The star formation law in Galactic high-mass star-forming molecular clouds

TL;DR

This study investigates the star formation law in 12 Galactic high-mass star-forming molecular clouds, revealing a broken power-law relation between star formation rate surface density and gas surface density, with implications for understanding star formation thresholds.

Contribution

It provides the first detailed analysis of the star formation law in high-mass star-forming clouds, identifying a break in the relation and linking high-mass YSO locations to dense gas regions.

Findings

Identified a break in the sigsfr-siggas relation at 150-360 Msun/pc^2.

Found the power-law index varies between 1.4 and 3.6 above the break.

High-mass YSOs are preferentially located in regions with densities >1200 Msun/pc^2.

Abstract

We study the star formation (SF) law in 12 Galactic molecular clouds with ongoing high-mass star formation (HMSF) activity, as traced by the presence of a bright IRAS source and other HMSF tracers. We define the molecular cloud (MC) associated to each IRAS source using 13CO line emission, and count the young stellar objects (YSOs) within these clouds using GLIMPSE and MIPSGAL 24 micron Spitzer databases.The masses for high luminosity YSOs (Lbol>10~Lsun) are determined individually using Pre Main Sequence evolutionary tracks and the evolutionary stages of the sources, whereas a mean mass of 0.5 Msun was adopted to determine the masses in the low luminosity YSO population. The star formation rate surface density (sigsfr) corresponding to a gas surface density (siggas) in each MC is obtained by counting the number of the YSOs within successive contours of 13CO line emission. We find a break in the relation between sigsfr and siggas, with the relation being power-law (sigsfr ~ siggas^N) with the index N varying between 1.4 and 3.6 above the break. The siggas at the break is between 150-360 Msun/pc^2 for the sample clouds, which compares well with the threshold gas density found in recent studies of Galactic star-forming regions. Our clouds treated as a whole lie between the Kennicutt (1998) relation and the linear relation for Galactic and extra-galactic dense star-forming regions. We find a tendency for the high-mass YSOs to be found preferentially in dense regions at densities higher than 1200 Msun/pc^2 (~0.25 g/cm^2).

Tables

Table 1. Table 1 General properties of MCs associated to IRAS sources with UCHII colors.

GMC	IRAS name	Lon	Lat	Field size	LIRAS	$V_{\mathrm{LSR}}^{\mathrm{CS}}$ ($\Delta V$)	D	Scale	Area	13CO Beam	$V_{13CO}^{\mathrm{peak}}$ ($\Delta V$)
		[∘]	[∘]	[${}_{}{}^{\prime }\times _{}^{\prime }$]	[${10}^3$L⊙]	[km s-1]	[kpc]	[pc/$\prime$]	[pc2]	[pc]	[km s-1]
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)
MC1	19230$+$1506	50.28	$-$0.39	15$\times$15	4.2 (1.2)	16.6 (3.7)	1.3 (0.2)	0.38	23	0.29	14.5 (3.6)
MC2	19236$+$1456	50.22	$-$0.61	30$\times$36	6.1 (1.8)	40.6 (4.5)	3.4 (0.4)	0.98	290	0.75	42.0 (8.6)
MC9	19139$+$1113	45.82	$-$0.28	30$\times$24	40.3 (13.5)	50.8 (4.9)	4.8 (0.6)	1.38	59	0.29	58.2 (9.5)
MC12	19132$+$1035	45.19	$-$0.44	18$\times$12	21.1 (6.8)	66.7 (2.2)	5.4 (0.6)	1.57	92	1.20	66.0 (5.1)
MC20	19074$+$0814	42.43	$-$0.26	15$\times$9	65.1 (20.3)	65.6 (5.0)	4.9 (0.6)	1.42	57	1.09	65.0 (7.8)
MC21	19074$+$0752	42.11	$-$0.44	36$\times$27	15.6 (4.6)	54.8 (3.0)	3.9 (0.5)	1.13	386	0.87	60.0 (10.7)
MC23	19048$+$0748	41.75	0.09	27$\times$12	0.7 (0.2)	13.2 (2.4)	1.0 (0.1)	0.29	17	0.22	16.5 (4.7)
MC75	18232$-$1154	19.49	0.15	9$\times$12	16.8 (5.2)	24.1 (4.0)	2.3 (0.3)	0.67	55	0.51	21.5 (8.0)
MC76	18236$-$1205	19.36	$-$0.02	25$\times$12	7.0 (2.1)	25.9 (7.6)	2.5 (0.3)	0.70	82	0.54	26.0 (4.8)
MC78	18223$-$1243	18.66	$-$0.06	27$\times$24	22.2 (6.8)	45.5 (2.9)	3.7 (0.4)	1.07	262	0.82	45.0 (3.4)
MC80	18205$-$1316	17.96	0.08	36$\times$24	1.3 (0.4)	21.8 (2.8)	2.2 (0.3)	0.64	171	0.49	23.0 (5.7)
MC81	18190$-$1414	16.94	$-$0.07	6$\times$9	5.4 (1.9)	$-$3.9 (2.7)	2.1 (0.3)	0.61	15	0.47	$-$4.1 (3.5)

Table 2. Table 2 Physical properties of the sample molecular clouds.

GMC	$N_{H_2}$	AV	Mvir	MLTE	MXF
	[${10}^{22}{\mathrm{cm}}^{-2}$]	[mag]	[${10}^4{M}_{\odot }$]	[${10}^4{M}_{\odot }$]	[${10}^4{M}_{\odot }$]
(1)	(2)	(3)	(4)	(5)	(6)
MC1	1.73$\pm$0.77	18.40$\pm$8.28	0.44$\pm$0.38	0.33$\pm$0.17	0.48$\pm$0.26
MC2	1.28$\pm$0.56	13.62$\pm$6.13	8.87$\pm$7.69	3.07$\pm$1.55	4.50$\pm$2.28
MC9	5.32$\pm$2.39	56.61$\pm$25.46	4.89$\pm$4.21	2.56$\pm$1.10	3.75$\pm$1.89
MC12	2.95$\pm$1.32	31.38$\pm$14.12	1.75$\pm$1.49	2.24$\pm$1.12	3.28$\pm$1.64
MC20	5.33$\pm$2.39	56.70$\pm$25.51	1.24$\pm$2.82	2.44$\pm$1.20	3.58$\pm$1.78
MC21	1.10$\pm$0.46	11.70$\pm$5.26	16.06$\pm$13.65	3.52$\pm$1.59	5.17$\pm$2.34
MC23	2.64$\pm$1.18	28.08$\pm$12.64	0.60$\pm$0.53	0.30$\pm$0.22	0.44$\pm$0.32
MC75	5.05$\pm$4.31	53.72$\pm$24.16	3.37$\pm$2.90	2.32$\pm$1.20	3.39$\pm$1.76
MC76	4.65$\pm$2.09	49.46$\pm$22.26	1.48$\pm$1.30	3.14$\pm$1.74	4.61$\pm$2.54
MC78	6.28$\pm$2.83	66.80$\pm$30.06	1.34$\pm$1.13	13.52$\pm$6.46	19.84$\pm$9.48
MC80	6.72$\pm$3.02	71.49$\pm$32.17	3.04$\pm$2.61	9.57$\pm$5.03	14.04$\pm$7.38
MC81	1.80$\pm$0.81	19.19$\pm$8.62	0.35$\pm$0.28	0.23$\pm$0.11	0.34$\pm$0.20

Table 3. Table 3 Selected MIPSGAL YSOs in our sample of MCs.

	MIPSGAL	Selected	Selected	NC	MS 2MASS	HLYSOs
GMC	Sources	Sources	Sources [%]	Sources	Sources	Sources
(1)	(2)	(3)	(4)	(5)	(6)	(7)
MC1	16	16	100	…	…	16
MC2	179	170	95 (5)	25	9	124
MC9	172	141	82 (18)	73	31	117
MC12	23	7	30 (70)	…	…	7
MC20	46	36	78 (22)	13	3	33
MC21	228	219	96 (4)	60	11	152
MC23	128	109	85 (15)	24	21	27
MC75	40	15	38 (62)	…	…	15
MC76	149	103	69 (31)	54	45	102
MC78	142	125	88 (12)	29	15	125
MC80	299	279	93 (7)	38	20	213
MC81	13	4	31 (69)	…	…	4
Avg	…	…	72 (28)	…	…	…
Median	…	…	82 (18)	…	…	…

Table 4. Table 4 Photometric and physical properties of the YSOs.

Name	Lon	Lat	[3.6]	e3.6	[4.5]	e4.5	[5.8]	e5.8	[8.0]	e8.0	[24]	e24	C	$\beta$	$L_{\mathrm{bol}}$	Mass	H
	∘	∘	mag	mag	mag	mag	mag	mag	mag	mag	mag	mag			L⊙	M⊙
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)	(13)	(14)	(15)	(16)	(17)	(18)
MC1-M1	50.26111	$-$0.49540	8.55	0.04	8.49	0.05	7.97	0.04	7.96	0.03	6.22	0.04	2	$-$2.2	0.33	1.5	0
MC1-M2	50.26936	$-$0.47162	14.87	0.14	13.19	0.12	11.67	0.10	10.91	0.05	7.68	0.10	2	$-$2.2	$-$0.26	1.0	0
MC1-M3	50.21999	$-$0.45542	12.50	0.18	12.50	0.18	5.28	0.03	5.32	0.02	5.15	0.03	3	$-$3.3	2.53	4.5	1
MC1-M4	50.32986	$-$0.44956	8.32	0.04	7.99	0.05	7.55	0.04	7.49	0.04	6.95	0.06	3	$-$3.3	1.81	3.0	1
MC1-M5	50.23466	$-$0.47592	12.45	0.07	12.23	0.09	12.05	0.18	12.50	0.18	7.22	0.07	1	$-$1.3	$-$0.07	1.0	0
MC1-M6	50.22746	$-$0.50346	13.51	0.07	13.18	0.10	12.50	0.18	12.50	0.18	7.05	0.07	1	$-$1.3	$-$0.01	1.0	0
MC1-M7	50.22213	$-$0.49915	12.93	0.07	12.17	0.07	11.65	0.10	10.95	0.10	4.04	0.02	1	$-$1.3	1.20	2.0	1
MC1-M8	50.23388	$-$0.47583	13.86	0.11	12.27	0.09	10.98	0.09	10.33	0.06	7.22	0.07	2	$-$2.2	0.71	1.5	0
MC1-M9	50.23447	$-$0.50454	14.39	0.13	12.69	0.11	11.88	0.10	11.17	0.07	8.07	0.12	1	$-$1.3	$-$0.41	0.9	0
MC1-M10	50.22111	$-$0.49915	12.80	0.07	12.18	0.08	11.51	0.10	11.14	0.13	4.04	0.02	1	$-$1.3	1.20	2.0	1
MC1-M11	50.22149	$-$0.49979	12.69	0.08	11.71	0.07	10.42	0.06	8.81	0.03	4.04	0.02	2	$-$2.2	1.98	3.0	1
MC1-M12	50.29696	$-$0.42889	12.50	0.18	6.65	0.06	6.25	0.03	6.08	0.03	5.09	0.03	1	$-$1.3	0.78	1.5	0
MC1-M13	50.30923	$-$0.42042	7.18	0.04	6.66	0.04	5.97	0.03	5.51	0.03	3.17	0.01	2	$-$2.2	2.33	4.0	1
MC1-M14	50.22479	$-$0.46537	13.01	0.06	11.48	0.07	10.38	0.06	9.51	0.03	4.98	0.03	1	$-$1.3	0.82	1.5	0
MC1-M15	50.25849	$-$0.50773	11.51	0.04	10.85	0.06	10.37	0.06	9.52	0.04	6.61	0.05	2	$-$2.2	0.95	1.5	0
MC1-M16	50.27218	$-$0.50201	11.54	0.15	11.09	0.13	12.50	0.18	12.50	0.18	3.59	0.01	3	$-$3.3	1.38	2.0	1

Table 5. Table 5 Class and mass distribution of YSOs in our sample of MCs

	High-Luminosity					Low-Luminosity					Stellar Mass
GMC	CI	C0	CII	CIII	HLYSOs	CI	CII	CIII	LLYSOs	Total	HLYSOs	LLYSOs	HLYSOs	LLYSOs
										YSOs	[$M_{\odot }$]	[$M_{\odot }$]	M⊙/N	M⊙/N
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)	(13)	(14)	(15)
MC2	17	16	71	15	119	74	352	11	437	556	440$\pm$13	219$\pm$5	3.7	0.5
MC9	22	42	21	32	117	93	633	18	744	861	630$\pm$19	372$\pm$5	5.4	0.5
MC20	11	10	6	2	29	40	211	12	263	292	144$\pm$9	132$\pm$3	4.9	0.5
MC21	17	46	30	59	152	49	448	23	520	672	665$\pm$17	260$\pm$5	4.4	0.5
MC23	6	3	14	4	27	162	398	5	565	592	102$\pm$7	283$\pm$5	3.8	0.5
MC76	19	15	57	8	99	121	433	13	567	666	371$\pm$12	284$\pm$5	3.7	0.5
MC78	33	14	11	61	121	149	435	16	600	721	795$\pm$24	300$\pm$5	6.6	0.5
MC80	29	18	52	112	211	118	325	7	450	661	882$\pm$20	225$\pm$5	4.2	0.5
MC1	7	0	3	2	12	14	91	13	118	130	25$\pm$3	59$\pm$3	2.0	0.5
MC12	2	0	1	4	7	51	106	6	163	170	28$\pm$3	82$\pm$3	3.5	0.5
MC75	5	0	4	5	14	131	428	23	582	596	62$\pm$6	293$\pm$5	4.6	0.5
MC81	1	0	1	2	4	3	2	5	10	14	13$\pm$2	5$\pm$1	3.3	0.5

Table 6. Table 6 Star formation related parameters in our MC sample.

GMC	$f_{\mathrm{II}}$	$t_{\mathrm{II}}$	fHL	Radio	Maser	SFR	${\overline{\Sigma }}_{\mathrm{SFR}}$	SFE	$\overline{\Sigma }{}_{\mathrm{gas}}$
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
MC2	0.80$\pm$0.06	1.98$\pm$0.13	0.21$\pm$0.03	Yes (1)	CH3O (2)	585$\pm$294	2.1$\pm$1.1	2.2$\pm$0.1	155$\pm$79
MC9	0.80$\pm$0.05	1.89$\pm$0.13	0.14$\pm$0.01	Yes (3)	H2O (4)	1105$\pm$556	19.1$\pm$9.6	3.9$\pm$0.2	646$\pm$332
MC20	0.78$\pm$0.07	2.02$\pm$0.14	0.10$\pm$0.02	Yes (5)	CH3O (6)	357$\pm$182	6.4$\pm$3.3	1.2$\pm$0.1	647$\pm$320
MC21	0.81$\pm$0.05	1.99$\pm$0.14	0.22$\pm$0.02	Yes (7)	…	916$\pm$461	2.3$\pm$1.2	2.6$\pm$0.2	133$\pm$60
MC23	0.70$\pm$0.05	1.93$\pm$0.14	0.05$\pm$0.01	Yes (8)	H2O (4)	370$\pm$186	26.8$\pm$13.5	11.9$\pm$0.7	320$\pm$240
MC76	0.76$\pm$0.05	1.99$\pm$0.14	0.15$\pm$0.02	No (8)	CH3O (9)	608$\pm$306	7.5$\pm$3.7	2.1$\pm$0.1	565$\pm$310
MC78	0.69$\pm$0.05	1.97$\pm$0.14	0.17$\pm$0.02	No (7)	…	1210$\pm$610	4.6$\pm$2.3	0.8$\pm$0.1	763$\pm$365
MC80	0.69$\pm$0.05	1.83$\pm$0.13	0.32$\pm$0.03	No (11)	H2O (4)	955$\pm$480	5.6$\pm$2.8	1.2$\pm$0.1	816$\pm$410
MC1	0.82$\pm$0.12	1.97$\pm$0.14	0.08$\pm$0.02	Yes (1)	No (4)	77$\pm$39	3.3$\pm$1.7	2.7$\pm$0.3	210$\pm$114
MC12	0.69$\pm$0.08	2.21$\pm$0.16	0.04$\pm$0.02	Yes (12)	No (4)	101$\pm$52	1.1$\pm$0.6	0.5$\pm$0.1	358$\pm$179
MC75	0.76$\pm$0.05	1.99$\pm$0.14	0.02$\pm$0.01	Yes (13)	No (4)	312$\pm$157	5.6$\pm$2.8	1.5$\pm$0.1	613$\pm$320
MC81	0.43$\pm$0.30	2.02$\pm$0.14	0.28$\pm$0.16	Yes (14)	No (4)	16$\pm$10	1.0$\pm$0.7	0.8$\pm$0.3	218$\pm$64

Table 7. Table 7 Parameters of the broken power-law model fit

GMC	$\alpha$	A	${\sigma }_{\alpha }$	${\sigma }_A$	${\chi }^2$	R	${\Sigma }_{th}$	$y\left({\Sigma }_{th}\right)$	$x_i$
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
MC2	0.49	$-$1.01	0.10	0.82	0.69	0.71	226.0	1.6	(1-4)
	2.34	$-$5.33	0.14	2.32	0.19	0.90	…	…	(3-6)
MC9	0.80	$-$1.65	0.10	0.50	0.30	0.59	211.0	1.0	(1-3)
	3.05	$-$6.76	0.05	4.06	0.10	0.92	…	…	(3-5)
MC12	0.54	$-$1.18	0.07	0.44	0.15	0.92	285.0	1.6	(1-4)
	2.80	$-$6.68	0.35	3.50	0.90	0.64	…	…	(3-6)
MC20	0.25	$-$0.56	0.13	0.31	0.91	0.82	255.0	1.0	(1-5)
	1.31	$-$3.14	0.10	1.05	0.27	0.98	…	…	(3-9)
MC21	0.73	$-$1.19	0.12	0.59	0.20	0.66	180.0	3.2	(1-3)
	1.54	$-$2.96	0.04	1.71	0.10	0.94	…	…	(2-5)
MC23	0.26	0.63	0.03	0.43	0.10	0.83	150.0	15.9	(1-3)
	1.94	$-$3.02	0.17	2.17	0.26	0.61	…	…	(2-4)
MC75	0.28	0.29	0.06	0.33	0.14	0.93	237.0	7.1	(1-4)
	2.68	$-$5.39	0.14	6.29	0.18	0.91	…	…	(3-6)
MC76	0.23	0.01	0.06	0.63	0.22	0.97	360.0	3.1	(1-5)
	3.01	$-$7.04	0.19	7.75	0.34	0.95	…	…	(4-8)
MC78	1.01	$-$2.01	0.47	0.27	2.71	0.98	…	…	(1-18)
Total YSOs	0.61	$-$1.38	0.15	0.07	5.11	0.26	445.0	1.8	(1-30)
	2.69	$-$6.09	0.33	0.44	3.65	0.35	…	…	(20-48)
Class I YSOs	0.55	$-$1.70	0.36	0.47	6.68	0.96	380.0	0.5	(1-20)
	3.35	$-$8.96	1.18	3.85	8.45	0.98	…	…	(16-37)

Equations

$$\left[\frac{{\mathrm{M}}_{\mathrm{vir}}}{10^4{\mathrm{M}}_{\odot }}\right]=1.58\times \left[\frac{{\mathrm{R}}_{\mathrm{e}}}{5\mathrm{pc}}\right]\times {\left[\frac{\mathrm{\Delta V}}{5\mathrm{km}{\mathrm{s}}^{-1}}\right]}^2,$$

$$\left[\frac{{\mathrm{M}}_{\mathrm{LTE}}}{10^4{\mathrm{M}}_{\odot }}\right]=3.25\times {\left[\frac{{\mathrm{R}}_{\mathrm{e}}}{5\mathrm{pc}}\right]}^2\times \left[\frac{\mathrm{N}{(}^{13}\mathrm{CO})}{10^{17}\mathrm{c}{\mathrm{m}}^{-2}}\right],$$

$$\left[\frac{\mathrm{N}({\mathrm{H}}_2)}{\mathrm{c}{\mathrm{m}}^{-2}}\right]=57\times 2.8\times 10^{20}\times \mathrm{W}{(}^{13}\mathrm{CO}).$$

$$\left[\frac{{\mathrm{M}}_{\mathrm{XF}}}{10^4{\mathrm{M}}_{\odot }}\right]=3.26\times {\left[\frac{{\mathrm{R}}_{\mathrm{e}}}{5\mathrm{pc}}\right]}^2\times \left[\frac{\mathrm{W}{(}^{13}\mathrm{CO})}{5\mathrm{K}\mathrm{km}{\mathrm{s}}^{-1}}\right].$$

Full text

To appear in ApJ

1. Introduction

The knowledge of physical processes driving the conversion of interstellar gas into stars is fundamental to the development of a predictive physical theory of star formation. A basic step in this direction is to obtain empirical relations between parameters that are related to star formation. Schmidt (1959) suggested a relation between the star formation rate (SFR) and the density of the gas, which in recent decades has been reformulated by Kennicutt (1998) as a relation between SFR surface density ($\Sigma_{\rm SFR}$) and the surface density of the gas ($\Sigma_{\rm gas}$). This relation, often referred to as Kennicutt-Schmidt (KS) relation, has a power-law form \Sigma_{\rm SFR}$$\propto$$\Sigma_{\rm gas}N with the index N=1.4. The relation was established over kiloparsec (kpc) scales, using the HI and CO lines to trace the gas content and holds over more than 4 orders of magnitude in $\Sigma_{\rm gas}$. The sampled regions encompass low density gas in disks of galaxies as well as high density gas in the infrared-bright circumnuclear regions.

Stars form predominantly in dense clumps, of sizes of about one parsec. These clumps themselves form part of larger molecular clouds (MCs). The CO line is the most commonly used tracer of the MC mass. Yet, it is a poor tracer of the high-density gas that resides in clumps. The denser regions are traced by high-density molecular lines such as NH3 or by mapping the dust continuum submm emission (Heyer et al., 2016). Gao & Solomon (2004) found a linear relation between SFR and the total mass of dense gas in luminous infrared galaxies. The non-linear KS relation and the linear relation between the SFR and the mass of the dense gas, jointly imply an increase in the fraction of total gas in dense form as \sim$$\Sigma_{\rm gas}0.4 (Heiderman et al., 2010). Systems with very high SFRs, such as ultraluminous galaxies, have almost 100% of their gas in the star-forming dense phase, whereas at lower SFRs, this fraction can be as small as 1% (Kennicutt & Evans, 2012).

In recent years, the relation between the SF and gas mass has been explored at the scale of clumps and down to core scale in Galactic star-forming  regions (e.g. Lada et al., 2010; Heiderman et al., 2010) using extinction maps at infrared (IR) wavelengths and/or dense gas tracers, to estimate the $\Sigma_{\rm gas}$, and number counts of Young Stellar Objects (YSOs) along with a mean value of stellar mass per YSO and lifetime of Class II phase to estimate the SFR. These studies found a linear relations, much like the relation found by Gao & Solomon (2004) using dense gas tracers in luminous external galaxies. Additionally, Wu et al. (2010) and Heyer et al. (2016) have found linear relationships for dense clumps using dense gas tracers. Indeed, Heyer et al. (2016) found for dense clumps a strong linear correlation between $\Sigma_{\rm SFR}$ and $\Sigma_{\rm gas}$ normalized by free-fall and clump crossing times, suggesting the star formation is regulated at local scales. Heiderman et al. (2010) found that the linear relation holds above a threshold gas density of $\Sigma_{\rm th}$=129 M${}_{\odot}~{}{\rm pc}^{-2}$. For densities below the threshold $\Sigma_{\rm th}$, the SFR drops steeply with a power index $N$ as large as 4.6. Heiderman et al. (2010) also found that the SFR at a given $\Sigma_{\rm gas}$ in Galactic star-forming regions lies above the KS-derived SFR by factors of up to 17. They argued that this difference might arise due to the kiloparsec-size beams used to determine $\Sigma_{\rm gas}$ in extra-galactic studies, which mostly contains non-star forming diffuse CO gas below $\Sigma_{\rm th}$. Recent simulations by Calzetti et al. (2012) on the effects of sampling scale on the KS law support this idea.

Galactic studies that obtained a linear relation between the SFR and the gas density, have restricted their analysis to clumps, that are expected to transform almost all their mass into stars. Typical MCs contain gas at a variety of column densities, covering the entire range of $\Sigma_{\rm gas}$ found in extra-galactic studies, with the clumps representing the high-end of the $\Sigma_{\rm gas}$ distribution. Hence, the star-formation law can be studied locally within individual MCs. Indeed the original conjecture of Schmidt (1959) pertains to SF law within the clouds that are actively forming stars (Lada et al., 2013). Gutermuth et al. (2011) studied eight nearby low-mass star-forming clouds, and found a power-law relation with N ranging between 1.37–3.8. Lada et al. (2013) carried out a similar study for a sample of four nearby clouds, finding N=2.04 for Orion A, Taurus, and California and N=3.3 for Orion B. More recently, Willis et al. (2015) studied six massive star-forming clouds, finding an average slope of $N=2.15\pm 0.41$. They also found that the dispersion of the relationship within individual clouds is much lower than the differences in the $N$ values from one cloud to another.

In summary, the value of $N$ within star-forming Galactic MCs varies more than the range of values found for extra-galactic star-forming regions (Kennicutt & Evans, 2012; Bigiel et al., 2008). In other words, there is no preferred value of $N$ within MCs. Lada et al. (2013) have pointed out that even clouds having similar $N$ values could have vastly different levels of star formation activity, as the latter depends on the density structure within clouds, which is found to vary significantly from cloud to cloud. Similar conclusions were drawn by Burkert & Hartmann (2013) using an analysis of the surface density structure within the Galactic star-forming clouds.

In the present study, we derive star formation laws for high-mass star-forming MCs at sub-pc to parsec scale spatial resolutions. Our approach differs from most previous explorations of this relationship for the Galactic MCs in three aspects: (1) the chosen MCs do not have a known optical nebula associated to them, indicating that SF activity has started recently in our sample of clouds. This ensures that the physical condition of the gas has not been altered by a previous generation of high-mass stars; (2) the chosen MCs contain at least one high-mass YSO, an IRAS source, which ensures that our MCs are high-mass star-forming regions, and (3) we use 13CO data, instead of extinction maps, to derive the gas density. This procedure allows the exploration of embedded SF, even at relatively low surface densities. Our approach also allows us to quantify the difference in SFRs at a given $\Sigma_{\rm gas}$ between Galactic and extra-galactic studies, that used the same tracer, namely CO.

In this work, we carry out a search for Young Stellar Objects (YSOs) in the MCs associated to 12 IRAS sources at distances from 1 kpc to 5 kpc. In §2, we discuss the criteria for sample selection, and the method we have followed for defining the MCs associated to the IRAS sources. The sample of YSOs is discussed in §3, and in §4, the mass function (MF) is presented. A detailed analysis of the star formation law for the sample of star-forming  regions is reported in §5. Our conclusions are summarized in §6.

2. The sample and Observational Data

Our star-forming MCs for the study of the YSO population were selected using the following criteria: (a) they contain an IRAS source with characteristics of an ultra-compact HII region (Wood & Churchwell, 1989), (b) the IRAS sources are associated with the dense cores detected by CS(2-1) line emission Bronfman et al. (1996), (c) the line of sight (LOS) for each cloud is devoid of any other foreground molecular component associated to other MCs as inferred by the 13CO(J=1-0) spectra, and (d) they have GLIMPSE (Churchwell et al., 2009) and 24 $\mu$m-MIPSGAL (Rieke et al., 2004) Spitzer public data. The first two criteria ensure us that the selected clouds contain high-mass star formation sites. The 3rd criterion is imposed to guarantee that all the YSOs associated geometrically to a MC are physically associated to it. The last criterion is the basis for the identification of the YSOs. Eighty clouds of the first quadrant of the Galaxy satisfied the first 2 criteria. However, the 3rd criterion was satisfied by only 12 of these clouds, thus restricting our sample size to 12 MCs. All these have GLIMPSE and MIPSGAL public data available. Typical Spitzer RGB images (3.6 $\mu$m, 8.0 $\mu$m and 24 $\mu$m) of the resulting sample of clouds are shown in Figure 2, where the position of IRAS source is identified by cross symbol. Table 1 lists the properties of the IRAS sources compiled from the literature such as IRAS name, galactic coordinates, bolometric luminosities, distance to the object etc.

2.1. Definition of the MC associated to an IRAS source

In order to define the parent molecular cloud that harbors the high-mass star-forming regions, we used 13CO(J=1-0) emission data from the Galactic Ring Survey (GRS) database (Jackson et al., 2006). The survey data have velocity resolution of 0.21 km s*-1*, a typical (1$\sigma$) rms sensitivity of $\sim$0.13 K, a main beam efficiency $\eta_{\rm mb}$=0.48, and a beam of 46″ (Jackson et al., 2006). The 13CO emission spectra for the Line of Sight (LOS) to the selected IRAS sources are shown in Figure 2, where the observed velocity of the CS(J=2-1) emission line (Bronfman et al., 1996) is marked with a dashed vertical line. In the inset we show the results of a Gaussian fit to the observed 13CO profile. The best fitting value of the central velocity and the 13CO line width (Full Width at Half Maximum - FWHM or $\rm\Delta$V) are given in the last column of Table 1. In majority of the cases (10/12), there is a peak in 13CO spectra within $\lesssim$3 km s*-1* with respect to the CS velocity, whereas in the two clouds (MC9 and MC21) the difference is $\sim$7 km s*-1*. Nevertheless in all clouds the difference between the 13CO and the CS velocities is less than the FWHM of the fitted 13CO profile. This association guarantees the coexistence of dense cores traced by the CS line with the dense molecular structures traced by the 13CO line. All clouds are located in the 1st quadrant, where the foreground molecular emission, if present, would have produced a prominent ($>3\sigma$) molecular component to the left of the CS velocities in the plotted spectra. We could verify the absence of such a prominent component in eleven out of our 12 clouds, thus ensuring that the selected cloud is the nearest cloud along the LOS. The exception is MC81, whose CS velocity lies at the lower extreme of the velocity range covered by the GRS, which prevents us to infer the presence/absence of foreground molecular clouds. In this particular case, we used the 12CO emission profile (See (Figure 1 and Figure 2 in Clemens et al., 1986; Dame et al., 2001, respectively) to ensure the absence of foreground molecular clouds along the LOS. For each MC, the integrated 13CO emission map was created by summing at each pixel all channels that have a velocity within 15 km s*-1* of the $\rm V^{{}^{13}{\rm CO}}_{\rm peak}$, and intensities $>3\sigma$.

The resulting column density map is shown by contours superposed on the RGB image in Figure 2. Only the portion of the map that has a molecular gas column density above N_{{\rm H}_{2}}$$\approx 1\times 10^{21} cm*-2* (the lowest plotted contour in Fig. 1) is considered part of the MC associated to the star-forming region. This limiting value corresponds to $\rm A_{\rm V}\sim 1$ mag, which is the value used in the literature as a physical threshold value to define MCs (Bolatto et al., 2013). We complemented our 13CO maps with the 12CO emission integrated maps of Sanders et al. (1986) to obtain $N_{{\rm H}_{2}}$, and thereby, the mass of the clouds.

2.2. Molecular cloud mass estimation

We computed the mass of the clouds of the sample using 3 independent methods: (a) the Virial theorem, (b) the Local Thermodynamical Equilibrium (LTE) approximation and (c) a constant conversion factor between CO and H2 column density or an X-Factor. To calculate the Virial mass (Mvir) of the MC sample, we use the expression (May et al., 1997),

[TABLE]

where $\rm\Delta$V13 is the 13CO line width (Full Width at Half Maximum - FWHM) obtained from a Gaussian fit to the observed spectra, $\rm R_{e}$ is the effective radius obtained from $\rm R_{e}=\sqrt{A/\pi}$, where $\rm A$ is the area of the cloud. The expression assumes spherical symmetry in density ($\rho$), with $\rm\rho\propto~{}r^{-2}$ (MacLaren et al., 1988).

The LTE mass (MLTE) is obtained from the 13CO column density using the expression (Simon et al., 2001):

[TABLE]

where $\rm N(^{13}CO$) is obtained by integrating the 13CO column density at each pixel. To obtain the 13CO column density map, we first obtained an excitation temperature for the cloud following the LTE approximation of Dickman (1978) and using the peak 12CO radiation temperature from the published b-V maps from Sanders et al. (1986) in the corresponding longitude range, and assumed that computed excitation temperature to be the same for all pixels in the 13CO map. The coupling between beams and antenna efficiencies of 12CO and 13CO were taken into account.

A third method for the determination of the molecular mass consists in using the X-Factor (XF). This factor is defined as the ratio between the column density of the molecular hydrogen ($N_{{\rm H}_{2}}$) and the CO luminosity W(${\rm{}^{12}CO}$) (${\rm XF}\equiv{N(\rm H_{2})}/{W({\rm{}^{12}CO});}$ Pineda et al., 2008). In order to compare with previous results of Galactic studies on the Schmidt Law (Heiderman et al., 2010), we adopted the value of XF=${\rm 2.8\pm 0.7\times 10^{20}}$ [$\rm cm^{-2}~{}K^{-1}~{}km^{-1}~{}s$] from (Bloemen et al., 1986). Following Goto et al. (2003), we assumed W(12CO)/W(13CO)=57, resulting in the following expression for the column density of $N_{{\rm H}_{2}}$:

[TABLE]

The molecular cloud mass (MXF) is then calculated using the equation:

[TABLE]

All the above equations are re-written from the original references by normalizing the quantities to their typical values in our MCs. The computed masses and other physical quantities of the clouds are listed in Table 2. The errors on the tabulated physical quantities are calculated by propogating the errors on $\rm R_{e}$ due to distance, $\rm~{}\Delta V~{}and~{}N(^{13}CO)$.

3. Identification of YSOs associated to the molecular clouds

We used the sources from GLIMPSE and MIPSGAL programs of the Spitzer111Spitzer Space Mission: https://www.spitzer.caltech.edu Space mission to identify the YSOs in our sample of MCs. GLIMPSE sources are directly extracted from the GLIMPSE PSC catalog (Benjamin et al., 2003). We selected only those objects in the PSC that have detectable emission in all the four GLIMPSE bands. MIPSGAL PSC (Gutermuth & Heyer, 2015) was not available at the time when we started this work. We extracted MIPSGAL sources using the 24 $\mu$m  images222The BCD MIPSGAL images are provided by MIPSGAL survey online database, http://irsa.ipac.caltech.edu/data/SPITZER/MIPSGAL/ (Carey et al., 2009) following the method described in the next subsection. The GLIMPSE and MIPSGAL sources are merged to form a complete catalog of all YSOs. Any GLIMPSE source that is within the beam of FWHM=6″ of the 24 $\mu$m image is considered a genuine counterpart of the 24 $\mu$m source. We aim to classify each detected YSO following the IR spectral index criterion of Lada (1987), see §3.4. The sensitivity of the MIPSGAL and GLIMPSE surveys ensures that our catalog contains all Class I sources of mass $\rm\gtrsim 0.3$ M*⊙, Class II sources more massive than $\sim$1 M⊙* and Class III sources more massive than $\sim$5 M*⊙* for a cloud at an average distance of 3 kpc. These detection limits are obtained by converting the flux sensitivities (5$\sigma$ corresponding to a point source) of GLIMPSE (Churchwell et al., 2006) and MIPSGAL333MIPS Instrument Handbook version 3: *https://irsa.ipac.caltech.edu/data/SPITZER/docs/

mipsinstrumenthandbook/MIPS-Instrument-Handbook.pdf* to bolometric luminosities at a distance of 3 kpc, and then using the Class-dependent luminsoity to mass relation described in §4.2.1. It may be noted that most of the bolometric luminosity of Class I sources in their very early phase is outside the GLIMPSE bands. These sources, are expected to be the brightest ones at 24 $\mu$m. Thus, the addition of MIPSGAL sources allows us to select the high-mass tail of the mass function, as will be discussed in §4.

3.1. Identification of 24 $\mu$m YSO candidates

We used SExtractor (Bertin & Arnouts, 1996) on the 24 $\mu$m images to detect all sources that have S/N$\geq$5/pixel within the previously-defined boundaries of the MCs. The IRAF/phot 444IRAF is distributed by the NAAO, which is operated by the Association of Universities for Research in Astronomy Inc. task was used to obtain the photometric magnitudes of the selected sources. We carried out aperture photometry of the SExtractor-selected point sources using an aperture radius of 5 pixels (1 pix=1.25″and FWHM of 24 $\mu$m MIPSGAL is 6″), and the sky annulus of inner radius of 18 pixels with a width of 5 pixels. The measured fluxes are corrected for the flux outside the aperture (infinite aperture correction) using a correction factor of 8.44 (average factor in all MIPSGAL images). Vega flux of 7.14 Jy as suggested in the MIPSGAL Data Handbook (Version 3.3) was used to convert the instrumental magnitudes to the 24 $\mu$m magnitudes. These sources are listed in the column 2 of the Table 3.

The SExtractor-selected source list is contaminated by sources such as foreground stars (bright main sequence and AGB) and background objects (galaxies and highly reddened main sequence stars). Use of bright 24 $\mu$m sources, where the infrared excess is more than two orders of magnitude above the photospheric emission, relatively small sizes ($\lesssim$30′) of our sample clouds, and the high column density towards the molecular clouds, minimizes the fraction of contaminating sources in our catalog. Nevertheless, we applied a Mid-Infrared (MIR) photometric filter to reject contaminating sources from our catalog.

3.2. 24 $\mu$m sources without GLIMPSE counterpart

Some of the 24 $\mu$m sources do not have a GLIMPSE counterpart. Genuine Class 0 and transitional Class 0/I sources are expected to be of this kind (André et al., 2010). However, the absence of a GLIMPSE counterpart doesn’t necessarily mean the absence of emission in the 3–8 $\mu$m region. Sources could be missed in the GLIMPSE data set due to the problems with the photometry in any/some band(s) of GLIMPSE due to crowding or (bright) diffuse emission in the band. The brightest of such sources are expected to be detected in the NIR by the 2MASS.555Two Micron All Sky Survey (2MASS): http://irsa.ipac.caltech.edu/Missions/2mass.html In order to identify these bright reddened main-sequence stars, we performed a positional cross-match of the MIPSGAL sources with the 2MASS point source catalog. All these 2MASS-detected objects that satisfy the photometric color criteria of reddened main-sequence stars ($(J-H)\geq 1.75\times(H-K)$), with the Main-Sequence as defined by Bessell & Brett (1988), were removed from the sample. In the column 6 of the Table 3 are listed the MS stars found in each cloud. The sources without 2MASS counterpart, or with $(J-H)\leq 1.75\times(H-K)$ and $(H-K)$ color excess, are considered genuine Class 0/I YSOs. These sources are listed in the column 3 of the Table 5 as Class 0 YSOs and are included in the final YSO selection (§ 3.3). However, a few of these sources could be highly embedded Class II YSOs in the high-density regions of the molecular clouds (Megeath et al., 2012).

3.3. Selection criteria of YSOs

In order to identify the candidates to YSOs from the 24 $\mu$m-detected sources, we used filters formed from a combination of GLIMPSE and MIPSGAL 24 $\mu$m colors as defined by Gutermuth et al. (2009). As a primary step, all GLIMPSE sources from the PSC with photometric errors $\sigma<0.2$ mag in all IRAC bands are selected. Foreground main sequence objects are expected to be bluer than $[X]-[24]=1.0$ mag (Gutermuth et al., 2009), where $[X]$ is the magnitude in the [4.5] and [5.8] bands. Such sources are also expected to have $[3.6]-[24]<1.0$ mag, which is the selection criterion we used to reject all foreground main sequence stars. Given the sensitivities of GLIMPSE and MIPSGAL images, all 24 $\mu$m sources without a GLIMPSE counterpart are redder than this criterion, and hence are genuine YSOs. Foreground AGBs are relatively bluer and brighter than YSOs,— and have colors $\rm[3.6]-[8.0]<1.5$ and $\rm[8.0]<6.0$ mag. (Marengo et al., 2008). The candidate sources that satisfied this criterion are removed from our sample. Given the relatively small field sizes and low Galactic latitudes of our sample sources, our sample of YSOs is not expected to be contaminated by background galaxies. In total, the rejected number of objects amounts to $\sim$28% (average; see column 4, Table 3). All objects remaining after applying these filters are considered “bona fide” YSOs. These sources, including the Class 0 YSOs, their galactic coordinates (columns 2 and 3) and GLIMPSE and 24 $\mu$m MIPSGAL photometric magnitudes (columns 4-8) are listed in the Table 4.

3.4. Classification of YSOs

In order to characterize the YSOs and describe their nature, we have used the infrared spectral index ($\alpha$) as was defined in early works by Lada & Wilking (1984) and Lada (1987). This index has been extensively used to study the nearby ($<$1 kpc), predominantly low-mass star-forming regions (Hartmann et al., 2005; Harvey et al. , 2006; Alcalá et al. , 2008; Gutermuth et al., 2008). Though originally defined to describe the evolutionary phases of low-mass stars, the $\alpha$ index is applicable for high-mass YSOs as well (Deharveng et al., 2012; Ellerbroek et al., 2013; Saral et al., 2015). We defined $\alpha$ for each 24$~{}\mu$m-source using the GLIMPSE and MIPSGAL data that cover a wavelength range from 3.6$~{}\mu$m to 24$~{}\mu$m. We followed the often-used definition of Greene et al. (1994) to carry out the classification. According to this classification, YSOs are Class I if $\alpha\geq-$0.3, Class II if $-$1.6$\leq\alpha<-$0.3, and Class III if $\alpha<-$1.6. The sources without GLIMPSE counterpart (Class 0/I YSOs) are considered Class I and a spectral index of 0.35 is assigned to them. In the original definition of Greene et al. (1994) the YSOs with $-$0.3$<\alpha<$0.3 are classified as Class FLAT. In this work, we included these $\alpha-$values into the Class I classification. The $\alpha-$values as defined by Greene et al. (1994) requires 24 $\mu$m detection and hence we used the slightly less reliable photometric color criteria of Gutermuth et al. (2008) to classify GLIMPSE YSOs without MIPSGAL counterpart. For this sample, 12–45% of the MIPSGAL sources are Class I with a median value of 22%, where the upper percentages are affected by the small statistics of YSOs in some clouds; whereas that only 9–31% (with a median value of 23%) is Class I YSOs in GLIMPSE sources. On the other hand, majority of the GLIMPSE sources are Class II (20–86%, with a median value of 77%). The classification of the YSOs are shown in the column 9 of the Table 4.

4. Mass of the YSOs

Having obtained a sample of YSOs, we now use their merged photometry in GLIMPSE and MIPSGAL data to obtain a Mass Function (MF). The first step for achieving this is to obtain the bolometric luminosities (Lbol) of the sample YSOs. Knowledge of Class is crucial in determining the bolometric luminosity of embedded YSOs. Even so, detection at 24 $\mu$m is key for a reliable determination of Lbol, as the bolometric correction factor for GLIMPSE-only sources is highly uncertain.

4.1. Mass of the GLIMPSE sources

The c2d Survey (Evans et al., 2003) is one of the legacy programs of the Spitzer Space Mission that has provided uniform photometric data in the IRAC and MIPS bands of Spitzer of YSOs in well-known nearby star-forming regions. These regions were part of follow-up studies at far-infrared (60 $\mu$m to 500 $\mu$m) by Herschel and submm/mm by ground-based facilities such as SIMBA/SEST, SCUBA/JCMT and MAMBO/IRAM. YSOs in Gould’s Belt (GB) also had been the target of multi-band surveys. Detailed analysis of the resulting Spectral Energy Distributions (SEDs) extending from the MIR to millimeter wavelengths has allowed the determination of stellar masses for individual YSOs. This in turn has allowed the construction of the mass function of YSOs (e.g. Enoch et al., 2006; André et al., 2010). Most of these low-mass star-forming regions are consistent with a log-normal Initial Mass Function (IMF) as defined by Chabrier (2003) with a characteristic mass of 0.5 M*⊙* (Kennicutt & Evans, 2012). Following these studies, we assume a log-normal mass function for the low luminosity YSOs, which are mainly GLIMPSE sources, and assign a mean YSO mass of 0.5 M*⊙* for each source in this population.

4.2. Mass of the 24 $\mu$m-detected sources

4.2.1 Bolometric luminosity

Stars are still embedded in the MC during their PMS phase. The material in the immediate vicinity of the star absorbs almost all of the radiation emitted from the PMS star, re-radiating it in the infrared, including at the 24 $\mu$m band. We define a quantity $\beta=L_{24}/L_{\rm bol}$, where $\beta$ is the fraction of the bolometric luminosity that is emitted in the 24 $\mu$m-band. The emission at 24 $\mu$m arises from the warm circumstellar dust and depends on the dust temperature and the mass of the heated dust envelope. During the evolution from Class 0/I (protostars) to Class II (PMS object), the dust temperature of the circumstellar material is expected to increase due to the increase of the “photospheric” temperature, which would lead to an increase of the fraction $\beta$. The accretion of the envelope material to the disk also contributes to the increase of the dust temperature (Stahler & Palla, 2005). However, the envelope mass decreases at a faster rate due to effects of outflows, radiative pressure and stellar winds, with the net result being a decrease of $\beta$ along the PMS phase. A detailed modeling of $\beta$ variation is beyond the scope of this work. Hence, we looked for an empirical relation between $\beta$ and the SED evolutionary phases of the YSOs using a recent complete IR plus submm photometric data set for YSOs.

We used the YSOs from the c2d and GB recent data sets, that have both the 24 $\mu$m and bolometric luminosities catalogued, to obtain $\beta$ empirically using the evolutionary Classes I–III. To start with, we classified the objects in this data sets into evolutionary SED Classes I, II and III following the criteria of Greene et al. (1994). In Figure 3, we show the $\beta-L_{\rm bol}$ diagram for the c2d+GB YSOs from Evans et al. (2003) and Dunham et al. (2015). This population is dominated by Class II YSOs (filled circles), although there is a considerable fraction of Class I (open circles) and Class III YSOs (triangle symbols). The Class I sources have the highest $\beta$-values, whilst the Class II and III span values between $10^{-1}$ and $10^{-5}$. The distribution of $\beta$ values for each Class is shown as histogram in the right panel, where the peak values traced by dashed lines are used as typical values for each Class. These $\beta$-values are $\rm 5.6\times 10^{-2},~{}6.0\times 10^{-3}~{}and~{}5.6\times 10^{-4},$ for Class I, II and III, respectively.

For our embedded sources of Class 0/I, which are detected only in 24 $\mu$m, we used the relation $L_{\rm bol}=3.31\times L_{\rm MIR}$ of Kryukova et al. (2012) for spectral index of 0.35. The mid infrared luminosity, $L_{\rm MIR}$, is the luminosity in the 1–24 $\mu$m range, which is calculated from the 24 $\mu$m flux ($S_{24}$ in Jy) using the relation, $L_{\rm MIR}/L_{\odot}=[0.05+0.81~{}S_{24}]\times D^{2}$, where $D$ is the distance to the cloud in kiloparsec. For deriving the latter relation, we have substituted the flux terms in the 2MASS and GLIMPSE bands by the 3$\sigma$ upper limits in these bands.

We constructed the bolometric luminosity function (LF) for all the 24 $\mu$m-detected YSOs of our sample using the $\beta$ value corresponding to its Class. The resulting bolometric LFs for eight of our MCs that contain at least 20 luminous YSOs are shown in Figure 4. These are plotted separately for Class I (shaded area), Class 0/I (hatched area) and all YSOs. The median luminosities of MIPSGAL-selected Class I YSOs (vertical dotted line) in different clouds varies between 0.3–80 L*⊙, with a global median of $\sim$8 L⊙*. It is interesting to note that the median luminosity for all sources (vertical dashed line) is higher than that for Class I YSOs in every cloud. This implies that the most luminous sources (and the most massive sources) in the MCs are systematically more evolved (Class II and III).

4.2.2 Luminosity-Mass relation

Observed bolometric luminosity of an accreting star comes from 3 physical processes: (1) accretion, (2) nuclear burning, and (3) Kelvin-Helmholtz contraction. The contribution from the last two processes is well understood and is calculated theoretically in traditional PMS evolutionary tracks (Palla & Stahler, 1993; Tognelli et al., 2011). For stars with mass $>$1.5 M*⊙, the luminosity from accretion is not dominating the observed 24 $\mu$m flux, and that the nuclear luminosity is a good approximation to the observed values (Hillenbrand & White, 2004). For example, Myers (2014) obtain, depending on the model assumptions, accretion luminosities between 250–1500 L⊙* for a protostar of 5 M*⊙, which is comparable to the 5 M⊙* ZAMS value of $\sim$800 L*⊙*  from Tognelli et al. (2011).

However, the relative contribution of accretion to the total luminosity, for low-mass stars (mass$<$1 M*⊙*) in their early Class I stage is still under debate. In general theoretically predicted accretion rates produce luminosities a factor of $\sim 5$ higher as compared to the median (observed) luminosity of accreting stars (Kenyon et al., 1990; Enoch et al., 2009). On the other hand, observations of high-luminosity variable sources, such as the prototype FU Ori, suggest that protostars undergo periods of high accretion, in events known as episodic accretion (e. g. Hartmann & Kenyon, 1996; Vorobyov & Basu, 2005). In the extreme case, protostars may spend most of their life in low-luminosity, low-accretion phase and accrete most of their mass during short, intense accretion bursts. Even with episodic bursts of accretion, protostars need to find a way to lose as much as 75% of the accretion energy in non-radiative winds (Offner & McKee, 2011).

The sensitivity of MIPSGAL at the distances of sample clouds corresponds to bolometric luminosity of $\sim$1 L*⊙. The median bolometric luminosity of Class I YSOs of our sample is $\sim$8 L⊙, which is similar to the value given for GLIMPSE-selected YSOs of Offner & McKee (2011) but exceeds by a factor of 5 from Kennicutt & Evans (2012). On the other hand, 10 L⊙*  corresponds to luminosity of a $\sim$2 M*⊙* ZAMS star in models of Tognelli et al. (2011). The above discussions clearly illustrate that reliable masses could be obtained for luminous MIPSGAL Class I YSOs using PMS evolutionary tracks. For Class II and Class III sources, the accretion contribution is expected to be even less. Hence we determined the high-mass end of the mass function using Tognelli et al. (2011) PMS evolutionary tracks.

YSOs spend different timescales in each of the evolutionary Classes I, II, and III. The timescale in each of these phases depends on the stellar mass. These timescales have been studied in nearby star-forming regions in the c2d and GBS (Evans et al., 2009; Dunham et al., 2015). The values used in this work are 0.5 Myr and 2.0 Myr for Class I and Class II, reported by Kenyon & Hartmann (1995) and Evans et al. (2009), respectively. These values are around to obtained in recent works by Heiderman et al. (2015) and Dunham et al. (2015), particularly the timescale for early Class I YSOs. For the Class III, a timescale of 3 Myr is used (although there are longer PMS timescales for low-mass stars).

In order to assign a mass to the bolometric luminosity of the YSOs, we have used the following procedure: (a) if the source is Class III, a mass from the PMS evolutionary track for 3 Myr is used, (b) if the source is Class II, we assign a mass from the evolutionary track for 2 Myr, while for Class I and Class 0/I YSOs, we assign a mass from the 0.5 Myr track.

Hereafter, we refer all YSOs satisfying $L_{\rm bol}>10\,L_{\odot}$ as High Luminosity sources (HLYSOs), with the rest of the sources, including the GLIMPSE-only detected YSOs referred to as Low Luminosity sources (LLYSOs). In Table 5, we give the Class distribution and masses of HLYSOs and LLYSOs. In general, HLYSOs are dominated by Class I sources, whereas the Class II sources are the most frequent among LLYSOs. The mean mass of HLYSOs in different MCs ranges from 2 to 6.6 M*⊙, with a median value of 4 M⊙* for the whole sample.

In Figure 5, we plot the fractions of Class I and Class II YSOs in high (top panel) and low (middle panel) luminosity samples for each MC. HLYSOs are predominantly Class I (63%), whereas LLYSOs are dominated by Class II (79%) sources. In the bottom panel, we show the contribution of HLYSOs to the total “stellar” mass for each MC. It can be noted that in 8 of our MCs, these HLYSOs, which are systematically more massive than the LLYSOs (see the mean mass in the last two columns of Table 5), contribute more than 50% of the total “stellar” mass. For the total sample, 62% of the total mass is contributed by the HLYSOs.

4.3. The mass function of the YSOs

Observational determination of the mass function (MF) of YSOs is critical for a detailed understanding of star formation. Studies of nearby star-forming regions have by now established the nature of the MF of YSOs in the $c2d$ and GB surveys (Enoch et al., 2006; André et al., 2010). Particularly, André et al. (2010) found that the Protostellar Core Mass Function resembles very much the log normal Stellar MF (Chabrier, 2003), but with the characteristic mass $\sim$3 times higher. It should be noted that the regions analyzed in these studies lack the high-mass star population, and hence the application of Chabrier function to star-forming regions containing high-mass stars, such as the regions studied in the present work, is still questionable. There is a growing evidence that such regions have an excess of high-mass stars above the Chabrier function, which can be well represented by the Salpeter (1955) IMF (see the review by Bastian et al., 2010). We here investigate whether this is the case for the star-forming regions studied in the present work.

The method we have followed for the MIPSGAL sources to obtain the YSO masses allows us the construction of the high-mass end of the MF in all MCs. The resultant MFs are plotted in Figure 6 by solid histograms. The MFs are binned in logarithmic mass intervals of 0.3 dex. The error in the masses due to the unaccounted contribution of the accretion in deriving the YSO masses, would over-estimate the masses at the most by one bin width. In Figure 6, we also show a log-normal Chabrier MF for the LLYSOs. This function is centered at a characteristic mass of 0.5 M*⊙* and normalized in such a way that the total number above 0.1 M*⊙* for the function is equivalent to the total number of observed LLYSOs in each cloud. The plotted Chabrier MF assumes that all the YSOs more massive than 0.1 M*⊙*  are detected in the GLIMPSE survey. For each MC, we show the sensitivity limit for Class I sources in the figure. It can be seen that almost all the Class I sources are indeed detected by GLIMPSE in our sample clouds. On the other hand, we may be missing the Class II sources with masses lower than $\rm g_{\rm II}$, the limiting GLIMPSE mass for Class II sources. In all cases $\rm g_{\rm II}$ is less than the characteristic mass, and hence in the extreme case, the actual numbers could be a factor of two higher than the detected YSOs.

Even after taking into account the possible error in the normalization of the Chabrier MF, the number of expected stars more massive than 3 M*⊙*  is less than 1 in our sample of MCs. Thus, the observed number of high-mass stars is clearly above those expected from the Chabrier MF in all clouds. On the other hand, the observed number of massive stars is consistent with a Salpeter IMF for stars more massive than $\sim$2 M*⊙. That is shown in the Figure 7, where the mass function for the total YSO population of the cloud sample is plotted. In this figure, the mass distribution of the HLYSOs is shown with the histogram while the dotted curve represents the mass distribution of the LLYSOs. A error weighted fit for the masses greater than 2 M⊙* is plotted with the dashed line, where the slope found ($\rm\alpha=-1.4\pm 0.4$) resembles well to the Salpeter index ($\rm\alpha=-1.3$). This hybrid Chabrier plus Salpeter mass function that we propose for our regions, is in fact the favored MF in other regions with massive SF (see the review by Bastian et al., 2010).

4.4. Age spread in the population of YSOs

More massive a star is, lesser is the time it spends as a YSO. For example stars more massive than 6 M*⊙* reach the zero age main sequence in $<$0.2 Myr (Tognelli et al., 2011). Thus, in a scenario where all stars formed over a period of time shorter than 0.2 Myr (instantaneous), more massive stars are expected to be in later stages of evolution. Given that our HLYSOs are systematically more massive than LLYSOs, we would expect a higher fraction of Class II and Class III sources among HLYSOs as compared to the LLYSOs.

The numbers of YSOs in each Class for HLYSOs and LLYSOs are given in Table 5. The most dominant population among both samples is the Class II, constituting more than 50% and 70% for the HLYSO and LLYSO samples, respectively. This is expected as stars spend most of their PMS lifetime as Class II sources. The fraction of Class III sources is marginally higher for the HLYSO sample as compared to that of LLYSO sample, as expected in the instantaneous SF scenario. However, in such a scenario, we won’t expect Class 0/I sources in the HLYSO sample. Presence of massive stars in these early stages suggests that the massive star formation process is not instantaneous in majority of our clouds. There are stars of 3 M*⊙*  in Class III stage in majority of these clouds, implying star formation, including the formation of massive stars, has started at least 2 Myr ago. Thus, YSOs in our clouds have an age spread of $>$2 Myr.

The MIPSGAL sources without GLIMPSE counterpart, namely Class I/0 YSOs are an interesting population, especially if these are massive sources. The nature of these massive (M$\rm{}_{YSO}>$3 M⊙) and bright sources could be linked to the earliest (and brightest) objects in the embedded population. These kind of sources are associated to (massive) Class 0 YSOs (André et al., 2010) or transitional Class 0/I YSOs. Due to their early evolutionary phase these objects don’t have (detectable) emission at NIR/MIR (André et al., 2010) and must be (only) detected/observed at submm/mm spectral range with a weak contribution at 24 $\mu$m. The Class I/0 sources also could be confused easily as Class I/II YSOs deeply embedded in the high column density regions of the molecular cloud. Due to the high fraction of the Class I/0 sources, these could be a sum of true Class I/0, and a deeply embedded Class I and Class II YSOs. Indeed, edge-on Class II YSOs with high extinction may be mis-classified as Class I objects (Robitaille et al. , 2006; Offner & McKee, 2011). Nonetheless, the Class I/0 objects without GLIMPSE counterpart are excellent candidates to embedded prestellar and protostellar core population and constitute the youngest population (from 0.1 Myr to 0.5 Myr age; Enoch et al. (2006); André et al. (2010)) of our sample of high-mass star formation regions. Their MF in a complete sample must give us a clue to the connection between the prestellar and protostellar MF.

4.5. Location of High Mass YSOs in the clouds.

The HLYSOs, for definition, are brighter than 10 L*⊙* (§ 4.2). This cut-off luminosity corresponds to a mass of 2 M*⊙* for a Class I YSO, whereas it is $\sim$3 M*⊙* for the Classes II and III YSOs. This implies that all the HLYSOs are of intermediate to high mass. For the purpose of the discussion in this section, we will refer these YSOs as high-mass YSOs. Recent works (Krumholz & Mckee, 2008; Lopez-Sepulcre et al., 2010) suggest that a high density environment is required to form high-mass stars. Thus, the relative fraction of HLYSOs is expected to increase with the surface density of the gas. In order to explore this idea, we plot the ratio $\rm f_{HLYSO}\equiv N(HLYSO)/(N({HLYSO})+N({LLYSO}))$ against $N_{{\rm H}_{2}}$ in Figure 8. This ratio contains all HMYSO population of the MC sample. The high-mass YSO fraction almost remains constant at $\sim$0.15 up to $\sim 3.0\times 10^{22}$ cm*-2*, above which the $f_{\rm HLYSO}$ starts to rise reaching unity value at $\sim 5.5\times 10^{22}$ cm*-2*. In this gas density range, the surface density of HLYSOs increases from $\sim$1 pc*-2* to more than 6 pc*-2*. In this plot, the fraction of LLYSOs ($\rm 1-f_{HLYSO}$) are shown with open circles. The error bars are obtained assuming a poisson error for each bin in the distribution for both the HLYSOs and total YSOs sample. These errors in the counts are propagated in the ratio defined previously ($\rm f_{HLYSO}$). The observed tendency agrees very well with the notion that high-mass stars require higher column densities for their formation.

Krumholz & Mckee (2008) addressed theoretically the issue of initial physical conditions for the formation of massive stars. They found a minimum density of 1.0 ${\rm g}~{}{\rm cm}^{-2}$ as necessary for the formation of the massive stars. However, Galactic star-forming regions are found to harbor high-mass stars at gas densities from 0.1 to 1.0 ${\rm g}~{}{\rm cm}^{-2}$ (Rathborne et al., 2009; Ragan et al., 2013). In our MCs, the high-mass fraction is 1 above $\sim~{}5.5\times 10^{22}$ cm*-2*, which corresponds to $\sim 0.25~{}{\rm g}~{}{\rm cm}^{-2}$. Thus our observed values are consistent with the values observed for other high-mass Galactic clouds.

4.6. Star formation rate and star formation efficiency

The values for the SFR in our sample of MCs lie between 16 and 1220 M${}_{\odot}~{}{\rm Myr}^{-1}$, with a median value of 585 M${}_{\odot}~{}{\rm Myr}^{-1}$. These values are comparable to the values of 715 and 159 M${}_{\odot}~{}{\rm Myr}^{-1}$ for Orion A and B, respectively (Lada et al., 2010). SFR varies over a wide range of values in the Galactic star-forming regions. Vutisalchavakul & Evans (2013) obtained SFR values from $\sim$1 to 2530 M${}_{\odot}~{}{\rm Myr}^{-1}$ for Galactic high-mass star-forming regions at distances D$>$700 pc. The values of SFE are in the interval from 0.5% to 3.9%, with MC23 being an outlier with a value of 11.9%. These values are in agreement with values of Orion B (0.6%) and Orion A (2%) star-forming regions (Megeath et al., 2012). However, these values are systematically smaller than the values of $\sim$3% to 6% obtained for nearby low-mass star-forming regions using c2d and GB data (Evans et al., 2009). Values of SFE $\lesssim$ 1% are commonly found for distributed star-forming regions (Bonell et al., 2011). Half of our clouds have SFE-values $\lesssim$1%, suggesting a distributed SF scenario. Recent studies of young stellar clusters have revealed that there is a time spread in the age of the young stars in stellar clusters (Myers, 2012; Foster et al., 2014). Instantaneous mode of cluster formation may apply to compact massive clusters such as Westerlund 1 with high SFE values (Kudryavtseva et al., 2012), but it is possible that low-mass stellar clusters and associations are built up over a long period of time. Thus, SFE is expected to increase with time as more and more stars form. We test this hypothesis below.

In the Figure 9, the radio continuum emission detection is denoted by the filled symbol, while no detection is denoted by the open symbol. The $\rm f_{II}$ and the timescale ($\rm t_{II}$) obtained based on $\rm 2.0\pm 1.0$ Myr (Evans et al., 2009), are shown in the columns 2 and 3 of the Table 6. The detection or not of radio continuum emission is also included in the same Table (column 5).

From the statistics of the YSO population and the molecular cloud mass for the sample of present study, we can determine some parameters of the star formation activity such as Star Formation Rate (SFR) and Star Formation Efficiency (SFE). The SFR is obtained from the simple expression (Evans et al., 2009), $SFR={\rm M}_{\rm YSO}/t_{\rm YSO}$, where the $M_{\rm YSO}$ and $t_{\rm YSO}$ are the “stellar” (YSOs) total mass and the timescale of the YSOs, respectively. The “stellar” masses are obtained by adding the mass of the HLYSO and LLYSO populations. The masses of the HLYSOs are individually obtained following the method already described. The total mass of LLYSOs is derived assuming a mean mass of 0.5 M*⊙* for every LLYSO. This assumption is equivalent to assuming that the sources follow a log normal mass distribution with a characteristic mass of 0.5 M*⊙*. We then used the expression $\rm M_{LLYSO}/{M_{\odot}}=0.5\times N_{LLYSO}$ (Evans et al., 2009), where $\rm N_{LLYSO}$ is the number of LLYSOs. A typical timescales of 2 Myr and 0.54 Myr for the Class II and Class I, respectively, are used for $\rm t_{\rm YSO}$ (Evans et al., 2009). The Star Formation Efficiency (SFE) is obtained from the expression ${\rm SFE}\equiv{\rm M}_{\rm YSO}/({\rm M}_{\rm YSO}+{\rm M}_{\rm cloud})$, where Mcloud is the X-Factor mass of the molecular cloud (column 6 of Table 2). The resultant SFR and SFE parameters are divided by the cloud area (column 10 of Table 1) to obtain average values of SFR ($\overline{\Sigma}_{\rm SFR}$) and gas density ($\overline{\Sigma}_{\rm gas}$) for our clouds. All these quantities are tabulated in Table 6.

5. Analysis of the Star formation law

5.1. Star formation law within the clouds

In order to study the star formation law within the molecular clouds, we need to obtain the surface density of star formation rate, $\Sigma_{\rm SFR}$, in successive bins of gas surface density, $\Sigma_{\rm gas}$. We used the $N_{{\rm H}_{2}}$ column density maps derived from the 13CO data in §2.1 in order to obtain $\Sigma_{\rm gas}$. The lowest $\Sigma_{\rm gas}$ value corresponds to the MC boundaries that are defined at column densities of $N_{{\rm H}_{2}}$=$1.0\times 10^{21}$ cm*-2* ($A_{v}=1$ mag; $\Sigma_{\rm gas}$=20 M${}_{\odot}~{}{\rm pc}^{-2}$). A linear bin in column density is used to define successive values of $\Sigma_{\rm gas}$, starting at $1.0\times 10^{21}$, and then taking the values $5.0\times 10^{21}$, $1.0\times 10^{22}$, $1.5\times 10^{22}$, etc. The number of pixels in the 13CO map between successive column density bins are counted to determine the area occupied by the gas at that surface density. The numbers of low and high luminosity YSOs between the column density contours are counted, which are then used to calculate the SFR following the methodology described in section § 4.6. The SFRs are divided by the area occupied by the contours to obtain $\Sigma_{\rm SFR}$ corresponding to each $\Sigma_{\rm gas}$.

The Schmidt law within each cloud in the range of column densities from $N_{{\rm H}_{2}}$=$1.0\times 10^{21}$ cm*-2* to $1.0\times 10^{23}$ cm*-2*, equivalent to $\Sigma_{\rm gas}$ range from 20 M${}_{\odot}~{}{\rm pc}^{-2}$  to 2000 M${}_{\odot}~{}{\rm pc}^{-2}$, is shown in Figure 10. For each cloud, data for each gas density bin are plotted with solid circles. The Kennicutt-Schmidt law with $N=1.4$ is shown by the solid line, whereas the dotted line shows the linear relation of Wu et al. (2005) for massive clumps. In six of the twelve MCs, our points lie between the two lines, whereas in the remaining six, most of our data points lie even above the linear relation of Wu et al. (2005). From a close inspection of the plot for 8 clouds (MC2, MC9, MC12, MC20, MC21, MC23, MC75 and MC76), it can be seen that our data points are not consistent with a power-law fit with a single index ($\alpha$) (linear in the log-log plane) over the entire range of gas densities. Instead, a broken power-law (BPL) relation seems a better fit to the data. Such relation is found previously by Heiderman et al. (2010) in their data, who used separate power-laws of the form $y=\alpha x+A$ ($y=\log_{10}$($\Sigma_{\rm SFR}$); $x=\log_{10}$($\Sigma_{\rm gas}$)), one in the low density regime (index $\alpha_{1}$ and coefficient $A_{1}$) and the other one in the high density regime (index $\alpha_{2}$ and coefficient $A_{2}$). We followed a similar procedure to fit our data points. We fitted two separate lines for the low density and high density regimes and obtained the slopes ($\alpha_{1}$ and $\alpha_{2}$) by following a weighted least-square fitting method, with the weights defined as the inverse square of the errors. The best-fitted lines are shown in the figure, where we also give the slopes $\alpha_{1}$ and $\alpha_{2}$ for each line. The regression coefficient for each line ($R_{1}$ and $R_{2}$), as well the range of data points used for fitting the low and high density regimes ($x_{1}$ and $x_{2}$) are given in Table 7.

A minimum of 3 points is used for the fit in both ranges of densities with the bin(s) at the transition used in either fits. Four MCs (MC1, MC23, M80 & MC81) have too few points and in addition do not offer the dynamical range in $\Sigma_{\rm gas}$ necessary to carry out a reliable fit (all points lie at $\Sigma_{\rm gas}$ below $\sim$200 M${}_{\odot}~{}{\rm pc}^{-2}$). We hence did not carry out power-law fitting to these clouds. For MC78, only one point lies in the low-density regime, and hence we show only the fit in the high-density regime. The relation is systematically flatter at the low density regime as compared to that at the high density regime, with values for the former in the range 0.2 to 1.0, and the latter 1.3 to 3.0. Note that the SFR is weakly dependent on gas density at low densities with a power-law index $<1$ in almost cases (with exception MC21). On the other hand, the power-law index at the high density regime varies from a similar value (1.3) to the canonical KS index of 1.4 to values as high as 3.6.

The intersection of the two lines in Figure 10 corresponds to the gas surface density value where the break occurs in the star formation relation. This value is marked by the dashed line in the figure for each cloud and their values are given in Table 7 under the column header $\Sigma_{\rm th}$. The break happens at $\Sigma_{\rm gas}$=150–360 M${}_{\odot}~{}{\rm pc}^{-2}$ in our sample of clouds. The density at the break $\Sigma_{\rm gas}$ value is often referred to as threshold gas density, $\Sigma_{\rm th}$, in the literature. Heiderman et al. (2010), doing a BPL fit similar to that described here, found a threshold value of 129 M${}_{\odot}~{}{\rm pc}^{-2}$ in nearby low-mass star-forming regions, while Lada et al. (2010) proposed a value of 116 M${}_{\odot}~{}{\rm pc}^{-2}$ (equivalent to $A_{V}\sim$8 mag). Willis et al. (2015) used a $\Sigma_{\rm th}$=200 M${}_{\odot}~{}{\rm pc}^{-2}$  to obtain the Schmidt law in galactic massive star-forming regions. Our lower threshold values are similar to those reported for other galactic star-forming clouds, whilst the higher values can be explained if is considered that the longer distances increase the value of the threshold density as was reported by Heiderman et al. (2010).

In order to explore if the resulting trend for a BPL fit to the data is statistically significant, we used a total YSOs sample for the clouds that show the trend and several fits were tested. This was made for two cases, for all Classes and Class I YSOs. First, in each cloud it was obtained the y-value ($\Sigma_{\rm SFR}$) (column 9, Table 7) related to the threshold value in $\Sigma_{\rm gas}$ (column 8, Table 7). Then each data point of the $\Sigma_{\rm SFR}$-$\Sigma_{\rm gas}$plot is divided by this y-value and the resulting data distribution is added in a total normalized sample. In this sample, the best error weighted fit is given by a broken power law relation with a slightly slope for lower densities ($\alpha$=0.6, 0.6) and steeper slope for higher densities ($\alpha$=2.7, 3.4), similarly as previously was found in the individual fits for clouds. Therefore, we suggested that the BPL fit is adequate to describe the data point distribution for each cloud with the BPL trend. The fitted parameters are shown in the Table 7 and BPL fit of the normalized samples (all Classes and Class I) is plotted in the Figure 11.

5.2. Global Star formation law

We now investigate the position of each cloud as a whole in the global SF law. For doing this, we compare the globally averaged values of $\overline{\Sigma}_{\rm SFR}$ and $\overline{\Sigma}_{\rm gas}$ for our clouds (given in Table 6), with the global values for Galactic clouds and clumps obtained in other studies in Figure 12. Our $\overline{\Sigma}_{\rm SFR}$–$\overline{\Sigma}_{\rm gas}$ values are shown by solid circles, while the data from the study of Heiderman et al. (2010) are plotted with open squares, and the massive dense clumps from Wu et al. (2010) and Heyer et al. (2016) are plotted with diamonds and cross symbols, respectively. The linear relation from Wu et al. (2005) and Kennicutt relation (Kennicutt, 1998) are shown by dashed and solid lines, respectively. Heiderman et al. (2010) had found that their global results for low mass star-forming regions lie above the KS relation by a factor of up to 17. They found the factors to be as high as 54 for regions containing the youngest sources (Class I and Class Flat). This latter value overlaps with values for the high-density massive clumps of Wu et al. (2010) and Heyer et al. (2016). Our clouds lie between the linear relation of Wu et al. (2005) and the canonical Kennicutt relation, and mostly related to the lower SFR dense clumps from Heyer et al. (2016) and Wu et al. (2010).

In recent work by Heiderman et al. (2010), it has been found that the use of 13CO in the calculation of mass of a molecular cloud results in an underestimation of the mass compared to if visual extinction is used, which is considered better tracer of the mass of the cloud. According to their results, the mass of the cloud is underestimated by a factor 4-5. If this correction factor is applied in calculating the mass of our sample of clouds plotted in Figure 12, the result is that the values of $\Sigma_{\rm gas}$ are increased by a value of 0.6 (in log scale) and our distribution follows well to distributions for dense clumps reported by Heyer et al. (2016); Wu et al. (2010). The corrected values of our clouds lie on the KS relationship.

It can be noted in the figure that the average gas densities of several MCs from our sample are clustered around $\log$($\overline{\Sigma}_{\rm gas}$)$=2.6\pm 0.3$, compared to clouds from Heiderman et al. (2010) clustered around $\log$($\Sigma_{\rm gas}$)$=2.0$. On the other hand, $\overline{\Sigma}_{\rm gas}$ for the dense clumps are distributed over a much wider dynamic range. Lada et al. (2013) also found almost constant average $\overline{\Sigma}_{\rm gas}$ for the MCs they analyzed, which they used to argue against the existence of a Schmidt law between MCs. They further argued that the clustering of global $\overline{\Sigma}_{\rm gas}$ is due to the well-known scaling law of Larson (1981) between cloud size and mass, where $\overline{\Sigma}_{\rm gas}$ is a constant for a given value of the limiting column density used to define the MCs. Heyer et al. (2009) found \overline{\Sigma}_{\rm gas}$$\sim 40 M${}_{\odot}~{}{\rm pc}^{-2}$ for MCs with a defining boundary of $A_{v}=1$ mag. Values obtained in our study are around a factor of 4–9 higher, and our lower values compare well with the \overline{\Sigma}_{\rm gas}$$\sim 170 M${}_{\odot}~{}{\rm pc}^{-2}$  obtained by Solomon et al. (1987).

5.3. Star formation below the threshold density

The threshold densities, $\Sigma_{\rm th}$ for our clouds are comparable to the values obtained in the studies of Heiderman et al. (2010) and Lada et al. (2012). However, there is a clear difference in the star formation relation below $\Sigma_{\rm th}$ in our clouds as compared to the clouds studied by Heiderman et al. (2010). In the latter study, SF decreases abruptly below $\Sigma_{\rm th}$, whereas in our case SF almost remains constant or decreases less steeply with index values being sub-linear below $\Sigma_{\rm th}$. This trend below $\Sigma_{\rm th}$ is also found in the recent study by Willis et al. (2015) (see their Figure 9). However, they suspected an increasing contamination from giant stars and background galaxies at decreasing densities as a possible reason for the apparent excess of YSOs below $\Sigma_{\rm th}$, and analyzed the star formation law only for the data points above \Sigma_{\rm gas}$$>200 M${}_{\odot}~{}{\rm pc}^{-2}$.

Is the apparent SF below $\Sigma_{\rm th}$ an artifact of the method used or is it real? The adopted filtering method (see §3) has effectively cleaned our sample of contaminants such as bright foreground/background stars or background galaxies, and hence the presence of YSOs at low densities is real. While calculating the $\Sigma_{\rm SFR}$, we have included all YSOs, including those of Class II and III. Heiderman et al. (2010) have used only Class I sources in obtaining the steeply decreasing $\Sigma_{\rm SFR}$ below $\Sigma_{\rm th}$. We now investigate whether the apparent constant $\Sigma_{\rm SFR}$ below $\Sigma_{\rm th}$ is due to the inclusion of Class II and III YSOs in our study. In Figure 13, we plot the number (upper panel) and fraction (lower panel) of YSOs separated by Class in each gas density bin. YSOs from all MCs are added at the corresponding gas densities to obtain this plot. It can be seen that Class II sources dominate the observed number of YSOs below $\Sigma_{\rm th}$. The Class III fraction remains almost constant at $\sim$10% at all gas densities. Thus, evolved Class III objects do not contribute significantly to $\Sigma_{\rm SFR}$ below $\Sigma_{\rm th}$.

The presence of Class I/II YSOs associated to gas below $\Sigma_{\rm th}$, could imply either (1) stellar migration or (2) effect of beam dilution or both. We discuss these two possibilities below:

1. Stellar migration from their birth sites: star-forming cores can migrate from their formation site in a dense environment to the current location. For typical velocities of dense cores of 0.33 to 0.55 km s*-1*(Muench et al., 2007; Kirk et al., 2010), Class I and II objects can be as far as 1–2 parsecs away from their formation sites. Our clouds measure typically $>$5 pc, implying the Class I and II YSOs in our clouds had no time to migrate from dense regions to low surface gas density regions.

2. Beam dilution: The $\Sigma_{\rm gas}$ in our study is obtained using 13CO beams (0.3–1.2 pc) that are in general larger than the typical core size (0.1 pc). The cores are expected to be surrounded by dense clump gas whose typical size (1 pc) is larger than our beam size. Under such a situation, the observed 13CO column densities are expected to be that of typical clump densities, which would be higher than $\Sigma_{\rm th}$. The observed low column densities would then imply that the star-forming cores at densities below $\Sigma_{\rm th}$ are isolated and are not be surrounded by the dense clump gas.

6. Conclusions

We here analyzed the local and global star formation law in a sample of 12 Galactic molecular clouds with signposts of ongoing high-mass star formation. The study is different from similar previous studies in that (1) we investigate the SF relation in the whole cloud, and not just in the dense clumps, and (2) Class-dependent masses are obtained for each MIPS source. The number of high luminosity sources ($L_{\rm bol}>$10 M*⊙) is found to be clearly in excess of that expected for a Chabrier mass function, suggesting a combined Chabrier-Salpeter mass function for the YSOs in our clouds. YSOs with $L_{\rm bol}>$10 M⊙* contribute more than 30% of the total mass in each of our clouds, with a mean value of 55%. The SFR is obtained by the standard technique of counting the YSOs enclosed within contours of gas density as traced by the 13CO emission above $A_{v}=1$ mag ($\Sigma_{\rm gas}$=20 M${}_{\odot}~{}{\rm pc}^{-2}$). The relation between $\Sigma_{\rm SFR}$ and $\Sigma_{\rm gas}$ has a break at $\sim$150–360 M${}_{\odot}~{}{\rm pc}^{-2}$for almost of the clouds, with power-law forms on either side of the break, which is also found using the total YSO population of the clouds with the trend. The power-law index above the break lies between 1.3—3.0 in different clouds, which is in general higher than the Kennicutt value of 1.4 for extra-galactic regions, but is consistent with the values of Galactic regions found in recent studies. At densities below the break, we find $\Sigma_{\rm SFR}$ almost independent of density. The density at the break is consistent with the threshold density $\Sigma_{\rm th}$ found in Galactic and extra-galactic star-forming regions. Globally averaged $\overline{\Sigma}_{\rm gas}$ for our sample of molecular clouds are clustered around a value of 350 M${}_{\odot}~{}{\rm pc}^{-2}$within a factor of 2, a range too small to explore a relation with these data alone. The global $\overline{\Sigma}_{\rm SFR}$ lies above the Kennicutt values for the observed $\overline{\Sigma}_{\rm gas}$ by factors between 1 to 60, but agrees within a factor of 2 with the linear relation for massive clumps.

This work was supported by the CONACyT (Mexico) research fellowship 199495 granted to RR, and research grants CB-2010-01-155142-G3 (PI:YDM) and 182841 (PI:AL).