HIERARCHICAL FRAGMENTATION OF THE ORION MOLECULAR FILAMENTS

In order to discuss the physical properties of the detected SMA continuum sources, their measured source sizes are plotted as a function of the estimated hydrogen number density. Jeans analysis can provide a useful way to discuss the gravitational stability of the observed sources. With the assumption of an infinite and homogenous medium, the Jeans length is described as follows (Jeans 1902):

$$\begin{equation} \lambda _{{\rm frag.}} = \sqrt{ \frac{{\pi }c_s^2}{G{\rho _0}}}, \end{equation} \tag{ 2 }$$

where G is the gravitational constant, ρ₀ is the mean density, and c_s is the sound speed (the relation between the sound speed and the temperature of the gas is described as $c_s = \sqrt{{kT_{{\rm gas}}}/{{\mu }m_{{\rm H}}}}$), where T_gas is the gas temperature and k is the Boltzmann constant. The Jeans length corresponds to the critical scale when the gas cloud becomes gravitationally unstable, i.e., if the source size is larger than the Jeans length (D_core  >  λ_frag.), the core becomes gravitationally unstable and starts to collapse. The Jeans critical number density is described as a function of the Jeans length and gas temperature as

$$\begin{equation} n_{{\rm H_2}} (\lambda _{{\rm frag.}}, T_{{\rm gas}}) = \frac{{\pi }k}{G({\mu }{m_{{\rm H}}})^2} \frac{T_{{\rm gas}}}{\lambda _{{\rm frag.}}^2}. \end{equation} \tag{ 3 }$$

In Figure 4(c), the Jeans critical number density is plotted as a function of source size for different gas temperatures (T = 10 K, 20 K, and 30 K) and is overlaid with the observed results. Here, we consider that the gas temperature within OMC filaments is approximately 20 K (e.g., Cesaroni & Wilson 1994). The plotted results clearly show that most SMA sources are located very close to the critical line (T_gas = 20 K) and several sources have a larger size than the Jeans length at their measured density. This result suggests that most of the detected continuum sources are gravitationally unstable and could be collapsing if we assume effect only from thermal support. No significant differences are detected between prestellar cores and protostellar cores.

The Jeans criterium indicates that most of our SMA sources are gravitationally unstable. Here, we examine other observational evidence of their protostellar nature. In Figure 2, we compare the positions of infrared sources, i.e., protostars (denoted by orange open circles), and T Tauri stars (denoted by white open circles) based on Peterson & Megeath (2008), and our 850 μm continuum sources; these comparisons are of similar spatial resolutions (2''–5'' corresponding to 830–2100 AU at Orion). Multi-wavelength data and outflow survey results are summarized in Columns 2–7 of Table 5. Spitzer 8 μm sources were detected toward eight out of twelve SMA sources (∼67%), six out of twelve SMA sources (∼50%) have CO bipolar outflow, while three out of twelve SMA sources (∼25%) are associated with radio jets (Reipurth et al. 1999; Aso et al. 2000; Stanke et al. 2002; Williams et al. 2003; Takahashi et al. 2006, 2008a, 2009; Takahashi & Ho 2012). These results suggest that approximately half of the SMA sources in this region are protostellar.

Eight out of sixteen Spitzer sources (within the SMA observing region) are associated with the SMA continuum sources, which are identified as protostars (Peterson & Megeath 2008; Megeath et al. 2012). Six out of sixteen Spitzer sources, which are categorized as T Tauri stars, have no counterparts in the SMA continuum emission. Two out of sixteen Spitzer sources identified as protostars (Peterson & Megeath 2008; Megeath et al. 2012) are not associated with SMA continuum sources. That we did not detect submillimeter continuum emission from the T Tauri source as well as two protostars is likely due to the mass detection limit of our SMA observations. Evolutionary status of the individual sources is summarized in the last column of Table 5. Here, the classification between prestellar and protostellar cores is based on the detection of either an infrared source, and/or molecular outflow, and/or radio jet. The evolutionary status of a protostellar core was determined based on its SED. Since the spatial resolution of submillimeter observations is limited, we could not use the original definition for identifying Class 0 sources (i.e., L_bol/L_submm < 200 by Andre et al. 1993). Instead of this, the classification of Class I (Class 0) sources was determined based on the detection (non-detection) of 2.2 μm emission.

Our SMA continuum observations with 45 angular resolution have revealed a chain of density enhancements within the OMC-3 filament. The large-scale filamentary structure and their fragmentation properties can be a key to understanding the initial conditions of embedded clusters such as core spatial distribution, mass distribution, number distribution, and evolutionary sequence, all of which may govern the initial CMF of young stellar clusters and initial stellar multiplicity.

Certain fragmentation length scales within the OMC filaments have been suggested by several previous studies (Dutrey et al. 1991; Hanawa et al. 1993; Wiseman & Ho 1998; Johnstone & Bally 1999). The largest spacing is shown in the ¹²CO image by Maddalena et al. (1986), which corresponds to the Orion A and Orion B GMCs, and has a separation of ∼47 (35 pc). Within the Orion A cloud, periodical structures have been reported by Dutrey et al. (1991) in the C¹⁸O (1–0) image with peak separations about 9' (1.1 pc) toward the northern part of the Orion A cloud, while similar structures, with ∼10' (1.2 pc) separation, have also been reported in the ¹³CO (1–0) image at the southern part of the Orion A cloud by Hanawa et al. (1993). The emission peaks correspond roughly to significant star-forming regions such as OMC-1S, OMC-1, OMC-2, and OMC-3 within the Orion A cloud (Johnstone & Bally 1999). Single-dish observations in the NH₃ (1, 1) and (2, 2) and single-dish (sub)millimeter continuum have shown the 0.1 pc scale dense clumps embedded within the OMC filaments (Cesaroni & Wilson 1994; Chini et al. 1997; Lis et al. 1998; Johnstone & Bally 1999). Johnstone & Bally (1999) pointed out that these clumps are equally separated along the filaments with separation of ∼150'' (0.31 pc). Moreover, smaller size scale density enhancements have been found in the high angular resolution VLA NH₃ observations of the OMC-1 region (Wiseman & Ho 1998). Their mean separation is ≈50'' (0.1 pc).

Our SMA continuum observations revealed fragmentation within the 0.1 pc clumps. Nearest neighbor distances for each source are calculated, and listed in the sixth column of Table 4. The median separation between nearest neighbors is measured to be ∼17'' or 0.035 pc.⁸ Furthermore, we have performed 1.3 mm continuum observations with the SMA toward the OMC-1 north region, and found similar source separation within their filaments (≈0.06 pc). The details of the OMC-1 north observations will be presented and discussed in P. S. Teixeira et al. (2013, in preparation).

In summary, our SMA observations have resolved the continuum sources with quasi-periodical separation as small as ∼0.035 pc. A comparison of the SMA continuum observations with previous single-dish observations suggests multi-spatial scale fragmentation within the Orion A GMC with size scales between 37 pc and 0.035 pc.

The fragmentation of molecular clouds is related to gravitational instabilities. Jeans analysis can provide a useful way to discuss which scale can grow and produce high-density regions. As described in Section 4.1.1, the critical Jeans length is described as Equation (4) with the assumption of an initially uniform background density (e.g., Jeans 1902; Binney & Tremaine 1987). For other geometric configurations, such as cylindrical and sheetlike structures, sufficiently large perturbations can also result in gravitational instability (e.g., Nakamura et al. 1993; Burkert & Hartmann 2004). In the case of an infinitely long static and cylindrical isothermal cloud, the perturbation wavelength for maximum instability (hereafter we simply call this the Jeans length) is described as follows (e.g., Nakamura et al. 1993; Wiseman & Ho 1998):

$$\begin{equation} {\lambda _{{\rm frag.}}} \sim {\frac{20c_s}{{\sqrt{4{\pi }G{{\rho }_0}}}}}. \end{equation} \tag{ 4 }$$

In order to understand the physical mechanism of fragmentation processes over a wide range of size scale, the measured source separation is plotted as a function of the number density of their respective parental molecular clouds/clumps/cores in Figure 5. Note that the inclination of the filaments should be taken into account in this discussion. According to Genzel & Stutzki (1989), a randomly oriented filament will have a line-of-sight mean (median) inclination angle of 45° (60°). To display this projection effect (λ_obs = λ_frag  ·  sin  i), core separations in the case of i = 45° are plotted in Figure 5 as open symbols. Table 6 summarizes the number densities, structure separations, and gas temperatures of different structures. These structures are listed by size: GMCs (⩾1 pc), large-scale clumps (0.5–1.0 pc), small-scale clumps (0.1–0.5 pc), and dense core (⩽0.1 pc). Here, we assume that dense cores are formed via the fragmentation of parental small-scale clumps, which in turn are formed via the fragmentation of parental large-scale clumps, and that these latter structures are formed via the fragmentation of parental GMCs. The measured separations were plotted as a function of the number density of the parental large scale structure. To further discuss the smaller scale fragmentation properties, the separation of the visible binary stars studied in the Orion Nebula Cluster (ONC; Reipurth et al. 2007) is also plotted assuming that the number density of their parental core is similar to that of our SMA continuum source.

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Table 6. Hierarchical Structure of the OMC Filaments

	$n_{{\rm H_2}}$	Tgas	Measured Separation	Corresponding Size Scale
	(cm−3)	(K)	(arcmin)
ISMs	50–100(1)	50–100(1)	...	ISMs observed in H i 21 cm observations
GMCs	200–3000(2)	...	282(3)	Ori A and Ori B (Separation between Orion BN/KL and NGC 2024)
Large-scale clumps	9.0e03–3.0e04(4)	...	9–10(5, 6, 7)	Significant star-forming regions such as OMC1,2,3, etc.
Small-scale clumps	8.8e04–1.1e06(5, 8)	15–28(8)	2.5(5)	Identified with single-dish NH3 and continuum observations
Dense cores	3.2e06–3.3e08(9)	...	0.15–0.55(9)	Individual star-forming sites detected with the SMA observations
Binaries	...	...	0.0025–0.025(10)	Binary separation range measured in ONC (Hα observations).

References. (1) Ferrière 2001; (2) Sakamoto et al. 1994; (3) Wilson et al. 2005; (4) Dutrey et al. 1993; (5) Johnstone & Bally 1999; (6) Dutrey et al. 1991; (7) Hanawa et al. 1993; (8) Cesaroni & Wilson 1994; (9) This work; (10) Reipurth et al. 2007.

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We overplot in Figure 5 the perturbation wavelength, λ_frag., given in Equations (2) and (4) for various gas temperatures. The observed fragmentation spacings (denoted in Figure 5 as filled symbols) roughly follow a single power law over almost four orders of magnitude, suggesting that the hierarchical fragmentation could be governed by a single physical process. Figure 5 indicates that the data are described better by a shallower index, −1.4 (dashed line), than that predicted by maximum instability size as a function of density of isothermal clouds, −2.0 (solid line). This shallower index can be explained by differences of cloud temperatures over different size scales. Parsec-scale GMCs are exposed to ionized gas from H ii regions, cosmic rays, large-scale turbulence, etc., which work as heating processes. Consequently, the mean temperature of parsec-scale clouds tends to be relatively high (e.g., 50–100 K in the interstellar mediums (ISMs); e.g., Ferrière 2001). Intermediate regions (both large- and small-scale clumps) of the OMC molecular cloud (0.1–1.0 pc) are also most likely exposed to the strong interstellar radiation field from nearby star-forming clusters, e.g., strong heating from the exterior, corresponding to 1000 times stronger interstellar radiation field as suggested toward a few small-scale clumps in the OMC-2/3 region from the CO isotopes observations with Monte Carlo line radiative transfer models (e.g., Jørgensen et al. 2006). This strong radiation is able to produce the higher gas temperature (>30 K) at the outer most part of the envelope gas. Finally, deeply embedded dense cores within filaments are shielded from these heating processes. Thus, the innermost parts of clouds have low temperatures (e.g., ≈20 K; Cesaroni & Wilson 1994; Wiseman & Ho 1998; Tatematsu et al. 2008). This decrease in temperature with size scale means the fragmentation length will also decrease more sharply than the isothermal case. This leads to a shallower power law in Figure 5.

Moreover, the dissipation of the magnetic flux on the smaller scale can also be responsible for the observed shallower power-law index. Since large-scale diffuse structures are always exposed to cosmic ray and/or ionized gas from the surrounding H ii region, the fractional ionization in large-scale molecular clouds is higher than in the small-scale dense gas core. Therefore, ambipolar diffusion will be more efficient in the dense cores, leading to less magnetic support, and a smaller fragmentation length (e.g., Nakano & Tademaru 1972). Finally, turbulence, which is often observed in GMC size scales (Larson 1981; Solomon et al. 1987), can also be responsible for the observed shallower power-law index. Detailed discussion for the OMC case will be described in Section 4.2.3.

Observational results suggest that the fragmentation spacings, especially at the scales of clumps (0.1–1 pc), are roughly consistent with the maximum instability size derived from either the initially uniform density cloud (orange solid line in Figure 5) or cylindrical clouds (blue solid line in Figure 5), assuming increasing temperatures with size scale. This indicates that the fragmentation may be occurring through the thermal fragmentation process. The SMA detected continuum sources (denoted in Figure 5 by triangle shaped data points) are embedded within a filament. Hence the core separation should be around the maximum instability size of the cylindrical clouds model (denoted in Figure 5 by the blue line). However, the observed core separation is a factor of 2.0–7.3 smaller than the thermal fragmentation spacing with T_gas = 20 K, $n_{{\rm H_2}}=8.8\times 10^4\hbox{--}1.1\times 10^6$ cm⁻³ (Table 6), and i = 90°. In the next subsection, we explore different physical processes that can affect the fragmentation size scale.

From our SMA observational result, the observed fragmentation (λ_frag = 0.035 pc) is smaller than the Jeans length of the clump. Projection effects are not sufficient to explain this difference: λ_frag = λ_obs/sin i < λ_J for 45° ⩾ i ⩾ 90°. In order to further discuss the fragmentation properties within the Orion A GMC, we summarize different possible mechanisms that can affect the fragmentation size scale.

Turbulence. In this case, the cloud fragmentation length is described by Equations (2) and (4) with the effective sound speed c_eff instead of the sound speed c_s. The c_eff is defined as c²_eff = (Δv_obs²/8ln 2), where Δv_obs is the observed velocity dispersion and is given by Δv²_obs = Δv_t² + Δv²_nt, where Δv_t and Δv_nt correspond to the thermal and non-thermal velocity dispersion, respectively. Turbulence will therefore produce a fragmentation scale larger than the Jeans length. Figure 5 and Table 6 present that observed clump separations are roughly consistent, or smaller, than the thermal fragmentation lengths for scales of a few parsec. These results suggest that turbulence has already decayed for size scales ⩽1 pc. For scales of a few parsec or larger, turbulent pressure may in fact be dominant. For example, Jackson et al. (2010) studied a filament that spans 80 pc and conclude that its observed fragmentation scale (≈5 pc) can only be described by an isothermal filamentary cloud determined by turbulent pressure, or equivalently, by a self-gravitating incompressible fluid cylinder. Our data indicate that the OMC 3 filament fragmentation cannot be explained by this scenario.

Magnetic field and cloud rotation. Other physical processes that might affect the fragmentation length of filaments are helical magnetic field and cloud rotation (Pudritz & Fiege 2000; Matsumoto et al. 1994). In fact, Matthews et al. (2001) use the helical field geometry of a magnetic field to interpret their JCMT/SCUBA polarization observations toward OMC-3. Their data indicate that the magnetic field in the northern part of OMC-3 is toroidal. Hanawa et al. (1993) have performed magnetohydrodynamical simulations for filament fragmentation, consisting of an isothermal rotating cloud + helical magnetic field, and found a dispersion relation (Equation (11), Hanawa et al. 1993). We can write the fragmentation wavelength for the toroidal magnetic field case as

$$\begin{equation} \lambda _{{\rm frag}} = \frac{2{\pi }H}{0.72[(1+{{\alpha }/2}+{\beta })^{1/3}-0.6]}, \end{equation} \tag{ 5 }$$

where H represents a length scale of the parental clump, and is given by

$$\begin{equation} H^2 = \frac{c_s^2}{4{\pi }G{\rho _c}} \left(1+\frac{1}{2}{\alpha }+{\beta } \right), \end{equation} \tag{ 6 }$$

where ρ_c is the peak density of the parental clump, and the effect from magnetic field and rotation are described by α and β as

$$\begin{equation} \alpha = \frac{B^2}{8{\pi }{\rho }_c{c_s^2}},\quad {\beta } = \frac{2{\Omega }^2H^2}{c_s^2}, \end{equation} \tag{ 7 }$$

where Ω is the rate of rotation and B is the magnetic field strength. Equation (5) shows that if (α + β)  ⩾  1, the maximum instability size will become smaller than the thermal fragmentation length. If α + β ≪ 1, the effects from the magnetic field and rotation are negligible.

Global collapse of the parental cloud. Recent theoretical studies present how cloud structures and global collapse significantly affect local fragmentation properties (e.g., Burkert & Hartmann 2004; Pon et al. 2011, 2012). If the original parental cloud is not infinitely long or spherical, gravitational focusing results in enhanced concentrations of mass inducing global collapse unless magnetic fields, cloud rotation, and/or external pressure dominate (Burkert & Hartmann 2004). As we can see from the JCMT/SCUBA 850 μm continuum observations (Figure 2, right panel), the parental filamentary clump (e.g., 1.5 Jy beam⁻¹ denoted by thick contour) shows a relatively well-defined edge, so according to Burkert & Hartmann's model, global collapse could potentially be underway in this clump. A recent paper by Pon et al. (2011) pointed out that filamentary geometry is the most favorable situation for the smallest perturbation to grow before global collapse overwhelms the cloud. Their following paper (Pon et al. 2012) suggested that filaments with large aspect ratio would take more time to collapse. For example, the global collapse timescale for a cloud aspect ratio of 10 (similar to the case of the observed clumps) is roughly seven times slower than the spherical free-fall time. This can explain how local fragmentation at the size scale of dense cores and global collapse of their parental small-scale clump could be concomitant. Global collapse will eventually shorten the spacing between the fragmented cores, and this scenario could therefore potentially explain the observed shorter fragmentation length compared to Jeans length.

In summary, there are three physical mechanisms that can increase (turbulence) or decrease (magnetic fields and rotation, or global collapse) the fragmentation length relative to the Jeans length. The fragmentation lengths obtained from the data presented in this paper are consistent (0.3 pc–1 pc scale) or smaller (⩽0.1 pc scale) than the Jeans fragmentation length, suggesting that turbulence does not play a significant role in the fragmentation process for these size scales. Note that some of their core are larger than their associated Jeans length, and therefore, they could fragment further. We discussed how the magnetic field and cloud rotation, or global collapse may decrease the observed fragmentation length. However, at the moment we are unable to quantitatively constrain α + β, or the inflow from the global collapse for our case—only future direct comparisons between higher-dynamic range kinematic data and more realistic models can properly address these issues.

Core fragmentation. In addition to the large-scale clump fragmentation properties discussed above, we would like to explore further the smaller core-scale fragmentation properties, which is related to binary star formation (⩽1000 AU). These binary stars observed in the ONC have a wide range of separations from 0.0027–0.027 pc (67.5–675 AU; Reipurth et al. 2007). The plotted results in Figure 5 suggest that wide binary systems (⩾200 AU) might follow the same power-law index extrapolated from the observed large size scale cloud fragmentation, i.e., these wide binaries could be formed through the next level of the hierarchical process. This lends support to the theoretically predicted binary formation scenario through the hierarchical fragmentation of the parental core occuring in the isothermal collapse phase (e.g., Goodwin et al. 2007; Machida et al. 2008). Close binary systems (⩽100 AU) might be formed through other mechanisms; hydrodynamic simulations suggest that angular momentum within the rotating and collapsing hydrostatic first cores is a crucial parameter to determine the later fragmentation properties (e.g., Boss 1989; Bonnell & Bate 1994; Bate 1998; Saigo & Tomisaka 2006; Machida et al. 2008). Furthermore, star–disk encounters could also play an important role in redistributing angular momentum in protoplanetary disks and forming binary systems (e.g., Larson 2002; Pfalzner 2004; Pfalzner et al. 2005).

One interesting issue related to filament fragmentation is whether the fragmentation occurs simultaneously in the entire filament, or instead propagates along it, producing sequential star formation along the filament. With the SMA continuum observations, we found the following interesting tendencies among the sources: (1) a chain of continuum sources is detected in the northern part of the OMC-3 filament from SMM 2 to SMM 8 with a similar spatial separation, (2) a relatively extended fluffy source (elongated structure), SMM 5, is located at the center of the chain of the continuum sources, and this fluffy structure still seems to be in the process of core formation (or fragmentation). No star formation signatures such as outflow/jet association or infrared source are found, which supports the idea that SMM 5 is probably the youngest star formation site within the filament. Why does SMM 5, which is located in the middle of the filament, have a fluffy structure? A fragmentation process which propagates along the filament from each end inward can be one possibility to explain this.

Previous numerical calculations give us a hint that the finite geometry of gas clouds promotes concentrations of material at the ends of the filament, at the beginning of the filament fragmentation. Hanawa et al. (1994) performed one-dimensional hydrodynamic simulations of a finite long filamentary cloud. These simulations suggest that dense cores are produced with a semi-regular interval in space and time from one edge to the other. The gas near the edge is attracted inward by gravity and the accumulation of the gas makes a dense core near the edge. When the dense core grows in mass up to a certain amount, it gathers gas from the other direction. Accordingly the dense core becomes isolated from the main cloud and the parent filamentary cloud has a new edge. Via the same process, the next accumulation of gas at the new edge makes a new dense core. The fragmentation is considered to propagate at approximately the effective sound speed (1.23c_eff derived from the simulation by Hanawa et al. 1994; 1.1c_eff derived from a linear stability analysis by Nakamura et al. 1993). Signatures of the fragmentation propagating down the filament are actually observed in the Taurus Molecular Cloud −1 region (Hanawa et al. 1994).

Our SMA results also show that the chain of continuum sources, from SMM 2 to SMM 8, is embedded within the most northern part of the OMC-3 filaments where both ends show edge-like density contrast (finite filamentary clouds with two edges; Figure 2). The mean value of the observed velocity width in FWHM is approximately 0.88 km s⁻¹ in this region (the mean velocity width derived from NH₃ observations toward the northern OMC cloud by Cesaroni & Wilson 1994) and the effective sound speed is estimated to be $c_{{\rm eff}} \sim \sqrt{{{\Delta }v^2}/{8 \,{\ln}\,2}}\sim 0.37$ km s⁻¹. The mean projected distance from the sources at the two edges of the filament (i.e., SMM 2 and SMM 8) to SMM 5 is measured to be 0.15 pc. Therefore, the timescale of the fragmentation propagation is estimated to be approximately 0.15 pc/0.37 km s⁻¹ ∼ 3.0 × 10⁵ yr. The detected continuum source in the filament has similar to the protostellar phase (Peterson & Megeath 2008; Megeath et al. 2012). The typical evolutionary timescale of such sources is 10⁴–10⁵ yr (Beichman et al. 1986; Shu 1977). Estimated propagation timescale of the filament fragmentation is similar to, or slightly longer than, the star formation timescales of the detected sources. This may suggest that the protostars formed within the fragmented cores, where star formation began almost at the same time as the clump fragmentation. More complete fragmentation timescale analysis (including other parts of the OMC filaments) will be presented in following papers.