Star formation in IRDC G31.97+0.07

TL;DR

This study investigates the physical and dynamical properties of the IRDC G31.97+0.07 using multi-wavelength data, revealing its potential for high-mass star formation and evidence of gravitational collapse in its clumps.

Contribution

It provides detailed analysis of the IRDC's structure, mass distribution, and star formation activity, including new estimates of its age and the identification of infall motions in clumps.

Findings

Total mass of the filament is about 2.5×10^5 M_sun.

Most clumps are gravitationally bound and likely to collapse.

Region is suitable for high-mass star formation studies.

Abstract

We utilize multiple-waveband continuum and molecular-line data of CO isotopes, to study the dynamical structure and physical properties of the IRDC G31.97+0.07. We derive the dust temperature and H$_2$ column density maps of the whole structure by SED fitting. The total mass is about $2.5\times10^5\,M_{\odot}$ for the whole filamentary structure and about $7.8\times10^4\,M_{\odot}$ for the IRDC. Column density PDFs produced from the column density map are generally in the power-law form suggesting that this part is mainly gravity-dominant. The flatter slope of the PDF of the IRDC implies that it might be compressed by an adjacent, larger \HII region. There are 27 clumps identified from the 850\,\micron \ continuum located in this filamentary structure. Based on the average spacing of the fragments in the IRDC, we estimate the age of the IRDC. The age is about $6.4\,$Myr assuming inclination angle $i = 30^\circ$. For 18 clumps with relatively strong CO and $^{13}$CO (3-2) emission, we study their line profiles and stabilities. We find 5 clumps with blue profiles which indicate gas infall motion and 2 clumps with red profiles which indicate outflows or expansion. Only one clump has $\alpha_\mathrm{vir} > 2$, suggesting that most clumps are gravitationally bound and tend to collapse. In the Mass-$R_\mathrm{eq}$ diagram, 23 of 27 clumps are above the threshold for high-mass star formation, suggesting that this region can be a good place for studying high-mass star-forming.

Tables

Table 1. Table 1: Physical properties of JPS compact sources derived from SED fitting. The columns are as follows: (1)-(6): JPS source name, coordinates information and source size come from Eden et al. ( 2017 ) ; (7) equivalent radius, calculated by: d ​ A / π 𝑑 A 𝜋 d\sqrt{\mathrm{A}/\pi} , where d = 7.07 ​ kpc 𝑑 7.07 kpc d=7.07\,\mathrm{kpc} and A is the area of the source; (8)-(14): dust temperature, column density, mass surface density, mass, luminosity and luminosity-mass ratio obtained from the SED fitting result.

ID	Name	${\ell }_{\mathrm{peak}}$	$b_{\mathrm{peak}}$	maj	min	PA	$r_{\mathrm{eq}}$	$T_{\mathrm{dust}}$	$N_{H_2}$	$n_{H_2}$	${\Sigma }_{\mathrm{mass}}$	Mass	Luminosity	L/M ratio
				″	″	°	pc	K	${10}^{22}{\mathrm{cm}}^{-2}$	${10}^4{\mathrm{cm}}^{-3}$	$g{\mathrm{cm}}^{-2}$	$M_{☉}$	$L_{☉}$	$L_{☉}/M_{☉}$
	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)	(13)	(14)
1	JPSG031.925+00.083	31.925	0.083	17	7	185	0.37	16.53	2.50	2.56	0.12	3.88e+02	4.35e+02	1.12
2	JPSG031.929+00.056	31.929	0.056	12	8	151	0.34	18.23	1.71	2.42	0.08	2.66e+02	4.35e+02	1.64
3	JPSG031.933+00.090	31.933	0.090	17	8	244	0.40	16.38	2.62	2.19	0.12	4.06e+02	4.94e+02	1.22
4	JPSG031.945+00.076	31.945	0.076	17	12	211	0.49	14.99	3.92	2.97	0.18	1.01e+03	6.45e+02	0.64
5	JPSG031.951+00.071	31.951	0.071	10	5	148	0.24	15.31	3.60	9.01	0.17	3.72e+02	1.66e+02	0.45
6	JPSG031.961+00.066	31.961	0.066	19	11	97	0.50	15.58	3.49	3.07	0.16	1.08e+03	7.45e+02	0.69
7	JPSG031.965+00.062	31.965	0.062	9	6	116	0.25	15.60	3.53	7.87	0.17	3.64e+02	1.95e+02	0.54
8	JPSG031.970+00.060	31.970	0.060	14	10	255	0.41	15.67	3.43	4.59	0.16	8.87e+02	5.08e+02	0.57
9	JPSG031.984+00.065	31.984	0.065	14	12	231	0.44	15.36	4.32	3.51	0.20	8.93e+02	6.76e+02	0.76
10	JPSG031.991+00.067	31.991	0.067	12	6	260	0.29	15.15	4.78	6.93	0.22	4.94e+02	2.94e+02	0.60
11	JPSG031.997+00.067	31.997	0.067	10	10	110	0.34	15.12	4.94	4.37	0.23	5.11e+02	4.17e+02	0.82
12	JPSG032.012+00.057	32.012	0.057	17	13	205	0.51	14.00	10.85	8.76	0.51	3.36e+03	1.23e+03	0.37
13	JPSG032.019+00.065	32.019	0.065	22	16	144	0.64	14.92	10.02	7.39	0.47	5.69e+03	2.64e+03	0.46
14	JPSG032.027+00.059	32.027	0.059	20	11	95	0.51	15.39	10.89	8.86	0.51	3.37e+03	2.13e+03	0.63
15	JPSG032.037+00.056	32.037	0.056	17	10	262	0.45	17.59	11.49	13.76	0.54	3.56e+03	3.75e+03	1.05
16	JPSG032.044+00.059	32.044	0.059	16	13	109	0.49	22.32	8.08	7.15	0.38	2.50e+03	1.33e+04	5.30
17	JPSG032.062+00.071	32.062	0.071	10	6	132	0.27	19.62	2.03	3.86	0.09	2.09e+02	4.96e+02	2.37
18	JPSG032.087+00.076	32.087	0.076	21	7	237	0.42	20.53	1.55	1.54	0.07	3.20e+02	1.22e+03	3.83
19	JPSG032.097+00.076	32.097	0.076	11	7	154	0.30	20.66	1.58	3.10	0.07	2.44e+02	6.78e+02	2.78
20	JPSG032.116+00.090	32.116	0.090	25	18	150	0.73	29.94	1.74	1.05	0.08	1.17e+03	3.92e+04	33.49
21	JPSG032.141+00.172	32.141	0.172	14	8	240	0.36	18.85	1.67	2.50	0.08	3.46e+02	6.06e+02	1.75
22	JPSG032.150+00.133	32.150	0.133	22	17	262	0.66	22.69	4.62	3.11	0.22	2.62e+03	1.59e+04	6.07
23	JPSG032.152+00.122	32.152	0.122	16	12	125	0.47	20.09	3.42	2.84	0.16	8.83e+02	3.01e+03	3.41
24	JPSG032.160+00.113	32.160	0.113	17	11	209	0.47	19.73	2.00	1.73	0.09	5.15e+02	1.57e+03	3.05
25	JPSG032.164+00.171	32.164	0.171	9	6	216	0.25	17.86	2.12	2.37	0.10	1.10e+02	2.67e+02	2.43
26	JPSG032.170+00.135	32.170	0.135	9	6	163	0.25	19.23	2.19	2.44	0.10	1.13e+02	4.27e+02	3.77
27	JPSG032.179+00.137	32.179	0.137	11	5	147	0.25	18.54	1.97	2.13	0.09	1.02e+02	3.16e+02	3.10

Table 2. Table 2: Physical properties statistics of JPS clumps

	$r_{\mathrm{eq}}$	$T_{\mathrm{dust}}$	$N_{H_2}$	$n_{H_2}$	${\Sigma }_{\mathrm{mass}}$	Mass	Luminosity	L/M ratio
	pc	K	${10}^{22}{\mathrm{cm}}^{-2}$	${10}^4{\mathrm{cm}}^{-3}$	$g{\mathrm{cm}}^{-2}$	$M_{☉}$	$L_{☉}$	$L_{☉}/M_{☉}$
Starless clumps
Min	0.24	10.8	1.43	1.31	0.07	6.51e+01	3.53e+01	0.14
Max	0.5	18.1	6.22	6.24	0.29	6.11e+02	2.32e+02	3.07
Median	0.34	12.72	2.86	1.73	0.13	2.57e+02	1.03e+02	0.37
Mean	0.33	13.48	3.12	2.42	0.15	2.67e+02	1.17e+02	0.80
Protostellar clumps
Min	0.25	11.68	0.64	0.38	0.03	4.8e+01	7.16e+01	0.22
Max	0.73	38.18	11.6	6.31	0.54	1.99e+03	1.30e+05	233.60
Median	0.46	17.05	2.875	1.66	0.14	4.66e+02	7.88e+02	2.00
Mean	0.45	19.64	3.98	2.14	0.19	6.71e+02	1.42e+04	24.47

Table 3. Table 3: Line profiles parameters of JPS clumps. Optically thick line: CO (3-2), Optically thin line: 13 CO (3-2). ‘*’ labels the clumps of which C 18 O (3-2) are used as optically thin lines in the following discussion; (2): Evolutionary type of clumps (see in Section  3.4 ); (3)-(9): parameters used in Equation  19 to select blue/red profile; (7): line profiles, BP: blue profile, RP: red profile. ‘**’ labels the clumps which may be misclassified by multiple velocity components; (11): Infall velocities of the clumps with blue profiles, obtained from Equation  20 .

ID	Name	Type	$V_{\mathrm{peak},\mathrm{thick}}$	$T_{r,\mathrm{thick}}$	$V_{\mathrm{peak},\mathrm{thin}}$	$T_{r,\mathrm{thin}}$	${\tau }_{{}_{}{}^{13}\mathrm{CO}}$	FWHMthin	${\delta }_V$	Profile	$V_{\mathrm{in}}$
			km s-1	K	km s-1	K		km s-1		km s-1
	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)
4	JPSG031.945+00.076	Protostellar	93.1	8.29	96.00	2.45	0.35	2.96	-0.98	BP	0.21
9	JPSG031.984+00.065	Protostellar	92.1	11.85	93.68	3.88	0.40	2.52	-0.63	BP**
10	JPSG031.991+00.067	Protostellar	92.1	14.43	94.62	4.01	0.33	4.09	-0.62	BP	0.47
11	JPSG031.997+00.067	Protostellar	92.1	13.51	95.10	4.46	0.40	3.68	-0.82	BP	0.21
12	JPSG032.012+00.057	Protostellar	94.1	13.75	96.31	5.03	0.45	3.31	-0.67	BP**
13	JPSG032.019+00.065	Protostellar	97.1	14.59	98.87	5.99	0.53	4.15	-0.43	BP**
14	JPSG032.027+00.059	Protostellar	93.1	19.47	94.73	6.34	0.39	3.80	-0.43	BP**
15	JPSG032.037+00.056	Protostellar	93.1	17.54	94.49	8.08	0.62	3.26	-0.43	BP**
16	JPSG032.044+00.059	Protostellar	92.1	16.45	94.82	10.36	0.99	3.45	-0.79	BP	0.13
17	JPSG032.062+00.071	Protostellar	95.1	13.42	95.42	6.32	0.63	2.44	-0.13	…
18	JPSG032.087+00.076	Protostellar	97.1	14.25	97.01	8.57	0.92	2.00	0.05	…
19	JPSG032.097+00.076	Protostellar	94.1	10.80	95.98	7.04	1.05	2.58	-0.73	BP	0.44
20	JPSG032.116+00.090*	Protostellar	97.1	22.46	96.18	6.40	0.96	2.65	0.35	RP
22	JPSG032.150+00.133*	Protostellar	96.1	19.77	94.45	3.35	0.72	3.94	0.42	RP
23	JPSG032.152+00.122	Protostellar	96.1	15.60	96.08	8.56	0.80	2.32	0.01	…
24	JPSG032.160+00.113	Protostellar	96.1	14.88	96.46	7.46	0.70	1.82	-0.20	…
26	JPSG032.170+00.135	Starless	95.1	8.72	94.83	3.61	0.53	1.94	0.14	…
27	JPSG032.179+00.137	Starless	95.1	5.92	94.94	2.70	0.61	1.67	0.09	…

Table 4. Table 4: The gas dynamical properties and viral parameters of JPS clumps derived from 13 CO (3-2) data. (3): The velocity dispersion which is derived from the optically thin line width in Table  3 . (5)-(8): Viral mass, Jeans mass, Jeans length and viral parameter calculated from velocity dispersion.

ID	Name	Req	${\sigma }_v$	Mclump	Mvir	MJeans	${\lambda }_{\mathrm{Jeans}}$	${\alpha }_{\mathrm{vir}}$
		pc	km s-1	M☉	M☉	M☉	pc
	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)
4	JPSG031.945+00.076	0.49	1.27	1.01e+03	9.24e+02	4.71e+02	0.76	0.92
9	JPSG031.984+00.065	0.44	1.09	8.93e+02	6.07e+02	2.71e+02	0.60	0.68
10	JPSG031.991+00.067	0.29	1.75	4.94e+02	1.03e+03	7.95e+02	0.68	2.08
11	JPSG031.997+00.067	0.34	1.57	5.11e+02	9.79e+02	7.32e+02	0.77	1.92
12	JPSG032.012+00.057	0.51	1.42	3.36e+03	1.19e+03	3.80e+02	0.49	0.36
13	JPSG032.019+00.065	0.64	1.77	5.69e+03	2.34e+03	8.08e+02	0.67	0.41
14	JPSG032.027+00.059	0.51	1.63	3.37e+03	1.57e+03	5.69e+02	0.56	0.47
15	JPSG032.037+00.056	0.45	1.40	3.56e+03	1.03e+03	2.92e+02	0.39	0.29
16	JPSG032.044+00.059	0.49	1.49	2.50e+03	1.26e+03	4.82e+02	0.57	0.50
17	JPSG032.062+00.071	0.27	1.06	2.09e+02	3.52e+02	2.38e+02	0.55	1.68
18	JPSG032.087+00.076	0.42	0.88	3.20e+02	3.80e+02	2.18e+02	0.73	1.19
19	JPSG032.097+00.076	0.30	1.12	2.44e+02	4.36e+02	3.12e+02	0.65	1.79
20	JPSG032.116+00.090	0.73	1.38	1.17e+03	1.61e+03	1.00e+03	1.38	1.38
22	JPSG032.150+00.133	0.66	1.77	2.62e+03	2.40e+03	1.23e+03	1.03	0.91
23	JPSG032.152+00.122	0.47	1.01	8.83e+02	5.59e+02	2.41e+02	0.62	0.63
24	JPSG032.160+00.113	0.47	0.81	5.15e+02	3.57e+02	1.58e+02	0.63	0.69
26	JPSG032.170+00.135	0.25	0.85	1.13e+02	2.12e+02	1.56e+02	0.56	1.87
27	JPSG032.179+00.137	0.25	0.74	1.02e+02	1.61e+02	1.11e+02	0.52	1.58

Equations

$$I_{\nu }=B_{\nu }(T_d)(1-e^{-{\tau }_{\nu }}),$$

$${\tau }_{\nu }={\mu }_{{\mathrm{H}}_2}{m}_{\mathrm{H}}{\kappa }_{\nu }{N}_{{\mathrm{H}}_2}/\mathrm{GDR}.$$

$${\kappa }_{\nu }={\kappa }_{{\nu }_0}{\left(\frac{\nu }{{\nu }_0}\right)}^{\beta },$$

$$M_{\mathrm{clump}}={\mu }_{{\mathrm{H}}_2}{m}_{\mathrm{H}}{d}^2{\Omega }_{\mathrm{pix}}\sum N_{{\mathrm{H}}_2},$$

$$N_{{\mathrm{H}}_2}=\sum N_{{\mathrm{H}}_2}/{\mathrm{N}}_{\mathrm{sour}},$$

$$n_{{\mathrm{H}}_2}=\frac{M_{\mathrm{clump}}}{(4/3)\pi r_{\mathrm{eq}}^3{\mu }_{{\mathrm{H}}_2}{m}_{\mathrm{H}}},$$

$${\Sigma }_{\mathrm{mass}}=\frac{M_{\mathrm{clump}}}{\pi r_{\mathrm{eq}}^2},$$

$$L_{\mathrm{clump}}=4\pi d^2{\Omega }_{\mathrm{pix}}\sum I_{\mathrm{int}},$$

$$\mathrm{L}/\mathrm{M}\mathrm{ratio}=L_{\mathrm{clump}}/M_{\mathrm{clump}}$$

$$p(\eta )=\frac{p_0}{\sqrt{2\pi {\sigma }_0^2}}\exp \left(\frac{-(\eta -{\eta }_0{)}^2}{2{\sigma }_1^2}\right),$$

$$p(\eta )\propto N_{{\mathrm{H}}_2}^m,$$

$$\lambda =22H=\frac{22c_{\mathrm{s}}}{(4\pi G\rho {)}^{1/2}},$$

$${\sigma }_v=\sqrt{{\sigma }_{{}^{13}\mathrm{CO}}^2+k_{B}T\left(\frac{1}{{\mu }_{{\mathrm{H}}_2}{m}_{\mathrm{H}}}-\frac{1}{m_{{}^{13}\mathrm{CO}}}\right)},$$

$${\tau }_{\mathrm{age}}\ge {\tau }_{\mathrm{crit}}\simeq \frac{\lambda }{2c_s},$$

$${\tau }_{\mathrm{crit}}=\frac{\lambda }{2{\sigma }_v}=\frac{M_{\mathrm{line},\mathrm{crit}}}{\dot{M}},$$

$${\tau }_{\mathrm{age}}=\frac{M_{\mathrm{line}}}{\dot{M}}.$$

$$T_r=\frac{h\nu }{k}[\frac{1}{\exp (h\nu /k{T}_{\mathrm{ex}})}-\frac{1}{\exp (h\nu /k{T}_{\mathrm{bg}})}]\times (1-e^{-\tau })f,$$

$$\frac{T_{r,\mathrm{CO}}}{T_{r{,}^{13}\mathrm{CO}}}\approx \frac{1-\exp (-{\tau }_{\mathrm{CO}})}{1-\exp (-{\tau }_{{}^{13}\mathrm{CO}})},$$

$${\delta }_V=(V_{\mathrm{thick}}-V_{\mathrm{thin}})/\Delta V_{\mathrm{thin}},$$

$$V_{\mathrm{in}}\approx \frac{{\sigma }_v}{v_{\mathrm{red}}-v_{\mathrm{blue}}}\ln \left(\frac{1+e{T}_{\mathrm{BD}}/T_{\mathrm{D}}}{1+e{T}_{\mathrm{RD}}/T_{\mathrm{D}}}\right),$$

$$M_{\mathrm{J}}=\frac{{\pi }^{5/2}{c}_{\mathrm{s}}^3}{6\sqrt{G^3\rho }},$$

$$\mathrm{and}{\lambda }_{\mathrm{J}}=c_{\mathrm{s}}{\left(\frac{\pi }{G\rho }\right)}^{1/2},$$

$${\alpha }_{\mathrm{vir}}\equiv \frac{M_{\mathrm{vir}}}{M_{\mathrm{clump}}}=\frac{5{\sigma }_v^2{R}_{\mathrm{eq}}}{G{M}_{\mathrm{clump}}},$$

Full text

††pubyear: 2018††pagerange: Star formation in IRDC G31.97+0.07–A

1 Introduction

Massive stars ($M\geq 8M_{\sun}$) and massive stellar clusters play an important role in moulding the galactic environment and determining the metal enrichment through their ionizing radiation, stellar winds, outflows and explosive deaths. However, much remains to be understood about the formation and protostellar evolution of high-mass stars. The initial conditions and the earliest evolutionary stages of massive star formation are still unclear.

Infrared dark clouds (IRDCs) have been proposed to be good candidates for the birthplaces of massive stars and clusters. IRDCs were first observed as dark silhouettes against the bright Galactic background infrared (IR) emission by the Midcourse Space Experiment (MSX; Carey et al. 1998; Egan et al. 1998; Simon et al. 2006), and the Infrared Space Observatory (Hennebelle et al., 2001). Previous molecular lines and continuum studies suggested that IRDCs were cold ($<20\,$K) and dense ($>10^{5}\,\mathrm{cm}^{-3}$) gas collections, typically arranged in filamentary and/or globular structures with compact cores. Their masses range from $10^{2}$ to $10^{5}\,M_{\sun}$ with a scale of several to tens of pc (Carey et al., 1998; Egan et al., 1998; Rathborne et al., 2006). All of these properties imply that IRDCs are an excellent mass reservoir for massive star formation.

The filamentary IRDC can be seen as a self-gravitating gas cylinder. Theoretical work predicted that the filament should fragment into multiple cores with quasi-regular spacing due to the “sausage" instability (Chandrasekhar & Fermi, 1953; Nagasawa, 1987; Inutsuka & Miyama, 1992; Tomisaka, 1995). That was approved by observational studies showing that many cores/clumps were embedded in filamentary IRDCs (Rathborne et al., 2006; Wang et al., 2008; Henning et al., 2010; Jackson et al., 2010; Wang et al., 2014). In addition, recent studies have revealed massive young stellar objects (YSOs) and cores in IRDCs (Rathborne et al., 2006, 2007, 2011; Beuther & Steinacker, 2007; Wang et al., 2008; Henning et al., 2010), indicating that IRDCs are active subjects with undergoing massive star formation.

G31.97+0.07 was first identified by MSX (Simon et al., 2006). The IR morphology has a long, thin filamentary structure, and it is located in the west side of an active star-forming complex containing several (UC)H II regions (see Figure 1 in Sec. 2.1). The central velocity of the IRDC is $96.7\,$ km s*-1* given by Simon et al. (2006). The central velocity and position are well consistent with the molecular cloud GRSMC G032.09+00.09 (Roman-Duval et al., 2009). So in this work, we use the distance of GRSMC G032.09+00.09 for our object. The distance is $7.07\,$kpc, using the Clemens rotation curve of the Milky Way (Clemens, 1985) and solving the kinematic distance ambiguity by H I self-absorption analysis.

Rathborne et al. (2006) observed nine dense clumps with masses ranging from $151\ \mathrm{to}\ 1890\,M_{\sun}$ using the IRAM 30 m single-dish telescope. Wang et al. (2006) reported water masers in G31.97+0.07, and Urquhart et al. (2009) identified an H II region in G31.97+0.07. Battersby et al. (2014) studied G31.97+0.07 in the NH${}_{3}\ (1,\ 1),\ (2,\ 2)\mathrm{and}\ (4,4)$ transitions with the VLA. They identified 11 dense cores less 0.1 pc in size, and found that those cores were virially unstable to gravitational collapse. They also reported that turbulence likely set the fragmentation length scale in the filament. Moreover, they note the existence of at least three large bubbles around the filament, which are likely older H II regions. They suggested that those large H II regions may have compressed the molecular gas to form and/or shape the IRDC and trigger more recent massive star formation. Using the IRAM 30 m and CSO 10.4 m telescopes, Zhang et al. (2017) performed observations with HCO+, HCN, N2H+, C18O, DCO+, SiO and DCN toward G31.97+0.07. They reported that C18O emission may be heavily depleted at the peak positions of some cold cores. And they suggest that some active cores are collapsing while their envelopes are expanding.

In this work, we utilize multiple wavebands of continuum data (from Mid-IR to 850 µm), and molecular line data of three CO isotopes at $J=1\rightarrow 0$ and $J=3\rightarrow 2$ transition, to study the dynamical structure of IRDC G31.97+0.07 and star formation processes in dense clumps embedded in the structure. The data used in this work are described in Sec. 2. We present results of large scale cloud and the dust and gas properties derived from continuum and molecular lines data in Sec. 3. Further discussion about gas dynamics and dense clumps properties is given in Sec. 4. We summarize our work in Sec. 5.

2 Observations and Data Reduction

2.1 IR to Sub-millimetre Continuum Data

We utilized near-IR and mid-IR data from the GLIMPSE and MIPSGAL survey. The GLIMPSE survey observed the Galactic plane with four IR bands (3.6, 4.5, 5.8, 8.0 µm) of the IRAC instrument on the $\it Spitzer$ Space Telescope, and the sky coverage is $|b|<1\degr$ for $10\degr<\ell<65\degr$ (Benjamin et al., 2003; Churchwell et al., 2009). MIPSGAL is a survey of the same area as GLIMPSE, using MIPS instrument at 24 and 70 µm on $\it Spitzer$ (Carey et al., 2009). The point-source catalogue from GLIMPSE (Spitzer Science, 2009) and 24 µm point-source catalogue from MIPSGAL (Gutermuth & Heyer, 2015) have been used to identify young stellar object (YSO) candidates .

Far-IR data were obtained from the Hi-GAL survey to investigate dust properties of our object. Hi-GAL ($\it Herschel$ Infrared Galactic Plane survey, Molinari et al. 2010) is a key project of $\it Herschel$ Space Observatory which mapped the entire Galactic plane with $|b|\leq 1\degr$ The Hi-GAL data were observed in the fast mode using the PACS (70 and 160  µm) and SPIRE (250, 350, 500 µm) instruments in the parallel mode. We used the $\it Herschel$ Interactive Processing Environment (HIPE) to download the continuum maps from the $\it Herschel$ Science Archive and to reduce the data using standard pipelines. The effective angular resolution is $10.\!\!\arcsec 2,\ 13.\!\!\arcsec 5,18.\!\!\arcsec 1,\ 24.\!\!\arcsec 9,\ \mathrm{and}\ 36.\!\!\arcsec 4$ at 70, 160, 250, 350, and 500 µm, respectively. In addition, the 70 µm point-source catalogues from Hi-GAL (Molinari et al., 2016a) were used to investigate evolutionary stages of the clumps in our object.

We extracted 850 µm continuum data from the James Clerk Maxwell Telescope (JCMT) Plane Survey (JPS) (Moore et al., 2015; Eden et al., 2017). JPS is a targeted, yet unbiased, survey of the inner Galactic Plane in the longitude range $7\degr<\ell<63\degr$ and latitude range $|b|<0.8\degr$ using Sub-millimetre Common-User Bolometer Array 2 (SCUBA-2; Holland et al. 2013) at 850 µm with an angular resolution of 14.5″. The survey observing strategy is to map six large, regularly spaced fields within the area. Each individual survey field is sampled using a regular grid of eleven circular tiles with a uniform diameter of one degree, observed using the pong3600 mode (Bintley et al., 2014). The data reduction process removes structures larger than 480 arcsec from the final maps, which are gridded to 3-arcsec per pixel. The average smoothed pixel-to-pixel noise is $7.19\,\mathrm{mJy\,beam}^{-1}$. The JPS 850 µm compact-source catalogue identified by Eden et al. (2017) using the Fellwalker algorithm (Berry, 2015) is also adopted in this study.

2.2 Molecular Line Data

13CO (1-0) emission data from the Boston University-Five College Radio Astronomy Observatory Galactic Ring Survey (BU-FCRAO GRS; Jackson et al. 2006) were used to identify large-scale structures. The survey covered a longitude range of $18\degr<\ell<55.\!\!\degr 7$ and a latitude range of $|b|<1\degr$, using the SEQUOIA multipixel array on the Five College Radio Astronomy Observatory 14 m telescope. The velocity resolution of the survey is 0.21 km s*-1*, and angular resolution is 46″. The typical rms sensitivity is $\sigma(T_{A}^{*})\sim 0.13\mathrm{K}$.

We used CO (3-2) data from the CO High Resolution Survey (COHRS; Dempsey et al. 2013) to study warm and high-velocity gas that may be excited by shocks and outflows from active star-forming regions. This survey covers $|b|<0.5\degr$ between $10\degr<\ell<65\degr$ using the HARP instrument on the JCMT. The final data cube were smoothed to the angular resolution of 16.6 arcsec and rebinned to 1 km s*-1*. The mean rms is about 1 K and a main-beam efficiency of $\eta_{\mathrm{mb}}=0.61$.

Data from the 13CO/C18O (3-2) Heterodyne Inner Milky-Way Plane Survey (CHIMPS: Rigby et al., 2016) were used to investigate the denser part (e.g. clumps embedded in the structure). CHIMPS was carried out using the HARP on the JCMT, observing 13CO and C18O (3-2) simultaneously. The survey covers $|b|\leq 0.5\degr$ and $28\degr\lesssim\ell\lesssim 46\degr$, with an angular resolution of 15 arcsec in 0.5 km s*-1*. The mean rms of the data is about 0.6 K, and a main-beam efficiency of $\eta_{\mathrm{mb}}=0.72$.

3 Results

3.1 IR and Sub-mm Continuum Emission

Figure 1 displays the three-colour image (red: JPS 850 µm, green: MIPSGAL 24 µm, blue: GLIMPSE 8 µm) of the filamentary IRDC G31.97+0.07 and its adjacent context. The filamentary structure is marked by the red-dashed line, and H II regions identified by Anderson et al. (2014) from WISE data are shown in magenta dashed circles. In Figure 1, the filamentary structure is divided into two parts: the eastern near/mid-IR dominant part and western sub-millimetre dominant part. The bright 8 µm emission is mainly attributed to polycyclic aromatic hydrocarbons (PAHs) excited by the UV radiation in the adjacent H II region (Pomarès et al., 2009). Thus, 8 µm emission can be utilized to delineate infrared bubbles. The eastern part, located on the rim of the large H II region G32.11+0.05, contains several 8 µm-bright H II regions: H II G32.16+0.13, H II G32.11+0.09 (or bubble N55 in Churchwell et al. 2006), and H II G32.06+0.08. This indicates that the eastern part hosts some young O or B type stars with strong stellar winds. Two UC H II regions located on the rim of N55 may suggest efficient star-forming in this region. The western part, mainly consisting of IRDC G31.97+0.07 and some other small filaments, displays a dark extinction feature. The green crosses show the positions of 9 cores, MM1-MM9, identified in 1.2 mm continuum data by Rathborne et al. (2006) using the IRAM 30 m telescope. The yellow crosses show the positions of 27 JCMT compact sources identified by Eden et al. (2017) from JPS 850 µm data.

The 24 µm continuum emission is mainly contributed by the thermal radiation of warm dust heated by protostars (Battersby et al., 2011; Guzmán et al., 2015). H II regions and infrared bubbles, distributed in the filamentary structure, exhibit strong emission in their inner regions in the 24 µm waveband. IRDC 31.97+0.07 mostly appears as dark extinction at 24 µm and 70 µm. Some of the mm continuum cores identified by Rathborne et al. (2006) and sub-mm compact sources identified by Eden et al. (2017) do not show 24 µm emission, indicating that they are cold and dense. Battersby et al. (2014) divided the IRDC into two parts, the ‘active part’ and the ‘quiescent part’. MM1, MM2, MM3, MM6 and MM9, belonging to the ‘active part’, are associated with mid-IR bright point sources. This suggests that these cores are undergoing star formation and heating the nearby dust.

Sub-mm continuum emission can be a good tracer of the cold and dense dust. It also can be an effective tool to search for clumps lacking signs of the star-forming process, or in early evolutionary stage. The long filament of IRDC G31.97+0.07 can be easily distinguished in the 850 µm image. The Bubble N55 and H II G32.06+0.08 are devoid of 850 µm emission, while other H II regions in the IRDC are associated with bright 850 µm clumps. This indicates that bubble N55 and H II G32.06+0.08 are more evolved than other H II regions in the filament. Considering the morphology in the continuum emission, we suggest that IRDC 31.97+0.07 and the H II regions on its east side may be one continuous structure, which is further confirmed by the continuity of velocity (see in Sec. 3.2).

3.2 CO Molecular Line Emission

We use GRS 13CO (1-0) molecular line emission to show the gas distribution of IRDC G31.97+0.07. Figure 2 shows the 13CO channel maps overlaid on 8 µm emission in steps of 1 km s*-1*, between 90 and 101 km s*-1*. Most 13CO emission is detected in the velocity interval of 93 to 99 km s*-1*, while UC H II regions and the ‘active’ clumps in the IRDC contain high-velocity components which may be driven by outflows from YSOs. The channel maps also show that IRDC G31.97+0.07 and the H II regions on its east side belong to the continuous filamentary structure also implied by continuum emission data, with a systemic velocity at 96.5 km s*-1*.

Figure 3 shows contours of GRS 13CO (1-0) integrated intensity overlaid on 8 µm and 850 µm continuum emission. The velocity interval is between 90 and 101 km s*-1*, with central velocity 96.5 km s*-1*. The red dashed line marks the filamentary structure. The 13CO (1-0) integrated map shows the molecular cloud GRSMC G032.09+00.09 identified by Rathborne et al. (2009) with central velocity 96.85 km s*-1* (Roman-Duval et al., 2009). The shape of the dense and gas-rich part of GRSMC G032.09+00.09 is consistent with the filamentary structure of the continuum. That suggests that IRDC 31.97+0.07 and the H II regions on its east side are one continuous structure embedded in the molecular cloud GRSMC G032.09+00.09, and the distance of the cloud is $7.07$ kpc (Roman-Duval et al., 2009). The length of the whole filamentary structure is about 20 arcmin, and $\sim 41$ pc at this distance. The radii of the JPS compact sources located in the structure range from 0.2 to 0.8 pc, hence they are molecular clumps.

3.3 Spectral Energy Distribution (SED) Fitting

To determine the properties of the dust, we fit the SED of our object pixel by pixel based on $\it Herschel$ data. As the most part of the IRDC is 70 µm dark, we just take emission at 160, 250, 350, 500 µm into account in our SED fitting.

In order to measure the flux of multi-waveband observations in different angular resolutions, we convolved our image to a common angular resolution by a Gaussian kernel with FWHM equal to $\sqrt{\theta_{500}^{2}-\theta_{\lambda}^{2}}$, where $\theta_{500}$ is FWHM beam size of Hi-GAL 500 µm band and $\theta_{\lambda}$ is the FWHM beam size of each Hi-GAL band. Then we re-gridded our data to be aligned pixel-by-pixel with the same pixel size of 14″.

We used the smoothed data to fit the SED for each pixel by using a grey-body model:

[TABLE]

where $I_{\nu}$ is the monochromatic intensity, $B_{\nu}(T_{d})$ is the Planck function and $\tau_{\nu}$, the optical depth at frequency $\nu$, is given by

[TABLE]

Here $\mu_{\mathrm{H}_{2}}=2.8$ is the mean molecular weight of molecular hydrogen (Kauffmann et al., 2008), $m_{H}$ is the mass of a hydrogen atom, $N_{\mathrm{H}_{2}}$ is the column density of $H_{2}$, GDR is the gas-to-dust ratio by mass and is assumed to be 100. The dust opacity $\kappa_{\nu}$ can be approximated by a power law,

[TABLE]

where $\nu_{0}=600\,\mathrm{GHz}$, $\beta$ fixed to be $2.0$ is the dust emissivity index and $\kappa_{\nu_{0}}=3.33\,\mathrm{cm}^{2}\,\mathrm{g}^{-1}$ (from column 5 of table 1 in Ossenkopf & Henning 1994), but scaled down by a factor of 1.5 as suggested in Kauffmann et al. (2010).

The fitting is performed by using curve_fit in the Python package scipy.optimization111https://docs.scipy.org/doc/scipy/reference/tutorial/optimize.html. For pixels with intensity in 160 µm lower than $60\,\mathrm{MJy}\,\mathrm{sr}^{-1}$ (about $3\sigma$), we found that the fitting result was not so reliable. So, for those pixels, only 250, 350, 500 µm data were used in fitting.

The dust temperature and column density maps produced by SED fitting are presented in Figure 4. The dust temperature of the whole filamentary structure ranges from $10$ to $30$ K, and the column density ranges from $0.5\times 10^{22}$ to $10^{23}\,\mathrm{cm}^{-2}$. Most of the IRDC is cold (lower than $18$ K) with relatively high column density (above $2\times 10^{22}\,\mathrm{cm}^{-2}$), while the H II and UC H II regions have higher dust temperature and lower column density.

We find that the contour of $1.1\times 10^{22}$ cm*-2* is similar to the CO integrated map of the cloud, and the contour of $1.8\times 10^{22}$ cm*-2* is consistent with the morphology of the IRDC G31.97+0.07 in IR maps. So, choosing those values as thresholds, we derive that the mass of the whole structure is $\sim 2.5\times 10^{5}\,M_{\sun}$ and the mass of the IRDC is $\sim 7.8\times 10^{4}\,M_{\sun}$.

The clump mass and dust temperature can be directly derived from the SED fitting result:

[TABLE]

here, $d=7.07\,\mathrm{kpc}$ is the distance of the filament structure, $\Omega_{\mathrm{pix}}$ is solid angular size of one pixel and $\sum N_{\mathrm{H_{2}}}$ is the sum of the column density value for all the pixels in the clump.

The source average H2 column densities are calculated as:

[TABLE]

where $\mathrm{N}_{\mathrm{sour}}$ is the number of pixels located in the clump. The average H2 volume densities are derived via:

[TABLE]

and the mass surface densities are given by:

[TABLE]

where $r_{\rm eq}$ is equivalent radius.

We also determined the integrated intensity in the SED fitting process, so luminosities of sources are derived from:

[TABLE]

where $\sum I_{\mathrm{int}}$ is the sum of the frequency-integrated intensity($I_{\mathrm{int}}=\int I_{\nu}\,d\nu$) for each pixel in the clump. And the luminosity-mass ratio is given by:

[TABLE]

Table 1 lists the resulting dust temperature, column density, volume density, mass surface density, mass, luminosity and luminosity-mass ratio of each source.

3.4 Distribution of YSOs

We selected sources with 3.6, 4.5, 5.8, 8.0 µm emission from the GLIMPSE I point-source catalogue (Spitzer Science, 2009) to identify young stellar objects (YSOs). Classification was performed using the method given in Appendix A of Gutermuth et al. (2009). In brief, the method uses flux ratios or colours, to classify YSO candidates into Class I and Class II, after excluding contamination like star-forming galaxies, broad-line active galactic nuclei (AGNs), and unresolved knots of shock emission. The result of the YSO identification is shown in Figure 5. Also, the 24 µm point sources and 70 µm point sources can be considered as the signature of early-stage star formation, hence, we plot them in Figure 5 to show the star-formation process in the area more completely. 24 µm point sources come from Gutermuth & Heyer (2015) and 70 µm point sources come from Molinari et al. (2016a).

We classify JPS clumps in the filament into two groups: protostellar clumps, which are associated with any YSOs, 24 µm or 70 µm point sources, and starless clumps, which are not. These two groups of clumps are shown in Figure 5, with protostellar clumps as green ellipses and starless clumps as yellow ellipses. We identify 9 starless clumps and 18 protostellar clumps. Most of starless clumps are located along the IRDC, while most of protostellar clumps are distributed in the active star-forming area. The distribution of JPS clumps indicates that, along the filament, the evolutionary stages of star formation vary from the east to the west. The H II regions and IR bubbles which are more highly evolved are located in the east of the filament, while the west of the filament is quiescent. The IRDC can be divided into two parts, the active part and the quiescent part as mentioned in Battersby et al. (2014). The active part, containing plenty of YSOs and mid-IR point sources, displays signs of active star formation. While the quiescent part, in which most of clumps are starless, shows weak star-forming activity at present. Most of the protostellar clumps are distributed in the highly evolved part (around H II regions or in active part of the IRDC). But, in our work, it is hard to distinguish whether the star formation in those clumps is triggered or pre-existing.

The physical properties statistics for both starless clumps and protostellar clumps are presented in Table 2. Starless clumps have lower temperature. And the average luminosity-mass ratio of starless clumps is less than 1, while that of protostellar clumps is about 24.5.

4 DISCUSSION

4.1 Dynamical Structure of IRDC G31.97+0.07

4.1.1 Column Density Probability Distribution Functions(PDFs)

The column density probability distribution function (PDF) is defined as the probability of finding gas within a bin [N,N+dN]. PDFs are a useful tool to study the properties of ISM (Padoan et al., 1997b; Burkhart et al., 2013), the core and stellar IMF (Padoan & Nordlund, 2002; Elmegreen, 2011; Veltchev et al., 2011; Donkov et al., 2012), and the SFR (Krumholz & McKee, 2005; Padoan & Nordlund, 2011; Federrath & Klessen, 2012).

Theoretical work and simulations (Vazquez-Semadeni, 1994; Padoan et al., 1997a; Kritsuk et al., 2007; Vázquez-Semadeni et al., 2008; Federrath et al., 2008, 2010) show that the density PDF of the gas dominated by isothermal supersonic turbulence is approximated well by a log-normal form. Deviations from the log-normal shape, which are mostly in the form of power-law tails, are present in simulations when compressible turbulence and/or self-gravity are considered (Passot & Vázquez-Semadeni, 1998; Klessen et al., 2000; Kritsuk et al., 2011; Federrath & Klessen, 2013).

As many previous studies did, due to the huge range of column densities, we switch to a logarithmic scale $\eta=\ln(N_{\mathrm{H_{2}}})/\langle N_{\mathrm{H_{2}}}\rangle$, where $N_{\mathrm{H_{2}}}$ is the column density and $\langle N_{\mathrm{H_{2}}}\rangle$ is the average column density of the cloud. So, the form of the PDF at low-densities which is characterized by being log-normal can be written as:

[TABLE]

where $\eta_{0}$, $p_{0}$ and $\sigma_{0}$ are the peak value, integral probability and standard deviation. And the power-law tail at high-densities:

[TABLE]

The models including self-gravity (Klessen et al., 2000; Kritsuk et al., 2011; Federrath & Klessen, 2013) indicate that the exponent $m$ can respond to an equivalent spherical density profile of exponent $\alpha$: $\rho(r)\propto r^{-\alpha}$, where $\alpha=1-2/m$. Those work also suggest that $\alpha$ of a spherical self-gravitating cloud value between 1.5 and 2 ($-4<m<-2$), which is supported by observational study of several IRDCs (Schneider et al., 2015; Yuan et al., 2018).

In this work, we determined the PDFs in two regions which are shown in Figure 6. Region 1 covers the whole filamentary structure including the H II regions and the IRDC. Region 2 only contains the IRDC in the west. Figure 7 shows the PDFs of those two regions. The error bars show the statistical Poisson error in each bin. The average column density $\langle N\rangle=1.68\times 10^{22}\,\mathrm{cm}^{-2}$. The black dashed line corresponds to the last closed contour which is regarded as the completeness limit (Lombardi et al., 2015; Ossenkopf-Okada et al., 2016; Alves et al., 2017). The completeness limit is $-0.42$ for Region 1, and $0.36$ for Region 2.

For the part above the completeness limit, neither PDFs of Region 1 nor Region 2 represent log-normal well. Thus we only discuss their power-law tails. One can easily notice that the PDF of Region 1 can be described by two power-laws with demarcation between 0.5 and 0.6. The slope $m$ is $-3.35(0.11)$ for the power-law at lower column densities and $-1.74(0.36)$ for the power-law at higher column densities. We fit the power-law for the PDF of Region 2 from 0.5 to 1.2, the slope $m$ is $-1.50(0.32)$. Comparing the PDFs of Region 1 and Region 2, we suggest that flat power-law at higher column densities is mainly contributed from the IRDC. That means more dense gas accumulate in the IRDC. The power-law of the PDF of the IRDC is flatter than theoretical prediction, which implies that this region may be compressed by the adjacent H II region at the south west according Figure 1. This suggests that when the ionized-gas pressure is higher than the turbulent ram pressure the ionized-gas would compress the material around the ionizing source, as shown in the numerical simulation of Tremblin et al. (2012). And the ionization compression can also play a role at high column densities which are expected to be gravity-dominant, as found by Tremblin et al. (2014) in Rosette nebula.

4.1.2 Investigating of the fragmentation of the IRDC

The model of a self-gravitating fluid cylinder can be used to describe a filamentary IRDC. The theoretical work describing the fragmentation of fluid cylinders due to the “sausage" instability has been established decades earlier (e.g., Chandrasekhar & Fermi 1953; Nagasawa 1987; Inutsuka & Miyama 1992; Tomisaka 1995). The theory predicts that the filament should fragment into multiple cores with quasi-regular spacing, that has been observed in several filamentary molecular clouds (e.g., Jackson et al. 2010; Wang et al. 2014; Henshaw et al. 2016). The characteristic spacing between cores corresponds to the wavelength of the fastest-growing unstable mode of the fluid instability.

For an incompressible fluid cylinder, the fragment spacing is $\lambda=11R$, where $R$ is the radius of the cylinder, given by Chandrasekhar & Fermi (1953). And for an infinite isothermal gas cylinder (Nagasawa, 1987):

[TABLE]

where $H$ is the isothermal scale height, $c_{\rm s}$ is the sound speed, $G$ is the gravitational constant, and $\rho$ is the gas density of the filament. For a finite isothermal cylinder embedded in external uniform medium, the fragments spacing depend on the ratio of cylinder radius R and isothermal scale height H. If $R\gg H$, $\lambda=22H$, and if $R\ll H$, $\lambda=11R$.

For IRDC G31.97+0.07, the sound speed $c_{\rm s}=[(k_{\rm B}T)/(\mu_{\rm p}m_{\rm H})]^{1/2}\approx 0.22\,\mathrm{km\,s}^{-1}$, where $k_{\rm B}$ is the Boltzmann constant, $T\approx 16\,\mathrm{K}$ is the mean temperature derived from SED fitting, $\mu_{\rm p}=2.33$ is the mean molecular weight per free particle, and $m_{\rm H}$ is the mass of atomic hydrogen. Thus, the isothermal scale height $H=c_{\rm s}/(4\pi G\rho)^{1/2}\approx 0.015\,\mathrm{pc}$, where $\rho=\mu_{\rm H_{2}}m_{\rm H}n_{\rm H_{2}}$, and $n_{\rm H_{2}}\sim 5.6\times 10^{4}\,\mathrm{cm}^{-3}$. Since the radius $R$ of the filament ($\sim 0.01\degr$, or $\sim 1.2\,\mathrm{pc}$ at $7.07$ kpc) is much larger than the isothermal scale height $H$, under the isothermal and thermally supported assumption, the predicted theoretical fragment spacing is $\lambda=22H=0.33\,\mathrm{pc}$.

The spacings of 15 clumps located in the IRDC range from $0.66$ to $1.31$ pc, with an average of $0.97$ pc and a median of $0.94$ pc. This agrees with The distribution of the clump spacings is roughly symmetrical and 5 clumps have spacing value in the interval of $[0.89,0.98]$ pc. Although the discussion of the clump spacing distribution is difficult for such a small sample, we can still consider that the clumps have a characteristic spacing of $0.97\pm 0.03$ pc and use it in the following discussion.

As we consider the effect of inclination, assuming the inclination angle $i=30\degr$, the observational fragments spacing should be corrected by: $\lambda_{\mathrm{obs},i}=\lambda_{\mathrm{obs}}/\cos(i)=1.12$ pc, which is much larger than the theoretical predicted $0.33$ pc. This discrepancy may be caused by the fact that the IRDC is turbulence and not thermal-pressure dominated.

When considering turbulence, the sound speed $c_{\rm s}$ in Equation 12 can be replaced by total velocity dispersion $\sigma_{v}$ (Fiege & Pudritz, 2000). To derive the total velocity dispersion of the IRDC, we fitted the spectrum of 13CO (3-2), observed by JCMT in the velocity interval between 90 and 100 km s*-1*, pixel-by-pixel with a Gaussian profile. The result is shown in Figure 8. The velocity dispersion $\sigma_{v}$ is given by Fuller & Myers (1992):

[TABLE]

where $\sigma_{{}^{13}\mathrm{CO}}$ is the 13CO line velocity dispersion derived from Gaussian fitting, $m_{{}^{13}\mathrm{CO}}=29m_{\rm H}$ is the mass of 13CO. For the IRDC, the average 13CO line velocity dispersion $\sigma_{{}^{13}\mathrm{CO}}\approx 1.09\pm 0.45\,\mathrm{km\,s}^{-1}$, hence, the total velocity dispersion $\sigma_{v}\approx 1.10\pm 0.45\,\mathrm{km\,s}^{-1}$. Replacing $c_{\rm s}$ in Equation 12 by $\sigma_{v}$, the expected fragment spacing is $\lambda\approx 1.7\pm 0.7\,\mathrm{pc}$, about 1.5 times larger than the observed value but consistent with the error range.

Kainulainen et al. (2013) suggests that at the scale smaller than Jeans’ length, the fragmentation more closely resembles Jeans’ fragmentation. According to the spherical Jeans’ instability, the clump separation is related to density via Jeans’ length, $\lambda_{\mathrm{J}}=c_{\mathrm{s}}(\pi/(G\rho))^{1/2}$. Here the sound speed $c_{\mathrm{s}}$ is replaced by velocity dispersion $\sigma_{v}$, and we obtain $\lambda_{\mathrm{J}}=0.48$ pc for the IRDC. That is smaller than all the observational clump spacings, which suggests that the IRDC fragmentation is roughly dominated by the collapse of the natal filament.

The previous discussion is based on the hydrostatic models, but actually, the filament may be non-equilibrium and fragmenting with accreting. Clarke et al. (2016) studied the fragmentation of non-equilibrium, accreting filaments and suggested that an age limit for a filament, which is fragmenting periodically, could be estimated by measuring the average core separation distance $\lambda$ (see also Williams et al., 2018):

[TABLE]

where sound speed $c_{s}$ should be replaced by velocity dispersion $\sigma_{v}$. And $\tau_{\mathrm{crit}}$ can also be related to the mass accretion rate $\dot{M}$:

[TABLE]

where $M_{\mathrm{line,crit}}=2\sigma_{v}^{2}/G=563\,M_{\sun}\,\mathrm{pc}^{-1}$ in this work ($G$ is the gravitational constant). Thus we can derive $\dot{M}$ as $1.3\times 10^{3}\,M_{\sun}\,\mathrm{pc}^{-1}\,\mathrm{Myr}^{-1}$ (with $i=0\degr$) or $1.1\times 10^{3}\,M_{\sun}\,\mathrm{pc}^{-1}\,\mathrm{Myr}^{-1}$ (with $i=30\degr$). If assuming mass accretion rate remained constant during the filament evolution, we derive the age $\tau_{\mathrm{age}}$ through:

[TABLE]

As the mass of the $11$ pc long filament is about $7.8\times 10^{4}\,M_{\sun}$ (mentioned at Sec. 3.3). The age of filament $\tau_{\mathrm{age}}$ is $\sim 5.5$ Myr (with $i=0\degr$) or $\sim 6.4$ Myr (with $i=30\degr$).

4.2 Properties of JPS Clumps

4.2.1 CO Line Profiles of Clumps

For each clumps, we extract 12CO (3-2) spectra from COHRS, 13CO (3-2) and C18O (3-2) from CHIMPS. The spectra of all clumps are shown in Appendix A. Some clumps contain multiple velocity components, which makes our analysis difficult. The peak values of 13CO (3-2) for some other clumps do not achieve $3\sigma$ threshold of main beam efficiency corrected CHIMPS data ($\sim 2.4\,$K). Thus, in the following discussion, we only consider the strongest component of which the 13CO (3-2) intensity must be higher than $3\sigma$ threshold for each clump, as some previous study did (Urquhart et al., 2007; Eden et al., 2012; Eden et al., 2013). There are 18 clumps that meet this requirement, 16 of which are protostellar and 2 are starless. And C18O (3-2) emission higher than $2.4\,$K are detected in only 9 clumps.

According to radiative transfer, brightness temperature ($T_{r}$) can be expressed as:

[TABLE]

where $T_{\mathrm{bg}}=2.73$ K is the temperature of cosmic background radiation, and $f$ which is the beam-filling factor can be consider to be $1$ here. Under the LTE condition, the optical depths of CO and 13CO can be obtain by:

[TABLE]

here, $\tau_{\mathrm{CO}}/\tau_{{}^{13}\mathrm{CO}}=[\mathrm{CO}]/[^{13}\mathrm{CO}]\approx 89$. According the equation we obtain the optical depths of 13CO, the results are listed in the Column 7 in the Table 3. The $\tau_{{}^{13}\mathrm{CO}}$ ranges from $0.33$ to $1.05$, so we can generally consider 13CO (3-2) as optically thin.

The asymmetry of line profiles can be used to probe different dynamic features. Generally, blue profiles are caused by collapse or infall motion, and red profiles are linked to expansion or outflow motion (Mardones et al., 1997). To select clumps with blue or red profiles, we adopted the normalized criterion set by Mardones et al. (1997):

[TABLE]

where $V_{\mathrm{thick}}$ is the velocity of the optically thick line (i.e., CO (3-2) in our discussion) at peak value, $V_{\mathrm{thin}}$ and $\Delta V_{\mathrm{thin}}$ are the systemic velocity and line width of the optically thin line, i.e., 13CO (3-2). For the clumps which may contain multiple components, their $\Delta V_{\mathrm{thin}}$ are measured by Gaussian fitting for the strongest component. If $\delta_{V}<-0.25$, the line profile is considered a blue profile, and if $\delta_{V}>0.25$, the line profile is considered to be red. For Clump 20 and Clump 22, comparing profiles of 13CO and C18O (3-2), we find that their 13CO (3-2) might be self-absorbed. As their C18O (3-2) emissions are strong enough, we choose their C18O (3-2) as optically thin lines in this part of discussion.

Our classification is shown in Table 3. For the 18 clumps, we find that 10 clumps show blue profiles, and 2 clumps show red profiles. After visual inspection we exclude five clumps as they may be misclassfied by multiple velocity components, and they are marked in Table 3.

According to Myers et al. (1996), we can estimate infall velocities of the clumps with blue profiles:

[TABLE]

where $T_{\mathrm{D}}$ is the brightness temperature of the dip of the optically thick line. $T_{\mathrm{BD}}$($T_{\mathrm{RD}}$) is the height of the blue (red) peak above the peak. The velocity dispersion $\sigma_{v}$ is obtained from the FWHM of a optically thin line.

The calculation results of five clumps with blue profiles are listed in Table 3. The infall velocities of them range from 0.13 to 0.47 km s*-1*, with a mean value of 0.31 km s*-1*.

4.2.2 The Stability of Clumps

Using the physical properties we obtain in the previous analysis, we can study the stability of JPS clumps, i.e., whether they are susceptible to gravitational collapse or expansion in the absence of pressure confinement.

First, we consider the thermal Jeans instability,

[TABLE]

[TABLE]

where $M_{\mathrm{J}}$ and $\lambda_{\mathrm{J}}$ are the Jeans mass and Jeans length, $c_{\mathrm{s}}$ is the sound speed and $\rho=\mu_{\mathrm{H_{2}}}m_{\mathrm{H}}n_{\mathrm{H_{2}}}$ is the density. When clumps are both thermally and turbulently supported, the sound speed $c_{\mathrm{s}}$ is replaced by the velocity dispersion $\sigma_{v}$.

Another important parameter to estimate the stability of clumps is virial parameter $\alpha_{\mathrm{vir}}$:

[TABLE]

similarly $\sigma_{v}$ is the velocity dispersion contributed from both thermal motions and turbulence, $R_{\mathrm{eq}}$ is the equivalent radius of the clump and $M_{\mathrm{clump}}$ is the clump mass. According to McKee & Holliman (1999) and Kauffmann et al. (2013), in a non-magnetized cloud, if $\alpha_{\mathrm{vir}}<2$, the cloud is gravitationally bound and will tend to collapse, otherwise, it may expand by lack of pressure confinement.

The calculation results are shown in Table 4. In our calculation, we adopt the velocity dispersion derived from the width of optically thin line, which have been obtained in Sec. 4.2.1. Only one clump has $\alpha_{\mathrm{vir}}>2$, suggesting that most the clumps of the filament are gravitationally bound and likely to collapse. There are 12 clumps with $\lambda_{\mathrm{Jeans}}<2R_{\mathrm{eq}}$, indicating that they are likely to split into smaller fragments while collapsing. In addition, we have to point out that the $\alpha_{\mathrm{vir}}$ of three clumps (Clump 10, 11 and 19) with blue profiles may be overestimate, as the line may be broadened by gas infall motion.

4.2.3 The Evolutionary Stage of Clumps

The luminosity-mass diagram is an effective tool to infer the evolutionary phases of the clumps. Saraceno et al. (1996) used this diagram to describe the evolutionary stages of low-mass objects, and Molinari et al. (2008) used it extendedly for the study of evolutionary tracks of high-mass regions.

The luminosity-mass diagram of the clumps is shown in Figure 9 with starless clumps denoted by red crosses and protostellar clumps by blue stars. The diagram is also superimposed with the evolutionary paths from Molinari et al. (2008) for cores with different initial envelope mass. The paths follow the two-phase model of McKee & Tan (2003). According to the model, in the first phase, when a clump gravitationally collapses, the mass slightly decreases due to accretion and molecular outflow, while its luminosity significantly increases, and the object moves along an almost vertical path in the $L_{\mathrm{clump}}-M_{\mathrm{clump}}$ diagram. At the end of the first phase, the object is surrounded by an H II region and begins to expel surrounding material through radiation and outflow. In the second phase, the object follows a horizontal track with a nearly constant luminosity and decreasing mass. Although these evolutionary tracks are initially modeled for single cores and some high-mass clumps may contain multiple cores in different evolutionary stages, the $L_{\mathrm{clump}}-M_{\mathrm{clump}}$ diagram has been used in the past to discuss the evolutionary stages of both low-mass and high-mass clumps (Hennemann et al., 2010; Elia et al., 2010; Traficante et al., 2015; Yuan et al., 2017).

The black solid and dashed lines in Figure 9 show the best log-log fit for Class I and Class 0 objects. We only find two protostellar clumps located between these two lines, and one protostellar clump lies above the log-log fit of Class I objects. Molinari et al. (2016b) suggest that $L_{\mathrm{clump}}/M_{\mathrm{clump}}\leq 1\,L_{\sun}/M_{\sun}$ is the characteristic for starless clumps, and we show this criterion by the green dashed line in Figure 9. However, Traficante et al. (2015) identify 667 starless clumps in IRDCs from $\it Herschel$ data, and suggest the mean $L_{\mathrm{clump}}/M_{\mathrm{clump}}\simeq 1.1$ for the starless clump distribution. Generally, the starless clumps in our sample are distributed along or under the green dashed line, with mean $L_{\mathrm{clump}}/M_{\mathrm{clump}}\simeq 1.4$, which indicates that these clumps are still in the very early stages of their evolution. We also find seven protostellar clumps with $L_{\mathrm{clump}}/M_{\mathrm{clump}}<1$. We suggest that they may be misclassified due to foreground mid-IR point sources. Another probable explanation is that they do contain YSOs, but the rest of the clump is still cold and quiescent. Thus on average, their $L-M$ ratios still keep low.

4.2.4 High-mass Star Forming Regions

To assess the potential of clumps to form massive stars, we have to consider their sizes and masses. Figure 10 shows the mass versus equivalent radius diagram on which starless clumps are shown by red crosses and protostellar clumps by blue stars. All the JPS clumps satisfy the threshold for “efficient" star formation of $116\,M_{\sun}\,\mathrm{pc}^{-2}$ ($\sim 0.024\,\mathrm{g\,cm}^{-2}$) given by Lada et al. (2010) or $129\,M_{\sun}\,\mathrm{pc}^{-2}$ ($\sim 0.027\,\mathrm{g\,cm}^{-2}$) given by Heiderman et al. (2010). The thresholds are shown as the lower solid lines in Figure 10. That means all of the clumps including starless clumps have sufficient material to form stars.

According to observations of nearby clouds, Kauffmann & Pillai (2010) give a more restrictive massive-star formation criterion of $M\geq 870\,M_{\sun}(r/\mathrm{pc})^{1.33}$. Only $4$ clumps don’t meet the criterion of massive-star formation, thus most clumps can potentially form high-mass stars. Six of the clumps above the threshold are starless clumps, so they are good candidates for studying massive star-forming regions in a very early evolutionary stage.

Moreover, mass surface density $\Sigma_{\mathrm{mass}}$ is another useful parameter to identify whether clumps have sufficient mass to form massive stars. Krumholz & McKee (2008) suggest that high-mass star formation requires a mass surface density larger than $1\,\mathrm{g\,cm}^{-2}$ to prevent fragmentation into low-mass cores through radiative feedback. Only one clump meets this criterion; however, this threshold is relatively uncertain, and it does not consider magnetic fields, which may play a significant role in preventing fragmentation. In addition, massive clumps and cores with $\Sigma$ less than $1\,\mathrm{g\,cm}^{-2}$ are reported in some observations (e.g. Butler & Tan 2012; Tan et al. 2013).

5 Summary

We present multiple infrared, sub-millimetre continuum and CO isotope rotation-line observations toward IRDC G31.97+0.07 on a large scale. From the continuum and spectral data, we get the dust temperature, column density, excitation temperature and velocity dispersion of this region. The main results of this work are summarized as follows:

The dust temperature and molecular-hydrogen column density maps are derived from fitting the SED of the $\it Herschel$ data pixel by pixel, using a grey-body model. The dust temperature of the IRDC is lower than the active star-formation region that contains UC H II  regions and IR bubbles, while the column density of the IRDC is higher. The total mass is about $2.5\times 10^{5}\,M_{\sun}$ for the whole filamentary structure and is about $7.8\times 10^{4}\,M_{\sun}$ for the IRDC, based on the results.

• 2.

From the column density map derived from SED fitting, we produce column density probability distribution functions (PDFs) toward two regions, Region 1: the whole filamentary structure; Region 2: the IRDC. The PDFs of both Region 1 and Region 2 mainly show power-law tails at the part above the completeness limit suggesting that this region is gravity-dominant.

Compared to other part, the power-law slope of the PDF of Region 2 is flatter, suggesting that more dense gas accumulate in the IRDC. For the PDF of Region 2, the power-law slope $m=-1.50(0.32)$ is flatter than the prediction of a spherical self-gravitating cloud model (-4<m<-2), which implies that this region may be compressed by the adjacent H II region.

• 3.

The theory of self-gravitating, hydrostatic cylinders shows that the filament would fragment into cores with a roughly constant spacing caused by the “sausage" instability. In this work, we derive the mean velocity dispersion by fitting the FWHM of the optically thin line, i.e., 13CO (3-2). Taking the average velocity dispersion $\sigma_{v}\sim 1.1\,\mathrm{km\,s}^{-1}$, number density $n_{\mathrm{\rm H_{2}}}\sim 2.2\times 10^{4}\,\mathrm{cm}^{-3}$ and inclination angle $i=30\degr$, we find the mean observed fragment spacing ($\sim 1.12$ pc) is about 1.5 times smaller than the prediction of the theoretical model ($\sim 1.7$ pc).

Using another non-equilibrium, accreting model (Clarke et al., 2016), we estimate the age of the IRDC, which is about $6.4\,$Myr when $i=30\degr$.

• 4.

There are 27 clumps which are identified from JPS 850 µm continuum located in the filamentary structure. Using the GLIMPSE point-source catalogue and the identification of the YSOs in the previous study, we get distributions of Class I and Class II YSOs. We classify them into two groups: protostellar clumps, which are associated with any YSOs, 24 or 70 µm point sources; and starless clumps which are not. We identify 9 starless clumps and 18 protostellar clumps. We derive physical properties of all the clumps from SED fitting. Compared to protostellar clumps, starless clumps have lower temperature and luminosity-mass ratios.

• 5.

Spectral analysis are performed for 18 clumps with relatively strong 13CO (3-2) emission. Using the optically thick line (i.e 12CO) and the optically thin (i.e 13CO or C18O) spectra simultaneously, we identify 10 clumps with blue profiles asymmetry, while 5 of them may be misclassified by multiple velocity components. Blue profiles indicate infall motion. We obtain the average infall velocity of the clumps with blue profiles, which is about $0.31\,$km s*-1*.

We also calculate the virial parameters of all the 18 clumps to investigate their stabilities. Only one clumps has $\alpha_{\mathrm{vir}}>2$, suggesting that most clumps are gravitationally bound and tend to collapse.

• 6.

All of the 27 JPS clumps fulfil the “efficient" star-formation threshold, thus they have sufficient mass to form stars. 23 clumps ($\sim$ 85%) are above the threshold for high-mass star formation proposed by Kauffmann & Pillai (2010), and 6 of them are starless clumps. According to the luminosity-mass diagram, all the starless clumps in our sample are in very early evolutionary stages. Hence, those 6 starless clumps are good candidates for studying high-mass star-forming regions in very early evolutionary stage.

Acknowledgements

This work was supported by the China Ministry of Science and Technology under the State R&D Program (2017YFA0402600), and by the NSFC grant no. U1531246, 11503035, and also supported by the Open Project Program of the Key Laboratory of FAST, NAOC, Chinese Academy of Sciences. J.H. Yuan is partly supported by the Young Researcher Grant of National Astronomical Observatories, Chinese Academy of Sciences.

The James Clerk Maxwell Telescope has historically been operated by the Joint Astronomy Centre on behalf of the Science and Technology Facilities Council of the United Kingdom, the National Research Council of Canada and the Netherlands Organisation for Scientific Research. JCMT continuum data were obtained by SCUBA-2. Additional funds for the construction of SCUBA-2 were provided by the Canada Foundation for Innovation. This work made use of data from the $\it Spitzer$Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. This work also used data from Hi-GAL which is one of key projects of $\it Herschel$spacecraft. The $\it Herschel$spacecraft was designed, built, tested, and launched under a contract to ESA managed by the Herschel/Planck Project team by an industrial consortium under the overall responsibility of the prime contractor Thales Alenia Space (Cannes), and including Astrium (Friedrichshafen) responsible for the payload module and for system testing at spacecraft level, Thales Alenia Space (Turin) responsible for the service module, and Astrium (Toulouse) responsible for the telescope, with in excess of a hundred subcontractors.

Appendix A CO spectra of all clumps

Figure 11 shows the spectra of 12CO, 13CO and C18O (3-2) of all 27 clumps. For clumps with 13CO/C18O (3-2) intensity higher than 2.4 K ($3\sigma$ of the main-beam-efficiency corrected CHIMPS data, shown by grey dashed lines), we fit the line profile with a Gaussian function and derive the centre velocity and line width. Blue dashed lines show the fitted centre velocities of 13CO and green dashed lines show the fitted centre velocities of C$18$O for each clump. Only for Clump 20 and Clump 22, centre velocities of 13CO deviate from centre velocities of C18O, indicates that 13CO might be self-absorbed. The evolutionary classes are labeled at the top right.