Gravitationally Unstable Condensations Revealed by ALMA in the TUKH122 Prestellar Core in the Orion A Cloud

To derive the dust temperature and mass of the ALMA-ACA observing region by using the Herschel 250, 350, and 500 μm data, we adopt a dust opacity of 5.5 cm² g⁻¹ at 350 μm and 2.7 cm² g⁻¹ at 500 μm modeled by Ormel et al. (2007). These values have been shown to reproduce well the observed spectral energy distribution (SED) from 2.2 to 850 μm in the Orion A cloud, as suggested by Lombardi et al. (2014). The mass M and dust temperature T_d are computed with

$$\begin{eqnarray}&&M=\displaystyle \frac{{F}_{\nu }{d}^{2}}{\kappa {B}_{\nu }({T}_{d})}{f}_{d},\end{eqnarray} \tag{ 1 }$$

where F_ν is the observed flux in Jy, d is the distance to the target, ${B}_{\nu }({T}_{d})$ is the Planck function, κ is the dust opacity, and f_d is the gas-to-dust mass ratio (assumed to be 100). By calculating the flux density within the ALMA observed field, the dust temperature of TUKH122 is derived to be 12 K, consistent with the previous results of ${T}_{\mathrm{kin}}=11$ K obtained by NH₃ spectra (Ohashi et al. 2016c). The mass is also derived to be ∼29 M_⊙. In spite of the observational studies that dust temperature is often higher than gas temperature because of averaging the line of sight of the dust, the similar temperatures between gas and dust may suggest that this region is widely isothermal.

The plane-of-the-sky orientation of the filament and elongated core is measured to be a position angle of P.A. = 1463 derived by 2D Gaussian fitting on the Herschel image. The directions parallel and perpendicular to the filamentary structure are shown in Figure 5.

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To investigate the density structure of this region, Figure 6 shows the column density profiles perpendicular to the filamentary structure. The column density profiles are derived with the ALMA-ACA and Herschel observations by using Equation (1), assuming the surface is of the ALMA-ACA beam size of ∼52 and the Herschel beam size of ∼18''. We measure a radial column density profile in the southwest direction (P.A. = 2363) perpendicular to the filament. Then, the profile is calculated by averaging the flux densities within 15'' width at each distance from the peak position (see also Figure 5). We do not use the data of the northeast side because several protostars may affect the density and temperature on this side (see Figure 2).

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Figure 6 shows that the Herschel observations recover extended emission at a distance larger than 0.1 pc, while the ALMA-ACA observations only detect dense gas within 0.1 pc. This is consistent with the results that the ALMA 12 m and ACA 7 m observations are not sensitive to the extended envelope for scales larger than 0.1 pc. We also find that the column densities of the ALMA-ACA observations are higher than those of the Herschel observations inside 0.02 pc. This would most likely be caused by the low resolution of the Herschel observation because the ALMA-ACA observations have three times better spatial resolution than Herschel. For example, Herschel does not resolve the 0.04 pc scale condensation found in this study.

To understand the density structure of the parent filament with the core, we plot the Plummer-like function as the red and black lines in the figure. The Plummer-like profile is defined as

$$\begin{eqnarray}&&\rho (r)=\displaystyle \frac{{\rho }_{c}}{{[1+{(r/{R}_{f})}^{2}]}^{p/2}},\end{eqnarray} \tag{ 2 }$$

and the corresponding surface density profile is

$$\begin{eqnarray}&&{\rm{\Sigma }}(r)=A\displaystyle \frac{{\rho }_{c}{R}_{f}}{{[1+{(r/{R}_{f})}^{2}]}^{(p-1)/2}},\end{eqnarray} \tag{ 3 }$$

where A is a numerical factor. The numerical factor of A depends on the density slope p: A = π corresponds to p = 2, while A = π/2 corresponds to p = 4 (Arzoumanian et al. 2011). We derive the column density assuming a mean molecular weight of 2.33. The profile is determined by the central density ρ_c, the radius R_f, and the density slope p (Nutter et al. 2008; Arzoumanian et al. 2011). Note that the isothermal hydrostatic filament has p = 4 (Stodólkiewicz 1963; Ostriker 1964). Recently, the Herschel observations suggested p ∼ 2 (e.g., Arzoumanian et al. 2011) and that the dynamical contraction makes a flatter density slope. In Figure 6, the red line shows the power law of p = 1.6, while the black line shows the power law of p = 4.0, nicely fitting the data plotted. The FWHM of the Herschel data fitting is 0.16 pc. On the other hand, the ALMA-ACA observations show a thinner and steeper filament due to the missing flux.

The slope of p = 4.0 is consistent with the isothermal hydrostatic filament or the density profile of prestellar cores (Whitworth & Ward-Thompson 2001). However, the ALMA-ACA observations only recover 8.4 M_⊙ out of a total of ∼29 M_⊙. It is difficult to distinguish the density structure of the TUKH122 core from the parent filament by these observations due to the missing flux. We cannot discuss the density profile or the power law of the ALMA-ACA observations. We only show that the missing flux is not the background emission but the surrounding gas associated with the filament.

It is possible that the power law of p = 1.6 indicates the dynamical contraction of the filament rather than the background effect. The filament actually forms the dense core and condensations. Therefore, we indicate that the power law of p = 1.6 is the radial profile of the parent filament that already forms the elongated core.

To identify dense condensations precisely, we apply the dendrogram method (Rosolowsky et al. 2008) to the combined 3 mm dust continuum image. The dendrogram traces local maxima and describes hierarchical structures, which have been shown to be more robust against noise and user-defined parameters (e.g., Goodman et al. 2009; Pineda et al. 2009). This method gives three types of structures: leaf, branch, and trunk. We define "leaves" as "condensations" in this paper because the leaf is the smallest structure. We adopt a threshold of 3σ and 1σ interval steps. The minimum number of pixels is 3 × θ_beam. In addition, we select the condensations as the robust ones when they can be identified as detection above 3σ with ALMA 12 m data. The identified condensations are shown in Figure 4 as the red contours and listed in Table 2.

The mass of the 3 mm dust condensations is computed with Equation (1) at a dust temperature T_d = 12 K. We adopt a dust opacity of 0.0755 cm² g⁻¹, assuming a dust emissivity index of β = 2 (Ormel et al. 2007). As a result, the condensation mass ranges from 0.1 to 0.4 M_⊙. The typical deconvolved radius is r ∼ 29. The radius is calculated from the surface area of the identified condensations. The H₂ densities are n ∼ 10^6–7 cm⁻³, which are of the order of or higher than those of typical low-mass prestellar cores (e.g., Onishi et al. 2002; Enoch et al. 2008). The condensation in the center of TUKH122, MM4, is the most massive and has ∼0.4 M_⊙. Note that the deconvolved radius of r ∼ 29 is very close to the beam-size radius. The condensations MM1, MM2, and MM6 show that the peak intensities have larger values than the total fluxes (see also Table 2). These results indicate that these three condensations (MM1, MM2, and MM6) are not well resolved with our observations. However, the distances between the condensations are larger than the beam size, indicating that we can determine the locations of the condensations with our observations. We find an interesting trend that the northern condensations (MM1, MM2, and MM3) have 0.1–0.2 M_⊙, while the southern condensations (MM4 and MM5) have ∼0.4 M_⊙. We discuss the different mass distribution along the filament in the following section.

The total mass of TUKH122 is derived to be 8.4 M_⊙ (2.5 M_⊙ for ALMA 12 m data) using the ALMA-ACA combined data. Taking into account the total mass of TUKH122 (∼29 M_⊙) estimated by Herschel, there is still a large fraction of flux that is not recovered by ALMA. In order to check how the condensation masses vary, we combine our ALMA and ACA data with the Herschel 250 μm data using feather in CASA. Note that we rescale the 250 μm flux to the 3 mm flux assuming the dust temperature of 12 K and ice-covered silicate-graphite conglomerate grains with a dust emissivity index of β = 2 modeled by Ormel et al. (2007). We confirm that the total mass is recovered to be ∼29 M_⊙. Then, we apply the dendrogram method to the 3 mm continuum image in the same way as the above. We find that the condensation mass ranges from 0.1 to 0.7 M_⊙. The MM1 to MM4 condensations increase in mass only ∼10%, but MM5 increases ∼60% in mass, suggesting that the condensation masses do not change even when we add the total power, except for MM5. However, we should also note that if condensations have relatively extended structures, as MM5 does, the masses may be underestimated due to the missing flux.

We also study the dynamical stability of these condensations. Assuming a uniform density structure, the virial mass can be estimated as

$$\begin{eqnarray}&&{M}_{\mathrm{vir}}=210\times \left(\displaystyle \frac{r}{\mathrm{pc}}\right)\times {\left(\displaystyle \frac{{\rm{\Delta }}v}{\mathrm{km}{{\rm{s}}}^{-1}}\right)}^{2}{M}_{\odot },\end{eqnarray} \tag{ 4 }$$

where r is the radius and Δv is the line width (MacLaren et al. 1988). The line width is derived by Gaussian fitting of the CH₃OH (${J}_{K}={2}_{0}{1}_{0}$ A⁺) emission because this line is optically thin. The centroid velocity, line width, and virial mass are shown in Table 2. As shown in the table, the condensation mass is much higher than the virial mass. Therefore, these condensations are not in virial equilibrium and may collapse immediately unless the magnetic field counterbalances gravity.

Figure 7 shows in color the velocity-integrated intensity maps of N₂H⁺ (J = 1–0, F₁ = 1–1, F = 0–1) and CH₃OH (${J}_{K}={2}_{0}-{1}_{0}$ A⁺) emission without the primary beam corrections. The ALMA 12 m and ACA 7 m data are combined. The F₁ = 1–1, F = 0–1 transition line is the weakest hyperfine component; hence, its optical depth is the smallest. The contours are the 3 mm dust continuum (same as Figure 4). The 3 mm dust continuum and the N₂H⁺ transition maps show that the intensities concentrate at the central part and have the same peak position. On the other hand, the CH₃OH emission is weak at the central part of the TUKH122 core, resembling a shell-like structure that surrounds the continuum and N₂H⁺ emission. These configurations are quite similar to the starless cores of L1498, L1517B, and L1544 (Tafalla et al. 2006; Vastel et al. 2014; Spezzano et al. 2016), indicating that the CH₃OH and N₂H⁺ molecules have chemical differentiation.

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The CH₃OH molecule is suggested to be formed on the surface of dust grains via successive hydrogenation of CO (e.g., Watanabe & Kouchi 2002) and is released by external heating and/or sputtering due to star formation activity. Therefore, the CH₃OH lines are often used as a tracer of star formation activity and shocks. However, even in the cold quiescent clouds including this object, the CH₃OH lines have also been observed with moderate intensities. Vastel et al. (2014) suggested that the nonthermal desorption mechanism is important for the observed emission of CH₃OH. Soma et al. (2015) also suggested that the CH₃OH formed on dust grains is liberated into the gas phase by nonthermal desorption, such as photoevaporation caused by cosmic-ray-induced UV radiation. According to this mechanism, the column density of gaseous CH₃OH is proportional to the path length of the emitting region along the line of sight. Because the UV radiation is shielded in the core interiors, the emitting region has a shell-like structure. The extended CH₃OH distribution seems to be consistent with this scenario. Note that the CH₃OH shell structure is mainly observed in the north and absent in the southeast. This may be because the CH₃OH emission is filtered out in the interferometric data if the CH₃OH emission is widely distributed, or the CO molecule (the precursor of CH₃OH) is less abundant on dust grains in the south if the southern part is chemically young (CO has not been formed yet).

Tatematsu et al. (2014a) also observed the TUKH122 core with the NH₃ and CCS molecules using the VLA and found that the CCS emission surrounds the NH₃ core. Figure 8 shows the velocity-integrated intensity map of the ALMA CH₃OH observations overlaid with the contours of the VLA CCS observations. In comparison with these distributions, we find that the CCS emission is located on the CH₃OH shell structure or slightly outside the shell.

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3.3.1. Analysis of N₂H⁺ Spectra

Because N₂H⁺ has hyperfine structure, we derive the optical depth (τ), local standard of rest (LSR) velocity (V_LSR), line width (dv), and excitation temperature (T_ex) assuming a uniform excitation temperature in the same way as Ohashi et al. (2014, 2016c) using the IDL fitting code. However, we were not able to estimate the optical depth values toward the dust peaks because the N₂H⁺ line profiles show some features of high optical depths. Such extreme optical depths have also been observed in infrared dark clouds (IRDCs; e.g., Sanhueza et al. 2013). Figure 9 shows an example of the N₂H⁺ profile of 1 pixel spectra obtained by the ALMA-ACA combined image at the dust peak position. The intrinsic intensity ratios of each hyperfine component are represented by the red segments at the bottom of Figure 9 in the optically thin limit (Tiné et al. 2000). The observed intensity profile does not follow the intrinsic intensity profile, indicating that most of the hyperfine intensities are saturated. Therefore, we only use the weakest and most optically thin component (J = 1–0, F₁ = 1–1, F = 0–1) to derive the physical parameters. The excitation temperature T_ex is assumed to be the maximum intensities of the main component, adding the cosmic microwave background of 2.7 K in each position, ranging from 4 to 6 K. Then, we derive the optical depth by using the radiative transfer formula, assuming a filling factor of unity. We also derive the LSR velocity and line width by Gaussian fitting of the optically thinnest line. The fitting is performed for the pixels above 5σ. We should note that the 3 mm dust emission is optically thin (${\tau }_{\mathrm{dust}}\lt 0.1$) and would not affect the line profile.

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Figure 10 shows the color maps of these parameters with the 3 mm dust continuum contours (from 3σ with 2σ steps). The color map of optical depth indicates that even the weakest N₂H⁺ hyperfine line is moderately optically thick (τ ≳ 1) toward the condensations. In particular, we were not able to calculate the optical depth due to the saturation of the lines in the southern dust local peak (MM5). Figure 11 shows the profile toward the MM5 peak position. This profile shows that the weakest hyperfine has the strongest intensity peak. Furthermore, we find two velocity components in the main hyperfine emission of F₁ = 2 − 1, F = 3 − 2. Figure 12 shows zoom up profiles in the same position, MM5. The black line represents the F₁ = 2–1, F = 3 − 2 emission at a rest frequency of 93.173777 GHz, and the green line represents the F₁ = 1–1, F = 0–1 emission at a rest frequency of 93.171621 GHz. The black line shows a "dip" in the emission profile at 3.7 km s⁻¹, while the green line reaches a peak in this gap. Thus, the double-peak feature seen in the strongest hyperfine component would be caused by a self-absorption effect. In a similar case, a self-absorption in NH₃ was found toward the IRDC G028.23–00.19 prestellar clump (Sanhueza et al. 2017). The self-absorption of the N₂H⁺ and NH₃ lines suggests that the inner region has a higher density and excitation temperature than the surrounding N₂H⁺ or NH₃ emission regions.

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We also find that the optical depth is not correlated well with the dust continuum emission. The MM4 is the peak position in the dust continuum, but the optical depth seems to be lower than that in the surrounding region, which may suggest a lower ionization degree in the dense part of the core or the depletion of N₂H⁺ at the densest parts. The N-bearing molecules of N₂H⁺ and NH₃ are suggested to be depleted at a density of ≳10⁶ cm⁻³ (e.g., Bergin et al. 2002; Pagani et al. 2005).

In the color map of V_LSR, there is no evident velocity variation along the filament. Systematic velocity fluctuations are also not found along the ridge of the continuum image. This could be due to the impossibility of tracing velocity fields using optically thick lines (most regions have τ ≳ 1) or the insufficient velocity resolution (0.098 km s⁻¹) of our observation to identify infall or fragment motions. On the other hand, in the line-width map, we find that the line width increases toward the condensations. Because the central condensation does not have a high optical depth (τ ∼ 1) in comparison with the surrounding part, a broad line width is not due to the saturation effect of the line but indicates motions (infall, expansion, or turbulent) toward the condensation. Considering that the N₂H⁺ molecule is partially depleted and the line width becomes broader toward MM4, this condensation may be beginning to collapse. However, our current velocity resolution and the molecular lines are not enough to resolve and investigate the motions in detail. It is highly necessary to investigate the nature of the enhanced line width, i.e., the presence of a blue-skewed profile suggesting infall, with higher spectral resolution.

Even though N₂H⁺ is highly optically thick, we roughly estimate the N₂H⁺ abundance at the dust peak position. In the dust peak, we assume an excitation temperature of T_ex = 5.8 K from the maximum intensity. The optical depth of the F₁ = 1–1, F = 0–1 line is derived to be τ = 0.9 ± 0.1. Then, the column density can be calculated by following Sanhueza et al. (2012) and Mangum & Shirley (2015). We derive ${N}_{{{\rm{N}}}_{2}{{\rm{H}}}^{+}}=(3.4\pm 0.4)\times {10}^{13}$ cm⁻². On the other hand, the 3 mm dust continuum shows $N({{\rm{H}}}_{2})=9.9\times {10}^{22}$ cm⁻². Therefore, the N₂H⁺ abundance is derived to be $X({{\rm{N}}}_{2}{{\rm{H}}}^{+})=(3.4\pm 0.4)\times {10}^{-10}$, which is similar to other low-mass star-forming regions (e.g., Caselli et al. 2002; Tafalla et al. 2002; Keto et al. 2004; Tafalla et al. 2006; Friesen et al. 2010).

3.3.2. Analysis of CH₃OH Spectra

The CH₃OH transitions observed are (${2}_{-1}-{1}_{-1}$ E), (${2}_{0}-{1}_{0}$ A⁺), and (${2}_{0}-{1}_{0}$ E). Assuming local thermodynamic equilibrium (LTE) and optically thin conditions, we can estimate the column density (${N}_{{\mathrm{CH}}_{3}\mathrm{OH}}$) and rotation temperature (${T}_{\mathrm{rot}}$) using the rotation diagram technique (e.g., Blake et al. 1987). However, CH₃OH (${2}_{0}-{1}_{0}$ E) has a low signal-to-noise ratio. In order to have a significant detection in all three lines, we average the profiles from the whole region inside the 5σ contour of the ACA 3 mm dust continuum image. The averaging region is shown by the magenta contour in Figure 13. Following the population diagram methods (e.g., Goldsmith & Langer 1999), we fit the three CH₃OH lines. We follow the procedure described in Sanhueza et al. (2013), which is optimized for cold gas. The obtained column density and rotation temperature are ${N}_{{\mathrm{CH}}_{3}\mathrm{OH}}=(1.1\pm 0.1)\times {10}^{13}$ cm⁻² and T_rot = 10.8 ± 0.4 K. This rotation temperature is consistent with T_rot = 10.6 K, derived by using the NH₃ (J, K = 1, 1) and (J, K = 2, 2) lines (Ohashi et al. 2016c). The rotation temperature is the same even if different molecular lines are used, suggesting the isothermal core with the LTE condition. The optical depth of the strongest transition (${2}_{0}-{1}_{0}$ A⁺) is ∼0.1; hence, it is optically thin. The average H₂ column density is derived to be $N({{\rm{H}}}_{2})=3.8\times {10}^{22}$ cm⁻² from the 3 mm dust continuum observations. Therefore, the CH₃OH abundance is $X({\mathrm{CH}}_{3}\mathrm{OH})\sim 2.9\times {10}^{-10}$. Soma et al. (2015) estimated the abundance of CH₃OH to be ∼10⁻⁹ toward the cold quiescent core, TMC-1 (CP). Tafalla et al. (2006) also estimated the abundance to be $X({\mathrm{CH}}_{3}\mathrm{OH})\sim 6\times {10}^{-10}$ in the L1498 and L1517B prestellar cores. TUKH122 has a similar CH₃OH abundance to these prestellar cores. In particular, the CH₃OH abundance can be even lower at the central part because this value is the average over the whole region. Assuming that the CH₃OH (${2}_{0}-{1}_{0}$ A⁺) line is optically thin and T_rot = 10.8 K, we estimate the column density, ${N}_{{\mathrm{CH}}_{3}\mathrm{OH}}$, at the dust peak position MM4 by using the rotation temperature derived above. By comparing the velocity-integrated intensities, the column density is derived to be ${N}_{{\mathrm{CH}}_{3}\mathrm{OH}}=1.3\,\times {10}^{13}$ cm⁻² at this dust peak position, and then the CH₃OH abundance is $X({\mathrm{CH}}_{3}\mathrm{OH})\sim 1.3\times {10}^{-10}$.

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Figure 13 shows color maps of the peak intensity, V_LSR, and line width of CH₃OH. The peak temperature map clearly shows the shell-like structure, and the dust condensations are located within this shell. The V_LSR map shows some systematic variations within the core. The northern edge is slightly redshifted in velocity, while the southern edge is blueshifted, suggesting core rotation or gas inflow motions along the filamentary structure. Furthermore, the southeastern side is blueshifted velocity, while the southwestern side is redshifted. The velocity structure is complicated, but the velocity difference might indicate the contraction or twisted motions of the core.

The line-width map shows no systemic variations along the filament. However, the dust peak position (MM4) has a broad line width of Δv ∼ 0.4 km s⁻¹, which is the same tendency as that in N₂H⁺. These broad line widths would thus suggest infall motions. Interestingly, the line-width color maps of N₂H⁺ and CH₃OH show a small value of dv ≲ 0.2 km s⁻¹ at the edge of the condensations, and our velocity resolution marginally resolve the lines. The line width of 0.2 km s⁻¹ is close to the thermal motion because thermal line width corresponds to 0.14 km s⁻¹ for N₂H⁺ and 0.13 km s⁻¹ for CH₃OH at 12 K. The narrow line width of the core edge indicates that the turbulence is almost dissipated in the filament, and the nonthermal motions within the core could indicate infall motions or some systemic velocity variations, such as rotation and fragmentation.

The important difference between TUKH122 and the other prestellar cores in nearby clouds is that TUKH122 consists of at least three dense condensations within the one CH₃OH shell. The TUKH122 core is more massive than low-mass dense cores (∼M_⊙; e.g., Onishi et al. 2002; Alves et al. 2007), even though the size is comparable, implying that the TUKH122 core is denser. The denser core would enhance the CH₃OH shell structure because UV radiation is only irradiated on the outer layer of the core.

To investigate the formation of the condensations, we study the density structure along the filament. Figure 14 shows the column density profiles taken from the maximum position along the filament direction (see also Figure 5). The reference position here is the MM4 condensation corresponding to the peak flux in the ALMA-ACA observations. The Herschel single-dish observations indicate an almost flat density structure due to the large beam size. On the other hand, the ALMA-ACA observations show column density variations. The green line represents a sinusoidal column density variation with an interval of 0.035 pc to visually match the observed variations. The southern part (MM3, MM4, and MM5) seems to be nicely fitted by this sinusoidal variation.

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The interval of 0.035 pc is consistent with the separations between neighbor sources in the OMC-2 and 3 regions (Takahashi et al. 2013; Kainulainen et al. 2017). In OMC-1, another part of the Orion A cloud, a separation of ∼0.01 pc was reported by Teixeira et al. (2016) and Palau et al. (2017). Since OMC-1 is the most massive part in the Orion A cloud, the fragmentation might occur at denser regions, and the separations would then become shorter. The separations and fragmentation processes have also been studied in other regions, including high-mass star-forming regions, and suggested to be governed by thermal Jeans processes (e.g., Kainulainen et al. 2013; Beuther et al. 2015; Palau et al. 2015; Busquet et al. 2016). Assuming that the fragmentation occurs by the Jeans instability, we can estimate the separation of the fragment at this region. With the assumption of an infinite size and a uniform density, the Jeans length is described as follows (Jeans 1902):

$$\begin{eqnarray}&&{\lambda }_{\mathrm{Jeans}}=\sqrt{\displaystyle \frac{\pi {c}_{s}^{2}}{G{\rho }_{0}},}\end{eqnarray} \tag{ 5 }$$

where G is the gravitational constant, ρ₀ is the mean density, and c_s is the sound speed of 0.2 km s⁻¹ at 12 K. Takahashi et al. (2013) also analyzed the Jeans length in the case of an infinitely long static and cylindrical isothermal cloud following Nakamura et al. (1993) and Wiseman & Ho (1998) and described as

$$\begin{eqnarray}&&{\lambda }_{\mathrm{Jeans}}\sim \displaystyle \frac{20{c}_{s}}{\sqrt{4\pi G{\rho }_{0}}}.\end{eqnarray} \tag{ 6 }$$

From the column density profile of the ALMA-ACA observations in Figure 6, we derive the central density, ρ_c ∼ 5.5 × 10⁵ cm⁻³, assuming the numerical factor of A = π/2. The Jeans length is derived to be ∼0.031 and ∼0.099 pc, respectively, following Equations (5) and (6), assuming a temperature of 12 K. The thermal Jeans instability with an assumption of an infinite and a uniform density, Equation (5), would be better explained to form the condensations. Therefore, the fragmentation occurs within the elongated core having a density of ∼10⁵ cm⁻³, and the denser condensations with a density of ∼10^6–7 cm⁻³ are formed by the thermal Jeans instability. Note that, assuming λ_Jeans = 0.035 pc, the density is derived to be $n({{\rm{H}}}_{2})\sim 4.4\times {10}^{5}$ cm⁻³ from Equation (5).

The northern part (MM1 and MM2) seems to have a shorter interval of ∼0.02 pc. An inclination might cause a shorter interval even if these condensations have the same interval of ∼0.035 pc, or the Jeans instability at higher densities may occur. We should also note that these northern condensations are located on the CH₃OH shell structure, suggesting the edge of the TUKH122 core. Therefore, the edge instabilities may also affect the separation of the condensations (Pon et al. 2011, 2012). The lower-mass condensations with shorter intervals may be formed in the edge region.

We also discuss the possibility of whether the condensations will merge or not. From the CH₃OH observations, the relative velocities of the surrounding condensations for MM4 are 0.02–0.05 km s⁻¹. To merge the condensations, we need at least 4 × 10⁵ yr, assuming a separation of 0.02 pc and velocity of 0.05 km s⁻¹. On the other hand, the freefall time at a density of 10⁶ cm⁻³ will be 4 × 10⁴ yr. Therefore, star formation can be started before these condensations merge. It is unlikely that these condensations merge to a larger scale. However, the magnetic field may be supporting the condensations and makes the star formation timescale longer (e.g., Crutcher et al. 2004; Ward-Thompson et al. 2007). It is suggested that the ambipolar diffusion timescale is about 10 times longer than the freefall time (Nakano 1998). In this case, the merging might be possible.

In the previous section, we identified the six condensations with a mass of ∼0.1–0.4 M_⊙ in the core, and they are gravitationally bound. However, the total mass of the condensations is only 1.2 M_⊙, and a large portion of the mass is located outside of them. One possibility for the small mass fraction of the condensations is that we may only detect the peak region of the condensations, and the condensations may have larger radius and mass. For example, if the condensations have an unstable Bonnor–Ebert sphere, they have a large contrast in density between the center and outer part. The condensations might be able to identify only the center part due to our spatial resolutions and sensitivities. Actually, the size of the condensations is derived to be r ∼ 0.006 pc even though we find a separation of 0.02–0.035 pc between them. If we use the separations as the size of these condensations, the mass is derived to be 0.3–1 M_⊙. Even if the condensations have masses of 0.3–1 M_⊙, they only have a small amount of the mass, and the virial parameters are still lower than unity. These condensations may collapse immediately. Another possibility is that very low-mass stars (∼0.03–0.1 M_⊙) are formed in these condensations in this region. In either case, a multiple-star system may be formed in the core along the filament. We suggest that fragmentation is still important in the dense core to form multiple stars.