Star formation occurs inside over-dense region in molecular clouds. The interstellar medium (ISM) is composed of gas and dust, and the primary element is hydrogen, followed by helium.
The ISM is affected by turbulence, gravity and magnetic field, forming complex structures.
In particular, the molecular clouds are formed in high-density regions of the ISM. Within the molecular cloud, complex structures also develop as consequence of interaction among the physical mechanisms mentioned above. Early observations found that stars are born in molecu-lar clouds. Moreover, in recent years, observations by space telescopes have found filamentary structures, which are highly associated with the prestellar cores (Andr´e et al. 2010) .
Figure 1.1 shows two molecular cloud regions in the Gould Belt. In Aquila, there are many high density regions, which are characterized as prestellar cores and protostars. However, the Polaris Flare region does not show any sign of star formation activities. Two regions both have filamentary structures. This suggests that filaments form prior to star formation.
Ph. André et al.: The Herschel Gould Belt Survey
Fig. 1.Column density maps of two subfields in Aquila (left) and Polaris (right) derived from our SPIRE/PACS data. The contrast of the filaments with respect to the non-filamentary background has been enhanced using a curvelet transform as described in Appendix A. Given the typical width
∼10 000 AU of the filaments, these column density maps are equivalent to maps of the mass per unit length along the filaments. The color scale shown on the right of each panel is given in approximate units of the critical line mass of Inutsuka & Miyama (1997) as discussed in Sect. 4.
The areas where the filaments have a mass per unit length larger than half the critical value and are thus likely gravitationally unstable have been highlighted in white. The maximum line mass observed in the Polaris region is only ∼0.45 × the critical value, suggesting that the Polaris filaments are stable and unable to form stars at the present time. The candidate Class 0 protostars and bound prestellar cores identified in Aquila by Bontemps et al. (2010) and Könyves et al. (2010) are shown as green stars and blue triangles, respectively. Note the good correspondence between the spatial distribution of the bound cores/protostars and the regions where the filaments are unstable to gravitational collapse.
Fig. 2.Core mass functions (blue histograms with error bars) derived from our SPIRE/PACS observations of the Aquila (left) and Polaris (right) regions, which reveal of total of 541 candidate prestellar cores and 302 starless cores, respectively. A lognormal fit (red curve) and a power-law fit (black solid line) to the high-mass end of the Aquila CMF are superimposed in the left panel. The power-law fit has a slope of −1.5 ± 0.2 (compared to a Salpeter slope of −1.35 in this dN/dlogM format), while the lognormal fit peaks at ∼0.6 M$and has a standard deviation of ∼0.43 in log10M. The IMF of single stars (corrected for binaries – e.g., Kroupa2001), the IMF of multiple systems (e.g., Chabrier2005), and the typical mass spectrum of CO clumps (e.g., Kramer et al.1998) are also shown for comparison. Note the remarkable similarity between the Aquila CMF and the stellar IMF, suggesting a ∼ one-to-one correspondence between core mass and star/system mass with M!sys= " Mcoreand " ≈ 0.4 in Aquila.
one-to-one basis, with a fixed and relatively high local efficiency, i.e., "core≡ M!/Mcore∼ 20−40% in Aquila. This is consistent with theoretical models according to which the stellar IMF is in large part determined by pre-collapse cloud fragmentation, prior to the protostellar accretion phase (cf. Larson1985; Padoan &
Nordlund2002; Hennebelle & Chabrier2008). There are sev-eral caveats to this simple picture (cf. discussion in André et al.
2009), and detailed analysis of the data from the whole GBS will be required to fully characterize the CMF–IMF relationship and, e.g., investigate possible variations in the efficiency "corewith
environment. It is nevertheless already clear that one of the keys to the problem of the origin of the IMF lies in a good understand-ing of the formation process of prestellar cores, even if additional processes, such as rotational subfragmentation of prestellar cores into binary/multiple systems (e.g., Bate et al.2003), probably also play an important role.
Our Herschel initial results also provide key insight into the core formation issue. They support an emerging picture (see also Myers2009) according to which complex networks of long, thin filaments form first within molecular clouds, possibly as a result Page 3 of7
Figure 1.1: Column density maps of two subfields in Aquila (left) and Polaris (right) derived from SPIRE/PACS data (Fig.1 fromAndr´e et al. 2010) . The green stars and blue triangles show the candidate Class 0 protostars and bound prestellar cores, respectively, which were identified in Aquila byBontemps et al.(2010).
Prestellar cores form from a sequence of gravitational collapse and fragmentations, that will eventually lead to the formation of stars. The following process describes how a prestellar core forms, and Figure 1.2 is a schematic diagram. First, the ISM contracts under gravity to form a clump. Second, inside a clump, gravitational instability leads to fragmentation into sheets. Third, inside a two-dimensional sheet, gravitational instability leads to fragmentation into filaments. Finally, the fragmentation of these latter results in the formation of prestellar cores.
Figure 1.2: A schematic diagram of the process of forming the prestellar core.
However, it is highly unlikely to form a high-mass star from fragmentation of one single filament, because there is not enough mass. Observations suggest that star formation occurs preferentially along the filaments, with high-mass stars forming in the highest density regions where several filaments converge, called ridges or hubs (Trevi˜no-Morales et al. 2019). This structure is usually called the hub-filament system, where high-mass stars and star clusters form.
In the following, we will give an example of such structure (Trevi˜no-Morales et al. 2019). The Monocerous R2 (Mon R2) is an excellent target to study hub-filament system, at a distance of 830 pc. Figure 1.3 is Mon R2, shows the cloud with several filaments converging into the central hub. In the next paragraph, we will briefly introduce the hub-filament system from the observations of the Mon R2.
Figure 1.3: Left: Three-color image of the Mon R2 cluster-forming hub-filaments system.Red: H2column density map derived from Herschel SPIRE and PACS observations (Didelon et al. 2015), green: 1.65 µm band of 2MASS (Two micron all survey;Skrutskie et al. 2006), and blue: 560 nm band of DSS (Digitalized Sky Survey;Lasker et al. 1990). (Fig.1 fromTrevi˜no-Morales et al. 2019)
In Figure1.4, the left column shows the integrated intensity maps over the whole surveyed area for the (1 → 0) transition lines of 13CO and C18O molecules. It is similar to the map of column density, indicating that the mass is concentrated in the central hub and the filaments.
The middle column shows the velocity gradient, in which we can see that the upper left is moving toward us and the lower right is moving away from us. This can be explained as a radially collapsing velocity field within a plane with an inclination angle with respect to the line of sight. The right column shows high velocity dispersion in the central hub. This is possibly a sign of gravitational collapse or outflow from massive stars. Figure1.5 shows the relationship between velocity and position along the filament. The velocity increases along the filament toward the central hub, suggesting that the gas is falling into the central hub. Overall, Fig.1.4 and Fig.1.5present the Mon R2 region with several filaments converging into the central hub, and the filaments lie within a 2D plane.
Figure 1.4: Left panels show the integrated intensity maps over the whole surveyed area for the (1→ 0) transi-tion lines of the 13CO and C18O molecules. Middle panels present the velocity centroid. Right panels show the linewidth. The yellow labels, and the dotted lines, indicate the main features identified in the region. (Fig.3 from Trevi˜no-Morales et al. 2019)
Figure 1.5: Position-velocity diagrams along the ‘skeleton’ of filament F3 obtained from the13CO (left) and C18O (right) data cubes. The vertical yellow dashed lines indicate the transition between the hub and the filaments, corresponding to radii 20000, 25000(Rhub), and 30000.( Fig.B.3 fromTrevi˜no-Morales et al. 2019)
In this study, we aim to describe the dynamics of this system with a physical model. There-fore, we first review the study related to self-similar collapse solutions have been studied for an isothermal spheres in the ISM (e.g.
Hunter 1977; Larson 1969; Penston 1969; Shu 1977),
poly-tropic cylinder/filament (e.g.Kawachi & Hanawa 1998), and magnetized molecular cloud cores
which are flattened (e.g.Contopoulos et al. 1998; Krasnopolsky & K¨onigl 2002). None of the
existing studies has considered the self-similar collapse of a thin sheet in the ISM. Therefore,we consider the radial collapse of a sheet with self-gravity and solve for the self-similar solu-tion.
Whitworth & Summers
(1985) found a family of solution (byHunter 1977; Larson 1969;
Penston 1969; Shu 1977), and did mathematical and physical analysis, as well as numerical
analysis. Therefore, we follow their procedure in our work, and compare the obtained surface density and velocity profiles to observations in Chapter2.In addition, we would like to study the development of filaments in this structure with per-turbative analysis. Therefore, we review the study related to perper-turbative analysis.