PHYSICAL PROPERTIES OF THE CIRCUMNUCLEAR STARBURST RING IN THE BARRED GALAXY NGC 1097

We observed NGC 1097 with the Submillimeter Array⁸ (SMA; Ho et al. 2004) at the summit of Mauna Kea, HI. The array consists of eight 6 m antennas. Four basic configurations of the antennas are available. With the compact configuration, two nights of ¹²CO(J = 2–1) data were obtained in 2004 (Paper I). To achieve higher spatial resolution, we obtained two further nights of ¹²CO(J = 2–1) data with the extended and the very extended configurations in 2005. To study the excitation of the gas, we also obtained one night of ¹²CO(J = 3–2) data in 2006 with the compact configuration. All the observations have the same phase center. We located the phase center at the 6 cm peak of the nucleus (Hummel et al. 1987). Detailed observational parameters, sky conditions, system performances, and calibration sources are summarized in Table 1.

Table 1. SMA Observation Parameters

	230 GHz	230 GHz	230 GHz	345 GHz
Parameters	Compact-N	Extended	Very Extended	Compact
Date	2004 Jul 23, 2004 Oct 1	2005 Sep 25	2005 Nov 7	2006 Sep 5
Phase center (J2000.0)
R.A.	α2000 = 02h46m18$\mbox{$.\!\!^{\mathrm s}$}$96
Decl.	δ2000 = −30°16'28 897
Primary beams	52''			36''
No. of antennas	8, 8	6	7	7
Project baseline range (kλ)	5–74, 10–84	23–121	12–390	7.2–80
Bandwidth (GHz)	1.989
Spectral resolution (MHz)	0.8125, 3.25	0.8125	0.8125	0.8125
Central frequency, LSB/USB (GHz)	219/228			334/344
τ225a	0.15, 0.3	0.06	0.1	0.06
Tsys, DSB (K)	200, 350	110	180	300
Bandpass calibrators	Uranus, J0423 – 013	3C454.3	3C454.3, 3C111	Uranus, Neptune
Absolute flux calibratorsb	Uranus (36.8, 34.2),	Uranus (37.2),	Uranus (34.9),	Neptune (21.1)
	J0423 – 013 (2.8, 2.5)	3C454.3 (21.3)	3C454.3 (21.3)
Gain calibrators	J0132 – 169	J0132 – 169	J0132 – 169	J0132 – 169, J0423 – 013

Notes. ^aτ₂₂₅ is the optical depth measured in 225 GHz. ^bThe numbers in the parentheses are the absolute flux in Jy.

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The SMA correlator processes two intermediate frequency sidebands separated by 10 GHz, with ∼2 GHz bandwidth each. The upper sideband (USB) and lower sideband (LSB) are divided into 24 slightly overlapping chunks of 104 MHz in width. With the advantage of this wide bandwidth of the SMA correlator, the receivers were tuned to simultaneously detect three CO lines in the 230 GHz band. The ¹²CO(J = 2–1) line was set to be in the USB, while the ¹³CO(J = 2–1) and C¹⁸O(J = 2–1) lines were set to be in the LSB. For the ¹²CO(J = 3–2) line, we placed the redshifted frequency of 344.35 GHz in the USB.

We calibrated the SMA data with the MIR-IDL software package. The detailed calibration procedures of the data of the compact configuration were described in Paper I. For the extended and very extended configurations, where Uranus is resolved quite severely, we observed two bright quasars (3C454.3 and 3C111) and adopted a bandpass/flux calibration method similar to the one described in Paper I. At 345 GHz, we used both Uranus and Neptune for the flux and bandpass calibrators.

Mapping and analysis were done with the MIRIAD and NRAO AIPS packages. The visibility data were CLEANed in AIPS with the IMAGR task. We performed the CLEAN process to deconvolve the dirty image to a clean image. The deconvolution procedure was adopted with the Högbom algorithm (Högbom 1974) and the Clark algorithm (Clark 1980). We used a loop gain of 10% and restricted the CLEAN area by iterative examinations. The CLEAN iterations were typically stopped at the 1.5σ residual levels. The number of CLEAN components in individual channel maps was typically about 300.

All of the 230 GHz visibility data were combined to achieve a better uv coverage and sensitivity. Our 230 GHz continuum data were constructed by averaging the line-free channels. Due to the sideband leakage at 230 GHz, a limited bandwidth of 1.3 GHz and 0.5 GHz were obtained respectively in the USB and LSB to make a continuum image. Weak continuum emission at 230 GHz with a peak intensity of 10 mJy beam⁻¹ was detected at about 4σ at the southern part of the ring. The 1σ noise level of the continuum emission at 345 GHz averaged over the line-free bandwidth of 0.4 GHz is 6 mJy beam⁻¹, and there is an ∼3σ detection in the nucleus and in the ring. In this paper, we did not subtract the continuum emission since it is too faint as compared to the noise level in the line maps. The spectral line data of ¹²CO(J = 2–1) and ¹²CO(J = 3–2) were binned to 10 km s⁻¹ resolution. As the ¹³CO(J = 2–1) emission is fainter, those data were binned to 30 km s⁻¹ resolution to increase the signal-to-noise ratio (S/N). The C¹⁸O(J = 2–1) line emission was detected in the LSB of the 230 GHz data. However, the leakage from the ¹²CO(J = 2–1) line in the USB was significant. Therefore, in this paper we will not make use of the C¹⁸O(J = 2–1) data for further analysis. In this paper, we present maps with natural weighting. The angular resolution and rms noise level per channel are 15 × 10 and 26 mJy beam⁻¹ (400 mK) for the ¹²CO(J = 2–1), 18 × 14 and 30 mJy beam⁻¹ (320 mK) for the ¹³CO(J = 2–1) data, and 35 × 21 and 35 mJy beam⁻¹ (47 mK) for the ¹²CO(J = 3–2) data. We used the AIPS task MOMNT to construct the integrated intensity-weighted maps. The task would reject pixels lower than the threshold intensity, set to be 2.5σ–3σ in the CLEANed cube after smoothing in the velocity and spatial directions. The smoothing kernels are 30 km s⁻¹ in the velocity direction, and a factor of two of the synthesized beam in the spatial direction in the maps we made.

In the following line ratio analysis, we use maps truncated to the same uv coverage as in Section 3.4.1.

As one of the sample galaxies in the HST NICMOS survey of nearby galaxies (PI: D. Calzetti, GO: 11080), NGC 1097 was observed on 2007 November 15 by HST equipped with the NIC3 camera. NIC3 images have a 51'' × 51'' field of view and a plate scale of 02 with an undersampled point-spread function (PSF). In this survey, each observation consists of images taken in two narrowband filters: one centered on the Paα recombination line of hydrogen (1.87 μm) (F187N) and the other on the adjacent narrowband continuum exposure (F190N), which provides a reliable continuum subtraction. Each set of exposures was made with a seven-position small (<1'' step) dither. Exposures of 160 and 192 s per dither position in F187N and F190N, respectively, reach a 1σ detection limit of 2.3 × 10⁻¹⁶ erg s⁻¹ cm⁻² arcsec⁻² in the continuum-subtracted Paα image of NGC 1097.

Using the STSDAS package of IRAF, we removed the NICMOS Pedestal effect, masked bad pixels and cosmic rays, and drizzled the dithered images onto a finer (01 per pixel) grid frame. The resultant drizzled F190N image was then scaled and subtracted from its F187N peer after carefully aligning with the latter using foreground stars. The residual shading effect in the pure Paα image is removed by subtracting the median of each column. This strategy works well for the NGC 1097 data which contain relatively sparse emission features. The PSF of the final Paα image has a 026 FWHM. Further details of the data reduction and image processing are described in G. Liu et al. (2011, in preparation).

The integrated intensity maps of ¹²CO(J = 2–1), ¹²CO(J = 3–2), and ¹³CO(J = 2–1) lines are shown in Figure 1. The maps show a central concentration and a ring-like structure with a radius of 700 pc (∼10''). The gas distribution of the ¹²CO(J = 3–2) map is similar to that of the ¹²CO(J = 2–1) map, where the central concentration has a higher integrated intensity than the ring. The ¹³CO(J = 2–1) map, on the other hand, shows comparable intensity between the ring and the nucleus. Compared with the previous observations (Paper I), the molecular ring and the central concentration have been resolved into individual clumps, especially for the twin-peak structure in the molecular ring, with the higher resolution ¹²CO(J = 2–1) map. We also show the peak brightness temperature map of the ¹²CO(J = 2–1) emission in Figure 1 made by the AIPS task SQASH to extract the maximum intensity along the velocity direction of each pixel. Note that the ring and the nucleus have comparable brightness temperatures.

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The position of the active galactic nucleus (AGN; 6 cm radio continuum core; Hummel et al. 1987), which is assumed to be the dynamical center of the galaxy, seems to be offset by 07 northwest to the central peak of the integrated CO maps. This is also seen in our previous results and we interpreted it as a result of the intensity weighting in the integration (Paper I). To confirm if the dynamical center as derived from the ¹²CO(J = 1–0) emission (Kohno et al. 2003) is consistent with that of ¹²CO(J = 2–1), we will perform additional analysis in Section 3.5.

In Figures 2 and 3, we show the ¹²CO(J = 2–1) and ¹²CO(J = 3–2) channel maps overlaid on the archival HST I-band image (F814W) to show the gas distribution at different velocities. We show these images at a resolution of 40 km s⁻¹, but we use a resolution of 10 km s⁻¹ for the actual analysis. The astrometry of the HST I-band image was corrected by the known positions of the 19 foreground stars in the USNO-A 2.0 catalog. After astrometry correction, we found that the intensity peak of the nucleus of the HST image has a 08 offset northeast of the position of the AGN, which may be due to extinction or inaccurate astrometry. We cannot rule out these factors. However, this offset is within the uncertainty of the CO-synthesized beam. Both ¹²CO transitions show emission with a total velocity extent of ∼550–600 km s⁻¹ at the 2σ intensity level. The western molecular ridge/arms (coincident with the dust lane in the optical image) joins the southwestern part of the ring from −162 km s⁻¹ to 37 km s⁻¹, and the eastern molecular ridge joins the northeast ring from −42 km s⁻¹ to 157 km s⁻¹. We find that the velocity extent of these ridges is ∼200 km s⁻¹ at the 2σ intensity level. In between the molecular ring and the nuclear disk, there is also extended ridge emission connecting the nuclear disk and the ring in both lines. This ridge emission is more significant in the ¹²CO(J = 2–1) map than in the ¹²CO(J = 3–2) map. However, this may be an angular resolution effect. The ridge emission looks more significant in the ¹²CO(J = 2–1) map, but the ridge emission does not have enough flux density (Jy beam⁻¹) to reach the same S/N in the high-resolution image, which means it has an extended structure, i.e., the faint ridge emission in the high-resolution map is more or less resolved.

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In Figure 4, we show the ¹²CO(J = 2–1) integrated intensity map overlaid on the HST I-band (F814W) archival image to compare the optical and radio morphologies. The Paα image (near-IR) is also shown in Figure 4. The optical image shows a pair of spiral arms in the central 1 kpc region. The arms consist of dusty filaments and star clusters. Two dust lanes at the leading side of the major bar connect to the stellar arms. Dusty filaments can also be seen filling the area between the spiral arms and the nucleus. The ¹²CO(J = 2–1) map (15 × 10) shows a central concentration, a molecular ring, and molecular ridges with good general correspondence to the optical image. However, a detailed comparison shows significant differences. Although the starburst ring looks like spiral arms in the optical image, the molecular gas shows a more complete ring, and the molecular ridges are seen to join the ring smoothly. Inside the stellar ring, the major dust lanes are offset from the molecular ring. Assuming that the ¹²CO emission is faithfully tracing the total mass, the prominent dark dust lanes are not significant features while they join the molecular ridges as the edge of the ring. The stellar ring or star formation activity is then correlated with the peaks in the total mass distribution. The central molecular concentration corresponds in general with the stellar nucleus, but is offset to the south of the stellar light. The nuclear molecular distribution is quite asymmetric with lots of protrusions in the lower intensity contours, which may correspond to gas and dust filaments that connect the nuclear concentration to the molecular ring.

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The molecular ring has been resolved into a complex structure of compact sources immersed in a diffuse emission with lower surface brightness as shown in Figure 1. At our resolution (100 pc), the contrast between clumps and diffuse emission is not high. Hence, the clumps are possibly connected to each other via the diffuse emission. With such a physical configuration, it is difficult to isolate uniquely the individual clumps with a clump-finding algorithm (e.g., Williams et al. 1994). Moreover, the detection of clumps is dependent on the available angular resolution. In this paper, we select the peaks of the main structures in the ¹²CO(J = 2–1) integrated intensity map in order to locate the individual clumps within the ring and the molecular ridges. We define the clumps located outside the 10'' radius as dust lane clumps, and those within the 10'' radius as ring clumps.

The typical size of giant molecular clouds (GMCs) is on the order of a few to a few tens of parsecs (Scoville et al. 1987). The size of the clumps in the ring of NGC 1097, as detected with our synthesized beam, is at least ∼200 pc. We are therefore detecting molecular clumps larger than GMCs, most likely a group of GMCs, namely, giant molecular cloud associations (GMAs; Vogel et al. 1988). Here, we still use the term "clumps" to describe the individual peaks at GMA scale. We expect that the clumps are resolved into individual GMCs when higher angular resolution is available. In Table 2, we show the quantities measured within one synthesized beam to study the kinematic properties with high resolution (i.e., one synthesize beam) in the following sections. The observed peak brightness temperatures of individual clumps are in the range of ∼2–8 K.

Table 2. Physical Parameters of the Peaks of the Molecular Clouds

ID	δR.A.	δDecl.	ICO	δVobs	δVint	$N_{\rm H_2}$	$\Sigma _{\rm H_2}$	$M_{\rm H_2}$	Tb
	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)
	('')	('')	(Jy beam−1 km s−1)	(km s−1)	(km s−1)	(1022 cm−2)	(M☉ pc−2)	(106 M☉)	(K)
N1	9.0	1.6	17.2	54 ± 3	42 ± 4	7.9	1270	10.7	4.1
N2	8.6	4.8	27.9	67 ± 3	61 ± 4	12.8	2070	17.3	5.5
N3	3.6	8.4	38.3	69 ± 1	64 ± 1	17.6	2840	23.8	7.5
N4	0.6	8.2	18.5	62 ± 4	56 ± 4	8.5	1370	11.5	5.8
N5	−4.8	7.8	21.3	38 ± 1	31 ± 1	9.8	1580	13.2	6.1
N6	−6.4	5.6	19.6	41 ± 2	37 ± 2	9.0	1460	12.2	4.1
N7	−8.6	−2.6	40.3	61 ± 2	52 ± 3	18.5	2990	25.0	8.0
N8	−5.4	−8.4	28.4	57 ± 2	51 ± 2	13.1	2110	17.6	5.8
N9	−2.8	−9.2	24.6	62 ± 3	57 ± 3	11.3	1820	15.2	4.1
N10	4.8	−7.8	19.7	43 ± 2	37 ± 2	9.1	1460	12.2	6.1
N11	7.8	−6.6	27.7	52 ± 4	49 ± 4	12.7	2050	17.2	5.8
B1	10.2	3.6	27.8	113 ± 10	109 ± 11	12.7	2060	17.2	2.4
B2	−7.4	−6.8	34.8	99 ± 5	96 ± 5	16.0	2580	21.6	4.6
B3	−5.4	−6.6	31.3	94 ± 8	90 ± 8	14.4	2320	19.4	4.1
D1	16.8	−1.4	18.9	100 ± 10	97 ± 10	8.7	1400	11.7	1.8
D2	12.8	1.6	21.1	86 ± 17	82 ± 18	9.7	1560	13.1	2.0
D3	−17.8	2.6	16.2	81 ± 7	78 ± 8	7.5	1200	10.1	1.8
D4	−13.0	−1.0	24.4	84 ± 5	80 ± 6	11.2	1810	15.1	3.5
D5	−10.0	−4.0	38.6	118 ± 4	115 ± 4	17.7	2860	24.0	4.6
Nu	0	0	50.4	52 ± 5	...	16.1	3740	23.2	...
Nu	0	0	...	57 ± 5	...	...	...	...	...
Nu	0	0	...	186 ± 20	...	...	...	...	...

Notes. We define the peaks based on their location and their velocity dispersions. The clumps in the dust lanes (molecular spiral arms) are named D1, ..., D5. The clumps in the ring are further designated by their velocity dispersion being broader or narrower than 30 km s⁻¹, and named, respectively, as B1, ..., B3, and N1, ..., N11. Nu is the ID of the nucleus. (1) R.A. offsets from the phase center. (2) Decl. offsets from the phase center. (3) Integrated CO(J = 2–1) intensity. The uncertainty is 2.2 Jy beam⁻¹ km s⁻¹. (4) Fitted FWHM for the observed line width. The nucleus has a multiple-Gaussian profile, and we list the fitted line widths with three Gaussians. Note that I_CO, $N_{\rm H_{2}}$, and $\Sigma _{\rm H_2}$ of the nucleus are the sum value of the three components. (5) FWHM line width of intrinsic velocity dispersion. (6) H₂ column density with the uncertainty of 1.1 × 10²² cm⁻². (7) $\Sigma _{\rm H_{2}}$ with the uncertainty of 170 M_☉ pc⁻². (8) Mass of molecular H₂ within the synthesized beam (15 × 10). The uncertainty is 1.4 × 10⁶ M_☉. (9) Peak brightness temperature.

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To show the general properties of the GMAs, we also measured the physical properties integrated over their sizes in Table 3. We define the area of the GMA by measuring the number of pixels above the threshold intensity of 5σ in the ¹²CO(J = 2–1) integrated intensity map. We calculated the "equivalent radius" if the measured areas are modeled as spherical clumps. The results are reported in Table 3 together with the resulting $M_{\rm H_{2}}$ integrated over the area. The derived values of $M_{\rm H_2}$ are therefore larger than that in Table 2, which only measured the mass within one synthesized beam at the intensity peak. The method for deriving $M_{\rm H_{2}}$ will be described in Section 3.2.2. Several factors are essential for considering fair comparisons of our GMAs with other galaxies such as beam size, filling factor, etc. A rough comparison shows that our GMAs in the starburst ring have a physical extent of ∼200–300 pc and have a similar order as the GMAs in other galaxies (e.g., Rand & Kulkarni 1990; Tosaki et al. 2007; Muraoka et al. 2009).

Table 3. Physical Parameters of the GMAs

ID	Diameter	$M_{\rm H_{2}}$	Mgas	Mvir	Mvir/Mgas
	(1)	(2)	(3)	(4)	(5)
	('')	(106 M☉)	(106 M☉)	(106 M☉)
N1	2.9	32.4 ± 5.3	44.1 ± 7.2	44.9 ± 10.2	1.0 ± 0.3
N2	3.7	71.2 ± 8.7	96.9 ± 11.8	118.0 ± 16.3	1.2 ± 0.2
N3	3.5	73.6 ± 7.9	100.1 ± 10.8	125.4 ± 5.9	1.3 ± 0.1
N4	2.2	17.7 ± 3.0	24.0 ± 4.1	58.6 ± 9.6	2.4 ± 0.6
N5	2.9	37.5 ± 5.5	51.0 ± 7.4	25.1 ± 2.7	0.5 ± 0.1
N6	2.9	33.8 ± 5.6	46.0 ± 7.6	34.6 ± 5.2	0.8 ± 0.2
N7	3.4	79.1 ± 7.6	107.5 ± 10.3	79.3 ± 9.9	0.7 ± 0.1
N8	2.5	51.5 ± 4.2	70.0 ± 5.7	57.5 ± 5.6	0.8 ± 0.1
N9	3.7	94.5 ± 8.8	128.5 ± 12.0	104.2 ± 14.1	0.8 ± 0.1
N10	2.8	32.7 ± 5.2	44.5 ± 7.1	34.3 ± 4.1	0.8 ± 0.2
N11	4.1	133.1 ± 11.1	181.0 ± 15.0	85.8 ± 15.1	0.5 ± 0.1
B1	3.1	48.6 ± 6.3	66.1 ± 8.6	316.6 ± 71.0	4.8 ± 1.2
B2	3.3	80.0 ± 7.2	108.8 ± 9.9	267.4 ± 29.8	2.5 ± 0.4
B3	3.0	77.0 ± 5.7	104.7 ± 7.7	208.8 ± 43.4	2.0 ± 0.4
D1	3.3	56.5 ± 6.9	76.9 ± 9.4	269.1 ± 62.8	3.5 ± 0.9
D2	3.5	58.6 ± 8.2	79.7 ± 11.1	207.7 ± 102.0	2.6 ± 1.3
D3	2.9	30.1 ± 5.5	41.0 ± 7.4	152.3 ± 34.2	3.7 ± 1.1
D4	5.4	117.7 ± 18.9	160.0 ± 25.7	303.0 ± 47.2	1.9 ± 0.4
D5	4.1	112.6 ± 11.0	153.1 ± 15.0	474.8 ± 36.8	3.1 ± 0.4

Notes. (1) Diameter of the clumps. (2) H₂ mass integrated over the diameter of the clumps. (3) Gas mass integrated over the diameter of the clumps. The M_gas is the $M_{\rm H_2}$ corrected with the He fraction of 1.36 in Section 3.2.2. (4) Virial mass of the clumps. (5) Ratio of the virial mass to M_gas. We have not corrected for the beam convolution effect as the derived diameters are sufficiently large. We think the assumption of Gaussian shapes may be the greater source of error.

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3.2.1. Spectra of the Molecular Clumps

We show in Figure 5 the spectra of the individual clumps selected above at their peak positions (i.e., within a synthesized beam). Most of the clumps show single Gaussian profiles. The Gaussian FWHM line widths of the clumps are determined by Gaussian fitting, and are listed in Table 2.

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The line widths we observed are quadratic sums of the intrinsic line widths and the galactic motion across the synthesized beam, and the relation can be expressed as

$$\begin{equation} \sigma _{\rm obs}^{2} = \sigma _{\rm rot}^{2} + \sigma _{\rm int}^{2}, \end{equation} \tag{ 1 }$$

where σ_obs is the observed FWHM line width, σ_rot is the line width due to the galactic rotation and the radial motions within the beam, and σ_int is the intrinsic line width. The systematic galactic motion would be negligible if the spatial resolution were small enough. We subtract σ_rot as derived from a circular rotation model by fitting the rotation curve in Section 3.5. The derived intrinsic line width of the clumps is reported in Table 2. In general, there is a ∼10%–20% difference between the observed and intrinsic line widths, which suggests that the galactic circular motions do not dominate the line broadening at this high resolution. However, the large "intrinsic" velocity dispersion thus derived by subtracting circular motions could still be dominated by non-circular motions, especially in the twin-peak region and in the dust lanes. We will mention this effect in the discussion section.

In Table 2, we define the clumps based on their location and their velocity dispersions. The clumps in the ring are further designated by their velocity dispersion being broader or narrower than 30 km s⁻¹, and are named, respectively, B1, ..., B3, and N1, ..., N11. The clumps located in the dust lanes are named D1, ..., D5.

3.2.2. Mass of the Clumps

Radio interferometers have a discrete sampling of the uv coverage limited by both the shortest and longest baselines. Here, we estimate the effects due to the missing information. In Figure 6, we convolved our newly combined SMA data to match the beam size of the James Clerk Maxwell telescope (JCMT) data (21'') and overlaid the spectra to compare the flux. The integrated ¹²CO(J = 2–1) fluxes of the JCMT and SMA data are ∼120 K km s⁻¹ and ∼71 K km s⁻¹, respectively. The integrated ¹³CO(J = 2–1) fluxes of the JCMT and SMA data are ∼15 K km s⁻¹ and ∼7 K km s⁻¹, respectively. Therefore, our SMA data recover ∼60% and ∼47% of the ¹²CO(J = 2–1) and ¹³CO(J = 2–1) fluxes measured by the JCMT, respectively. If the missing flux is attributed to the extended emission, then the derived fluxes for the compact clumps will remain reliable. However, the spectra of the SMA data seem to have similar line profiles as that of the JCMT. Part of the inconsistency could then possibly be due to the uncertainty of the flux calibration, which is 15% and 20% for the JCMT and SMA, respectively. We note therefore that the following quantities measured from the flux will have uncertainties of at least 20% for the flux calibration.

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There are several ways to calculate the molecular gas mass. The conventional H₂/I_CO (X_CO) conversion factor is one of the methods to derive the gas mass assuming that the molecular clouds are virialized. First, we use the ¹²CO emission to calculate the molecular H₂ column density, $N_{\rm H_{2}}$, as

$$\begin{equation} N_{\rm H_{2}} = X_{\rm CO} \int {{T_{\rm b}} dv} = X_{\rm CO} I_{\rm CO}, \end{equation} \tag{ 2 }$$

where $N_{\rm H_{2}}$ for each clump is calculated by adopting the X_CO conversion factor of 3 × 10²⁰ cm⁻² (K km s⁻¹)⁻¹ (Solomon et al. 1987), T_b is the brightness temperature, and dv is the line width. The values are listed in Table 2. Note that the X_CO has a wide range of (0.5–4) × 10²⁰ cm⁻² (K km s⁻¹)⁻¹ (Young & Scoville 1991; Strong & Mattox 1996; Dame et al. 2001; Draine et al. 2007; Meier & Turner 2001; Meier et al. 2008). We adopted 3 × 10²⁰ cm⁻² (K km s⁻¹)⁻¹ to be consistent with Paper I. The surface molecular gas mass density $\Sigma _{\rm H_2}$ and the molecular gas mass $M_{\rm H_2}$ are thus calculated as

$$\begin{equation} \Sigma _{\rm H_{2}} = N_{\rm H_{2}} m_{\rm H_{2}}, \end{equation} \tag{ 3 }$$

$$\begin{equation} {M}_{\rm H_{2}} = \Sigma _{\rm H_{2}} {\rm d}\Omega, \end{equation} \tag{ 4 }$$

where $m_{\rm H_{2}}$ and dΩ are the mass of a hydrogen molecule and the solid angle of the integrated area, respectively. Since this conversion factor is used for the ¹²CO(J = 1–0) emission, we assume the simplest case where the ratio of ¹²CO(J = 2–1)/¹²CO(J = 1–0) is unity. The H₂ mass of the clumps measured within one synthesized beam is listed in Table 2. We would obtain the molecular gas mass (M_gas) by multiplying the H₂ mass by the mean atomic weight of 1.36 of the He correction.

The ¹²CO(J = 2–1) is usually optically thick and only traces the surface properties of the clouds. The optically thinner ¹³CO(J = 2–1) is a better estimator of the total column density. We therefore calculate the average $M_{\rm H_{2}}$ in the nucleus and the ring to compare the $M_{\rm H_{2}}$ derived from both ¹²CO(J = 2–1) and ¹³CO(J = 2–1) lines. In the case of ¹²CO(J = 2–1), the average $M_{\rm H_{2}}$ of the ring is about 1.7 × 10⁷ M_☉ within one synthesized beam, and the corresponding value for the nucleus is 3.4 × 10⁷ M_☉. However, in Paper I we derived an intensity ratio of ¹²CO(J = 2–1)/¹²CO(J = 1–0) ∼2 for the nucleus in the lower resolution map. Therefore, the $M_{\rm H_{2}}$ of the nucleus is possibly smaller than that derived above. The conventional Galactic X_CO of 3 × 10²⁰ cm⁻² (K km s⁻¹)⁻¹ is often suggested to be overestimated in the Galactic center and the starburst environment by a factor of 2–5 (e.g., Maloney & Black 1988; Meier & Turner 2001). Therefore, our estimated $M_{\rm H_{2}}$ in the ring and the nucleus might be smaller by at least a factor of two.

Assuming the ¹³CO(J = 2–1) emission is optically thin, and the ¹³CO/H₂ abundance of 1 × 10⁻⁶ (Solomon et al. 1979), we calculate the $M_{\rm H_{2}}$ with an excitation temperature (T_ex) of 20 K and 50 K in the LTE condition. The $M_{\rm H_{2}}$ of the nucleus averaged over one synthesized beam are (7.0 ± 2.5) × 10⁶ M_☉ and (1.3 ± 0.4) × 10⁷ M_☉ for 20 K and 50 K, respectively. The $M_{\rm H_{2}}$ of the ring averaged over one synthesized beam are (5.8 ± 2.5) × 10⁶ M_☉ and (1.1 ± 0.4) × 10⁷ M_☉ for 20 K and 50 K, respectively. With an assumption of constant ¹³CO/H₂ abundance, if the T_ex is ⩽20 K, then the conversion factor we adopted for the ¹²CO line is overestimated by a factor of ∼3–5. The overestimation is smaller (by a factor of 2–3) if the gas is as warm as 50 K. We measured the ¹²CO(J = 2–1) flux higher than 3σ to derive the total H₂ mass of the nucleus and the ring. The total flux of the nucleus and the ring are 490.2 ± 59.3 Jy km s⁻¹ and 2902.2 ± 476.8 Jy km s⁻¹, respectively. If we adopt intensity ratios of ¹²CO(J = 2–1) and ¹²CO(J = 1–0) of 1.9 ± 0.2 and 1.3 ± 0.2 for the nucleus and the ring (Paper I), the $M_{\rm H_2}$ of the nucleus and the ring are (1.6 ± 0.3) × 10⁸ M_☉ and (1.4 ± 0.3) × 10⁹ M_☉, respectively. Thus, the nucleus and the ring account for 10% and 90% of the $M_{\rm H_2}$ within the 2 kpc circumnuclear region, respectively.

We calculate the virial mass of individual clumps by

$$\begin{equation} M_{\rm vir} = \frac{2r\sigma _{\rm rms}^{2}}{G}, \end{equation} \tag{ 5 }$$

where r is the radius of the clump derived in Section 3.2 and σ_rms is the three-dimensional intrinsic root-mean-square velocity dispersion, which is equal to $\sqrt{(3/8ln2)}$σ_int. Since we observe σ_int in one dimension, we need to multiply the observed σ²_int by 3. The radius is adopted with the size of the clumps assuming that σ_int is isotropic. The results are shown in Table 3. Note that the virial mass is for GMAs not GMCs, and we assume that the GMAs are bound structures. The ratio of the virial mass M_vir to M_gas is shown in Table 3. We found that the narrow line clumps have M_vir/M_gas that are more or less about unity, but are larger than unity for the broad line and dust lane clumps.

We plot the general properties of the molecular gas mass of individual molecular clumps in Figure 7. In Figure 7(a), the histogram of the gas mass integrated over their size seems to be a power law with a sharp drop at the low-mass end. This is due to the sensitivity limit, since the corresponding 3σ mass limit is ∼27 × 10⁶ M_☉ for a clump with a diameter of 33 (the average of the clumps in Table 3). In Figure 7(b), the FWHM intrinsic line widths have a weak correlation with gas mass in the ring. In Figure 8(a), we show the azimuthal variation of $\Sigma _{\rm H_{2}}$ calculated at the emission peaks of the clumps integrated over one synthesized beam. If we assume that NGC 1097 has a trailing spiral, then the direction of rotation is clockwise from east (0°) to north (90°). The $\Sigma _{\rm H_{2}}$ in the orbit-crowding region is roughly from 0° to 45°, and from 180° to 225°. Note that the dust lane clumps typically have $\Sigma _{\rm H_{2}}$ similar to the narrow line ring clumps, which are lower than the broad line ring clumps in the orbit-crowding region. The average $\Sigma _{\rm H_{2}}$ of the narrow line ring and dust lane clumps is ∼1800 M_☉ pc⁻², and ∼2300 M_☉ pc⁻² for the broad line ring clumps. In Figure 8(b), the velocity dispersion of clumps located in the orbit-crowding region and the dust lane is larger than that of the narrow line ring clumps. The average velocity dispersion of the narrow line clumps is ∼50 km s⁻¹, and ∼90 km s⁻¹ for the broad line ring/dust lane clumps. Given the similar line brightnesses (Table 2) of the peaks, the increased $\Sigma _{\rm H_{2}}$ values are probably the results of increased line widths as indicated in Equation (2). On the other hand, since the $N_{\rm H_2}$ is proportional to the number density of gas and the line-of-sight path (i.e., optical depth), the higher $\Sigma _{\rm H_2}$ of the broad line clumps may be due to either a larger number density, or a larger line-of-sight path. We will discuss these effects in Section 4.

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To check how the star formation properties are associated with the molecular clumps in the ring, we compare the 6 cm radio continuum, V-band, and Paα images with the molecular gas images. The 6 cm radio continuum sources are selected from the intensity peaks (Hummel et al. 1987) at an angular resolution of 25. The V-band clusters (<13 mag) are selected from the HST F555W image (Barth et al. 1995). These V-band-selected clusters have a typical size of 2 pc (003), and are suspected to be super star clusters by Barth et al. (1995). Paα clusters are identified in our HST F187N image described in Section 2. The Paα clusters are identified by SExtractor with main parameters of a detection threshold of 15σ (DETECT_THRESH) and a minimum number of pixels above a threshold of 1 (DETECT_MINAREA). The number of deblending sub-thresholds is 50 for DEBLEND_NTHRESH and 0.0005 for DEBLEND_MINCONT. The parameters were chosen by a wide range of verifying and visual inspections.

We used 6 cm radio continuum and V-band-selected clusters for the phenomenological comparison with the molecular clumps. To get a high resolution and uniform sample of the star formation rate (SFR), we use the Paα clusters in the following analysis.

In the ¹²CO(J = 2–1) integrated intensity map (Figure 1), the star clusters and radio continuum sources are located in the vicinity of molecular clumps within the synthesized beam. The distribution of the massive star clusters is uniform in the ring instead of showing clustering in certain regions. The star clusters do not coincide with most of the CO peaks. The spatial correlation seems to be better in the peak brightness temperature map in Figure 1. Furthermore, there are no detected star clusters and radio continuum sources in the dust lane clumps, namely, clumps D1, D2, D3, and D4.

We corrected the extinction of the Paα emission by the intensity ratio of Hα (CTIO 1.5 m archived image) and Paα. The PSFs of Hα and Paα are ∼10 and ∼03, respectively, and an additional convolving Gaussian kernel has been applied to both Hα and Paα images to match the CO beam size. The observed intensity ratio of Hα and Paα, (I_Hα/I_Paα)_o, and the predicted intensity ratio (I_Hα/I_Paα)_i are multiplied by the extinction as follows:

$$\begin{equation} \left(\frac{I_{\rm H\alpha }}{I_{\rm Pa\alpha }}\right)_{\rm o} = \left(\frac{I_{\rm H\alpha }}{I_{\rm Pa\alpha }}\right)_{\rm i} \times 10^{-0.4{E(B-V)}(\kappa _{\rm H{\alpha }}-\kappa _{\rm Pa\alpha })}, \end{equation} \tag{ 6 }$$

where E(B − V) is the color excess, κ_Hα and κ_Paα are the extinction coefficients at the wavelengths of Hα and Paα, respectively.

The predicted intensity ratio of 8.6 is derived in the case B recombination with a temperature and an electron density of 10,000 K and 10⁴ cm⁻³, respectively (Osterbrock 1989). The extinction coefficients of Hα and Paα are adopted from Cardelli's extinction curve (Cardelli et al. 1989), with κ_Hα = 2.535 and κ_Paα = 0.455, respectively. We derive the color excess E(B − V) using Equation (6), and derive the extinction of Paα using the following equation,

$$\begin{equation} A_{\rm \lambda } = \kappa _{\rm \lambda } E(B-V), \end{equation} \tag{ 7 }$$

where A_λ is the extinction at wavelength λ. We show the derived values for A_Paα and A_V in Table 4, where κ_V is 3.1 in the V band. The average extinction of the clumps at a wavelength of 1.88 μm (redshifted Paα wavelength) is about 0.6 mag, which is quite transparent compared with the extinction at the Hα line of ∼4 mag. We calculate SFRs using the Paα luminosity based on the equation in Calzetti (2007):

$$\begin{equation} {\rm SFR} (M_{\odot } \, \rm yr^{-1}) = 4.2\times 10^{-41} {L}_{\rm Pa\alpha } (\rm erg \, s^{-1}). \end{equation} \tag{ 8 }$$

The SFR surface density (Σ_SFR) is thus calculated within the size of the CO-synthesized beam (15 × 10). Note that SFR cannot be determined in the dust lane clumps since we do not detect significant star clusters in both Paα and Hα images. It seems unlikely that the low Σ_SFR in the broad line ring and dust lane clumps is due to extinction, since the extinction of the broad line ring clumps is similar to that of the narrow line ring clumps, and the $\Sigma _{\rm H_{2}}$ of the dust lane clumps is similar to that of the narrow line ring clumps. To compare the star formation activity with the properties of the molecular gas, we measured Σ_SFR at the position of each clump (Table 4). We show the correlation of Σ_SFR and $\rm \Sigma _{H_{2}}$ of the molecular clumps in Figure 9(a). This plot shows very little correlation. In Figure 10, we overlay our data on the plot of Σ_SFR and $\rm \Sigma _{H_{2}}$ used in Kennicutt (1998) to compare the small- and large-scale star formation. The average number follows the Kennicutt–Schmidt correlation closely. However, we have either lower Σ_SFR or higher $\rm \Sigma _{H_{2}}$ than the global values in Kennicutt (1998). This might be because our spatial resolution is smaller than that for their data. Recent investigations have shown that the power scaling relationship of the spatially resolved Schmidt–Kennicutt law remains valid in the sub kiloparsec scale (Bigiel et al. 2008), to ∼200 pc in M51 (Liu et al. 2011) and M33 (Verley et al. 2010; Bigiel et al. 2010), but becomes invalid at the scale of GMC/GMAs (Onodera et al. 2010) because the scaling is overcome by large scatter. The absence of a correlation in our 100 pc study is thus not surprising because even if the Schmidt–Kennicutt law is still valid, the scatter is expected to be as large as ∼0.7 dex (Liu et al. 2011), larger than the dynamical range of the gas density in Figure 9(a). Another possibility for explaining the inconsistency is the uncertain conversion factor mentioned in Section 3.2.2. The conversion factor is likely to be overestimated in the galactic center and starburst region. Moreover, given that the Schmidt–Kennicutt law was derived from the global galaxies that might be dominated by disk GMCs, our nuclear ring might not follow the same relation for its particular physical conditions.

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As for the distribution of Σ_SFR in the ring, we find that Σ_SFR is low in the broad line clumps in Figure 9(b), but do not have an obvious systematic azimuthal variation such as $\rm \Sigma _{H_{2}}$ or intrinsic line width in Figure 8. In general, Σ_SFR is higher in the northern ring than in the southern ring. This distribution, as an average quantity, shows no strong dependence on location within the ring.

3.4.1. Intensity Ratio of Multi-J CO Lines

Figure 11(a) is the intensity-weighted isovelocity map of ¹²CO(J = 2–1). The gas motion in the ring appears to be dominated by circular motion, while it shows clear non-circular motions in the ¹²CO(J = 1–0) map as indicated by the S-shape nearly parallel to the dust lanes. As we discussed in Paper I, the non-significant non-circular motion in the ¹²CO(J = 2–1) maps is perhaps because the dust lanes are not as strongly detected in the ¹²CO(J = 2–1) line, along with the fact that they are closer to the edge of our primary beam, or the non-circular motion is not prominent at high spatial resolution. The circumnuclear gas is in general in solid body rotation. The velocity gradient of the blueshifted part is slightly steeper than the redshifted part. We also show the intensity-weighted velocity dispersion map in Figure 11(b). As we mentioned above, the velocity dispersion is larger in the twin-peak region, and lower in the region away from the twin-peak region.

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The dynamical center of NGC 1097 was derived by Kohno et al. (2003) in their low-resolution ¹²CO(J = 1–0) map. With our high-resolution ¹²CO(J = 2–1) map, we expect to determine the dynamical center more accurately. We use the AIPS task GAL to determine the dynamical center. In the task GAL, the ¹²CO(J = 2–1) intensity-weighted velocity map (Figure 11) is used to fit a rotation curve. The deduced kinematic parameters are summarized in Table 6. We use an exponential curve to fit the area within 7'' in radius. The observed rotation curve and the fitted model curve are shown in Figure 12. From the fitted parameters, we find that the offset (∼03) of the dynamical center with respect to the position of the AGN is still within a fraction of the synthesized beam size. The derived V_sys has a difference of ∼5 km s⁻¹ between ¹²CO(J = 1–0) and ¹²CO(J = 2–1) data, which is less than the velocity resolution of the data. Upon examining the channel maps of the ¹²CO(J = 2–1) data, we find that the peak of the nuclear emission is almost coincident with the position of the AGN with an offset of 03. We therefore conclude that the position offset in the integrated intensity map, as mentioned in Section 3.1, is due to the asymmetric intensity distribution.

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Table 6. Dynamical Parameters Fitted by GAL

Parameters	Value
R.A.	02h46m18$\mbox{$.\!\!^{\mathrm s}$}$95
Decl.	−30°16'2913
Position angle	1330 ± 01
Inclination	417 ± 06
Systemetic velocity (km s−1; Vsys)	1249.0 ± 0.5
Vmax (km s−1)	387.6 ± 4.3
Rmax	95 ± 01

Note. The 6 cm peak of the nucleus is R.A. = 02^h46^m18$\mbox{$.\!\!^{\mathrm s}$}$96, decl. = −30°16'28897.

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4.1.1. Twin-peak Structure

In the low-resolution CO maps (Paper I, Kohno et al. 2003), NGC 1097 shows bright CO twin-peak structure arising at the intersection of the starburst ring and the dust lanes. The ⩾300 pc resolution CO data show that the barred galaxies usually have a large amount of central concentration of molecular gas (Sakamoto et al. 1999). Kenney et al. (1992) found that in several barred galaxies that host circumnuclear rings (M101, NGC 3351, NGC 6951), the central concentrations of molecular gas were resolved into twin-peak structures when a resolution of ∼200 pc was attained. In these cases, a pair of CO intensity concentrations is found in the circumnuclear ring at the intersection of the ring and the dust lane. Their orientation is almost perpendicular to the major stellar bar. The twin-peak structure can be attributed to the orbit crowding of inflowing gas stream lines. The gas flow changes from its original orbit (the so-called x₁ orbit) when it encounters the shocks, which results in a large deflection angle, and migrates to a new orbit (the so-called x₂ orbit). The gas then accumulates in the family of the x₂ orbits in the shape of a ring or nuclear spirals (Athanassoula 1992a; Piner et al. 1995). Intense massive star formation would follow in the ring/nuclear spiral once the gas becomes dense enough to collapse (Elmegreen 1994).

In our 100 pc resolution CO map, the starburst molecular ring is resolved into individual GMAs. In the orbit-crowding region, we resolve the twin peak into broad line clumps associated with the curved dust lanes. The narrow line clumps are located away from the twin peak and are associated with star formation. This kind of "spectroscopic component" was also shown in several twin-peak galaxies such as NGC 1365 (Sakamoto et al. 2007), NGC 4151 (Dumas et al. 2010), NGC 6946 (Schinnerer et al. 2007), and NGC 6951 (Kohno et al. 1999) at the intersection of dust lanes and circumnuclear ring. However, most of the spectra at these intersections show blended narrow/broad line components, which is perhaps due to insufficient angular resolution. For the first time our observations spatially resolved these two components toward the twin-peak region of NGC 1097.

It is interesting to note that the circumnuclear ring is nearly circular at ∼42° inclination, which indicates its intrinsic elliptical shape in the galactic plane. The schematic sketch is shown in Figure 13. The loci of dust lanes are invoked to trace the galactic shock wave, and their shapes are dependent on the parameters of the barred potential. In the case of NGC 1097, the observed dust lanes resemble the theoretical studies (Athanassoula 1992b), with a pair of straight lanes that slightly curve inward in the inner ends. These findings are consistent with the predicted morphology from bar-driven nuclear inflow. The physical properties of these clumps are discussed in the following subsection.

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4.1.2. The Nature of the Molecular Clumps in the Ring

In Figure 1 and Table 2, the peak brightness temperatures of individual clumps are from 2 to 8 K. These values are lower than the typical temperatures of molecular gas as expected in the environment of a starburst (∼100 K), and as estimated by our LVG results in Paper I and this work. This lower brightness temperature may be due to the small beam filling factor:

$$\begin{equation} f_{\rm b} = \left(\frac{\theta ^{2}_{\rm s}}{\theta ^{2}_{\rm s}+\theta ^{2}_{\rm b}}\right) \sim \left(\frac{\theta ^{2}_{\rm s}}{\theta ^{2}_{\rm b}}\right), \end{equation} \tag{ 9 }$$

$$\begin{equation} f_{\rm b} = \left(\frac{T_{\rm b}}{T_{\rm c}}\right), \end{equation} \tag{ 10 }$$

where f_b is the beam filling factor and θ_s and θ_b are the source size and the beam size, respectively. We assume θ_s ≪ θ_b. T_b is the brightness temperature and T_c is the actual temperature of the clouds. We assume here that the source size is compact and much smaller than the beam. If we assume the LVG predicted T_b ∼ 100 K (Section 3.4.1) for narrow line ring clumps, and an average observed T_b ∼ 5 K, then f_b ∼ 0.05. This suggests that θ_s is ∼20 pc or an association of much smaller clumps for the narrow line ring clumps. In the case of the broad line ring clumps, with an average observed T_b of ∼5 K and LVG predicted T_b of ∼20 K, then f_b is ∼0.25 and θ_s ∼ 44 pc. The size of the broad line ring clumps is a factor of two higher than the narrow line ring clumps. Given the estimated low volume number density of the broad line clumps in Section 3.4.1, the inconsistency between high $\Sigma _{\rm H_2}$ and low number density is possibly due to the large line-of-sight path, or the larger scale height of the broad line clumps. If the assumption of a spherical shape of the clumps holds, the larger size estimated above seems to be consistent with this scenario. However, considering the higher opacity of the broad line clumps as suggested by the high $\Sigma _{\rm H_2}$, it is likely that we are tracing the diffuse and cold gas at the surface of the GMA rather than the dense and warm gas. On the other hand, the narrow line clumps show the opposite trend of tracing the relatively warm and dense gas.

4.1.3. Azimuthal Variation of the Line Widths and $\Sigma _{\rm H_{2}}$

How do the GMAs form in the starburst ring? In the ring of NGC 1097, we consider gravitational collapse due to the Toomre instability (Toomre 1964). We estimate the Toomre Q parameter (Σ_crit/Σ_gas) to see if the rotation of the ring is able to stabilize against the fragmentation into clumps. Here the Toomre critical density can be expressed as

$$\begin{equation} \Sigma _{\rm crit} = \alpha \left(\frac{\kappa \sigma _{\rm int}}{3.36{\rm G}}\right), \end{equation} \tag{ 11 }$$

$$\begin{equation} \kappa = 1.414 \left(\frac{V}{R}\right)\left(1+\frac{R}{V}\frac{\it dV}{\it dR}\right)^{0.5}, \end{equation} \tag{ 12 }$$

where the constant α is unity, G is the gravitational constant, κ is the epicycle frequency, V is the rotational velocity, and R is the radius of the ring. If $\Sigma _{\rm H_{2}}$ exceeds Σ_crit, then the gas will be gravitationally unstable and collapse. We approximate the velocity gradient dV/dR as close to zero in Figure 12, because of the flat rotation curve of the galaxy at the position of the starburst ring. Therefore κ ∼ 1.414V/R. V is ∼338 km s⁻¹ in the galactic plane assuming an inclination of ∼42°. We find that the ratio of Σ_crit/$\Sigma _{\rm H_{2}}$ is less than unity in the ring (∼0.6). Since Σ_gas consists of H₂, H i, and metals, this ratio of 0.6 is an upper limit. However, H i is often absent in centers of galaxies, and NGC 1097 also shows an H i hole in the central 1' (Higdon & Wallin 2003). Hence, the H i gas does not have an important contribution in the nuclear region. A ratio less than unity suggests that the ring is unstable and will fragment into clumps.

As for whether or not the GMA itself is gravitationally bound, in Section 3.2.2 we found that M_vir/M_gas is around unity in the narrow line clumps and larger than unity by a factor of more than two in the broad line clumps. This seems to suggest that the broad line clumps are not virialized, probably because of the larger turbulence. However, several factors will reduce the ratio. First, since we do not subtract the non-circular motion in the broad line clumps, the observed M_vir/M_gas should be an upper limit. We can estimate how large the non-circular motion is if we assume that the M_vir/M_gas of the broad line clumps is unity. The magnitude of the non-circular motion is from 50 to 100 km s⁻¹ for the broad line clumps under this assumption. Athanassoula (1992a) showed that there is a correlation between the bar axial ratio and the velocity gradient (jump) across the shock wave. NGC 1097 has a bar axial ratio of ∼2.6 (Menéndez-Delmestre et al. 2007), which indicates the maximum velocity jump is ∼70 km s⁻¹ (Figure 12; Athanassoula 1992b). However, this number is supposed to be an upper limit since it was measured at the strongest strength of the shock, where the strength of the shock is a function of position relative to the nucleus. The shock strength seems to be weaker at the intersection of the circumnuclear ring than for the outer straight dust lanes, as suggested in the model. Hence we expect the velocity gradient caused by the shock front is smaller than ∼70 km s⁻¹, based on this correlation. In this case, the broad line clumps still have 20% larger velocity dispersion than narrow line clumps. Second, the size of the clumps is also be a possible factor for reducing M_vir/M_gas to unity. In Section 4.1.2, we estimate that the filling factor of the broad line clumps is ∼0.25, and therefore the intrinsic radius will be smaller by a factor of 0.25^1/2 ∼ 0.5. Third, in Section 3.2.2, we point out that the molecular gas mass derived from ¹²CO(J = 2–1) might be underestimated by at least a factor of ∼2. These factors can also lower M_vir/M_gas to roughly unity in most of the broad line clumps and hence the GMA could also be gravitationally bound in the broad line clumps.

4.3.1. Extinction

4.3.2. Molecular Gas and Star Formation

In NGC 1097, the mechanism of the intensive star formation in the ring are still uncertain. It could be stochastically induced by the gravitational collapse in the ring (Elmegreen 1994). The other possible mechanism is that the stars form downstream of the dust lane at the conjunction of the ring, and the star clusters continue to orbit along the ring (e.g., Böker et al. 2008). The major difference is that the latter scenario will have an age gradient for the star clusters along the ring while it is randomized in the previous case. Several papers have discussed these mechanisms in the galaxies that have star-forming rings and there is no clear answer so far (e.g., Mazzuca et al. 2008; Böker et al. 2008; Buta et al. 2000). Sandstrom et al. (2010) tested the above pictures by examining if there is an azimuthal gradient of dust temperature in the ring of NGC 1097, assuming that the younger population of massive star clusters will heat the dust to higher temperatures than older clusters. There seems to be no gradient, though it is difficult to conclude since a few rounds of galactic rotation might smooth out the age gradient. We do not aim to solve the above question in this paper since the most direct way is to measure the age of the star clusters, and this needs detailed modeling. It is interesting to compare the properties of the molecular clumps with star clusters since the molecular clouds are the parent site of star formation. The one relevant result here is that the line widths are narrower further away from where the molecular arms join the ring. This could be related to the dissipation of turbulence which may allow cloud collapse to proceed.

In Figure 9(b), we found that Σ_SFR, compared with velocity dispersion (Figure 8), has no significant azimuthal correlation. Furthermore, R₃₂ shows a trend similar to Σ_SFR as a function of azimuthal direction (Figure 9(d)), where R₃₂ and Σ_SFR have a correlation in Figure 9(c). This suggests that Σ_SFR and R₃₂ are physically related, but might not be associated with the large-scale dynamics in this galaxy although Σ_SFR seems to be suppressed in the broad line clumps. Nevertheless, in Figures 9(b) and (d) we consider that the standard deviations of the measured Σ_SFR do not significantly differ among global variations. The standard deviations are 0.7 M_☉ yr⁻¹ kpc⁻² and 0.4 M_☉ yr⁻¹ kpc⁻² northeast and southwest of the ring, respectively. The mean values of the Σ_SFR are 3.3 M_☉ yr⁻¹ kpc⁻² and 1.4 M_☉ yr⁻¹ kpc⁻² of northeast and southwest of the ring, respectively. With the limited points, our results suggest that the star formation activities are randomly generated on the local scale instead of a systematic distribution.

In Figure 14, we show how the R₃₂ ratio varies for different densities and kinetic temperatures. It shows that when R₃₂ varies from 0.3 to 0.9, the required number density of the molecular gas changes from 10² cm⁻³ to 5 × 10³ cm⁻³. R₃₂ seems to be dependent on the density more than the temperature when it is below unity. Hence the variation of R₃₂ indicates different density among clumps. It is interesting to note that there is a correlation between R₃₂ and SFR, and some clumps (N1, N2, N7) which have higher SFR values are spatially close to the HCN(J = 1–0) peaks (Kohno et al. 2003). Higher values of R₃₂ will select denser gas associated with higher SFR, as has been shown in the large-scale observations of HCN and FIR correlation (Gao & Solomon 2004). In the smaller GMC scale, Lada (1992) also showed that the efficiency of star formation is higher in the dense core rather than in the diffuse gas. The ratio map of R₃₂ can be useful for determining the location of star formation.

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