The Physical and chemical structure of Sagittarius B2 -- IV. Converging filaments in the high-mass cluster forming region Sgr B2(N)

TL;DR

This study uses high-resolution ALMA observations and a new spectral line stacking method to analyze the small-scale filamentary structure and accretion processes in the high-mass star-forming region Sgr B2(N), revealing high velocity gradients and potential for super stellar cluster formation.

Contribution

The paper introduces a novel line stacking technique to analyze kinematics in line-rich regions and provides detailed insights into filamentary accretion in Sgr B2(N).

Findings

Eight converging filaments identified with high accretion rates.

Velocity gradients along filaments are 10-100 times larger than in other regions.

Potential for Sgr B2(N) to evolve into a super stellar cluster.

Abstract

We have used an unbiased, spectral line-survey that covers the frequency range from 211 to 275 GHz and was obtained with ALMA (angular resolution of 0.4 arcsec) to study the small-scale structure of the dense gas in Sagittarius B2 (north). Eight filaments are found converging to the central hub and extending for about 0.1 pc. The spatial structure, together with the presence of the massive central region, suggest that these filaments may be associated with accretion processes. In order to derive the kinematic properties of the gas in a chemically line-rich source like Sgr B2(N), we have developed a new tool that stacks all the detected transition lines of any molecular species. This permits to increase the signal-to-noise ratio of our observations and average out line blending effects, which are a common problem in line-rich regions. We derive velocity gradients along the filaments of about 20-100 km s$^{-1}$ pc$^{-1}$, which are 10-100 times larger than those typically found on larger scales (1 pc) in other star-forming regions. The mass accretion rates of individual filaments are about 0.05 M$_\odot$ yr$^{-1}$, which result in a total accretion rate of 0.16 M$_\odot$ yr$^{-1}$. Some filaments harbor dense cores that are likely forming stars and stellar clusters. The stellar content of these dense cores is on the order of 50% of the total mass. We conclude that the cores may merge in the center when already forming stellar clusters but still containing a significant amount of gas, resulting in a "damp" merger. The high density and mass of the central region, combined with the presence of converging filaments with high mass, high accretion rates and embedded dense cores already forming stars, suggest that Sgr B2(N) may have the potential to evolve into a super stellar cluster.

Tables

Table 1. Table 1: Kinematic and physical properties of the filaments in Sgr B2(N)

	Velocity gradient (km s-1 pc-1)					Filament parametera𝑎aa𝑎a$a$Mass accretion rates have been computed with the median of the velocity gradients of the four species.
ID	\ceCH3OCHO	\ceCH3OCH3	\ceCH3OH	\ce^13CH3OH		$M$ ($M_{\odot }$)	$L$ (pc)	$M/L$ ($M_{\odot }$ pc-1)	$\dot{M}$ ($M_{\odot }$ yr-1)
F01	0$+$31.5$\pm$40	0$+$40.0$\pm$30	0$+$37.6$\pm$20	0$+$34.9$\pm$20		1050–490	0.14	7500–3500	0.039 – 0.019
— I	$+$129.0$\pm$10	0$+$60.3$\pm$11	$+$100.3$\pm$90	0$+$29.6$\pm$11		0690–320	0.03	23000–10800	0.057 – 0.027
— II	00$+$1.2$\pm$20	0$+$13.5$\pm$20	0$+$11.0$\pm$20	0$+$26.4$\pm$20		0380–180	0.08	4800–2300	0.005 – 0.003
— III	0$+$96.4$\pm$12	0$+$95.3$\pm$70	0$+$88.9$\pm$60	$+$152.2$\pm$20		16–7	0.03	540–250	0.002 – 0.001
F03	0$+$20.0$\pm$10	0$+$19.9$\pm$20	00$+$9.0$\pm$30	0$+$28.6$\pm$20		0620–290	0.13	4800–2200	0.013 – 0.006
— I	00$+$8.4$\pm$30	00$+$6.8$\pm$20	00$+$6.5$\pm$10	0$+$10.4$\pm$20		200–95	0.03	6800–3200	0.002 – 0.001
— II	00$+$8.8$\pm$20	0$+$16.8$\pm$10	0$+$17.7$\pm$10	0$+$18.5$\pm$10		040–18	0.05	800–380	0.001 – 0.001
F04	0$-$22.1$\pm$20	0$-$12.8$\pm$20	0$-$12.4$\pm$20	00$-$9.7$\pm$10		0570–270	0.14	4100–1900	0.008 – 0.004
F05	0$-$17.0$\pm$20	0$-$20.7$\pm$10	0$-$28.0$\pm$10	0$-$18.1$\pm$10		0490–230	0.13	3800–1800	0.010 – 0.005
F06	0$-$54.7$\pm$20	0$-$41.0$\pm$20	0$-$46.0$\pm$20	0$-$39.4$\pm$20		0410–190	0.16	2500–1200	0.019 – 0.009
— a	0$-$18.1$\pm$20	0$-$13.6$\pm$10	0$+$67.1$\pm$10	0$-$24.3$\pm$10		0500–240	0.02	25000–12000	0.009 – 0.004
— b	0$-$43.1$\pm$20	$-$103.3$\pm$17	0$-$46.4$\pm$31	0$-$40.7$\pm$70		0500–240	0.02	25000–12000	0.023 – 0.011
F07	0$+$52.3$\pm$40	0$+$38.9$\pm$30	0$+$61.5$\pm$20	0$+$70.2$\pm$70		0970–460	0.10	9700–4600	0.057 – 0.027
F08	0$+$18.0$\pm$20	0$+$16.1$\pm$20	0$+$11.0$\pm$10	0$+$18.2$\pm$20		0810–380	0.30	2700–1300	0.015 – 0.007
— I	0$-$17.3$\pm$20	0$-$14.9$\pm$10	00$-$6.1$\pm$40	00$-$8.1$\pm$10		200–95	0.03	6700–3200	0.003 – 0.002
— II	0$+$94.5$\pm$60	0$+$85.2$\pm$50	0$+$48.5$\pm$70	0$+$77.7$\pm$30		0210–100	0.01	21000–99000	0.018 – 0.009
— III	00$+$5.8$\pm$40	00$-$6.2$\pm$30	0$+$23.5$\pm$10	00$-$9.9$\pm$20		150–70	0.03	5100–2400	0.001 – 0.001

Table 2. Table 2: Properties and timescales of the dense cores located within filaments in Sgr B2(N)

	${\theta }_{\mathrm{core}}$a𝑎aa𝑎a$a$Angular radius (${\theta }_{core}$) of the cores as listed in Paper II	$d_{\mathrm{hub}}$b𝑏bb𝑏b$b$The distance between the core and the hub ($d_{\mathrm{hub}}$) is determined from a straight line connecting both, and can, therefore, be a lower limit to the real distance when considering projeciton effects.	$T$c𝑐cc𝑐c$c$The temperatures ($T$) have been determined for different molecular species (Möller et al. in prep) resulting in an error of 10%. After propagation of these errors, we have determined the uncertainties in the gas mass and timescales.	$L$	$M_{d+g}$	$M_{\mathrm{stellar}}$d𝑑dd𝑑d$d$The lower value of the stellar mass is obtained from the luminosity listed in the table after decreasing it by a factor of eight. This aims to reproduce: (i) a core radius two times smaller than the measured size, and (ii) a contribution of 50% to the total luminosity coming from accretion and not from stellar radiation.	$M_{\mathrm{stellar},\mathrm{ratio}}$	$t_{\mathrm{core},\mathrm{ff}}$	$t_{\mathrm{fil},\mathrm{ff}}$
ID	(″)	(pc)	(K)	($\times$103 $L_{\odot }$)	($M_{\odot }$)	($M_{\odot }$)	(%)	(kyr)	(kyr)
F01 A02e𝑒ee𝑒e$e$Cores A02, A03, and A06 can also be connected to the central hub via filament F02.	0.64	0.21	1200	61	0300 $\pm$ 450	090 – 220	20 – 45	08 $\pm$ 0.3	10 – 16
A03e𝑒ee𝑒e$e$Cores A02, A03, and A06 can also be connected to the central hub via filament F02.	0.70	0.20	1950	510	0140 $\pm$ 200	230 – 650	60 – 85	13 $\pm$ 0.6	09 – 15
A06e𝑒ee𝑒e$e$Cores A02, A03, and A06 can also be connected to the central hub via filament F02.	0.90	0.19	8000	24	0255 $\pm$ 350	060 – 140	20 – 40	14 $\pm$ 0.6	09 – 13
A17	0.53	0.12	1100	29	0015 $\pm$ 100	070 – 160	80 – 90	27 $\pm$ 1.0	4 – 7
F05 A20	0.44	0.16	50f𝑓ff𝑓f$f$Temperatures are assumed, rather than obtained by fitting.b𝑏bb𝑏b$b$The distance between the core and the hub ($d_{\mathrm{hub}}$) is determined from a straight line connecting both, and can, therefore, be a lower limit to the real distance when considering projeciton effects.	0.87	0020 $\pm$ 500	20 – 35	40 – 70	18 $\pm$ 0.9	07 – 11
F06 A10	0.64	0.11	1050	36	0040 $\pm$ 500	070 – 170	60 – 80	22 $\pm$ 0.9	4 – 6
F08 A04	0.77	0.10	1700	360	0140 $\pm$ 200	070 – 170	30 –60	15 $\pm$ 0.6	3 – 5
A05	0.84	0.12	1000	51	0200 $\pm$ 300	080 – 200	25 – 55	15 $\pm$ 0.6	4 – 7
A07	0.44	0.12	8500	7.3	0070 $\pm$ 100	35 – 85	30 – 60	09 $\pm$ 0.4	4 – 7
A08	0.56	0.40	1100	33	0055 $\pm$ 100	065 – 170	50 – 80	15 $\pm$ 0.7	27 – 42
A11	0.84	0.18	50f𝑓ff𝑓f$f$Temperatures are assumed, rather than obtained by fitting.b𝑏bb𝑏b$b$The distance between the core and the hub ($d_{\mathrm{hub}}$) is determined from a straight line connecting both, and can, therefore, be a lower limit to the real distance when considering projeciton effects.	3.2	0100 $\pm$ 150	30 – 60	20 – 40	21 $\pm$ 0.9	08 – 13
A15	0.80	0.27	50f𝑓ff𝑓f$f$Temperatures are assumed, rather than obtained by fitting.b𝑏bb𝑏b$b$The distance between the core and the hub ($d_{\mathrm{hub}}$) is determined from a straight line connecting both, and can, therefore, be a lower limit to the real distance when considering projeciton effects.	2.9	0060 $\pm$ 100	30 – 60	30 – 55	25 $\pm$ 1.1	15 – 23
A19	0.83	0.45	50f𝑓ff𝑓f$f$Temperatures are assumed, rather than obtained by fitting.b𝑏bb𝑏b$b$The distance between the core and the hub ($d_{\mathrm{hub}}$) is determined from a straight line connecting both, and can, therefore, be a lower limit to the real distance when considering projeciton effects.	3.1	0050 $\pm$ 500	30 – 65	35 – 60	29 $\pm$ 1.1	32 – 50
Hub A01	1.45	–	250f𝑓ff𝑓f$f$Temperatures are assumed, rather than obtained by fitting.	5900	1960 $\pm$ 280	0800 – 3000	25 – 65	11 $\pm$ 0.4	–

Equations

$$\dot{M}=v_{||}\cdot \frac{M_{\mathrm{fil}}}{L_{\mathrm{fil}}}=\nabla v_{||}\cdot M_{\mathrm{fil}},$$

$$L_{\mathrm{fil},\mathrm{obs}}=L_{\mathrm{fil}}\cos (\alpha )\text{and}{v}_{||,\mathrm{obs}}=v_{||}\sin (\alpha ),$$

$$\dot{M}=\frac{\nabla v_{||,obs}\cdot M_{\mathrm{fil}}}{\tan (\alpha )}.$$

$${\left(\frac{M_{\mathrm{fil}}}{L_{\mathrm{fil}}}\right)}_{\mathrm{crit}}\approx \frac{2{\sigma }^2}{G}$$

$$L=4\pi r^2\sigma T^4$$

$$\log \left(\frac{M/M_{\odot }}{L/L_{\odot }}\right)=-0.6\times \log \left(\frac{L}{L_{\odot }}\right)+\mathrm{0.5.}$$

$$M_{\mathrm{stellar},\mathrm{fraction}}=\frac{M_{\mathrm{stellar}}}{M_{\mathrm{d}+\mathrm{g}}+M_{\mathrm{stellar}}}.$$

$$t_{\mathrm{ff}}=\sqrt{\frac{4{\pi }^2{r}^3}{32G{M}_{d+g}}}$$

$$B_{\nu ,\text{mod}}(T)=B_{\nu }(T)(1-e^{-{\tau }_{\nu }}),$$

$${\tau }_{\nu }={\int }_{\text{line of sight}}{\kappa }_{\nu }\rho dl,$$

$$B(T)={\int }_0^{\infty }{B}_{\nu }(T)d\nu .$$

$$B(T)=\sigma T^4,\sigma =5.67\times 10^{-5}{\text{erg cm}}^{-2}{\text{s}}^{-1}{\text{K}}^{-4},$$

$$\xi (M)\propto \{\begin{array}{ll}{M}^{1.3}& \text{if }0.1\text{leavevmode}{M}_{\odot }<M\le 0.5\text{leavevmode}{M}_{\odot }\\ M^{2.3}& \text{if }M>0.5\text{leavevmode}{M}_{\odot }\end{array}$$

$$\log \left(\frac{L}{L_{\odot }}\right)=a\cdot \log \left(\frac{M}{M_{\odot }}\right)-b,$$

$$0.1<\mathrm{M}/M_{\odot }\le 0.45\log L=2.028\times \log \mathrm{M}-0.976$$

$$0.45<\mathrm{M}/M_{\odot }\le 0.72\log L=4.572\times \log \mathrm{M}-0.102$$

$$0.72<\mathrm{M}/M_{\odot }\le 1.05\log L=5.743\times \log \mathrm{M}-0.007$$

$$1.05<\mathrm{M}/M_{\odot }\le 2.40\log L=4.329\times \log \mathrm{M}+0.010$$

$$2.40<\mathrm{M}/M_{\odot }\le 7.0\log L=3.967\times \log \mathrm{M}+0.093$$

$$7.0<\mathrm{M}/M_{\odot }\le 100\log L=2.865\times \log \mathrm{M}+\mathrm{1.105.}$$

$$\log \left(\frac{M/M_{\odot }}{L/L_{\odot }}\right)=-0.6\times \log \left(\frac{L}{L_{\odot }}\right)+0.5,$$

Full text

*Aims. *We aim to characterize the filamentary structure of Sgr B2(N) and its kinematic properties using multiple molecular dense gas tracers.

*Methods. *We have used an unbiased, spectral line-survey that covers the frequency range from 211 to 275 GHz and obtained with ALMA (angular resolution of 0$\aas@@fstack{\prime\prime}$4, or 3300 au) to study the small-scale structure of the dense gas in Sgr B2(N). In order to derive the kinematic properties of the gas in a chemically line-rich source like Sgr B2(N), we have developed a python-based tool that stacks all the detected line transitions of any molecular species. This allows us to increase the signal-to-noise ratio of our observations and average out line blending effects, which are common in line-rich regions.

Results. A filamentary network is visible in Sgr B2(N) in the emission maps of the molecular species \ceCH3OCHO, \ceCH3OCH3, \ceCH3OH and \ceH2CS. In total, eight filaments are found that converge to the central hub (with a mass of 2000 $M_{\odot}$, assuming a temperature of 250 K) and extending for about 0.1 pc (up to 0.5 pc). The spatial structure, together with the presence of the massive central region, suggest that these filaments may be associated with accretion processes, transporting material from the outer regions to the central dense hub. We derive velocity gradients along the filaments of about 20–100 km s-1 pc*-1*, which are 10–100 times larger than those typically found on larger scales ($\sim$1 pc) in other star-forming regions. The mass accretion rates of individual filaments are $\lessapprox$ 0.05 M⊙ yr*-1*, which result in a total accretion rate of 0.16 $M_{\odot}$ yr*-1*. Some filaments harbor dense cores that are likely forming stars and stellar clusters. We determine an empirical relation between the luminosity and stellar mass of the clusters. The stellar content of these dense cores is on the order of 50% of the total mass. The timescales required for the dense cores to collapse and form stars, exhausting their gas content, are compared with the timescale of their accretion process onto the central hub. We conclude that the cores may merge in the center when already forming stellar clusters but still containing a significant amount of gas, resulting in a ”damp” merger.

*Conclusions. *The high density and mass of the central region, combined with the presence of converging filaments with high mass, high accretion rates and embedded dense cores already forming stars, suggest that Sgr B2(N) may have the potential to evolve into a super stellar cluster.

1 Introduction

The star-forming complex Sagittarius B2 (hereafter Sgr B2) is the most massive cloud in our Galaxy with a mass of $\leavevmode\nobreak\ 10^{7}$ M⊙ and H2 densities of 103–105 cm*-3* (Schmiedeke et al. 2016, Hüttemeister et al. 1995, Goldsmith et al. 1990), located at a distance111For consistency with previous findings in the literature, for our calculations we used the distance $8.34\pm 0.16$ kpc (Reid et al., 2014) of $8.127\pm 0.031$ kpc (Gravity Collaboration et al., 2018) in the vicinity of the Galactic center (at a projected distance of 107 pc) within the Central Molecular Zone (CMZ). The CMZ provides an extreme environment in terms of pressure, turbulent Mach number, and gas temperature ($\sim$60 to ¿100 K), which are much higher than those found in star-forming regions distributed throughout the Galactic disk (Ginsburg et. al 2016, Morris & Serabyn 1996), but comparable to the physical conditions found in starburst galaxies. Therefore, the CMZ and Sgr B2  are perfect targets to study star formation under extreme conditions in our local environment. The complex Sgr B2 harbors two main sites of active high-mass star formation, Sgr B2 Main (M) and North (N), with masses of 104–105 M⊙ and \ceH2 densities of $\sim$107 cm*-3*. Moreover, active high-mass star formation and a huge number of different (complex) molecules have been observed in both of them (e.g. De Pree et al., 2014; Belloche et al., 2013; Schmiedeke et al., 2016; Sánchez-Monge et al., 2017; Ginsburg et al., 2018; Mills et al., 2018). We have begun a project characterizing the physical and chemical properties of Sgr B2  covering all the spatial scales from au up to tens of parsec. For this, we combined multiwavelength observations with 3D radiative transfer modeling (see Schmiedeke et al. 2016; Sánchez-Monge et al. 2017; Pols et al. 2018). In this paper, the fourth of the series, we make use of high-spatial resolution observations obtained with ALMA222Atacama Large Millimeter/submillimeter Array, ALMA partnership et al. 2015 (0$\aas@@fstack{\prime\prime}$4, or 3300 au) to study the small-scale structure of the dense gas in Sgr B2(N). The dust continuum emission studied in Sánchez-Monge et al. (2017, herafter Paper II) revealed a filamentary network converging to a central hub, which has a total mass of about 2000 $M_{\odot}$ (assuming a temperature of 250 K) and densities $\sim$109 cm*-3*. In this paper, we characterized the kinematic properties of the filaments in more detail by making use of the molecular line emission observed with ALMA. Filaments have been found, in part thanks to the Herschel satellite, to be ubiquitous entities in molecular clouds at many different scales. Their presence in almost all the environments suggest that they play an important role during the early phases of the star formation process. It has been found in local low-mass star forming regions at scales of a few pc that filaments are hatcheries of prestellar cores (e.g. Taurus B211/B213 , Palmeirim et al. 2013; Marsh et al. 2016). Similarly, molecular clouds forming high-mass stars (O and B spectral type stars) are found to be pervaded by large and complex filamentary networks, when observed at large scales of a few pc (see e.g. Orion: Suri et al. 2019; Hacar et al. 2018; MonR2: Trevino-Morales et al. in prep). In these cases, stellar clusters are found to form in dense hubs with densities $>$106 cm*-3*. Denser high-mass star forming regions like W33A ($\sim$ 107-108 cm*-3*), when observed with high enough angular resolution, show filamentary structures at scales of a few 1000 au and converging toward the most massive objects forming in the clusters (Maud et al. 2017; Izquierdo et al. 2018). Sgr B2(N) provides us, thanks to the high densities of 108-109 cm*-3*, with an ideal target to characterize filaments in even more extreme environments. Moreover, the high densities and masses found in the central region of Sgr B2(N) suggest that this region may evolve into a super stellar cluster or young massive cluster (YMC), similar to the Arches and Quintuplet clusters. YMCs are considered to be predecessors of globular clusters, which are formed at the earliest epochs of the universe. Their formation process is still poorly understood, in particular, the mechanism of mass accumulation, during which filaments may play an important role. This paper is structured as follows. In Sect. 2 we describe our ALMA observations. In Sect. 3 we present the physical and kinematic properties of the filaments found in Sgr B2(N). In Sect. 4 we discuss our results of the converging filaments and argue that Sgr B2(N) may form a super stellar cluster. Finally, in Sect. 5 we summarize the main results.

2 ALMA observations

In June 2014 and June 2015, Sgr B2 was observed with ALMA (Atacama Large Millimeter/submillimeter Array) during its Cycle 2. 34–36 antennas were used in an extended configuration with baselines in the range from 30 to 650 m, which provided an angular resolution of 0$\aas@@fstack{\prime\prime}$3–0$\aas@@fstack{\prime\prime}$7. The chosen spectral scan mode surveyed the whole ALMA band 6 (211 to 275 GHz) with 10 different spectral tunings, providing a spectral resolution of 0.5–0.7 km s*-1*. Sgr B2(N), with phase center at $\alpha$(J2000)=17h47m19s.887, $\delta$(J2000)=$-$28∘22*′*15$\aas@@fstack{\prime\prime}$76, was observed in track-sharing mode together with Sgr B2(M). The calibration and imaging processes were carried out with CASA333The Common Astronomy Software Applications (CASA; McMullin et al. 2007). Downloaded at http://casa.nrao.edu version 4.4.0. All the images were restored with a common Gaussian beam of 0$\aas@@fstack{\prime\prime}$4. Further details of the observations, calibration, and imaging are provided in Paper II.

3 Filamentary structure in Sgr B2(N)

The line survey toward Sgr B2(N) revealed an extremely rich chemistry in many of the compact sources reported in Paper II, with more than 100 lines per GHz. While the detailed chemistry is the subject of another study (Möller et al. in prep), in the current work we focus on species with bright and clearly identified444The assignment of transitions to molecular species is done using the software XCLASS: eXtended CASA Line Analysis Software Suite: Möller et al. (2017), which make uses of the Cologne Database for Molecular Spectroscopy (CDMS, Müller et al. 2001, 2005) and Jet Propulsion Laboratory (JPL, Picket et al. 1991) catalogs, via the Virtual Atomic and Molecular Data Centre (VAMDC, Endres et al. 2016). transitions. Among them, S-bearing species like \ceH2CS and \ceOCS, N-bearing species like \ceCH3CN and \ceC2H5CN, and O-bearing species like \ceCH3OH, \ceCH3OCHO and \ceCH3OCH3. We used the most isolated lines of these molecular species and produced peak intensity maps. These maps reveal two main types of structures which are illustrated in Fig. 1. The dust continuum emission at 242 GHz (see Paper II) and species like \ceCH3OCH3 (dimethyl ether), \ceCH3OCHO (methyl formate), \ceCH3OH, \ce^13CH3OH (methanol) and \ceH2CS (thioformaldehyde) show a filamentary structure (green and red in Fig. 1), whereas other molecules like \ceC2H5CN (ethylcyanid) or \ceOCS (carbonylsulfid) trace instead a spherical or bubble-like shape (blue in Fig. 1). The existence of two different morphologies in different molecular species hints at opacity effects, excitation conditions or chemical variations as possible origins. Opacity effects are most likely to be discarded, because we detect a large number of species including rare isotopologues (i.e. less affected by opacity effects) tracing both structures. The chemical and excitation properties of both structures will be discussed in a forthcoming paper (Schwörer et al. in prep). Here, we have focussed on the four molecular species \ceCH3OCHO, \ceCH3OCH3, \ceCH3OH, and \ce^13CH3OH included in the spectral-line survey that trace the filamentary structure and study its kinematic properties. The analysis of spectral-line surveys with thousands of lines is challenging because of line blending effects of known and unknown molecular species, particularly in a chemically-rich region like Sgr B2. For any single transition it is thus not straightforward to accurately determine its spatial distribution and the kinematic properties (e.g., velocity, linewidth, lineshape). However, taking advantage of the existence of 100s of lines for each species, we developed a tool that stacks all transitions of a certain molecule. Line stacking improves the signal-to-noise ratio, as well as averaging out line blendings. This permits us to derive the real spatial distribution of the molecular species, as well as the true line shape and number of (velocity) components. A more detailed description of the method is presented in Appendix A. We used the stacked data cubes for different species to characterize the properties of the filaments in Sgr B2(N).

3.1 Filament identification and properties

The identification of the filaments has been done tracing their peak intensity, starting at a position close to the dense hub and following the intensity ridge outwards by visual inspection. Since the central region is optically thick and most lines appear in absorption against the bright continuum, we exclude this region. We used peak intensity maps created from stacked cubes (see Appendix A), because their higher signal-to-noise ratios allow for a better determination of faint structures. The identified filaments were also confirmed in peak intensity maps created from individual molecular transitions. In total, we identify eight arms or filaments (see Fig. 2) which are visible in emission of different species. While arms F01, F03, F04, F05, F06, F07, and F08 emit in the molecules \ceCH3OCH3, \ceCH3OCHO, \ceCH3OH and \ce^13CH3OH, arm F02 is only visible in \ceH2CS . Arm F08 is the most extended and seems to connect the central hub with regions located $\sim 0.5$ pc to the west. However, the outer (western) part is mainly visible in continuum and not in molecular lines which prevents an analysis of its velocity structure (see Section 4). This arm leads to a chalice-shaped area (cf. bright red emission in Fig. 1) when approaching the central hub, and contains the dense cores A04, A05, A07 (from Paper II), and the H ii region H02 (De Pree et al., 2014). On the contrary, filament F07 appears the shortest, probably due to strong projection effects. Some filaments (F01, F05, F06, and F08) harbor embedded cores, while others appear more homogeneous and without clear hints of fragmentation along them, suggesting the existence of different physical conditions (Schwörer et al. in prep). The northern satellite core (containing dense cores A02, A03, and A06, and being one of the main targets in the search of complex molecular species, e.g. Belloche et al. 2013) seems to be connected by arms F01 and F02 to the main hub (see Section 4 for further detail). However, arm F01 seems broken, maybe caused by feedback of the neighbor H ii region H01. The area to the southwest, in between filaments F05 and F07 looks disordered, and the possibility of another underneath filament can not be excluded (see gray dashed lines in Fig. 2, see also Fig. 1).

3.2 Filament kinematics

The analysis of the velocity structure of each filament has been done using the stacked cubes, excluding arm F02 which is only visible in \ceH2CS. This molecule has, in the frequency range of our line survey, only four transitions, thus the statistical method of line-stacking offers no significant advantages. The peak velocity map of the stacked transitions of \ceCH3OCHO (see Fig. 3) reveals already a clear velocity gradient from red to blue-shifted velocities between the northern satellite core and the main hub. Overall, we see a smooth velocity structure, except at the interface between the northern and main cores, where several velocity components are present. A more detailed study of the velocity structure is obtained by producing position-velocity (pv) cuts along the filaments and by fitting Gaussian functions to the spectra. In Appendix B we show the pv-cuts for different molecular species. Most of the filaments are well described by a single velocity component, with the exception of filament F06, for which we consider two velocity components. The Gaussian fits of each spectrum along the filament allow us to derive the velocity and linewidth of each species, and better explore possible variations with distance along the filaments (see Appendix D). For all the filaments, we see velocity gradients in all four species, and we fit them with a linear function. In Table 1, we list the velocity gradients of seven filaments for three species. The velocity structure of each filament is consistent between different molecular species, suggesting that these species trace gas with similar kinematic properties. We determine an average velocity gradient of roughly 20 km s*-1* pc*-1*. Arm F07 reaches the highest values of 52–70 km s*-1* pc*-1*, which together with its short length, suggests that F07 is a filament oriented close to the line of sight. While the gradients in filaments F04, F05, and F06 are negative, meaning that they go from red to blue-shifted velocities when moving from the main hub outwards, the gradients in filament F03 and F07 are positive. The pv-cut along the most extended filament F08 shows a connection in its velocity structure between the main hub and the far away western dense cores A15 located at a distance of $\sim$0.27 pc and A08/A09 located at a distance of $\sim$0.5 pc (see Fig. 4). The kinematic structure of some filaments shows clear variations in the trend of their velocity structure (see e.g. F03, F08, Fig. 14, and Fig. 15). This is likely due to the filaments not being perfect straight lines, but having some curvature in the 3D space, which may result in different velocity gradients (along the filaments) when seen in projection. We used these variations to define subsections (labeled with roman numbers) along these filaments (see Appendix 14). We divide filament F03 in two sections from 0.02 to 0.055 pc, and 0.07 to 0.12 pc, and filament F08 in three sections from 0.01 to 0.04 pc, 0.055 to 0.07 pc, and 0.08 to 0.11 pc. Filament F01 has a weak additional velocity component, best visible in \ceCH3OH (see Fig. 12), but also in \ceCH3OCHO, connecting the northern satellite core with the main hub. We divided F01 into three sections from 0 to 0.03 pc, 0.032 to 0.11 pc, and 0.115 to 0.018 pc. Moreover, as discussed above (see also Fig. 12), F06 has two velocity components in the range 0 to 0.04 pc, that we labeled as F06a and F06b. In addition to the velocity gradient derived for each filament as a single entity, we calculate the velocity gradient for every identified segment (see Table 1). In Fig. 5 (top panel), we present an overview of the velocity gradients of each filament and subfilament. The variation of the velocity linewidth along the filament for each species is shown in Fig. 16, while the mean linewidths, evaluated as the average Gaussian width along the filament, are shown in Fig. 5 (middle panel). Typical linewidths range from 3 to 8 km s*-1*, with a decreasing trend when moving from the central hub outwards. Along filaments F03, F04, F05, and F08, the linewidth of \ceCH3OCHO and \ce^13CH3OH behave quite similar, while the linewidth of \ceCH3OH is in most cases twice the linewidth of the other species. This is further studied in a forthcoming paper (Schwörer et al. in prep). Filament F05 has a local linewidth minimum at $\sim$0.075 pc, coincident with the position of another filament, which is not unambiguously identified in our current data (see Fig. 2 and Sect. 3.1).

3.3 Mass accretion rates and filament stability

We evaluated the mass accretion rate of each filament using its mass and the derived velocity gradients. The mass of each filament was estimated from the dust continuum emission at 242 GHz (see Paper II). For optically thin emission, a dust opacity of 0.899 cm2/g (Ossenkopf & Henning, 1994), a dust-to-gas mass ratio of 100, and dust temperatures between 50–100 K, we derive masses of the filaments in the range 200–1000 $M_{\odot}$ (see Table 1). The higher mass values for filaments F01 and F08 are due to the elongation of F01 into the central hub and the presence of the bright chalice-shaped structure surrounding F08. Since filaments F03 and F04 are located close together in the plane of the sky, part of their mass might be double-counted, although the effect should be negligible, since the overlap of the filaments occurs only in about one fourth of their extend. Following Kirk et al. (2013), we calculate the mass accretion rate implied by our velocity gradients assuming a cylindrical model for our filaments. The mass accretion rate ($\dot{M}$) is derived as

[TABLE]

where $M_{\mathrm{fil}}$ is the mass of the filament, $L_{\mathrm{fil}}$ the length of the filament and $v_{||}$ the velocity parallel to the filament. Considering an inclination $\alpha$ between the filament and the plane of the sky, the parameters that we observe are given by

[TABLE]

which modify the mass accretion rate to

[TABLE]

For the calculation of the mass accretion rates, we assumed a projection angle $\alpha$ of 45∘. An angle of 25∘ roughly doubles our results, while an angle of 65∘ halves them. For the velocity gradient, we used the median of the values determined for all the species (see Table 1). In Fig. 5 (bottom panel), we plot the mass accretion rates for all the filaments and subfilaments. The mass accretion rates along the filaments of Sgr B2(N) are between 0.004–0.04 M⊙ yr*-1* at the scales of 0.1–0.2 pc (see Table 1), thus $10$–$100$ times larger than rates usually found in star-forming filaments at larger scales of $\sim$1 pc (e.g., Peretto et al. 2013, Lu et al. 2018, Treviño-Morales et al. in prep). Altogether, considering the average accretion rates listed in Table 1, the filaments in Sgr B2(N) accrete at a rate of 0.08–0.16 $M_{\odot}$ yr*-1*. This results in a total of 80–160 $M_{\odot}$ accreted onto the dense hub in about 1000 yr, suggesting a timescale for the formation of the hub of about 60–300 kyr, assuming current accretion rates. In Section 3.1, we show that some filaments appear fragmented and harbor embedded cores, which raises the question of how stable the filaments are. The mass-to-length ratio ($M/L$) is a measure of the stability of the filament (e.g., Ostriker 1964, Fischera & Martin 2012). If this value exceeds a critical limit, the filament will gravitationally collapse under its weight perpendicular to its main axis. Assuming that turbulence can stabilize the filament, the critical value is calculated as

[TABLE]

where $\sigma$ is the velocity dispersion and $G$ the gravitational constant. Since the measured linewidths are in the range of 3 to 8 km s*-1* (see Fig. 5), we can neglect the thermal contribution, which corresponds to a few 0.1 km s*-1* for the considered species at $50$–$100$ K. This results in a value of $(M/L)_{\mathrm{crit}}$ about ($4$– $30)\times 10^{3}$ $M_{\odot}$ pc*-1*. According to our measured mass-to-length ratio (see Table 1), the filaments in Sgr B2(N) seem to be stable under gravitational collapse. The variation of mass-to-length ratio along the filaments by assuming different temperatures is presented in Fig. 22 and show a decreasing trend when moving from the central hub outwards.

3.4 Dense core properties and accretion timescales

We determined the temperatures of the dense cores (from Paper II) located within the filaments by fitting the spectral lines of different molecular species with the software XCLASS (Möller et al. in prep). All values are in the range of 50–200 K and listed in Table 2. These high temperatures suggest the presence of already formed stars inside the cores777Other mechanisms such as external or shock heating can be excluded. For a typical density of $10^{6}$ cm*-3* and temperature of 100 K, a heating source would need to see the core as optically thin around the peak ($\sim 30$ $\mu$m) of the black body emission. The dust absorption coefficient $\kappa_{\nu}$ at $30$ $\mu$m is about 300 cm2g*-1*, which yields a distance of only 50 au. Therefore, the heating source has to be effectively embedded in order to heat the gas to the measured temperatures. Heating through shocks is also unlikely since for infall rates of $\leq 0.16\leavevmode\nobreak\ M_{\odot}$ yr*-1* and a assumed accretion shock velocity of $\sim 8$ km s*-1*, the derived luminosities are only $1700$ L⊙, less than the average luminosities derived for our cores. . The stellar content can be derived from the luminosity of the dense cores. Considering them as spherical black bodies, the stellar luminosity is given by the Stefan-Boltzmann equation

[TABLE]

where $r$ is the radius of the dense cores, $\sigma$ the Stefan-Boltzmann constant and $T$ the temperature. We present more details regarding the black body assumption and the validity of the Stefan-Boltzmann law in Appendix E. The derived luminosities are listed in Table 2. The large gas masses of the dense cores (see Paper II and Table 2) suggest that not one single star, but a stellar cluster is forming in each core. While the relation between stellar mass and luminosity is known for single stars (e.g., Eker et al. 2018), there is no direct relation for star clusters. Thus, we have simulated $10^{5}$ clusters, and determined the final stellar luminosity and mass (see Appendix F for details). We find that the cluster luminosity and stellar mass of a cluster follow the relation

[TABLE]

The stellar mass content, Mstellar, for each core within the filaments is listed in Table 2. With this, we calculated the ratio between the stellar mass and total mass as

[TABLE]

All values are summarized in Table 2 and visualized in Fig. 6. For all cores, the stellar mass is within the range 20-90% of the total mass, with a mean (median) value of 50% (44%). We have also investigated the relation of the mass of the different cores with respect to their distance to the central hub. In Fig. 7, we plot the core mass (top panel) and the stellar mass ratio (bottom panel) as a function of the distance to the hub. While no striking correlation is found, we see a bimodal distribution. Cores with masses above 200 $M_{\odot}$ are located preferentially at distances $\textless$0.25 pc, while less massive cores can be found up to distances of about 0.5 pc. Similarly, cores located closer to the central hub can have a higher stellar mass fraction than those located farther away. In Sect. 4.2 we discuss these results in the context of converging filaments and the formation of a cluster in Sgr B2(N). One further question of interest is whether the dense cores will be accreted onto the central hub before they are disrupted by internal star formation. The timescale for forming stars can be estimated by assuming that the dense cores will collapse on a free-fall timescale

[TABLE]

where $M$ is the mass of the dense core and $r$ its radius (values reported in Paper II). The $t_{\mathrm{core,ff}}$ values are listed in the last column of Table 2, and range from $10^{3}$ to $10^{4}$ yr . We compared the $t_{\mathrm{core,ff}}$ with the time required by the dense cores to travel from their position to the central dense hub assuming a free-fall scenario along the filaments, $t_{\mathrm{fil,ff}}$ (see Table 2). To estimate the gravitational acting central mass (see Equation 8), we integrated the flux density over the entire dense hub and assume an averaged temperature of $50$–$100$ K that result, in a mass between ($25$– $10)\times 10^{3}$ $M_{\odot}$. The distance is given by the separation between the core and the central hub. The $t_{\mathrm{fil,ff}}$ ranges from $10^{3}$ to $10^{4}$ yr, comparable to the timescale over which a core collapses and forms a stellar cluster.

4 Discussion

4.1 Converging filaments

The analysis of transitions of different molecular species revealed a network of filaments in Sgr B2(N), converging toward a central massive core. We derived velocity gradients along the filaments, which are 10–100 times larger than is usually found in other star-forming regions at larger scales (e.g. Peretto et al. 2013, Lu et al. 2018). We have considered here if these large velocity gradients can be caused by other mechanisms rather than accretion to the center, for example expanding motions associated with outflows or explosive events like the one seen in Orion KL (e.g. Bally et al. 2017). For outflows, typical linewidths are 30–100 km s*-1* (e.g. Wu et al. 2004, López-Sepulcre et al. 2010, Sánchez-Monge et al. 2013), roughly ten times larger than what we measure. Furthermore, common outflow tracers are \ceCO, \ceHCO+ and \ceSiO (e.g. Wu et al. 2004, Sánchez-Monge et al. 2013, but see also Palau et al. 2011, 2017) and not complex molecules like \ceCH3OCHO or \ceCH3OCH3 which we find tracing the filamentary structure of Sgr B2(N). Explosion events such as that in Orion KL are also a doubtful explanation since they produce clear and straightforward radial structures. Contrary to this, the structures that we identified in Sgr B2(N) appear curved and bent, which leaves us with accreting filaments as the most plausible explanation. For cylindrical, spatially tilted filaments, the transport of material along them toward the dense hub (i.e. velocities close to the center are higher than outwards) would result in different signs of the velocity gradient, as we find in Sgr B2(N) (see Fig. 5 and Table 1). In this scenario, positive velocity gradients correspond to filaments extending backwards, and vice-versa, allowing us to have an idea of the possible 3D structure of the region. In Fig. 8 we show the distribution of filaments in Sgr B2(N) colored by velocity gradient, in which red corresponds to positive velocity gradients (filaments located in the back) and blue to negative velocity gradients (filaments located in front). A remarkable case in Sgr B2(N) are the neighboring filaments F03 and F04. Filament F03 would be located in the back while F04 in front of the hub. Overall, filaments F03, F07, and F08 would be located in the back, and F04, F05, and F06 would be in front (see Fig. 8). In addition to the velocity field, we find that the linewidths for many filaments (e.g. F01, F03, F05, F06, and F08, see Fig. 16) increase when approaching the center888We note that the increase of linewidth toward the center may be partially caused by the line stacking method. The positions along the filament close to the center show a richer chemistry (i.e., more line features). This excess of lines, compared to the outer positions, may result in a artificial increase of the linewidth (see Appendix A for further detail).. This effect, combined with the increasing mass-to-length ratio (derived independently from the continuum emission) when moving inwards, suggests that the accretion or infall rates onto the filaments may be increasing in the vicinity of the main hub.

Finally, some filaments appear extended, connecting multiple dense cores with the main hub. In particular, filament F08 extends up to the distant western cores A08 and A19, located at distances $\sim$0.5 pc. Although the molecular line emission is not bright in the filament between core A05 and cores A08 and A19, the velocity in A19 fits the velocity gradient obtained in the region of the filament closer to the hub (see Fig. 4). This suggests that these cores, even if being far away, are connected to the central hub. Similarly, the northern core seems to be connected to the main hub trough filaments F01 and F02. Despite filament F01 being affected by the H ii region H01, the emission of \ceCH3OH reveals, through a second velocity component, a link between the northern core and the hub (see Fig. 12). Filament F02, which is only visible in the emission of \ceH2CS, shows a smooth velocity gradient from highly red to blue-shifted velocities when moving from the northern core to the central hub (see Fig. 13). This suggests that the distant cores as well as the northern core may fall onto the hub, increasing the gas and stellar density, and favoring Sgr B2(N) to become a super stellar cluster.

4.2 High-mass cluster formation

The high densities found in Sgr B2(N) (109 cm*-3*, or 107 $M_{\odot}$ pc*-1*, see Paper II), the large mass reservoir of the envelope, and high star formation activity in a number of dense cores suggest that Sgr B2(N) may gather enough mass to form a super-stellar cluster or young massive cluster (YMC), therefore constituting a good candidate YMC progenitor. The formation of YMCs is still poorly understood, and different scenarios have been proposed (e.g. Portegies Zwart et al. 2010; Longmore et al. 2014 for a review, and Bressert et al. 2012; Fujii 2015; Fujii & Portegies Zwart 2016; Howard et al. 2018). The ”in-situ formation” scenario proposes that the required amount of gas mass is gathered into the same volume of the final stellar cluster before star formation sets in. The extreme densities required to form a YMC, entail that the accumulated gas has a short free-fall time, which in turn implies that the time for gathering the final mass has to be short or, alternatively, the star formation activity is delayed until the accumulation of the total mass. In a second scenario, the ”conveyor belt formation” theory proposes that the mass reservoir extends farther away than the final stellar cluster size. Thus, stars can form in regions with much lower densities and be transported toward the center of the cluster, increasing the final density of the YMC. Our analysis of the properties of Sgr B2(N) suggests a mix of the two scenarios. On one side, the dense cores located within the filaments already harbor stars, according to our simulations in the ballpark of 2000 (see Appendix F), before merging into the main hub. This supports the idea of the conveyor belt models (”dry” merger), in which star forming clusters are transported throughout the filaments toward the center of the region. On the other hand, the stellar mass ratio is 50%, that is, the dense cores are still surrounded by dense gas while reaching the central region. Although our statistics are small, we see the trend that dense cores located closer to the central hub can be more massive than in the outer regions, (see Fig. 7) suggesting that they have acquired more mass on their way to the main hub. In addition, the comparison of timescales listed in Table 2 suggests that the dense cores can reach the center before they have exhausted all the gas to form stars. Finally, the measured high mass accretion rates ($0.08$–$0.16$ M⊙ yr*-1*) point to a short time necessary to accumulate a large mass reservoir, as proposed by the in-situ (”wet” merger) formation scenarios. Assuming constant accretion rates, the central dense hub, with ($25$–$10)\times 10^{3}$ $M_{\odot}$, has been assembled in only 60–300 kyr. However, the fraction of stars is large ($\sim 50$%) compared to the amount of gas mass in the cores. As a result we cannot consider this process as a clear wet merger, but as a ”damp” merger. Overall, Sgr B2(N) has the potential to become a YMC or super stellar cluster similar to the Arches and Quintuplet clusters, which are also located in the CMZ near the Galactic center.

5 Summary

We have studied the spatial distribution and kinematic properties of a number of molecular species detected in the high-mass star-forming region Sgr B2(N). We used ALMA observations with an angular resolution of 0$\aas@@fstack{\prime\prime}$4 (corresponding to 3300 au) and covering the frequency range from 211 to 275 GHz. The rich chemistry of Sgr B2(N) results in the detection of thousands of lines, often overlapping and therefore difficult to analyze. In order to overcome this problem, we have developed a python-based tool that stacks lines of any given molecular species resulting in an increase of the final signal-to-noise ratio and averaging out line blending effects. This has permitted us to study the distribution of the molecular emission and its kinematic properties. Our main results are summarized in the following:

• •

We have identified two different spatial structures related to the molecular emission in Sgr B2(N). Molecular species like \ceCH3OCH3, \ceCH3OCHO, \ceCH3OH, or \ceH2CS show a filamentary network similar to the distribution of the dust continuum emission at 242 GHz. Contrary to that, species like \ceOCS or \ceC2H5CN show a spherical or bubble-like distribution that will be further analyzed in forthcoming papers.

• •

We have identified eight filaments, which converge toward the central massive core or hub. All but one filament are traced in multiple molecules, including rare isotopologues. Filament F02 is detected in our dataset in only one molecule, \ceH2CS. The filaments extent for about 0.1 pc (some up to 0.5 pc) and have masses of a few hundred $M_{\odot}$. The structure and distribution of the filaments, together with the presence of a massive central region ($\sim$2000 $M_{\odot}$), suggest that these filaments play an important role in the accretion process and transport mass from the outer regions to the central hub.

• •

From the line emission, we measure velocity gradients along the filaments on the order of 20–100 km s*-1* pc*-1* at scales of $\sim$0.1 pc. This is 10–100 times larger than typical velocity gradients found in other star forming regions at larger scales ($\sim$1 pc). We derive mass accretion rates in individual filaments of up to 0.05 $M_{\odot}$ yr*-1*, which add to a total of 0.16 $M_{\odot}$ yr*-1* when considering the accretion of all the filaments into the central hub.

• •

The dense cores identified in the 242 GHz continuum emission map are found distributed along these filaments. We have determined the stellar mass content of these cores, and compare the timescale for the dense cores to collapse and form stars or clusters, with the timescale required for them to be accreted onto the central hub. Considering simple approximations, the free-fall timescale for the core collapse is longer than the time of accretion onto the hub. This suggests that although stars and clusters can form within the cores while being accreted, they will not exhaust all the gas. This is consistent with the stellar mass fractions of 50% (with respect to the total mass) that we derive. This suggests a scenario of a damp merger Sgr B2(N).

• •

In summary, Sgr B2(N) contains a central hub with large densities (\sim$$10^{9} cm*-3*, or $10^{7}$ $M_{\odot}$ pc*-3*), a series of massive dense cores ($\sim$200 $M_{\odot}$) with the potential to form stellar clusters, and a network of filaments converging toward a central hub, with large velocity gradients and high mass accretion rates (up to 0.16 $M_{\odot}$ yr*-1*). The large mass already contained in the central hub ($\sim$2000 $M_{\odot}$ in about 0.05 pc size) together with the merging process of dense cores already harboring stellar clusters, suggest that Sgr B2(N) has the potential to become a super stellar cluster like the Arches or Quintuplet clusters.

Acknowledgements.

This work was supported by the Deutsche Forschungsgemeinschaft (DFG) through grant Collaborative Research Centre 956 (subproject A6 and C3, project ID 184018867) and from BMBF/Verbundforschung through the projects ALMA-ARC 05A11PK3 and 05A14PK1. This paper makes use of the following ALMA data: ADS/JAO.ALMA#2013.1.00332.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada) and NSC and ASIAA (Taiwan) and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. DCL acknowledges support of the Scientific Council of the Paris Observatory. Part of this research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. In order to do the analysis and plots, we used the python packages scipy, numpy, pandas, matplotlib, astropy, aplpy and seaborn.

Appendix A Line stacking method

The analysis of line-rich surveys with often thousands of lines has its pitfalls due to an unknown number of spectral line components and blending of multiple species. Particularly, deriving velocity information is therefore rather complicated in line-crowded spectra. In chemically rich regions like Sgr B2, it is not easily possible to identify well isolated transitions, which are usable to determine the kinematic structure. Moreover, large velocity gradients across the observed region as well as different chemical compositions throughout the source are obstructive. In spite of these adversities, we convert the difficulty of the presence of innumerable lines within a blind survey into an advantage. For this, we have developed an automatized (python based) tool to stack all transitions of a certain species to increase the signal-to-noise ratio and to average out line blending effects. The process of stacking is commonly used in Astronomy to increase the signal-to-noise ratio of for example, weak radio galaxies, recombination lines and even faint molecular line transitions (see e.g., Beuther et al. 2016; Lindroos et al. 2015; Loomis et al. 2018). In our case, stacked data also simplifies the structure and shape of the spectral lines, thus making it possible to analyze the velocity and kinematic properties. The method is carried out in three main steps, illustrated in Fig. 9. First (step A in the figure), we correct our observed data by the source LSR velocity, corresponding to about 64 km s*-1* for Sgr B2(N). Subsequently, we produce a synthetic spectrum with XCLASS (Möller et al., 2017) for any molecular species assuming similar conditions (i.e., temperatures, column densities) to those representing the region (Möller et al. in prep). An accurate fit of the observed data to derive the best temperature, column densities and velocities is helpful but not mandatory. Reasonable assumptions are enough to determine which frequency transitions are detected in the observations, since the synthetic spectrum is used for the identification of bright transitions. For example, for the case of \ceCH3OCHO, we identified 250 bright lines above a 10 $\sigma$ threshold in the frequency range 211 to 275 GHz. This selection prevents us from including transitions that are included in the catalog entries of the CDMS and JPL databases, but are too weak and most likely non detected, in our source of interest. Moreover, the exclusion of the weak transitions, reduces the addition of components highly dominated by noise or by the presence of (bright) transitions from other species. Second (step B in Fig. 9), we cut out all the identified transitions and produce subspectra centered at them. These spectra are transformed from frequency to velocity by applying the Doppler equation. The velocity at 0 km/s (see Fig. 9) corresponds to the rest frequency listed in the database, since we have corrected the data by the source LSR velocity. The spectra are uniformly resampled and rebined, using the python packages signal and numpy, to ensure that the channels in the velocity frame have the same width. Finally (step C in Fig. 9), all spectra are averaged by taking the the arithmetic mean. In the final stacked spectra, the signal-to-noise ratio has increased, while at the same time line blendings are averaged out. This simplifies ensuing the analysis of the kinematic properties like velocity, linewidth, lineshape or number of components. The stacking process can be applied to a subsample of transitions, selected for example, by excitation energy level ($E_{\mathrm{upper}}$) or Einstein $A$ coefficient, parameters included in the molecular line databases CDMS and JPL. This selection of transitions will be used in a forthcoming paper to characterize the excitation and physical conditions of the filaments in Sgr B2(N). We performed tests with synthetic spectra to demonstrate the reliability of the line stacking method. We have used the XCLASS software to generate a synthetic spectrum of the molecular species C2H3CN (Acrylonitrile), which is a good candidate since it has approximately 300 clearly detectable transitions in the observed frequency range (211–275 GHz). The synthetic spectrum of C2H3CN is produced including two different velocity components with velocities of [math] and $-5$ km s*-1*, and linewidths of $4$ and $2$ km s*-1*. In Fig. A.2, we show the results. In the top panel we have stacked the transitions of the synthetic spectrum. The stacked spectrum shows a clear two-velocity component profile. We fit Gaussian distributions and recover the velocities and linewidths with an accuracy of 98%. In a second run (see Fig. A.2, bottom panel) we combine the synthetic spectrum with the observed data of Sgr B2(N). We note that the molecule C2H3CN is not detected in our data and therefore the only transitions that should be found are those include in the synthetic spectrum. After performing line stacking, we recovered the velocities of the two components with a high accuracy (about 98%). The derived linewidths are slightly larger than the original ones, resulting in an accuracy of 82%. This increase in the linewidth is likely produced by the presence of neighboring transitions of other molecular species, which act like noise and result in the broadening effect. Therefore, line-stacking of chemically rich regions results in slightly broader lines. We also note that the presence of this additional transitions artificial increases the continuum level, while not affecting the velocity information of the lines.

Appendix B Peak intensity maps

In Fig. 11 we present peak intensity maps of selected transitions of different molecular species. The path of the filaments is overlaid on all of them. The filamentary structures are clearly visible in molecular species \ceCH3OCHO, \ceCH3OCH3, \ceCH3OH, and \ce^13CH3OH. For other species such as \ceC2H5CN and \ceOCS, the bubble-shape structure dominates the emission.

Appendix C Position-velocity plots along the filaments

Figure 12 shows the position-velocity plots produced along the main spines of the filaments as depicted in Fig. 2. We have used the stacked cubes of the molecular species \ceCH3OCHO, \ceCH3OCH3, \ceCH3OH and \ce^13CH3OH, shown in gray, green, blue, and orange, respectively. The position-velocity plots are obtained following a similar approach to what is implemented in the python tool pvextractor, and considering an averaging width across the filament of 0$\aas@@fstack{\prime\prime}$4, corresponding to the synthesized beam. Similarly, Fig. 13 shows the position-velocity plot along filament F02, which is only visible in \ceH2CS.

Appendix D Velocity gradients and linewidth variation

In order to constrain the velocity properties, we fit Gaussian functions to the spectra along the filaments to study the variation of velocity and linewidth. In Figs. 14 and 15 (right panels), we plot the variation of velocity with distance, while in Fig. 16, we plot the variation of the linewidth (see Sect. 3 for a discussion on the kinematic properties of the filaments). As discussed in Sect. 3.2, all filaments show velocity gradients in the molecular species \ceCH3OCHO, \ceCH3OCH3, \ceCH3OH and \ce^13CH3OH. We study the variation of the velocity gradient along the filament by performing linear fits in 0.014 pc size sections (corresponding to 0$\aas@@fstack{\prime\prime}$4, or a beam size) along the filament. The distribution of the velocity gradients are shown in the left panels of Figs. 14 and 15 in the form of violin plots, which are a combination of box plots and a kernel density plot. The distribution of velocity gradients for the different filaments has, in most cases, a dominant peak, which is generally in agreement with the average velocity gradient determined for the whole filament (listed in Table 1). The second components in the distributions confirm the presence more than one velocity gradient value along the filaments, suggesting a complex velocity gradient structure which could be due to curvatures and projection effects in the filaments.

Appendix E Blackbody assumption

We derived the stellar luminosity of the dense cores assuming that they are spherical black bodies and we use the Stefan-Boltzmann law. However, these dense cores are not perfect black bodies, and the dust opacity has to be taken into account. The modified blackbody or Planck function is

[TABLE]

where $B_{\nu}(T)$ is the Planck function at temperature $T$ and $\tau_{\nu}$ is the optical depth at frequency $\nu$, which is proportional to the density along the line of sight, given by

[TABLE]

with $\kappa_{\nu}$ being the absorption coefficient (opacity) per unit of total mass density and $\rho$ the density. We modeled the modified Planck function for different hydrogen column densities $N_{\mathrm{H_{2}}}$ between $10^{20}$–$10^{25}$ cm*-2* and a temperature of $100$ K (see Fig. 18). The absorption coefficients are taken from Ossenkopf & Henning (1994) according to dust grains with thin ice mantles and gas densities of $10^{6}$ cm*-3*. We assume a dust-to-gas mass ratio of 100. The total brightness of an emitting source can be obtained by integrating over the (modified) Planck function

[TABLE]

For a perfect black body, this results in

[TABLE]

where $\sigma$ is the Stefan-Boltzmann constant. The luminosity calculated from a numerical integration over the Planck function and multiplied by a spherical surface of a given radius matches the luminosity calculated with the Stefan-Boltzmann law. For a more detailed derivation see Wilson et al. (2009). In Fig. 18, we present the ratio between the brightness of a perfect black body and a modified black body as function of the column density. For hydrogen densities above roughly 5$\times$1023 cm*-2*, the total brightness obtained for a modified Planck function is in agreement with the brightness of a perfect black body, with deviations of less than 6%. At the observed high column densities, the emission is optically thick already at lower frequencies compared to lower column densities (see deviations from the Planck function in Fig. 18), and the variation in the total integrated brightness is not significant. In Sgr B2(N) we have column densities about 1024 cm*-2* (Sánchez-Monge et al., 2017). Hence, the assumption of a blackbody is reasonable and the luminosity, which is proportional to the total brightness, can be calculated with the Stefan-Boltzmann law (see Eq. 5).

Appendix F Stellar luminosity and mass relation for clusters

In order to determine the stellar mass and luminosity of clusters, we simulated 105 clusters with a random amount of stars. The number of stars varies between 1 and $50000$, with masses from $0.1$ and $100$ M⊙. The mass distribution of stars for each cluster follows the Initial Mass Function (IMF, Kroupa 2001),

[TABLE]

where $\xi$ is the number of stars of a certain mass. The stellar luminosity ($L$) for a given star of mass $M$ follows the relation (see Eker et al. 2018)

[TABLE]

where the constants $a$ and $b$ vary depending on the stellar mass. For the range of masses considered, we use (based on Eker et al. 2018):

[TABLE]

The masses and luminosities of all individual stars are summed, and result in a total mass and luminosity per cluster. The result for all simulated clusters is shown in Fig. 20, presenting the relation between the stellar mass of the cluster and the cluster luminosity. In Fig. 20, we show the relation between $(M/L)_{\mathrm{cluster}}$ and the cluster luminosity. We fit a linear function to the relation and find

[TABLE]

where $M$ is the mass of the cluster and $L$ the cluster luminosity. In the particular case of the cores of Sgr B2(N), we determine the stellar mass of the clusters that form inside by making use of the luminosities listed in Table 2. For this, we extract all the simulated clusters with luminosities corresponding to the core luminosity, within a range of $\pm$10%. The stellar mass distributions of the selected clusters are shown as histograms in Fig. 21 We finally determined the stellar mass $M_{\mathrm{stellar}}$ as the median value of the distribution. We note that the number of stars within the dense cores of Sgr B2(N) is found to be below 2000 objects (see also Fig. 20)

Appendix G Mass-to-length ratio

In Fig. 22 we plot the variation of mass-to-length ration along the filament. The mass has been computed for segments of 0.011 pc in length and for three different dust temperatures: 50 K, 100 K, and 300 K. Taking the 50 K case as an example, we find for all filaments an increase of mass when approaching the center of the region. This may be due to a higher mass content in the inner regions of the filaments. Alternatively, this might be an artifact produced by a wrong temperature assumption. If the filament is hotter in the inner regions (e.g., six times hotter, from 50 K to 300 K) the increase in mass toward the center is less significant, although still existing in many filaments.