Morphological and Kinematical Analysis of the Double-Barred Galaxy NGC 3504 Using ALMA CO (2–1) Data

MNRAS 000,1–18 (2021) Preprint 16 April 2021 Compiled using MNRAS LATEX style ﬁle v3.0

Morphological and Kinematical Analysis of the Double-barred Galaxy NGC 3504 Using ALMA CO (2–1) Data

Yu-Ting Wu,1? Alfonso Trejo,2 Daniel Espada3,4 and Yusuke Miyamoto1 1National Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588, Japan 2Institute of Astronomy and Astrophysics, Academia Sinica, Taipei 10617, Taiwan 3SKA Organisation, Lower Withington, Macclesﬁeld, Cheshire SK11 9DL, UK 4Departamento de F´ısica Te´orica y del Cosmos, Campus de Fuentenueva, Universidad de Granada, E18071, Granada, Spain

Accepted 2021 April 13. Received 2021 April 12; in original form 2020 July 5

ABSTRACT We present results obtained from ALMA CO (2–1) data of the double-barred galaxy NGC 3504. With three times higher angular resolution (∼ 000. 8) than previous studies, our observations reveal an inner molecular gas bar, a nuclear ring, and four inner spiral arm-like structures in the central 1 kpc region. Furthermore, the CO emission is clearly aligned with the two dust lanes in the outer bar region, with differences in shape and intensity between them. The total molecular gas mass in the observed 00 00 9 region (50 ×57 ) is estimated to be ∼ 3.1×10 M , which is 17 per cent of the stellar mass. We used the Kinemetry package to fit the velocity field and found that circular motion strongly dominates at R = 0.3−0.8 kpc, but radial motion becomes important at R < 0.3 kpc and R = 1.0−2.5 kpc, which is expected due to the presence of the inner and outer bars. Finally, assuming that the gas moves along the dust lanes in the bar rotating frame, we derived the pattern speed of the outer bar to be 18±5 km s−1 kpc−1, the average streaming velocities on each of the two dust lanes to be 165 and 221 km s−1, −1 and the total mass inflow rate along the dust lanes to be 12 M yr . Our results give a new example of an inner gas bar within a gas-rich double-barred galaxy and suggest that the formation of double-barred galaxies could be associated with the existence of such gas structures. Key words: Galaxies: kinematics and dynamics – Galaxies: structure – Methods: observational – Radio lines: galaxies

1 INTRODUCTION et al. 2014). In such case, knowing the mass inflow rate of gas in barred galaxies is essential (Regan et al. 1997; Sormani Both external processes, such as galaxy collisions and merg- & Barnes 2019). ers, and internal processes in galaxies, such as the actions of bars, play important roles in the evolution of galaxies. For On large scales, bars, spiral arms or dynamical pertur- example, galaxy collisions can not only change the shape of bations, such as galaxy collisions, mergers and other minor arXiv:2104.07227v1 [astro-ph.GA] 15 Apr 2021 galaxies but also enhance their star formation rate (Toomre accretions, can effectively drive gas inflow down to kpc scales & Toomre 1972; Teyssier et al. 2010). On the other hand, (Heckman et al. 1986; Combes 1988; Hernquist 1989; Barnes regarding the internal processes, many studies have been & Hernquist 1992). However, gas may settle into a nuclear conducted to understand how non-axisymmetric potentials, ring, which is usually associated with resonance phenom- such as bars or spirals, redistribute the angular momentum ena, and cannot easily reach the nucleus of a galaxy (Patsis within galaxies and then change the distribution of stars and & Athanassoula 2000). To further reduce the angular mo- gas. For example, the gas inflow driven by bars may lead mentum and allow the inflow of gas to the galactic center to gas concentration near the galactic center, such as the (∼ 100 pc scale), other mechanisms are needed, such as the Central Molecular Zone in our Galaxy, and change the star existence of an inner bar or dissipation in a warped nuclear formation rate (Schwarz 1981; Athanassoula 1992; Kruijssen disk (Shlosman et al. 1989; Pringle 1996; Schinnerer et al. 2000; Shu et al. 1990). As mentioned in Shlosman et al.(1989), the importance ? E-mail: [email protected] of the inner bar in double-barred galaxies is that the inner

© 2021 The Authors 2 Wu et al. bar inside the inner Lindblad resonance of the outer bar the outer bar to be 77 km s−1 kpc−1 and the two inner Lind- could possibly drive gas inflow to fuel active galactic nu- blad resonances at r ∼ 200 and 500–1200 (0.2 and 0.5−1.2 kpc, clei. However, the formation mechanism of the inner bar is respectively). On the other hand, Kuno et al.(2000) derived still unclear. Different numerical simulations have been con- an upper limit of the pattern speed of 41 km s−1 kpc−1 by ducted to understand the formation of double-barred galax- assuming that the gas on the dust lanes in the outer bar ies and can be classified into two major categories: either mainly flows along the dust lanes. This assumption implies with or without gas. In the first case, the well-known study that the velocity component perpendicular to the outer bar by Friedli & Martinet(1993) showed that a gaseous inner is much smaller than the component parallel to the bar and bar can be formed either simultaneously or a few 100 Myr thus allows to obtain an upper limit of the pattern speed of after the outer bar, and then decouple from the outer bar the outer bar. If this upper limit is close to the pattern speed with a different pattern speed. Note that the required gas of the outer bar, Kuno et al.(2000) suggested the existence mass in this scenario is about 10 per cent of the stellar mass of two inner Lindblad resonances located at r ∼ 100 and 1600 of the galaxy in their simulations. On the other hand, in (0.1 and 1.6 kpc, respectively). the second case, pure collisionless simulations demonstrate Since NGC 3504 is rich in molecular gas emission and the feasibility of forming double-barred galaxies under par- classified as a double-barred galaxy, this galaxy is unique to ticular conditions, such as with a rapidly rotating pseudo- investigate the properties of molecular gas in its type. In this bulge (which can also be referred to as circumnuclear stel- paper, we present high resolution CO (2–1) data and study lar disk) (Shen & Debattista 2009) or with a dynamically its morphology and kinematics. Our paper is organized as cooled inner disk embedded in a hotter outer disk (Du et al. follows. In Section 2, we present our ALMA CO (2–1) ob- 2015). Therefore, the first key step to distinguish these two servations and adopted calibration. The CO channel maps, competing scenarios is to resolve the gaseous inner bar us- distribution, and velocity field are presented in Section 3. In ing high-resolution observations and determine the gas mass Section 4, we describe the kinematic modelling and the fit- in double-barred galaxies. In addition, studying the pattern ting results, including the basic galactic parameters, circular speed of the outer bar can be used to examine the location and non-circular velocity fields. We calculate the gas stream- of the inner Lindblad resonance and compare it with the ing velocity along the dust lanes and the pattern speed of locations of the nuclear ring and the inner bar. the outer bar in Sections 5 and 6. Finally, we present our Our target NGC 3504 has been classified as an early discussion and conclusions in Sections 7 and 8. type barred spiral galaxy, (R)SAB(s)ab (de Vaucouleurs et al. 1991), as well as a potential double-barred galaxy. Although the isophotes on the near-infrared (NIR) image show clear evidence for an inner stellar bar, this has also 2 OBSERVATIONS AND DATA REDUCTION been interpreted as a possible secondary bright source near the galactic nucleus (Pérez-Ram´ırezet al. 2000; Erwin 2004). Our ALMA CO (2–1) observations presented here (project In addition, NGC 3504 has been classified as a ”circumnu- 2016.1.00650.S) were carried out on 21 November 2016 and clear starburst”, with a higher star formation rate and star 18 March 2017 with the two 12m array configurations C43-4 formation rate per mass unit of molecular hydrogen than and C43-1, and on 12 and 17 November 2016 with the ACA most other barred galaxies reported in Jogee et al.(2005). 7m array. Table1 shows the summary of the observational Fig.1 shows the SDSS g-band, HST/WFPC2 F606W, and parameters. The mosaic area was covered with five point- HST/WFC3 F160W images of NGC 3504. The two dust ings for each of the two 12m arrays and three pointings for lanes in the outer bar can be seen more clearly in the SDSS the 7m array, as shown in Fig.2. Note that the Interna- g-band image, and are indicated with two white arrows in tional Celestial Reference System (ICRS) is adopted in this Fig.1 (a). Also, with the higher angular resolution of the paper. The total on-source time for the C43-4, C43-1, and HST F606W optical image shown in Fig.1 (e), we see some 7m array observations were about 34, 10, and 100 minutes, nuclear spiral structures and dust lanes in the circumnuclear respectively. The four spectral windows were centered at the region. Furthermore, Fig.1 (f) shows the inner stellar bar in rest frequencies 230.538, 231.901, 244.936, and 246.819 GHz the HST F160W NIR image. The contours in Fig.1 (f) are with bandwidths of 1.875 GHz and spectral resolutions of used to emphasize the change of the position angle from the 5.10, 5.07, 4.80, and 38.15 km s−1. The primary beams of inner bar to the outer bar. The blue line denotes the ma- the 12m and 7m antennas are 2500. 4 and 4300. 5 at the observed jor axis of the inner bar with a length of 500. 2 and position frequency of 229.367 GHz, which is used to observe the CO ◦ angle of 172 . The size (Lbar) and the position angle were (2–1) line. Furthermore, in our combined data, given the 5th estimated using the ellipticity and position angle curves as percentile of the uv-distance L5 ≈ 9 m, the maximum recov- 0.983λ 00 functions of radius on the K-band image (Pérez-Ram´ırez erable scale per pointing is θMRS ≈ ≈ 30 , where λ L5 et al. 2000) by following the procedure of Erwin(2004). is the observed wavelength. CO line observations of NGC 3504 have been obtained Data reduction was performed using the Common As- with single-dish (Young et al. 1984; Planesas et al. 1997; tronomy Software Applications (CASA) package (McMullin Kuno et al. 2000) and interferometric (Kenney et al. 1993) et al. 2007), version 4.7.0 for the C43-4 and ACA 7m data telescopes. With the best resolution of 200. 5, achieved at that and version 4.7.2 for the C43-1 data. The calibrated visibil- time by OVRO, Kenney et al.(1993) found that the CO dis- ities were concatenated using the task CONCAT in CASA tribution in the galactic central region is nearly azimuthally 5.4 and the continuum subtraction was done using line-free symmetric and can be described by an exponential profile. channels with the task UVCONTSUB. As a result, using Furthermore, by assuming the corotation radius is located the TCLEAN task with Briggs weighting (robust = 0.5), at the end of the outer bar, they derived the pattern speed of the achieved synthesized beam size of the CO (2–1) im-

MNRAS 000,1–18 (2021) Double-barred Galaxy NGC 3504 3

(a) SDSS g-band (d) SDSS g-band 150 150 100 100 30" 45" 60 60

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Figure 1. (a) to (c): The SDSS g-band, HST/WFPC2 F606W, and HST/WFC3 F160W images of NGC 3504. The white arrows indicate the dust lanes seen in the SDSS g-band image. (d) to (f) are the same as panels (a) to (c), but showing the central 2.4 kpc region. The contour levels in panel (f) are 10, 20, 30, 40, 60, 100, 200, and 600 electrons/s. The blue line represents the major axis of the inner bar with a length of 500. 2 and position angle of 172◦. age is 000. 79 × 000. 64 (PA=26◦. 3) and the rms noise is ∼ IRAM 30m telescope. Planesas et al.(1997) estimated an 1 mJy beam−1 for the velocity resolution of 6.5 km s−1. integrated flux of CO (2–1) of 1343 Jy km s−1 within 2700 9 and derived a molecular gas mass of 2.1 × 10 M , assum- To assess whether there is missing flux in the CO (2– ing the CO-to-H2 conversion factor X10 = N(H2)/ICO(1−0) 20 −2 −1 −1 1) ALMA data, due to the lack of short-spacing informa- is 3.0 × 10 cm (K km s ) , a line intensity ratio R21= tion in the uv-plane, we compared the integrated line flux ICO(2−1)/ICO(1−0) of 0.8, and that the distance to the galaxy with the result reported by Planesas et al.(1997) using the

MNRAS 000,1–18 (2021) 4 Wu et al.

Table 1. ALMA Observational Parameters

Parameter 12m Array ACA (Morita Array)

Conﬁguration C43-4 C43-1 7m Observation date 21 Nov 2016 18 March 2017 12 Nov 2016 17 Nov 2016 Number of antennas 41 43 12 11 Number of pointings 5 5 3 3 Time on source (mins) 34 10 50 50 Flux calibrator J1058+0133 Ganymede J1058+0133 Ganymede Bandpass calibrator J1058+0133 J1058+0133 J1058+0133 J1058+0133 Phase calibrator J1103+3014 J1150+2417 J1159+2914 J1159+2914

confined in a radius of 300. 5 (∼ 0.35 kpc). The peak flux 150 is 1.45 mJy beam−1, which translates to a ∼ 36σ signifi- 100 cance. The location of the galactic center (R.A., DEC) = h m s s ◦ 0 00 00 45" 60 (11 03 11.24 ± 0.002, +27 58 21. 41 ± 0. 03) is derived by two dimensional Gaussian fitting of the compact nucleus, as

30 indicated by the plus sign in Fig.3. That position can be h m s ◦ 0 00 30" compared with (R.A., DEC) = (11 03 11.25, +27 58 21. 0),

15 measured with 1.4 GHz continuum emission by Condon et al. J y (1990). The positions diﬀer by ∼ 000. 4, smaller than the syn- 00 00 15" 8 thesized beam of 1. 5. However, this position is about 0. 83 Dec (ICRS) north of the galactic center as given by the SDSS, and is 4 indicated by the blue cross sign in Fig.3. This diﬀerence 58'00" is most likely due to the extinction by gas and dust at the 2 galactic center in the SDSS data. 1 kpc +27°57'45" 1 13s 12s 11s 10s 11h03m09s RA (ICRS) 3.2 CO (2–1) Distribution

Figure 2. Observed ALMA pointings overlaid on the SDSS g- The CO (2–1) channel maps are shown in Fig.4. There is band image of NGC 3504. The red and yellow circles show the no detection about 2800 (2.8 kpc) away to the southeast of five pointings of each 12m array and the three pointings of the the galactic center because the mosaic observation did not 7m array, respectively. The size of the red and yellow circles are fully cover this region. However, the observations did cover 00 00 the primary beams of the 12m and 7m antennas, 25. 4 and 43. 5. the dust lane in the outer bar region, which is within 2800 southeast of the galactic center. Fig.5 shows the CO (2–1) channel maps at the central ∼1.7 kpc region, from where we is 20.7 Mpc. From our ALMA data, the integrated flux ob- identified four spiral features and one nuclear ring structure tained within 2700 is 1235 Jy km s−1. Given that the absolute by tracking the molecular gas, as shown by the white dotted flux density accuracy of ALMA is about 5 to 10 per cent at and dashed lines, respectively. ∼230 GHz (Section 10.2 in ALMA Technical Handbook), the Fig.6 (a) and (b) present the CO (2–1) integrated in- results above may imply that most of the flux within 2700 is tensity (moment 0) map and the CO (2–1) contours super- recovered by ALMA. However, the lack of a quoted uncer- imposed on the SDSS g-band image, respectively. It is clear tainty in the IRAM result prevents us from stating a more that the CO (2–1) emission lies along the two dust lanes. We definite answer. For example, an error of 10 per cent in the also notice that the CO (2–1) emission on the two dust lanes IRAM flux estimate would make the two results compatible are asymmetric in shape, and there are more massive clumps within the uncertainties. (integrated intensity larger than 2.0 Jy beam−1 km s−1) located on the northern dust lane. Fig.7 (a) to (f) show the CO (2–1) integrated intensity 3 RESULTS (moment 0) map, together with the HST/WFPC2 F606W and HST/WFC3 F160W images, at the central 2.4 kpc re- 3.1 Continuum Emission gion. An inner gas bar is present in NGC 3504, as shown The 230 GHz continuum image of NGC 3504 was made in Fig.7 (a). The black dashed and dotted lines in Fig.7 using all four spectral windows after the lines were sub- (b) indicate the identified nuclear ring and four spiral struc- tracted. The available bandwidth is about 6234 MHz, yield- tures, as in Fig.5. In Fig.7 (d) we compare the CO (2–1) ing an rms noise level of 0.04 mJy beam−1, with a 000. 75 emission and HST/WFPC2 F606W optical image. It can be × 000. 60 (PA=25◦. 2) synthesized beam. Fig.3 displays the seen that the four identified spirals trace dust regions in the 230 GHz continuum image and the continuum contours su- HST optical image, especially more clear for the spiral 4. perposed on the SDSS g-band image. It clearly shows that In addition, the comparison of the CO (2–1) emission and the continuum emission is asymmetric and is characterized HST/WFC3 F160W NIR image in Fig.7 (f) shows that the by a bar-like structure with a compact nucleus, and mostly position angle and the length of the inner gas bar are about

MNRAS 000,1–18 (2021) Double-barred Galaxy NGC 3504 5

(a) (b) 1.50 200.0 1.20 150.0 28" 1.00 28" 100.0 0.90 80.0 0.80 60.0 0.70 0.60 40.0 30.0 24" 0.50 24"

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y 15.0

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Figure 3. (a) 230 GHz continuum emission with contour levels of 5.0, 7.5, 10.0, 12.5, and 25.0σ, with σ=0.04 mJy beam−1. (b) 230 GHz continuum overlaid on the SDSS g-band image of NGC 3504. Each image has a size of 1800× 1800 with north up and east left. The synthesized beam (000. 75 × 000. 60, PA=25◦. 2) is shown on the lower-left corner of the map. The yellow plus sign in each panel marks the center of the galaxy at (R.A., DEC) = (11h03m11s.24 ± 0s.002, +27◦5802100. 41 ± 000. 03) which is derived by two-dimensional Gaussian ﬁtting. The blue cross sign indicates the galactic center location (R.A., DEC) = (11h03m11s.24, +27◦5802000. 58) given by SDSS. the same as the inner stellar bar. In Fig.7 (g) we compare 3.2.2 Gas Mass the CO (2–1) emission and the 230 GHz continuum emission. It is apparent that the location of the continuum emission is The total mass of molecular gas is calculated using the consistent with the position of the strong CO (2–1) region. equations (3) and (4) in Solomon & Vanden Bout(2005) with the assumptions that the CO-to-H2 conversion fac- Because the 230 GHz continuum mainly originates from the 20 −2 −1 −1 tor X10 is 2 × 10 cm (K km s ) (corresponding to dust thermal emission, our result would imply that the dust −2 −1 −1 is mixed with the gas near the galactic center, including at αCO = 3.2 M pc (K km s ) ), the line intensity ratio the inner gas bar. Fig.7 (h) presents the CO (2–1) isoveloc- R21 = ICO(2−1)/ICO(1−0) is 0.8, and the correction for the ity contours superposed on the CO (2–1) integrated intensity second most abundant element Helium is 1.36. Therefore, map. It can be seen that the velocity ﬁeld slightly twists due the total molecular gas mass is to the inner gas bar. More details are described in Section 3.3. 0 Mgas[M ] = 1.36 αCO LCO(2−1)/R21, (1)

0 7 −2 2 −3 where L = 3.25 × 10 SCO(2−1) ∆v ν DL (1 + z) . 3.2.1 The Double-horned Profile CO(2−1) obs The velocity-integrated flux SCO(2−1) ∆v is measured in −1 The total detected flux is 1480 Jy km s−1 with ∼ 24 per cent Jy km s , the observed frequency νobs in GHz, the lumi- within the inner bar region. Fig.8 (a) shows the spectrum nosity distance DL in Mpc, and z is the redshift. Given the −1 over the inner bar region (in grey color). The emission ranges total estimation of flux of 1480 Jy km s , the total molec- 00 00 from ∼ 1350 km s−1 to ∼ 1715 km s−1. There are two peaks ular gas mass within our mapping area (∼ 50 × 57 ) is 9 at velocities of 1488.5 and 1573.0 km s−1, corresponding to 3.1 × 10 M , which could be 5–10 per cent uncertain given peak fluxes of 2.0 and 2.2 Jy, respectively. This double- the absolute flux accuracy alone, as mentioned in Section 2. horned profile was also found by Planesas et al.(1997) using In addition, we calculated the concentration factor the IRAM 30m telescope and noted that it could be due to (fcon) of NGC 3504. The fcon is defined as the ratio of the an unresolved rotating ring-like structure or the concentra- surface density of molecular gas averaged over the central tion of molecular gas located at the end of a bar. However, kiloparsec to the surface density of total molecular gas aver- Planesas et al.(1997) could not confirm the origin of this aged over the whole optical disk at R < R25 (the radius at −2 double-horned profile because the angular resolution of their a B-band surface brightness of 25 mag arcsec )(Sakamoto 0 data was insufficient to show a ring or a bar feature. On the et al. 1999). Note that R25 is 2.5 in NGC 3504 (Kenney et al. other hand, with the higher angular resolution of the ALMA 1993). Given the averaged surface density at R < 500 pc and 0 −2 −2 data, we are able to see the double-horned profile from the R < 2.5 of 1360.3 M pc and 4.4 M pc , respectively, inner bar region, as shown in Fig.8. In addition, within the the concentration factor fcon of NGC 3504 is 309. This value inner bar region, about 50 per cent of the flux at veloci- is larger than 100.2±69.8, the average of 10 barred galaxies ties of 1488.5 and 1573.0 km s−1 come from regions 1 and 2, reported in Sakamoto et al.(1999). This result agrees with respectively. Jogee et al.(2005), which shows that NGC 3504 has a large central molecular gas concentration, comparable to other starburst galaxies and larger than typical barred galaxies.

MNRAS 000,1–18 (2021) 6 Wu et al.

1 1 1 0.20 48" 1378.0 km s 1404.0 km s 1430.0 km s

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Figure 4. Channel maps of the CO (2–1) line of NGC 3504. The velocities are shown at the upper-left corner of each panel. The synthesized beam (000. 79 × 000. 64, PA=26◦. 3) is shown on the lower-left corner of the map. Contour levels are 10, 50 and 200σ, which corresponds to 0.01, 0.05 and 0.20 Jy beam−1. The plus sign in each panel marks the galactic center, as in Fig.3.

3.3 Kinematics dust lanes, we made position-velocity (PV) diagrams along four slices (A to D) perpendicular to the dust lanes, as seen The CO (2–1) velocity field and velocity dispersion are in Fig. 10. The velocity gradient across the northern dust shown in Fig.9. In Fig.9 (a) and (b), the black con- lane can be up to ∼50 km s−1 over ∼ 100, as shown in Fig. 10 tour shows the CO (2–1) integrated intensity at a level of (b) and (c). On the other hand, panels (d) and (e) show that −1 −1 0.35 Jy beam km s , which helps to indicate the location the velocity gradient can be up to about 30 km s−1 over ∼ of the dust lanes as shown in Fig.6 (a). As can be seen in 100 across the southern dust lane. After correcting for the Fig.9 (a), the velocity field crossing the dust lanes shows a inclination angle of 25◦, which is derived in Section 4, the velocity gradient, which is predicted due to gas shocks along corresponding velocity gradients are 1.2 and 0.7 km s−1 pc−1 the leading edges of bars (Sanders & Tubbs 1980; Athanas- across the northern and southern dust lane, respectively. soula 1992). The velocity dispersion along the southern and −1 northern dust lanes is about 10 and 20 km s , respectively, In addition to the molecular gas along the dust lanes, as shown in Fig.9 (b). the CO (2–1) velocity field and velocity dispersion over the To show more clearly the velocity gradient across the central 2.4 kpc region are presented in Fig.9 (c) and (d).

MNRAS 000,1–18 (2021) Double-barred Galaxy NGC 3504 7

1 1 1 0.20 28" 1430.0 km s 1462.5 km s 1475.5 km s spiral 4 spiral 1 J 24" y 0.10 b e a m 20" 0.05 1 Dec (ICRS)

+27°58'16" spiral 3 spiral 2 0.5 kpc

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11.60s 11.20s11h03m10.80s 11.60s 11.20s11h03m10.80s 11.60s 11.20s11h03m10.80s RA (ICRS) RA (ICRS) RA (ICRS)

Figure 5. Channel maps of the CO (2–1) line of NGC 3504 at the central ∼ 1.7 kpc region. The velocities are shown at the upper-left corner of each panel. The dashed and dotted lines mark the identiﬁed nuclear ring and the four spiral structures, respectively. The synthesized beam is shown on the lower-left corner of the map. The plus sign in each panel marks the galactic center, as in Fig.4.

The velocity contours in Fig.9 (c) are also overlaid on the well as in the regions indicated by E and F in Fig.9 (d). CO (2–1) integrated intensity map in Fig.7 (h). We find To probe the cause of the higher velocity dispersion in re- that the velocity field is twisted, likely due to the inner gas gions E and F, where the CO emission is relatively low bar, and also slightly irregular due to the inner spiral struc- (∼ 3 Jy beam−1 km s−1), PV diagrams along them are shown tures and the dust lanes. No strong twist of the velocity field in Fig. 11 (a) and (b), respectively. We can see that there is appears 300 North of the galactic center, which implies that a discrete cloud with a velocity offset of ∼ 70 km s−1 from the possibility to have a massive second nucleus mentioned most of the gas in region E. This indicates that there is in Pérez-Ram´ırezet al.(2000) is low. In addition, the steep molecular gas at two distinct velocities at the same projected velocity gradients of about 50 km s−1 at the intersection of location, as opposed to one component with a broad line. the dust lanes and the inner molecular region at R ≈ 1100 On the other hand, Fig. 11 (b) shows an asymmetric broad would imply the existence of shock fronts there. component centered on the same velocity ∼ 1530 km s−1 in region F. To reveal the origin of these two features in Fig. 11 −1 The velocity dispersion increases to ∼ 30 km s in the (a) and (b), Fig. 11 (c) and (d) show the CO (2–1) channel central disk and to ∼ 55 km s−1 at the galactic center as

MNRAS 000,1–18 (2021) 8 Wu et al.

(a) (b) 50.0 150

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Figure 6. (a) CO (2–1) integrated intensity (moment 0) map. Contour levels are 0.35, 0.8, 2.0, 4.0, 10.0,15.0 and 30 Jy beam−1 km s−1. (b) CO (2–1) contours from the moment 0 map overlayed on the SDSS g-band image of NGC 3504. The synthesized beam is shown on the lower-left corner of the map and the plus sign marks the center of the galaxy, same as in Fig.3. maps at the central ∼ 1.3 kpc region around ∼ 1470 km s−1 tion as well as non-circular perturbations. The error of the −1 and ∼ 1600 km s , respectively. From Fig. 11 (c), we can see velocity field, M1,err, was calculated as that there is localized emission at ∼ 1470 km s−1 in panel v (a) that is not connected to the main structure at higher ve- u  n 2 u P locities. Such emission is almost completely absent at other n wvi − (vj I(vj )) n u j=1 2 uX   2 X I (vi) 2 velocity frames and probably not related to the motion of the M1,err = u   σ + σv, t  w2  I w2 bar and spirals. Also, Fig. 11 (d) shows that the asymmet- i=1 i=1 ric distribution of the molecular gas around ∼ 1600 km s−1 in panel (b) is associated with a spur-like substructure at (2) (11h03m11s.4, +27◦5802300. 2), ∼ 200. 5 away to the northwest of where n is the number of channels, I(vi) is the intensity of the drawn circle. The feature is more clear in the channel a given pixel at a channel with velocity vi, w is the sum −1 n maps at 1605.5 and 1625.0 km s . P of the intensity I(vi), σI is the standard deviation of Fig. 12 shows the PV diagrams along the major and i=1 minor axes of the galaxy. The position angle of the galaxy the intensity at line-free channels, and σv is the velocity is derived using Kinemetry (Krajnovićet al. 2006) and will uncertainty, which is assumed to be one half of our velocity be described in Section 4. It is very close to the position resolution. angle derived from the optical isophotes in the outer disk Under the assumptions that the velocity field can be (Grosbøl 1985). From Fig. 12 (b), we can see a tilted bow-tie decomposed in a number of ellipses and the velocity profiles feature in the PV diagram along the major axis. This feature along the ellipses can be expressed as a Fourier series (see indicates the presence of two kinematically distinct gaseous Equation (6) in Krajnovićet al.(2006)), Kinemetry derives components, which could be caused by a black hole or a non- the system velocity, the position angle and the inclination axisymmetric potential, such as a bar (Binney et al. 1991; of each ellipse by minimizing the coefficients of the Fourier Kuzio de Naray et al. 2009; Funes et al. 2002). The velocity series which are used to describe the non-circular velocity, as peaks located at offset positions of +300 and +600 are due to shown in Equations (7) and (8) in Krajnovićet al.(2006). the presence of the nuclear ring and spiral 1 in the north, In our case, we only considered six odd terms during the 00 respectively. The emission at offset position of −4 is from fitting, i.e. An sin(ψ) and Bn cos(ψ) with n=1, 3, 5, where spiral 2 located in the south. Along the minor axis, as shown ψ is the eccentric anomaly. The even terms are usually very in Fig. 12 (c), it can be seen that there exist non-circular small and can be negligible in the velocity field (Krajnović motions. The non-circular motions within the central ∼ 100. 5 et al. 2006, 2008). are an indication of the presence of an inner bar. Outside In the first Kinemetry run we only fixed the galactic cen- the radius of 100. 5, the wiggly patterns are mainly due to the ter to (R.A., DEC) = (11h03m11s.24, +27◦5802100. 41), which presence of the nuclear ring and spirals. was derived in Section 3.1, and obtained the system velocity of each ellipse. Then, in the second Kinemetry run, we fixed the velocity of the galaxy to the median value of the system velocities calculated from the first run and obtained the po- 4 KINEMATIC MODELLING sition angle (PA) and the inclination of each ellipse. Finally, To investigate the kinematics in NGC 3504, we used we fixed both PA and inclination to their median values and Kinemetry (Krajnovićet al. 2006) to fit our velocity (mo- derive the coefficients of the Fourier series of each ellipse. ment 1) field and quantify the contribution of circular rota- Using the procedure described above, our best fit re-

MNRAS 000,1–18 (2021) Double-barred Galaxy NGC 3504 9

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15" 1 0.4 0.5 kpc 0.5 kpc +27°58'10" 0.2 12s 11h03m11s 12s 11h03m11s RA (ICRS) RA (ICRS) Figure 7. (a) and (b): CO (2–1) integrated intensity map of the central 2.4 kpc region with the same contours as in Fig.6. The black dashed and dotted lines in panel (b) are the identiﬁed nuclear ring and spiral structures, as in Fig.5. The synthesized beam is shown on the lower-left corner of the map. (c) to (f): The HST/WFPC2 F606W and HST/WFC3 F160W images, together with the CO (2–1) contours. (g) Superposition of the 230 GHz continuum contours on the CO (2–1) integrated intensity map. The contour levels are the same as in Fig.3. (h) CO (2–1) isovelocity contours superposed on the CO (2–1) integrated intensity map. The contour levels are from 1410 to 1670 km s−1 in steps of 20 km s−1. The grey line denotes the major axis of the inner bar, as in Fig.1. The plus signs mark the center of the galaxy, as in Fig.3.

MNRAS 000,1–18 (2021) 10 Wu et al.

(a) (b) 2.5 1573.5 km s 1 50.0 1488.5 km s 1

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Figure 8. (a) CO (2–1) spectrum within the inner bar region (grey), region 1 (blue) and region 2 (red), as marked in panel (b). The vertical line indicates the systemic velocity of 1532.2 km s−1, as derived in Section 4. (b) The black ellipse, with a major axis of 500. 5, a minor axis of 300. 5, and a position angle of 172◦, indicates the inner bar region used for plotting the spectrum in panel (a). The green circles mark regions 1 and 2 with a radius of 100. sults yield a systemic velocity of the galaxy of 1532.2 ± to the lack of data points at R = 1000 − 1500 in Kuno et al. −1 ◦ ◦ 0.2 km s , an inclination angle of 25 ± 1 , and a PA of (2000) and the use of different tracers, Hα and CO (1–0). ◦ ◦ 153 ± 2 , where the errors are the statistical errors cor- Using the derived parameter B1 as well as the systemic responding to 1σ confidence levels estimated by Kineme- velocity, the inclination and PA, we can generate the mod- try using Monte Carlo simulations. Our fit results for the elled velocity field from the derived rotation curve, as shown galaxy velocity, inclination and PA are consistent with the in Fig. 15 (a) and (c). The difference between the ALMA values found by Grosbøl(1985) and Kenney et al.(1993), CO (2–1) velocity field and the modelled velocity field is 1535 ± 2 km s−1, 22◦, PA=149◦ derived from the optical presented in Fig. 15 (b) and (d). Assuming that the dust isophotes and PA=147◦ ± 2◦ derived from the nuclear CO lanes are located at the leading side of the bar (Athanas- (1–0) velocity field, respectively. soula 1992), we can identify the far and near sides of the galaxy from Fig. 15 (a). From Fig. 15 (b), we also see that Fig. 13 shows the Fourier decomposition coefficients of the residual velocities can reach |v| ≈ 50 km s−1 along the our fitting model as a function of the semi-major axis length two dust lanes and are different across them. The difference of the ellipses. The errors indicate the variance caused by the of the residual velocity across the dust lanes on the leading uncertainty of the inclination and PA. The B term, which 1 and trailing side of the bar can be up to ≈ 30 km s−1 when represents pure circular motion, dominates in most regions we compare the residual velocity at the locations indicated at R < 2500. The A term, which indicates the existence of 1 by two circles in Fig. 15 (b). Furthermore, the absolute value radial motion, is the second dominant mode in the inner- of the residual velocity is larger across the dust lanes on the most ∼ 300 and at R > 1000. Furthermore, the A and B 3 3 leading side of the bar. This implies that the velocity devi- terms, which are more sensitive to the center, inclination ation from circular motion is larger after crossing the dust and position angle of the ellipses, and might also indicate lanes and there is gas streaming motion along the dust lanes complex velocity structures, are relatively small in our fit as predicted by hydrodynamic models (Athanassoula 1992). results. The A and B terms are also small, and indicate 5 5 The black dashed and dotted lines in panel (c) and (d) show the absence of strong multiple kinematic components, such the identified nuclear ring and spiral structures as presented as a counter-rotating disk, in the inner regions of NGC 3504. in Fig.5. The non-zero residual velocity along the spiral The rotation curve and the angular velocity Ω are pre- structures may indicate gas inflows toward the galactic cen- sented by the black and blue lines in Fig. 14. The rotational tral region. velocity was calculated from the B1 term and corrected for inclination, i.e. vφ(R) = B1(R)/ sin i. The angular velocity was calculated as Ω(R) = vφ(R)/R. As can be seen from Fig. 14, the rotational velocity increases to ∼ 240 km s−1 5 STREAMING MOTION at R ∼ 0.5 kpc and becomes nearly flat at R > 1.7 kpc. This is to be compared with the rotation curve derived by We calculate the gas streaming velocity along the dust lanes 00 Kuno et al.(2000) using CO (1–0) (at R < 10 ) and Hα (at assuming that (1) the gas moves along the dust lanes in R > 1500) data, and corrected here for our adopted inclina- the rotating frame of the bar, and (2) the circular motion angle of i = 25◦, as shown by the red line in Fig. 14. We tions are well described by the rotation curve derived from found that the two rotation curves are generally consistent Kinemetry. The first assumption is predicted by hydrody- with each other within ∼10 km s−1. Between R ∼ 1000 and namic models and revealed in observations, especially in ex- 1400, our rotation velocities are slightly higher than those de- plaining the velocity gradient across dust lanes and the neg- rived by Kuno et al.(2000). This discrepancy could be due ative/positive residual velocity on the far/near side of the

MNRAS 000,1–18 (2021) Double-barred Galaxy NGC 3504 11

(a) (b) 1700 50 40 48" 48" 30 1650

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Figure 9. (a) CO (2–1) velocity field (moment 1) with the contour of the CO (2–1) integrated intensity at the level of 0.35 Jy beam−1 km s−1. (b) CO (2–1) velocity dispersion image (moment 2) with the same contours as in panel (a). (c) Same as panel (a), but showing the central 2.4 kpc region with contours from 1410 to 1670 km s−1 in steps of 20 km s−1. (d) Same as panel (b), but showing the central 2.4 kpc region. The circles with a radius of 000. 5 indicate regions E and F, which have higher velocity dispersion. The dotted rectangles represent the slices used for extracting the position–velocity diagrams shown in Fig. 11 (a) and (b). The synthesized beam is shown on the lower-left corner of the map and the plus sign marks the center of the galaxy, as in Fig.3. dust lanes (Athanassoula 1992; Benedict et al. 1996; Regan because the PA of the bar is similar to the PA of the galaxy et al. 1997, 1999; Schinnerer et al. 2002; Pérezet al. 2004; (Kuno et al. 2000) (i.e. the major axis of the bar and that Pérez 2008; Fanali et al. 2015). It also implies that the ve- of the galaxy are similar), we assume that they are identi- R φ locity of the gas in the dust lanes on the galactic plane con- cal in our calculation. Vst and Vst in Equation (3) can be sists of the rotational velocity of the bar RgalΩb and the gas expressed as streaming velocity Vst along the dust lanes. The stream- R Vst = Vst cos(π − α + θ) = −Vst cos(α − θ), (4) ing velocity Vst can be decomposed into the radial and az- R φ imuthal directions, Vst and Vst, as shown in the scheme in π Fig. 16. Therefore, the observed velocity in the dust lanes is Vφ = V cos(α − − θ) = V sin(α − θ), (5) st st 2 st

where α is the angle between the streaming velocity Vst and φ π R Vobs = Vsys+ Vst + RgalΩb sin( +θ) sin i+Vst sin(π+θ) sin i, the major axis of the bar. 2 Furthermore, under the second assumption, the veloc- (3) ity component in the azimuthal direction can be subtracted from the velocity ﬁeld, so that the residual velocity V in where V is the velocity of the galaxy and θ is the angle be- res sys the dust lanes in Fig. 15 (b) only consists of the third term tween the major axis of the bar and the position R . Note gal on the right-hand side of Equation (3): that the second and third terms represent the azimuthal and R radial components, respectively. In the case of NGC 3504, Vres = Vst sin(π + θ) sin i. (6)

MNRAS 000,1–18 (2021) 12 Wu et al.

(b) (c) (a) 1700 30 1700 50 1700 region A region B 25 1650 1650 40 48" m m J J ) 1600 20 ) 1600 y y s s

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Figure 10. (a) Selected slices (A to D) perpendicular to the dust lanes with a width of 0.2 kpc overlaid on the CO (2–1) velocity ﬁeld (moment 1). The black contours of the CO (2–1) integrated intensity is at a level of 0.35 Jy beam−1, same as in Fig.9 (a). (b) to (e): Position-velocity diagrams along slices (regions) A to D.

(a) (c) 28" 0.200 1700 60 1462.5 km s 1 1469.0 km s 1 1475.5 km s 1 1482.0 km s 1 50 0.100 1650 40

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Figure 11. (a) and (b): PV diagrams along two slices across regions E and F, as marked by the dotted rectangles in Fig.9 (d). (c) Channel maps of the CO (2–1) line of NGC 3504 at the central ∼ 1.3 kpc region. The velocity range is chosen to illustrate the origin of the CO (2–1) emission at ∼ 1470 km s−1 in panel (a). The white circle and the dotted rectangle mark region E. The synthesized beam is shown on the lower-left corner of the map. The plus sign marks the galactic center, as in Fig.4. (d) Same as panel (c), but the velocity range is chosen to show the origin of the asymmetric CO (2–1) distribution around ∼ 1600 km s−1 in panel (b). The white circle and the dotted rectangle indicate region F.

Thus, combining Equation (4) and (6), we obtain the superimposed on the ALMA CO (2–1) integrated intensity streaming velocity (moment 0) map. Fig 17 (b) shows the gas streaming velocity (solid lines) Vres Vst = . (7) and the integrated intensity (dashed lines) along the dust cos(α − θ) sin θ sin i 00 lanes and toward the galactic center region to Rsky ≈ 11 . The average streaming velocities on Dust Lane 1 and 2 are 0 −1 −1 tan θ about 165 and 221 km s , respectively. The letters 1a to 2d Note that θ and α can be calculated by θ = tan cos i indicate the radii where the gas streaming velocity curves −1 tan α0 0 0 and α = tan cos i , where θ and α are defined in the have a local minimum or change significantly. Their corre- same way as θ and α but in the plane of the sky. sponding locations are also marked on the integrated inten- Using Equation (7), we can calculate the gas streaming sity map, as shown in Fig. 17 (a). Comparing the informa- velocity along the dust lanes. The location of the dust lanes tion of the gas streaming velocities and the CO (2–1) inte- are identified on the SDSS g-band image (Fig.6 (b)). Fig 17 grated intensities in Fig 17 (b), we find that the gas stream- (a) shows the dust lanes, marked as Dust Lane 1 and 2, ing velocities decrease when the gas intensities increase in

MNRAS 000,1–18 (2021) Double-barred Galaxy NGC 3504 13

(a) (b) (c) 50.0 1700 350 1700 300 + major minor 30" 20.0 300 250 m m J y

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Figure 12. (a) CO (2–1) integrated intensity map of the central 2.4 kpc region with the same contours as in Fig.6. The brown solid and dashed lines indicate the slices along the major and minor axis of the galaxy, respectively. The plus and minus signs denote the direction of the oﬀset in panels (b) and (c). The magenta dashed and dotted lines are the identiﬁed nuclear ring and spiral structures as in Fig.5. (b) and (c): PV diagrams along the major and minor axes with a width of the slice of 000. 8, same as the major axis of the synthesized beam. The horizontal dashed line is drawn at the system velocity of 1532.2 km s−1, as derived in Section 4.

0" 5" 10" 15" 20" 25" A1 300 300 100 B1 IILR OILR A3 250 250 B3 ) ) A5 80 1 ( 1 B5 200 200 k s m

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0 0 0.0 0.5 1.0 1.5 2.0 2.5 0 R (kpc) 0 5 10 15 20 25 Figure 14. CO (2–1) rotation curve (black solid line) and angular R (arcsec) velocity Ω (blue line) derived from our Kinemetry analysis and corrected for the inclination of i = 25◦. The red line shows the Figure 13. Coefficients of the Fourier components obtained from rotation curve obtained by Kuno et al.(2000) from CO (1–0) the Kinemetry analysis as a function of the semi-major axis and Hα data and then corrected for the adopted inclination of ◦ length of the ellipses. i = 25 . The black dashed line represents the derived pattern −1 −1 speed of the bar ΩBar = 18 km s kpc in Section 6. With this pattern speed, an inner inner Lindblad resonance (IILR) is Dust Lane 1. The gas streaming velocity in Dust Lane 1 located at R ∼ 0.2 kpc and an outer inner Lindblad resonance starts to drop significantly at point 1a when the integrated (OILR) is predicted to exist between 1.9 and 2.3 kpc, and are intensity increases from 0.13 Jy beam−1 km s−1 at point 1a indicated by vertical black dotted lines. to 1.15 Jy beam−1 km s−1 at point 1b. Then, the streaming velocity increases toward point 1c when the integrated intensity decreases. The same phenomenon also happens to a where Σgas is the gas mass surface density, W is the width of smaller extent in Dust Lane 2. At point 2a, the increase of the streaming gas lane and Vst is the gas streaming velocity the streaming velocity slows down and the streaming veloc- as calculating using Equation (7). From Fig. 17 (c), it can ity keeps being relatively constant toward point 2b. Then, be seen that the average of the gas inflow rate along each of −1 the gas streaming velocity starts to decrease from point 2b the two dust lanes is about 6 M yr with a slight decline to point 2c, where the integrated intensity is higher. Finally, towards larger radius. Thus, the total inflow rate along two −1 the streaming velocity increases toward point 2d where the dust lanes is 12 M yr . This estimated mass inflow rate integrated intensity decreases. At the region where the ra- is comparable to the typical mass inflow rate of the order of −1 dius is smaller than that of point 1c and 2d, the motion of 0.1–10 M yr found in previous works for galaxies, such the gas might be affected by the inner molecular region and as NGC 1530, NGC 3368, NGC 4736 and NGC 5248 (Regan is not considered here. et al. 1997; Sakamoto et al. 1999; Haan et al. 2009). The Fig. 17 (c) shows the mass inflow rate in the dust lanes. declining trend of the mass inflow rate could be due to the Using Equation (1) to derive the gas mass and assuming the lower gas streaming velocity and/or the lower gas mass sur- width of the streaming gas lane is 100. 5, the mass inflow rate face density at larger radius. This phenomenon is also found (M˙ gas) at the position R in the dust lane can be calculated in the western dust lane in NGC 1530 (Regan et al. 1997). as in Regan et al.(1997):

M˙ gas(R) = Σgas(R)WVst(R), (8)

MNRAS 000,1–18 (2021) 14 Wu et al.

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Figure 15. (a) Modelled velocity field generated by the term A0 + B1 cos(ψ) derived from Kinemetry and then corrected for the inclination and PA. (b) Residual velocity map after subtracting the modelled velocity field from the CO (2–1) velocity field. The white lines indicates the two dust lanes identified on the SDSS g−band image (described in Section 5 and in Fig. 17 (a)) and the two white circles mark the locations where the difference of the residual velocities is ≈ 30 km s−1 across the northern dust lane. The plus sign marks the center of the galaxy, as in Fig.3. The brown solid and dashed lines represent the galactic major and minor axes, respectively. (c) Same as panel (a), but showing the central 2.4 kpc region. The black dashed and dotted lines are the identified nuclear ring and spiral structures as in Fig.5. (d) Same as panel (b), but showing the central 2.4 kpc region.

6 THE PATTERN SPEED OF THE BAR bar at different radius Rgal on the galactic plane. The errors indicate the variances caused by the uncertainty of the incli- Under the same assumptions described in Section 5, the ve- nation and PA. At R < 1200. 5, the estimated pattern speed locity in the dust lanes on the modelled velocity field in gal has lower confidence since the motion of the gas might be Fig. 15 (a) consists of the first and second terms of the right- affected by the inner molecular region, therefore not satisfy- hand side of Equation (3): ing our first assumption in Section 5, that is the gas moves φ along the dust lanes in the rotating frame of the bar. Fur- V = V + V + R Ω cos θ sin i, (9) mod sys st gal b thermore, in Dust Lane 1, the existence of large gas clumps could change the motion of the gas and therefore render the where R = R pcos2 θ0 + sin2 θ0/ cos2 i. Therefore, com- gal sky first assumption invalid and may cause a larger variation bining Equation (5) and (9), we can get the pattern speed of the streaming velocity and Ω . On the other hand, we of the bar: b note that the integrated intensity is nearly the same along Vmod − Vsys − Vst sin(α − θ) cos θ sin i Dust Lane 2, and the pattern speed is relatively constant at Ωb = . (10) 00 00 Rgal cos θ sin i 12. 5 < Rgal < 23 in Dust Lane 2. Therefore, we performed 00 00 −1 least-squares fitting of the data points at 12. 5 < Rgal < 23 Using the systemic velocity Vsys = 1532.2 km s and of the Dust Lane 2 to obtain the pattern speed of the bar, the derived Vst in Dust Lane 1 and 2 as mentioned in Sec- −1 −1 and obtained ΩBar = 18 ± 5 km s kpc . tion 5, Fig. 18 shows the estimated pattern speed of the

MNRAS 000,1–18 (2021) Double-barred Galaxy NGC 3504 15

Near side

dust lane

b major axis of the bar i Rgal R Vst line of sight

Vst Vst Far side

Figure 16. Schematic diagram of the dust lanes and the streaming velocity V~ st on the galactic plane. θ is the angle between the major axis of the bar and the location of a point in the dust lane. α is the angle between the major axis of the bar and the streaming velocity ~ ~ R ~ φ ~ Vst. Vst and Vst are Vst projected to the radial and azimuthal directions, respectively. Ωb represents the pattern speed of the bar and i is the inclination of the galaxy. Note that the major axis of the bar is identical to the major axis of the galaxy in this diagram, such as in NGC 3504 (Kuno et al. 2000).

(a) 50.0 (b) (c) 5 40" 350 30.0 Dust Lane 1 Dust Lane 1

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2b 1 J +27°58'10" (

0.4 50 2a I

0 0 0 Dust Lane 2 1 kpc 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 0.2 12.5s 12.0s 11.5s 11.0s 10.5s 11h03m10.0s R (arcsec) R (arcsec) RA (ICRS) sky sky

Figure 17. (a) Illustration of the gas streaming along the two dust lanes, marked as Dust Lane 1 and Dust Lane 2, as indicated by the magenta lines superimposed on the ALMA CO (2–1) integrated intensity (moment 0) map. The black contours of the CO (2–1) integrated intensity are at levels of 0.8 and 2.0 Jy beam−1. The letters 1a to 2d indicate the locations corresponding to the radii given in panel (b). (b) Gas streaming velocity (solid lines) and CO (2–1) integrated intensity (dashed lines) along the dust lanes as a function of ˙ radius Rsky. (c) Same as panel (b), but for the gas mass inﬂow rate, Mgas.

Although ΩBar is only derived from Dust Lane 2, from Kenney et al.(1993) estimated the pattern speed of the 00 00 −1 −1 Fig 18, we can see that at 17 < Rgal < 19 in Dust Lane bar in NGC 3504 to be ∼ 77 km s kpc , assuming the 1, where there are no large clumps and the motion of gas corotation radius is located at the end of the bar. On the is assumed to satisfy the ﬁrst assumption in Section 5, the other hand, Kuno et al.(2000) obtained the upper limit estimated pattern speeds are also consistent with our derived of the pattern speed of the bar, ∼ 41 km s−1 kpc−1, based pattern speed ΩBar. This implies that our derived pattern on the two assumptions that the gas ﬂows along the dust speed ΩBar could be valid for both sides of the bar, even lane toward the galactic center and that the misalignment when it is derived using only the Dust Lane 2. between the major axis of the bar and the major axis of With the derived pattern speed of the bar ΩBar = the galaxy is small. Furthermore, they suggested that the 18 km s−1 kpc−1 and the frequency curve Ω − κ/2, we found corotation radius is at least two times larger than the radius that the inner inner Lindblad resonance (IILR) is located of the bar if that upper limit is close to the pattern speed. −1 −1 at R ∼ 0.2 kpc, as shown in Fig. 14. The outer inner Lind- Our result of ΩBar = 18 ± 5 km s kpc is consistent with blad resonance (OILR) is predicted to be between 1.9 and the results of Kuno et al.(2000) and has a smaller value 2.3 kpc due to the uncertainty of Ω − κ/2 in this region. The due to the consideration of the gas streaming velocity Vst epicyclic frequency κ was calculated from in Equation (10). r dΩ2 κ = 4Ω2 + R . (11) dR

MNRAS 000,1–18 (2021) 16 Wu et al.

90 Dust Lane 1 80 Dust Lane 2 )

1 70 c

p 60 k

1 50 s

m 30 k (

b 20

0 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Rgal (arcsec)

Figure 18. Ωb along Dust Lane 1 and 2 as function of the radius on the galactic plane, as shown by the black and red solid lines, −1 −1 respectively. The red dashed line represents the derived pattern speed of the bar, ΩBar = 18 ± 5 km s kpc , which is the least-squares 00 00 ﬁt to the data points at 12. 5 < Rgal < 23 . The grey area is the 95 per cent conﬁdence interval.

7 DISCUSSION ular gas in their nuclei. NGC 3504, which was classified as a candidate double-barred galaxy (Pérez-Ram´ırezet al. 2000), 7.1 The Nuclear Ring presents an inner gas bar as can be seen in Fig.7 (a). The Regarding the identified nuclear ring as shown in Fig.5, position angle and the length of the inner gas bar are about we applied least-squares fitting to an ellipse to determine the same as the inner stellar bar, as shown in Fig.7 (f). From the center of the ring at (11h03m11s.29, +27◦5802100. 31) with the CO velocity field as shown in Fig.7(h), the twisted and semi-major and semi-minor axes of about 300. 6 and 300. 2, and a tilted isovelocity contours relative to the minor axis of the PA of −44◦. 4. The derived ellipticity of the ring is 0.11, which galaxy at the central region R < 200. 5 strongly suggest the is in the range of 0–0.4 reported by Comerónet al.(2010), presence of an inner bar in NGC 3504. This is also supported based on a sample of more than 100 galaxies. In addition, by the indication of non-circular motion in the PV diagrams the difference between the center of the ellipse and the center in Fig. 12 and the clear evidence for radial gas motions in of the galaxy is about 000. 7, which is comparable to the size the central 300 shown in Fig. 13. 10 of our synthesized beam. Therefore, high resolution data is Furthermore, given an stellar mass of 1.84 × 10 M 9 needed to confirm whether this discrepancy is real or not. (Font et al. 2017) and a molecular gas mass ∼ 3.1×10 M , Furthermore, with the semi-major and semi-minor axes as mentioned in Section 3.2.2, the gas mass in NGC 3504 of the ring of ∼ 300. 6 and 300. 2, it seems that the inner ring is about 17 per cent of the stellar mass. This agrees with is not located at either IILR or OILR, but between the two the simulation result in Friedli & Martinet(1993) where the ILRs and close to the IILR, as can be seen from Fig. 14. Inner required gas mass to form a double-barred galaxy is at least rings located between the two ILRs are also found in other ∼ 10 per cent of the stellar mass of the galaxy. Therefore, galaxies, such as NGC 1097, IC 1438, IC 4214, NGC 1512 our result of NGC 3504 gives a new example of an inner gas and NGC 6753 (Piñol-Ferrer et al. 2014; Schmidt et al. 2019) bar within a gas-rich double-barred galaxy, and supports and could be explained by different scenarios. For example, the scenario where the formation of double-barred galaxies Combes(1996) proposed that a nuclear ring forms between could be associated with the existence of molecular gas, as the IILR and the OILR when there are two ILRs. Sormani mentioned in Section 1. et al.(2018) suggested that the gas follows the x2 orbits and forms a ring in the reversed shear region (see their Fig. 3). However, it is still hard to predict the location and the size of 7.3 Gas Flow in the Bar the nuclear ring as it depends on several factors, such as the Considering the average streaming velocity of 165 and random turbulent motion of interstellar clouds, the sound 221 km s−1 along Dust Lane 1 and 2 (see Section 5) and speed of the gas, the pattern speed of the bar, and the bar the deprojected bar radius of 3200 (3.2 kpc) (Kenney et al. strength (Patsis & Athanassoula 2000; Kim et al. 2012; Seo 1993), the time for the gas to fall in from the end of the et al. 2019; Schmidt et al. 2019). Nevertheless, our results 00 bar to the inner molecular region at Rsky < 11 (1.1 kpc) confirm the existence of the nuclear molecular ring between along Dust Lane 1 and 2 are ∼ 12 and 9 Myr, respectively. two ILRs and close to the IILR. Note that the these estimated times could be a factor of two longer because the streaming velocity becomes smaller when the local gas density increases. 7.2 The Inner Gas Bar At the central region R < 1000, since the molecular As mentioned in Section 1, to understand whether the gas gas mass is 75 per cent of the total molecular gas mass 9 is involved in the formation of the inner bar, the first crit- (3.1 × 10 M ) (see Section 3.2) and the star formation rate ical step is to investigate the morphology and the mass of estimated from the Lyman continuum photon rate in the −1 the molecular gas in double-barred galaxies. Petitpas & Wil- same region is 2.3 M yr (Kenney et al. 1993), the average son(2002) studied two candidate double-barred galaxies and gas consumption time would be about 1 Gyr. Furthermore, found that only NGC 2273 has an inner gas bar, although if we consider the additional 25 per cent of the total gas both NGC 2273 and NGC 5728 have large amounts of molec- mass at R > 1000 and assume it will accrete to the central

MNRAS 000,1–18 (2021) Double-barred Galaxy NGC 3504 17

00 −1 region at R < 10 at the total inflow rate of 12 M yr (See fore, to distinguish between these two scenarios, a major Section 5), this gas will take an additional ∼ 64 Myr to reach step is to resolve the inner gas bar and to determine the −1 the central region. At the star formation rate of 2.3 M yr , gas mass in double-barred galaxies. Based on our results, the gas consumption time for all total molecular gas would we conclude that the existence of the inner gas bar and the be about 1.3 Gyr. large amount of gas in NGC 3504 support the scenario for the formation of double-barred galaxies associated with the existence of molecular gas. 8 CONCLUSIONS We presented the morphology and kinematics of the CO ACKNOWLEDGEMENTS (2–1) line in the galaxy NGC 3504, as observed by ALMA. The achieved linear resolution of less than 80 pc allowed us This paper makes use of the following ALMA data: to reveal the non-axisymmetric symmetric structures in the ADS/JAO.ALMA#2016.1.00650.S. ALMA is a partnership central region and also to investigate the kinematics of the of ESO (representing its member states), NSF (USA) and molecular gas. Our main results are: NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in coop- (i) Both axisymmetric and non-axisymmetric structures, eration with the Republic of Chile. The Joint ALMA Ob- including the inner molecular gas bar, the nuclear ring, and servatory is operated by ESO, AUI/NRAO and NAOJ. The the nuclear spirals, are found in the central 1 kpc region. authors are grateful to Krajnović,Cappellari, de Zeeuw & These inner structures were not recognized in previous Copin for the Kinemetry code. The authors thank the ref- publications (Kenney et al. 1993; Kuno et al. 2000) due to eree, Prof. Jeffrey Kenney, for valuable comments and sug- the lack of sufficient angular resolution. gestions that improved this paper. The authors also thank the support of the Astronomical Data Center of the Na- (ii) The estimated total molecular mass is about tional Astronomical Observatory of Japan, and Anthony 9 Moraghan for assistance on the computational facilities and 3.1 × 10 M , corresponding to 17 per cent of the stellar mass. resources at Academia Sinica Institute of Astronomy and Astrophysics (ASIAA). YTW wishes to thank Dr. Yu-Nung (iii) Kinemetry was employed to fit the velocity field of Su and Dr. Sheng-Yuan Liu for hosting her visit at ASIAA NGC 3504. The fit results are the systemic velocity of the in 2020. galaxy of 1532.2 ± 0.2 km s−1, the inclination of 25◦ ± 1◦, and PA=153◦ ± 2◦, which are consistent with the values reported in the literature (Grosbøl 1985; Kenney et al. 1993). DATA AVAILABILITY

00 00 The ALMA data underlying this article is available online (iv) Circular motion strongly dominates at R = 3 − 8 in the ALMA archive at https://almascience.nao.ac.jp/aq/, (∼ 0.3 − 0.8 kpc), but radial motion becomes important in with project code #2016.1.00650.S. This study has also used the regions where the bars are present, corresponding to SDSS data provided at https://dr12.sdss.org/ﬁelds. R < 300 (0.3 kpc) and R = 1000 − 2500 (1.0 − 2.5 kpc).

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