission to be detected with the OVRO millimeter array.  The mass of the GMCs in NGC 5461 was calculated empirically and from the virial theorem, and we found that the appropriate $X$ factor toward the giant [[Hii]{}]{} regions is 5 times smaller than $X_{Gal}$. These data provide the first empirical demonstration that the value of $X$ may decrease in regions with intense star formation.  The corrected empirical masses of the large GMCs cover a range of $(2 \mbox{ -- } 8)\times 10^5$ , which is comparable to that observed in the Galaxy.  Using this determined value of $X$, we calculate the mass of the associations of GMCs in NGC 5461, NGC 5462, and NGC 5471. 

\(2) The molecular mass for the association of GMCs in NGC 5461 is calculated to be $(15 \mbox{ -- } 40) \times 10^6$ .  The higher value in the range comes from the LTE analysis, while the lower value comes from the empirical method using NRAO data of NGC 5461.  The discrepancy would be resolved if the $\eta_c$ was larger.  This efficiency probably is larger because there could be more GMCs in NGC 5461 that were not detected.  The interferometric data provide a lower mass ($6 \times 10^6$ ) because the flux from extended low-level emission is typically lost in an interferometer.  The molecular mass toward NGC 5462 is estimated to be (2.2–3.5)$\times 10^6$ .  CO emission was not detected toward NGC 5471; the upper limit to the mass is $6.5 \times 10^6$ . 

\(3) The gas emitting the CO is very dense, so dense that it cannot fill up more than 1% of the volume without exceeding the GMC mass limits.  It is possible that the intense radiation field in the vicinity of a giant [[Hii]{}]{} region partially dissociates or ionizes portions of the molecular gas leaving small dense cores in large sparse envelopes. 

\(4) The molecular mass of the NGC 5461 association of clouds is accompanied by 1–2 times as much atomic mass.  The fairly large ratio of atomic to molecular gas can be attributed either to slow formation of GMCs from the original atomic clouds or to efficient dissociation of the GMCs to atomic gas.  An even stronger presence of atomic hydrogen is observed in the vicinity of NGC 5471; in this region, the atomic mass is an order of magnitude larger than molecular mass.  Perhaps NGC 5471 has not converted much of its atomic gas to molecular gas yet, or the molecular gas has been dissociated to form atomic gas.  For both NGC 5461 and NGC 5471, the total gas mass can be extrapolated from the mass of the dust estimated from IRAS data.  The estimates of the total gas mass from the dust are consistent with the sum of the molecular, ionized, and atomic masses. 

\(5) The relatively normal masses observed toward the clouds in NGC 5461 reinforce the hypothesis that giant [[Hii]{}]{}regions in M101 are so much brighter than the [[Hii]{}]{} regions in our Galaxy because of the different properties of the natal clouds in M101.  In particular, this paper suggests that the high star-formation efficiency of the gas, and not the large mass of the cloud, is the key to the formation of giant [[Hii]{}]{} regions in M101. 

The authors would like to thank Lorne Avery for his LVG code, Robert Braun for providing his [[Hi]{}]{} maps of M101, David Frayer for helping with the NRAO figure, Jeff Kenney for sharing his unpublished CO map of M101, Robert Kennicutt for providing the H$\alpha$ image of NGC 5461, and the staff at the JCMT.  We are grateful to the National Research Council of Canada for supporting observing trips and to the Astronomy Department of the California Institute of Technology.  This work was partly supported by NSERC Canada. 

Calculating the Radiation Temperature, $T_R$, from Observed Temperatures
========================================================================

To calculate \[etac\] the radiation temperature, one needs $\eta_c$, the efficiency with which the antenna diffraction pattern couples to the source [@68]: $$T_R =  {T_A^* \over \eta_c \; \eta_{fss}} = {T_R^* \over \eta_c}, 
    \label{twentytwo}$$ where $\eta_{fss}$ is the efficiency of forward scattering and spillover.  The model used for the integration takes into account both the size (as determined from the high resolution OVRO data of NGC 5461) and the position of each GMC with respect to the center of the JCMT beam.  We found that the coupling efficiencies for NGC 5461 are $0.056 \pm 0.024$ and $0.037 \pm 0.016$ for the two transitions [$^{12}$CO $J=2\to1$]{} and $J=3 \to 2$ respectively. 

One has to assume that the coupling efficiency for the gas in NGC 5462 are similar to NGC 5461 because there were no interferometric detections of clouds in NGC 5462.  This assumption is probably not too bad because the values for the molecular mass obtained with the LTE method are within a factor of two or three from the values obtained from the empirical method, which does not utilize the coupling efficiency (Table \[tab:ansummary\]). 

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Young, J. S., & Scoville, N. Z. 1991, , 29, 581

[lrrccccc]{} NGC 5461a & 0 & 0 & $5.8  \pm 0.3$ & 276 & 24 & $228 \pm    11$ & 3000 NGC 5461b & 7 & 7 & $5.4  \pm 0.3$ & 277 & 24 & $206 \pm    11$ & 1800 NGC 5461c & $-7$ & $-7$ & $5.3  \pm 0.3$ & 273 & 24 & $207 \pm    10$ & 1800 NGC 5461d & $-7$ & 7 & $4.7  \pm 0.4$ & 275 & 23 & $181 \pm    11$ & 1800 NGC 5461e & 7 & $-7$ & $2.1  \pm 0.4$ & 273 & 24 & $\phn83 \pm \phn9$ & 1800 NGC 5461f & $-14$ & $-14$ & $5.6  \pm 0.3$ & 272 & 25 & $212 \pm    11$ & 1650 NGC 5461g & 14 & 14 & $3.5  \pm 0.4$ & 287 & 22 & $158 \pm    12$ & 1800 NGC 5461h & $-14$ & 0 & $5.9  \pm 0.3$ & 274 & 26 & $217 \pm \phn9$ & 1800 NGC 5461i & 14 & 0 & $1.7  \pm 0.4$ & 277 & 25 & $\phn68 \pm    10$ & 1800 NGC 5461j & 0 & 14 & $1.5  \pm 0.4$ & 280 & 17 & $\phn79 \pm    10$ & 1800 NGC 5461k & 0 & $-14$ & $5.3  \pm 0.3$ & 272 & 24 & $202 \pm    11$ & 1800 NGC 5461a & 0 & 0 & $4.1  \pm 0.2$ & 277 & 23 & $180 \pm    12$ & 7200 NGC 5461b & 7 & 7 & $3.3  \pm 0.3$ & 281 & 20 & $140 \pm    15$ & 3000 NGC 5461c & $-7$ & $-7$ & $5.7  \pm 0.2$ & 275 & 27 & $210 \pm    17$ & 3000 NGC 5461d & $-7$ & 7 & $<$$1.1  \pm 0.3$ & & & $<$$\phn51 \pm    18$ & 3000 NGC 5461e & 7 & $-7$ & $<$$1.1  \pm 0.3$ & & & $<$$\phn53 \pm    24$ & 3000 NGC 5461a & 0 & 0 & $0.49 \pm 0.05$ & 272 & 27.8 & $17 \pm     4.0$ & 10840

[lrrccccc]{} NGC 5462a & 0 & 0 & $<$$0.10 \pm 0.10$ & & & $<$$\phn6 \pm 8$ & 1800 NGC 5462b & 0 & 6 & $0.51 \pm 0.10$ & 300 & 17.6 & $   28 \pm 8$ & 1800 NGC 5462c & 0 & $-6$ & $<$$0.69 \pm 0.20$ & & & $<$$   19 \pm 7$ & 1800 NGC 5462d & 6 & 0 & $0.49 \pm 0.10$ & 304 & 17.3 & $   27 \pm 7$ & 1800 NGC 5462e & $-6$ & 0 & $0.57 \pm 0.10$ & 299 & 16.4 & $   32 \pm 7$ & 1800 NGC 5462f & 6 & 6 & $<$$0.41 \pm 0.10$ & & & $<$$   18 \pm 7$ & 1800 NGC 5462g & 0 & 12 & $0.69 \pm 0.05$ & 292 & 12.0 & $   53 \pm 7$ & 1800 NGC 5462a & 0 & 0 & $<$$0.45 \pm 0.4$ & & & $   21 \pm \phn6$ & 4800 NGC 5462b & 0 & 6 & $0.20 \pm 0.4$ & 295 & 12 & $   18 \pm \phn6$ & 4800 NGC 5462d & 6 & 0 & $0.56 \pm 0.4$ & 296 & 23 & $   25 \pm \phn7$ & 4800 NGC 5462f & 6 & 6 & $<$$0.66 \pm 0.2$ & & & $<$$   14 \pm \phn6$ & 6000 NGC 5462g & 0 & 12 & $1.13 \pm 0.2$ & 297 & 22 & $   50 \pm \phn6$ & 6000 NGC 5462h & $-6$ & 6 & $<$$0.41 \pm 0.3$ & & & $<$$   15 \pm    12$ & 1200 NGC 5462n & 0 & 18 & $1.00 \pm 0.3$ & 300 & 27 & $   36 \pm    12$ & 1200 NGC 5462p & 6 & 12 & $<$$0.94 \pm 0.3$ & & & $<$$   27 \pm    12$ & 1200 NGC 5462q & $-6$ & 12 & $<$$0.49 \pm 0.3$ & & & $   49 \pm    12$ & 1200 NGC 5462g & 0 & 12 & $0.10 \pm 0.01$ & 295 & 6 & $10 \pm 2.5$ & 15600

[ccccc]{} NGC 5461 & $2.50 \pm 0.48$ & 274 & 29 & $78 \pm \phn8$ NGC 5462 & $0.38 \pm 0.11$ & 296 & 30 & $12 \pm \phn2$ NGC 5471 & $<$$0.54 \pm 0.20$ & & & $<$$12 \pm    12$

[cccccccc]{} 1 & 263 & $\phn99\times 90$ & (19,20) & 7.8 & $ 2.6 \pm 0.9$ & 7.5 2 & 270 & $   108\times 90$ & ($-$2,$-$2) & 10.4 & $ 4.4 \pm 1.0$ & 5.3 3 & 270 & $   180\times 81$ & (0,$-$7) & 5.2 & $ 1.9 \pm 0.5$ & 5.5 4 & 271 & $\phn99\times 73$ & ($-$21,$-$19) & 7.8 & $ 3.0 \pm 0.9$ & 8.1 5 & 274 & $\phn94\times 73$ & ($-$6,$-$2) & 7.8 & $ 1.2 \pm 0.3$ & 4.5 6 & 276 & $   140\times 90$ & ($-$15,$-$13) & 7.8 & $ 4.7 \pm 1.2$ & 6.1 7 & 282 & $   126\times 81$ & (10,15) & 7.8 & $ 3.3 \pm 0.7$ & 6.6 8 & 285 & $\phn99\times 73$ & (17,28) & 5.2 & $ 2.1 \pm 0.6$ & 7.7 9 & 285 & $   108\times 90$ & ($-$9,$-$25) & 5.2 & $ 2.2 \pm 0.6$ & 6.0 10 & 298 & $   180\times 90$ & (11,15) & 5.2 & $ 2.8 \pm 0.9$ & 5.7

[cccccc]{} 1 & 7.8 & 42 & $<$$\phn6 \pm 2$ & $46 \pm    16$ & $5 \pm 2$ 2 & 10.4 & 75 & $<$$   11 \pm 3$ & $78 \pm    17$ & $8 \pm 2$ 3 & 5.2 & 19 & $<$$\phn3 \pm 1$ & $34 \pm \phn9$ & $3 \pm 1$ 4 & 7.8 & 42 & $<$$\phn6 \pm 2$ & $54 \pm    16$ & $5 \pm 2$ 5 & 7.8 & 42 & $<$$\phn6 \pm 2$ & $22 \pm \phn5$ & $2 \pm 1$ 6 & 7.8 & 42 & $<$$\phn6 \pm 2$ & $85 \pm    22$ & $8 \pm 2$ 7 & 7.8 & 42 & $<$$\phn6 \pm 2$ & $59 \pm    13$ & $6 \pm 1$ 8 & 5.2 & 19 & $<$$\phn3 \pm 1$ & $38 \pm    11$ & $4 \pm 1$ 9 & 5.2 & 19 & $<$$\phn3 \pm 1$ & $40 \pm    11$ & $4 \pm 1$ 10 & 5.2 & 19 & $<$$\phn3 \pm 1$ & $50 \pm    16$ & $5 \pm 2$

[lc]{} LTE & $  400 \; [350\mbox{ -- }500]$ Empirical (NRAO 12-meter) & $ 150 \pm 30$ Empirical (OVRO) & $  60 \pm 23$ Empirical (OVRO) & $   5$

LTE & $ >35$ Empirical (NRAO 12-meter) & $  22 \pm 7$ Empirical (OVRO) & $  <30$ Empirical (NRAO 12-meter) & $<65 \pm 24$

[cccc]{} NGC 5461 & $0.80 \pm 0.04$ & $0.56 \pm 0.11$ & $0.147 \pm 0.015$ NGC 5462 & $1.14 \pm 0.22$ & & $0.135 \pm 0.030$

[ccccc]{} NGC 5461 & 30 & 1& 15& 6 NGC 5471 & 14 & 0.8 & $<$6.5 & $>$11

[^1]: The JCMT is operated by the Joint Astronomy Centre in Hilo, Hawaii on behalf of the parent organizations Particle Physics and Astronomy Research Council in the United Kingdom, the National Research Council of Canada, and The Netherlands Organization for Scientific Research.