Distortion of Magnetic Fields in a Starless Core. VI. Application of Flux Freezing Model and Core Formation of FeSt 1–457

Observational data for the hourglass-like magnetic field toward the starless dense core FeSt 1–457 were compared with a flux freezing magnetic field model. Fitting of the observed plane-of-sky magnetic field using the flux freezing model gave a residual angle dispersion comparable to the results based on a simple 3D parabolic model. The best-fit parameters for the flux freezing model were a line-of-sight magnetic inclination angle of γmag = 35° ± 15° and a core center to ambient (background) density contrast of ρc/ρbkg = 75. The initial density for core formation (ρ0) was estimated to be ρc/75 = 4670 cm−3, which is about one order of magnitude higher than the expected density (∼300 cm−3) for the interclump medium of the Pipe Nebula. FeSt 1–457 is likely to have been formed from the accumulation of relatively dense gas, and the relatively dense background column density of AV ≃ 5 mag supports this scenario. The initial radius (core formation radius) R0 and the initial magnetic field strength B0 were obtained to be 0.15 pc (1.64R) and 10.8–14.6 μG, respectively. We found that the initial density ρ0 is consistent with the mean density of the nearly critical magnetized filament with magnetic field strength B0 and radius R0. The relatively dense initial condition for core formation can be naturally understood if the origin of the core is the fragmentation of magnetized filaments.


Introduction
The characteristics of newborn stars are thought to be determined by the physical properties of the nursing molecular cloud cores (dense cores). Revealing the formation mechanism of cores is important because it will help determine the initial conditions of star formation.
Cores are thought to develop and evolve in molecular clouds via a mass accumulation process involving gravity, thermal pressure, turbulence, and magnetic field. Several scenarios have been proposed for the formation mechanism of cores. One is the quasi-static contraction of material under a relatively strong magnetic field (Shu 1977;Shu et al. 1987). The other extreme is core formation through supersonic turbulence (e.g., Mac Low & Klessen 2004). In this scenario, supersonic turbulence produces cores that collapse dynamically, accompanied by highly supersonic infalling motion. However, these models do not match observations in several aspects. Many observations show a moderately supercritical condition in molecular clouds (e.g., Crutcher 2004), which is not the case for the first model. Also, quiescent kinematic gas motions are widely observed toward dense cores (e.g., Caselli et al. 2002), which does not match the second model. A core formation mechanism between those two extreme models may better account for the observations (e.g., Nakamura & Li 2005;Basu et al. 2009aBasu et al. , 2009b. Many observations of dense cores have been made, using various methods at various wavelengths, e.g., radio molecular line observations (e.g., Jijina et al. 1999;Caselli et al. 2002), dust emission/continuum observations (e.g., Kauffmann et al. 2008;Launhardt et al. 2010), Zeeman observations (e.g., Crutcher 1999;Crutcher et al. 2010), dust emission polarimetry (e.g., Ward- Wolf et al. 2003), and dust dichroic extinction polarimetry (e.g., Jones et al. 2015;Kandori et al. 2017aKandori et al. , 2017b. There is considerable observational data on the physical/chemical properties of cores, and important evidence has been reported (e.g., a tight geometrical relationship between the location of cores and filamentary structures; André et al. 2010). However, obtaining direct observational constraints of the core formation process is extremely difficult. For example, there are no observational results for the initial radius R 0 , initial density ρ 0 , or initial magnetic field strength B 0 as the starting conditions of core formation.
To investigate the elementary process of core formation, we focused on the 3D magnetic field structure of dense cores. Since the process must proceed from the accumulation of interstellar matter to create dense cores, and magnetic flux freezing is expected during the process, the most fundamental form of the magnetic field surrounding dense cores is expected to be hourglass shaped. The hourglass magnetic field is generated by core formation, and the history of mass condensation to create the core is reflected in the curvature of the hourglass field. Thus, a comparison of the appropriate flux freezing model with observations of the hourglass field can provide information on the initial conditions of core formation.
The object considered in the present study is the starless dense core FeSt 1-457. The fundamental physical parameters for FeSt 1-457 were determined based on density structure studies using the Bonnor-Ebert sphere model (Ebert 1955;Bonnor 1956). The radius, mass, and central density of the core are R=18,500±1460 au (144″), M core =3.55±0.75 M e , and ρ c =3.5(±0.99)×10 5 cm −3 (Kandori et al. 2005), respectively, at a distance of -+ 130 58 24 pc (Lombardi et al. 2006). The dimensionless radius parameter characterizing the Bonnor-Ebert density structure was ξ max =12.6±2.0, which corresponds to a center-to-edge density contrast of ρ c /ρ s =75. The subsequently measured background star polarimetry at nearinfrared (NIR) wavelengths revealed an hourglass-shaped magnetic field toward the core (Kandori et al. 2017a, hereafter Paper I). Through simple modeling based on a 3D parabolic function, the structure of the 3D magnetic field (the magnetic field inclination angle toward the line of sight γ mag = 35°±15°and the 3D field curvature C) was determined (Kandori et al. 2017b, hereafter Paper II; see also the Appendix). Note that γ mag is the line-of-sight inclination angle of the magnetic axis of the core measured from the plane of the sky. Since NIR polarization and extinction in FeSt 1-457 exhibit a linear relationship even in the dense region of the core, the above results reflect the overall dust alignment in the core (Kandori et al. 2018b, hereafter Paper III;Kandori et al. 2018a, hereafter Paper V).
From the γ mag information, the total magnetic field strength of the core was determined to be 28.9±15.4 μG using the Davis-Chandrasekhar-Fermi method (Davis 1951;Chandrasekhar & Fermi 1953), which reveals the core to be in a magnetically supercritical state with λ=1.64±0.44 (Paper II; see also the Appendix). Note that the total magnetic field strength at the core edge is 15 μG, estimated based on an analysis of the magnetic field scaling on density (Kandori et al. 2018c, hereafter Paper IV; see also the Appendix). The value of 15 μG is consistent with the recently measured magnetic field strength for the intercore regions of molecular clouds using the OH Zeeman effect (∼15 μG; Thompson et al. 2019). The stability of the core can be evaluated by comparing the observed mass of the core, M core , with the theoretical critical mass considering the magnetic and thermal/turbulent contributions in the core of  + M M cr mag M BE (Mouschovias & Spitzer 1976;Tomisaka et al. 1988;McKee 1989), where M mag is the magnetic critical mass and M BE is the Bonnor-Ebert mass. The critical mass of the core is M cr =3.35±0.83 M e , which is comparable to the observed mass (M core =3.55± 0.75 M e ) of the core, suggesting that the core is in a nearly critical state. In the present study, an analytic flux freezing magnetic field model (Myers et al. 2018; see also Mestel 1966;Ewertowski & Basu 2013) was employed for comparison with the FeSt 1-457 data. The results were compared with our previous results (Paper II and the Appendix) based on the axisymmetric parabolic function. The flux freezing model explained the FeSt 1-457 data well, and we derived the best-fit model parameters. With the obtained background density (ρ bkg ) parameter and known core density, the initial contraction radius for core formation (R 0 ) and the initial magnetic field strength (B 0 ) were determined. Using these quantities, we discuss the initial conditions of the core formation and core formation mechanisms.

Data and Methods
The NIR polarimetric data for FeSt 1-457 for the 3D magnetic field modeling were taken from Paper I. Observations were conducted using the JHK s -simultaneous imaging camera SIRIUS (Nagayama et al. 2003) and its polarimetry mode SIRPOL (Kandori et al. 2006) on the IRSF 1.4 m telescope at the South African Astronomical Observatory (SAAO). SIRPOL can provide deep-field (18.6 mag in the H band, 5σ in 1 hr exposure) and wide-field (7 7×7 7 with a scale of 0 45 pixel −1 ) NIR polarimetric data.
In the observed NIR polarimetric data, the polarization vectors toward FeSt 1-457 are superpositions of vectors arising from the core itself and from the core's ambient medium. The contribution from the ambient medium was removed in order to isolate the polarization vectors associated with the core (Paper I). A total of 185 stars located within the core radius (R144″) in the H band were selected for the polarization analysis. Figure 1 shows the result. The magnetic field lines pervading the core have a shape reminiscent of an hourglass, which can be approximately traced using parabolic functions.
The existence of the distorted hourglass-shaped magnetic field can be interpreted as evidence for the mass condensation process. The curvature of the magnetic field lines in the outer region seems steep, and the mass located outside the core should move across a large distance to create the current distorted magnetic field of the core. It is therefore clear that the core radius was previously larger than the current radius and that the core contracted by dragging the frozen-in magnetic field lines. field of view is the same as the diameter of the core (288″=0.19 pc). The white lines show the magnetic field direction inferred from fitting with a parabolic function = + y g gCx 2 , where g specifies the magnetic field lines and C determines the degree of curvature in the parabolic function. The scale of 5% polarization degree is shown at the top.
Since FeSt 1-457 is in a nearly kinematically critical state (Paper I; Paper II), the field distortion cannot be attributed to the dynamical collapse of the core. The observed distorted magnetic field is thus considered to be an imprint of the core formation process, in which mass was gathered and the magnetic field lines were dragged toward the center to create the dense core. These interpretations were presented in Paper I, and in the present study we quantitatively investigate core formation for FeSt 1-457 using a simple flux freezing model in an analytic form (Myers et al. 2018).
Examples of the distribution of the magnetic field lines using the flux freezing model (Myers et al. 2018) are shown in Figures 2 and 3. The model calculates the magnetic flux structures of spheroidal cores based on flux freezing and mass conservation. Since the projected shape of FeSt 1-457 is not elongated, we focus on the spherical case in the model. As initial conditions, we take a uniform magnetic field with a strength B 0 pervading the uniform medium with a density ρ 0 . After the initiation of mass accumulation, isotropic contraction takes place, preserving the shape of the cloud during contraction. For the density structure, a Plummer-like model (Myers 2017) with an index p=2 was used. The index p=2 was chosen to approximate the density structure of the Bonnor-Ebert sphere (Ebert 1955;Bonnor 1956). The problem of mass loading in a flux tube was solved to connect the initially uniform density and flux distribution with the stage of mass and flux condensation arising from the cloud contraction.
In the model, the shape of the magnetic field lines, as shown in Figures 2 and 3, is a function of the density contrast ρ c /ρ bkg , where ρ c is the density at the core center and ρ bkg (alternatively, ρ 0 ) is the initial uniform density. Solutions with larger density contrast can result in a higher degree of central condensation in the magnetic field lines.
The equations in Myers et al. (2018) to obtain the magnetic field structure for a spherical core are as follows: Here ξ c and ζ c are dimensionless coordinates (x and z normalized to the scale length s p º r Gmn 4 0 0 , where σ is the 1D thermal velocity dispersion, G is the gravitational constant, m is the mean particle mass 2.33m H , and n 0 is the peak density) representing the contours of the constant flux in the x-z plane (sky plane); ν 0 ≡n 0 /n u is the peak density normalized to the background value (density contrast); ω is the dimensionless radius of the sphere, which serves as a dummy variable increasing from 0 to ¥; and f c is the flux normalized to p F = r B u 0 0 2 , where B u is the initial magnetic field strength. Though the magnetic field structure of the flux freezing model looks similar to the structure derived using the parabolic model (2D: Paper I; 3D: Paper II and Appendix), they are not identical. Figure 4 shows a comparison between the magnetic field structure based on the flux freezing model (gray vectors, ρ c /ρ bkg =75) and the parabolic fit to the flux freezing model data (black lines, C=1.7×10 −6 pixel −2 for the function = + y g gCx 2 ). Though the general trend in the structure of both models is the same, the gray vectors and black lines clearly deviate. Thus, we need to check whether the  Myers et al. (2018). Results for a density contrast parameter of 75 are shown. The circle shows the core radius. The x-z plane of the core is shown, and both the x-and z-axes are normalized by the scale length s p r º r G 4 0 0 , where σ is the 1D thermal velocity dispersion, G is the gravitational constant, and r 0 is the background density.  Myers et al. (2018). Results for a density contrast parameter of 750 are shown. The circle shows the core radius. The x-z plane of the core is shown, and both the x-and z-axes are normalized by the scale length s p r º r G 4 0 0 , where σ is the 1D thermal velocity dispersion, G is the gravitational constant, and r 0 is the background density.
conclusions obtained using the parabolic model, especially in Paper II, can be reproduced for the flux freezing model.
The magnetic field structure shown in Figures 2 and 3 is the calculated result in the x-z plane (sky plane) of the spherical cloud core. To compare this with observations, we need to integrate the 3D polarization distribution toward the line of sight to derive the projected polarization map for various density contrast values. This process and the comparison with observations are described in the next section.

Application of Flux Freezing Model
3D polarization calculations of the flux freezing model (Myers et al. 2018) were made. Figures 2 and 3 show the calculation results on the x-z plane (sky plane). We assumed that the magnetic field lines are axisymmetric around the z-axis (radius r and the direction f around z-axis) in cylindrical coordinates. The model function ( ) f r r z r, , c bkg thus has no dependence on the parameter f, where r r c bkg shows the density contrast for the core. For comparison with observations, after generating the model function, the 3D model is rotated in the line-of-sight (γ mag ) and plane-of-sky (θ mag ) directions, and the axis of the cylindrical coordinates is set parallel to the direction of the magnetic axis (the orientation of the magnetic field pervading the core). The configuration of the coordinates and angles is shown in Figure 5.
For polarization modeling of the core, the 3D unit vectors of the polarization following the model function with a specific density contrast value were calculated using 750 3 cells. Assuming that the orientation of the polarization vectors is parallel to the direction of the magnetic field, the 3D orientation of the polarization was determined in each cell. These unit vectors were then scaled to describe both the polarization angle and degree in each cell, , . To determine the length of the polarization vector in each cell, we prepared the volume density value and the density-polarization conversion relationship. The volume density of molecular hydrogen in each cell, , can be obtained from the known Bonnor-Ebert density structure of FeSt 1-457 (ξ max =12.6; Kandori et al. 2005). The density-polarization conversion factor was estimated based on the slope of the P H versus -H K s diagram of 4.8% mag −1 (Paper I) as  (Bohlin et al. 1978), where N H 2 is the column density of molecular hydrogen.
The rotation of the polarization vector ( ) DP x y z , ,

H,model
around the x-axis with an inclination angle g mag can be written as follows: The data cube of ( ) DP x y z , ,

H,model
is also rotated around the x-axis by an angle g mag .
For sampling, 30 3 cells were used, and the integrations of the cubes of the Stokes parameters toward the line of sight (y-direction) were conducted as  ) and the parabolic model ( =´-C 1.7 10 6 pixel −2 for the function = + y g gCx 2 ). The comparison was done on the x-z plane. The circle shows the radius of the core.
is the position angle on the plane of the sky and ( ) g x y z , , cell is the inclination angle with respect to the plane of the sky in each cell. Since the magnetic field pervading the model core is distorted, the magnetic inclination angle in each cell ( ) g x y z , , cell is different from the inclination angle g mag , which is the magnetic axis for the whole field. The ( ) g x y z , , cell angle can be calculated using the following equation: The polarization degree and angle can be obtained as Finally, the orientation of the magnetic axis on the plane of the sky, q =  179 mag , was applied.
( ) q x z , H,model was rotated by q mag in both value and coordinates, and the P H,model array was also rotated. Figure 6 shows the polarization vector maps for the flux freezing model with a density contrast parameter of r r = 75 c bkg for several line-of-sight inclination angles g mag . In each panel of Figure 6, q mag is set to 0°for display. The white line shows the polarization vector, and the background color and color bar show the polarization degree of the model core. The applied viewing angle, g  -90 mag , is labeled in the upper left corner of each panel. Note that g  -90 mag is the angle between the direction toward the observer and the magnetic axis.
The features of the polarization vector maps in Figure 6 are similar to those in the 3D parabolic model described in Paper II, i.e., (1) a decrease of the maximum polarization degree from g =  90 view to g =  0 view , (2) an hourglass-shaped polarization angle pattern that converges to a radial pattern toward small g view , (3) depolarization in the polarization vector map, especially along the equatorial plane of the core, and (4) an elongated structure of the polarization degree distribution toward small g view . Figure 7 shows the c 2 distribution calculated using the model and observed polarization angle as where n is the number of stars (n = 185), q i obs, and q i model, denote the polarization angle from observations and the model for the ith star, respectively, and dq i obs, is the observational error. c q 2 values were obtained for each inclination angle g mag after determining the best magnetic curvature parameter C. The inclination angle that minimizes c q 2 is g =  35 mag , although the distribution of c q 2 for the range between g =  0 mag and~ 60 is relatively flat. Note that the reduced c 2 values obtained in this analysis are large, because the relatively large variance originating from the Alfvén wave cannot be included in the polarization angle error term, dq i 2 , in Equation (12). Figure 8 shows the distribution of c 2 calculated using the model and observed polarization degree as where P i obs, and P i model, represent the polarization degree from observations and the model for the ith star, and dP i obs, is the observational error. c P 2 values were calculated for each g mag after minimizing the difference in polarization angles.
It should be noted here that the model polarization degree for each star P i model, was rescaled before calculating c P 2 . Though the scaling of P i model, was initially performed using Equation (13), it was without knowledge of the true magnetic inclination angle of the core. In other words, the factor in Equation (13) is the value assuming that the magnetic axis of the core is on the plane of the sky. To correct this, we rescaled P i model, by the factor á ñ P P obs model determined using a robust least absolute deviation fitting. The mean values of P model and P obs are therefore always the same, and the deviation of the rescaled P model from P obs was calculated to evaluate c P 2 .
The minimization point for c P 2 is the same inclination angle, g =  35 mag . We further conducted the same analysis using the 3D parabolic model (Appendix). The minimization angles, g =  35 mag and 50°, were obtained for c q 2 and c P 2 , respectively. On the basis of these analyses, we selected to use the value g =    35 15 mag throughout this paper. Figure 9 shows the relationship between c q 2 and the density contrast r r c bkg when g mag is fixed to 35°. The minimization point of c q 2 is r r » 85 c bkg . This is consistent with the value r r » 75 c edge obtained based on the Bonnor-Ebert density profile analysis of FeSt 1-457 (Kandori et al. 2005). Two independent measurements, one based on the shape of the flux freezing magnetic field lines and the other based on the density profile, produce very consistent results. Hereafter we use a value of 75 for the density contrast of FeSt 1-457.
It is notable that the physical meaning of r edge is different from that of r bkg . r edge means the density at the core's boundary, which can be determined by comparing observations with the edge-truncated density profile model, such as the Bonnor-Ebert model. On the one hand, r bkg means the initial density for the core formation or the diffuse uniform density at a large distance from core region, which can be determined by comparing the observed magnetic field structure of the core with the flux freezing magnetic field model. We found r r r~= 75 4670 bkg edge c cm −3 for FeSt 1-457. Figure 10 shows the best-fit flux freezing model (g =  35 mag and r r = 75 c bkg ; white vectors) compared with observations (yellow vectors). The background image shows the distribution of the polarization degree. Figure 11 shows the same data but with the background image processed using the line integral convolution technique (LIC; Cabral & Leedom 1993). We used the publicly available interactive data language (IDL) code developed by Diego Falceta-Gonçalves. The direction of the LIC "texture" is parallel to the direction of the magnetic field, and the background image is based on the polarization degree of the model core. The standard deviation of the polarization angle difference between the model and observations is 8°.33. This is comparable to the value of 7°. 28 for the 3D parabolic model case.

Core Formation of FeSt 1-457
For an obtained core's density contrast, the initial density before core contraction (r 0 ) or the density of the interclump medium surrounding the core (r bkg ) can be derived to be r = 75 4670 c cm −3 . This is about one order of magnitude higher than we expected for the interclump medium of the Pipe Nebula dark cloud complex. Radio molecular line observations toward the Pipe Nebula showed that (1) the overall distribution of 12 CO ( = -J 1 0) that traces~10 2 cm −3 gas is similar to that of the optical obscuration, and (2) the distribution of 13 CO ( = -J 1 0) that traces~10 3 cm −3 gas is similar to that of 12 CO ( = -J 1 0) (Onishi et al. 1999). The density of the overall diffuse interclump gas in the Pipe Nebula seems to be 10 2 -10 3 cm −3 , while we expected a value of several× 10 2 cm −3 , in particular ∼300 cm −3 (Myers et al. 2018), for the density of the interclump medium in the Pipe Nebula.
The diffuse initial condition does not match the case for FeSt 1-457. If we assume this diffuse initial condition, the observed magnetic curvature should be steep, because in this case the magnetic curvature should follow the flux freezing model's solution of a density contrast one order of magnitude larger (see and compare Figures 2 and 3). The solution of the model provides a steeper magnetic curvature as the density contrast increases. To explain the consistency between observations and the flux freezing model, FeSt 1-457 should be formed from the accumulation of relatively dense gas of several × 10 3 cm −3 . The core formation of FeSt 1-457 can be started from a relatively dense initial condition pervaded by a uniform magnetic field. In fact, FeSt 1-457 is located in a relatively dense region of the Pipe Nebula, in which the average -H K s Figure 7. c 2 distribution of the polarization angle (c q 2 ). The best density contrast parameter (r r c bkg ) was determined for each inclination angle (g mag ). g =  0 mag and 90°correspond to the edge-on and pole-on geometries with respect to the magnetic axis. Figure 8. c 2 distribution of the polarization degree (c P 2 ). The calculations of c 2 in polarization degree were performed after determining the best density contrast parameter (r r c bkg ) that minimizes c 2 in the polarization angle. This calculation was carried out for each g mag . g =  0 mag and 90°correspond to the edge-on and pole-on geometries in the magnetic axis.  color of stars is 0.4 mag in the reference field of FeSt 1-457 (Paper V), andÃ 5 V mag is expected in the Pipe Bowl region. The cloud thickness toward the Pipe Bowl region is ∼0.5 pc (Franco et al. 2010). Dividing the background column density by the cloud thickness, we obtain ∼3000 cm −3 for the expected density for the Pipe Bowl region, which is comparable to the initial density (r 0 ) of FeSt 1-457 derived based on the magnetic field analysis. Thus, the suggestion of a relatively dense initial condition is observationally plausible. The Herschel observations of the Aquila Rift complex showed that ∼90% of the candidate bound cores are found above a background dust extinction (column density) of  A 8 V mag (André 2015; see also Onishi et al. 1998;Johnstone et al. 2004, for earlier ground-based studies). This is consistent with our scenario of relatively dense initial conditions for core formation.
The formation mechanism for such initial conditions is an open problem. A scenario of two colliding filamentary clouds in the Pipe Nebula region ) may explain the relatively dense initial condition. The magnetic field can be compressed and can dominate in the Pipe Bowl region in the scenario involving the collision of filaments. The combination of the existence of a relatively dense interclump medium and uniformly aligned magnetic field lines in the Pipe Nebula is not surprising. Alves et al. (2008) reported mass-to-flux ratio measurements of l~0.4 pos toward the Pipe Bowl region based on wide-field optical polarization observations. The existence of such a magnetically subcritical part is not special, because H I clouds are known to be significantly magnetically subcritical (Heiles & Troland 2005), and it is natural for molecular clouds, namely, assemblies of diffuse H I clouds, to have magnetically subcritical subregions. Since the magnetic field seems to dominate in the Pipe Bowl region, the field lines should be aligned even for the region of relatively high density. These results remind us of the classic ambipolar diffusion idea of slow drift of neutrals past nearly stationary field lines, followed by a more rapid supercritical collapse of an inner dense region (e.g., Mouschovias & Ciolek 1999). In this scenario, the rapid collapse with flux freezing may be started at a density of several × 10 3 cm −3 . Note that from Zeeman observations the density of 300 cm −3 was suggested as the point at which interstellar clouds become self-gravitating (Crutcher et al. 2010).
Since the initial density, r 0 , is known through the analysis of the flux freezing model, the initial radius (core formation radius), R 0 , can be obtained by , where m = B 28.9 G tot is the total magnetic field strength averaged for the whole core (Paper II and the Appendix). It is notable that there are few methods available to obtain a dense core's initial radius (R 0 ), initial density (r 0 ), and initial magnetic field strength (B 0 ).
On the basis of obtained physical quantities, we consider the formation of FeSt 1-457. The Jeans mass M J of the core calculated using the initial density r = 4670 0 cm −3 is 3.84  M at 10 K. This value is consistent with the observed core mass of =  M 3.55 0.75 core  M . Moreover, the Jeans length is l = 0.29 J pc, which is close to the diameter of the core formation radius » R 2 0.3 0 pc. Though these results do not preclude the possibility of external compression by turbulence or shocks to create the core, the results of the Jeans analysis match the observations. The strength of gravity inside the formation radius of the core seems sufficient for initiating the formation of FeSt 1-457.
In addition to the Jeans analysis, we considered interstellar filaments for the origin of FeSt 1-457. In the nonmagnetic case, an interstellar isothermal filament with gas temperature of 10 K has Figure 11. Same as Figure 10, but the background image was made using the LIC technique (Cabral & Leedom 1993). The direction of the LIC "texture" is parallel to the direction of the magnetic field, and the background image is based on the polarization degree of the model core. M pc −1 (Stodólkiewicz 1963;Ostriker 1964;Inutsuka & Miyama 1992). If we employ R 0 as a radius of the filament, the mean hydrogen molecule density of the critical filament is3.6 10 3 cm −3 . In the magnetized case, following Tomisaka (2014) -4.7 5.3 10 3 cm −3 , respectively. These densities are well consistent with the initial density r 0 of FeSt 1-457. Therefore, the fragmentation of a filamentary cloud with a nearly critical state can be the origin of FeSt 1-457. Figure 13 shows the Herschel column density map (André et al. 2010;Roy et al. 2019) covering the same spatial extent as Figure 12 ( ¢ 30 ) around FeSt 1-457. The column density was converted to A V using =Ń A 9.4 10 V H 20 2 cm −2 mag −1 (Bohlin et al. 1978). The resolution of the image is 18 2. In the map, there is a filamentary structure extending northward from FeSt 1-457, although the core seems relatively isolated especially toward the south. The Pipe Nebula dark cloud complex is well known for its filamentary shape, and the filamentary structure around FeSt 1-457 is small in scale compared with the global filament of the Pipe Nebula. Note that a network of subfilaments within a large filament has been reported in the B59 region and the "stem" region in the Pipe Nebula (Peretto et al. 2012).
The mean density of the magnetized critical filament is slightly greater than r 0 . The initial condition of the formation of FeSt 1-457 may be in a slightly magnetically subcritical state. It is notable that the magnetized cylinder is unstable even when the magnetic field is extremely strong (Hanawa et al. 2017(Hanawa et al. , 2019. The nearly critical filament was naturally derived from the analysis of the initial conditions of the formation of FeSt 1-457. This may be the result of supporting the "interstellar filament paradigm" (e.g., André et al. 2014) from the core side.
However, the initial diameter (2R 0 ) of FeSt 1-457 is ∼0.3 pc, which is larger than the 0.1 pc width obtained based on the Herschel data for a number of molecular clouds (e.g., Arzoumanian et al. 2011Arzoumanian et al. , 2019. A problem to employ this scenario is that there is no evidence of the infalling gas motion in FeSt 1-457 (Aguti et al. 2007). If the fragmentation of an interstellar filament can be the initial condition of core formation and the unstable condition evolves in a "runaway" fashion, the motion of gas moving inward of the core should be detected in observations, because FeSt 1-457 has been shrinking in radius from the initial radius = R R 1.64 0 to the current radius R. We speculate that the physical properties of the core born from the fragmentation of a magnetically subcritical filament may be a key to explain the physical state of FeSt 1-457, because such a core can evolve in a quasi-static way until the mass-to-flux ratio of the core exceeds the critical value through the ambipolar diffusion. This scenario naturally explains rather static gas kinematics of FeSt 1-457. The model that best describes the structure of the core is the magnetohydrostatic model (e.g., Tomisaka et al. 1988 (Mouschovias & Spitzer 1976;Tomisaka et al. 1988;McKee 1989). M cr decreases with decreasing magnetic critical mass M mag through ambipolar diffusion, whereas there is a thermal support, which is represented in the equation by the Bonnor-Ebert mass M BE . Thus, if the thermal support is strong enough, the core can be stable even if the magnetic condition turns into supercritical. In this case, magnetically supercritical but quasi-static evolution continues until the thermal and magnetic support is defeated by gravity. This scenario matches the physical conditions of FeSt 1-457, because the core is currently magnetically supercritical but kinematically nearly critical with additional support from the thermal pressure (Paper I, II; see also the Appendix).
This scenario is also useful in explaining the hourglass structure of the magnetic field in FeSt 1-457. If the core is magnetically subcritical from birth to the present, the curvature of hourglass magnetic fields should be shallow, whereas the supercritical model can have more curvature in magnetic field lines (Basu et al. 2009b). We expect that most of the field curvature of FeSt 1-457 can be made during the magnetically supercritical phase of the core, and this should be investigated by comparing the observations of hourglass-like fields with theoretical simulations of dense core formation that include the ambipolar diffusion process.
The freefall time, t ff,ini , obtained based on the initial density r 0 of FeSt 1-457 is~5 10 yr 5 . The sound-crossing time,t 1.5 10 yr sc,ini 6 , can be inferred from the initial core diameter 2R 0 and nearly sonic internal velocity dispersion. These quantities, about 1 million yr, serve as a lower limit value for the duration of the starless phase of the core and are a factor of ∼2-6 longer than the freefall time calculated using the mean density of the current core ( =t 2.4 10 ff,core 5 yr). The obtained factor, ∼2-6, is consistent with the value of ∼2-5 (Ward- Thompson et al. 2007) estimated based on the number ratios of cores with and without embedded young stellar objects (e.g., Beichman et al. 1986;Lee & Myers 1999;Jessop & Ward-Thompson 2000).
It is known that the ambipolar diffusion timescale t AD is about one order of magnitude longer than t ff (e.g., McKee & Ostriker 2007). The timescale of several times t ff is short for the evolution of the core with a highly magnetically subcritical condition (e.g., Shu 1977). However, in a turbulent medium, the efficiency of ambipolar diffusion can be accelerated (e.g., Fatuzzo & Adams 2002;Zweibel 2002;Nakamura & Li 2005;Kudoh & Basu 2014), and this may make t AD a reasonable length in timescale. Note that the estimated starless timescale for FeSt 1-457 serves as a lower limit, and it is still possible that FeSt 1-457 is a long-lived object.
The initial magnetic field strength B 0 is as weak as a typical interclump magnetic field in a molecular cloud (Crutcher 2012). The B 0 value was estimated by dividing the core's mean magnetic field strength B tot by a geometrical dilution factor of 1.64 2 . The actual initial magnetic field strength may be much larger, because the effect of ambipolar diffusion is not taken into account in the present calculation. The total magnetic field strength at the core boundary was estimated to be 14.6 μG (Paper IV; see also the Appendix). We thus consider the initial magnetic field strength B 0 to be in the range from 10.8 to m 14.6 G. Note that the value is consistent with the recently measured magnetic field strength for the intercore regions of molecular clouds using the OH Zeeman effect (∼15 μG; Thompson et al. 2019).
Finally, we emphasize the importance of comparing observational (polarimetry) data with the theoretical flux freezing magnetic field model (e.g., Myers et al. 2018), with which we can obtain information on the initial conditions of core formation. A relatively dense initial condition may be common for core formation. Table 5 of Kandori et al. (2005) shows that the external pressure of dense cores is on the order of 10 4 K cm −3 based on Bonnor-Ebert density structure analyses. Assuming a gas temperature of 10 K, we find a relatively high value of~10 3 cm −3 for the density of the medium surrounding the dense cores, which is consistent with the case for FeSt 1-457 presented in this study. In order to determine common properties and regional property variations of dense cores, it is important to analyze a greater number of cores with the flux freezing magnetic field model.

Summary and Conclusion
In the present study, the observational data for an hourglasslike magnetic field toward the starless dense core FeSt 1-457 were compared with a flux freezing magnetic field model (Myers et al. 2018). The flux freezing model gives a magnetic field structure consistent with observations. The best-fit parameters for the flux freezing model were a line-of-sight magnetic inclination angle of g =  35 mag and a core center to ambient (background) density contrast of r r = 75 c bkg . Note that the same density contrast value was obtained through independent measurements based on a Bonnor-Ebert density structure analysis (Kandori et al. 2005). The initial density for core formation (r 0 ) was estimated to be r = 75 4670 c cm −3 , which is about one order of magnitude higher than the expected density (∼300 cm −3 ) for the interclump medium of the Pipe Nebula. FeSt 1-457 is likely to have formed from the accumulation of relatively dense gas. The picture of a relatively dense initial condition for the formation of the core is supported by the relatively dense background column density (  A 5 V mag) around FeSt 1-457. The initial radius (core formation radius) R 0 and the initial magnetic field strength B 0 were obtained to be = R 1.64 0.15 pc and 10.8 μG, respectively, where R is the current radius of the core. It is notable that there are few methods to obtain a dense core's initial physical parameters. The B 0 value is roughly consistent with a magnetic field strength measured at the core boundary of 14.6 μG (Paper IV). We thus conclude that the B 0 value is in the range from 10.8 to 14.6 μG. We found that the initial density r 0 is consistent with the mean density of the nearly critical magnetized filament with magnetic field strength B 0 and radius R 0 . The relatively dense initial condition for core formation can be naturally understood if the origin of the core is the fragmentation of magnetized filaments.
We thank Takahiro Kudoh and Kate Pattle for helpful discussions. We are grateful to the staff of SAAO for their kind help during the observations. We with to thank Tetsuo Nishino, Chie Nagashima, and Noboru Ebizuka for their support in the development of SIRPOL, its calibration, and its stable operation with the IRSF telescope. The IRSF/SIRPOL project was initiated and supported by Nagoya University, National Astronomical Observatory of Japan, and the University of Tokyo in collaboration with the South African Astronomical Observatory under the financial support of Grants-in-Aid for Scientific Research on Priority Area (A) No. 10147207 and No. 10147214, and Grants-in-Aid No. 13573001 and No. 16340061 of the Ministry of Education, Culture, Sports, Science, and Technology of Japan. M.T. and R.K. acknowledge support by the Grants-in-Aid (Nos. 16077101, 16077204, 16340061, 21740147, 26800111, 19K03922).

Appendix Physical Properties of FeSt 1-457
Here we summarize the physical properties of FeSt 1-457, measured by our group and others, for reference when referring to the series of FeSt 1-457 papers (Kandori et al. 2005; Papers I, II, III, IV, V; this work). The FeSt 1-457 physical parameters are shown in Table 1 in Appendix A.3. In addition, we report revised parameters and figures from the papers, especially Papers II and III (see Appendix A.1) and Paper V (see Appendix A.2). The Stokes parameters (q and u) determined through integration of the numerical cubes of the polarization parameters are shown in Equations (7) and (8) in Section 3.1. Though the same analysis was intended to be made in Paper II, the square in the g cos 2 cell factor was absent in the calculations, and thus we evaluated the effect of this and updated the physical parameters and figures. The line-of-sight inclination angle of the magnetic axis was revised from g =    45 10 mag (Paper II) to    35 15 (this paper). g mag is mainly used in the inclination correction of the physical parameters as the factor g 1 cos mag , which changes by about 15% through the revision. Though this change is not large, it is not negligible. The revised figures from Papers II and III are presented in Appendix A.1, and the revised parameters are shown in Table 1 in Appendix A.3. In Appendix A.1, the parameters derived using the parabolic magnetic field model are compared with the results based on the flux freezing model. In Appendix A.2, we present the reanalyzed submillimeter polarimetry data (Alves et al. 2014 of Paper V. The data were reanalyzed using a recently proposed method (Pattle et al. 2019), and the updated parameters are shown in Table 1    Mass-to-flux ratio (pos) 2.00 L r (7) l tot Mass-to-flux ratio (total) 1.64 L r (8), (13) l tot,edge l tot at core edge »1 L r (10), (13) l tot,center l tot at core center »2 L r (10), (13) M mag Magnetic critical mass 2.16±0.65 Alfven

A.1. 3D Parabolic Model and Polarization-Extinction Relationship
A 3D polarization calculation of the simple parabolic magnetic field model was conducted (Paper II; this work). A 2D version of the model, = + y g gCx 2 , was employed in Paper I, and we further assumed that the magnetic field lines are axisymmetric around the z-axis. The 3D function can be expressed as ( ) f = + z r g g gCr , , 2 in cylindrical coordinates ( ) f r z , , , where g specifies the magnetic field line, C is the curvature of the lines, and f is the azimuth angle (measured on the plane perpendicular to r). This 3D function has no dependence on the parameter f. After generating the model function, for comparison with observations, the 3D model is virtually observed after rotating in the line-of-sight (g mag ) and plane-of-sky (q mag ) directions. For this analysis, we followed the procedure described in Section 3.1 of this paper. The resulting polarization vector maps of the 3D parabolic model are shown in Figure 14. The white lines show the polarization vectors, and the background color and color bar show the polarization degree of the model core. The density structure of the model core was assumed to be the same as the Bonnor-Ebert sphere with a solution parameter of 12.6 (the same parameter as obtained for FeSt 1-457; Kandori et al. 2005). The 3D magnetic curvature was set to =´-C 2.0 10 4 arcsec −2 for all the panels. The applied viewing angle ( g  -90 mag ), i.e., the angle between the line of sight and the magnetic axis, is labeled in the upper left corner of each panel.
The model polarization vector maps change depending on the viewing angle (g view ). As described in Paper II, there are four characteristics: (1) a decrease of maximum polarization degree from g =  90 view to g =  0 view ; (2) an hourglass-shaped polarization angle pattern for large g view converges to a radial pattern for small g view ; (3) depolarization occurs in the polarization vector map, especially along the equatorial plane of the core; and (4) an elongated structure of the polarization degree distribution toward small g view . Compared with the case of the flux freezing model (Figure 6), there are some differences in Figure 14, especially for the low-g view regions. However, both models have the above four characteristics, showing a similar dependence of the polarization features on g view . For details of these characteristics, see Section 3.1 of Paper II. Figures 15 and 16 show the c 2 distributions with respect to the polarization angle and degree (c q 2 and c P 2 ). The calculation methods are the same as those described in Section 3.1, and the minimization points are 35°for c q 2 and 50°for c P 2 . Since we obtained 35°for both c q 2 and c P 2 using the flux freezing model in Section 3.1, we concluded that the line-of-sight inclination angle g mag is 35°±15°. Jeans length (initial)5.8 10 4 au 0.29 pc (13) t ff,ini Freefall time4.5 10 5 yr z (13) t sc,ini Sound-crossing time1.5 10 6 yr z (13) Note. (a) The centroid center of the core measured on the A V map. (b) Alves & Franco (2007) estimated the distance to the Pipe Nebula to be 145±16 pc based on optical polarimetry. Dzib et al. (2018) estimated the distance to the Barnard 59 (B59) cloud in the Pipe Nebula to be 163±5 pc based on the GAIA data. (c) The P ext value was taken from Table 5 of Kandori et al. (2005). The value was determined based on the assumption that the Bonnor-Ebert equilibrium is maintained. However, =Ṕ 1.1 10 ext 5 K cm −3 is larger than ( ) r +´»T T 75 4.9 10 kin turb c . We chose the former value in the present study. If we use the latter value, the Bonnor-Ebert mass of ( ) ) ( ) 10 10 BE kin turb 2 ext 5 1 2 (McKee 1999) is 1.79  M , and M cr increases to 3.95  M . Comparing the observed core mass with M cr , the core is still located in a nearly critical state, and the conclusions of this paper do not change. (d) This parameter serves as a stability criterion of the Bonnor-Ebert sphere (Ebert 1955;Bonnor 1956). (e) The density contrast is the value of the central density r c divided by the surface density r s . (f) This value was measured on the A V map with a resolution of  33 (Kandori et al. 2005). (g) Measured using the rotation temperature of the NH 3 molecule (Rathborne et al. 2008). (h) Measured using the N 2 H + ( = -J 1 0) molecular line (Kandori et al. 2005). (i) The ratio of rotational energy and gravitational energy. (j) Aguti et al. (2007) suggest the existence of oscillation in the outer gas layer of FeSt 1-457. (k) Forbrich et al. (2009Forbrich et al. ( , 2010 searched young stars in the Pipe Nebula region in the mid-infrared and X-ray wavelengths, and no young sources were found toward the FeSt 1-457 core. (l) Measured using the N 2 H + ( = -J 1 0) molecular line (Aguti et al. 2007). (m) The plane-of-sky inclination angle of the core's magnetic axis was measured after subtracting the ambient polarization vector component (Paper I). (n) Though the line-of-sight inclination angle of the core's magnetic axis (measured from the plane of the sky) was previously estimated to be    45 10 , the value was updated in this paper to    35 15 . (o) The magnetic curvature term C was used in the simple parabolic magnetic field model, = + y g gCx 2 , and its 3D version (Papers I, II). (p) The plane-of-sky magnetic field strength estimated using the Davis-Chandrasekhar-Fermi method (Davis 1951;Chandrasekhar & Fermi 1953). (q) The total magnetic field strength obtained by dividing B pos by g cos mag . (r) The mass-to-flux ratio is defined as the observed ratio divided by the theoretical critical value: obs critical . We used pG 1 2 1 2 (Nakano & Nakamura 1978) for the critical value. (s) The speed of sound at 9.5 K (C s ), the turbulent velocity dispersion (s turb ), and the Alfvén velocity were used to estimate the ratios between the thermal, turbulent, and magnetic energies.   Figure 18 shows the same data but with the background image processed using the LIC technique (Cabral & Leedom 1993). The direction of the LIC "texture" is parallel to the direction of the magnetic field, and the background image is based on the polarization degree of the model core. The results look similar to the flux freezing model case (Figures 10 and 11). Figures 19-22 show the polarization-extinction (P-A) relationship measured at NIR wavelengths. The linearity in the P-A relationship is important in two respects: it shows that the observed polarization vectors trace the magnetic field structure inside the core, and it can be used to compare the relationship with theories of dust grain alignment (e.g., grain alignment with radiative torque; Dolginov & Mitrofanov 1976;Draine & Weingartner 1996, 1997Lazarian & Hoang 2007). Comparing Figure 19 with Figure 4 of Paper III, panels (a) and (b) are the same, and the shapes of the plots in panel (c) are very similar except for the slope. Note that in panel (c) we corrected the effects of depolarization and the line-of-sight inclination at the same time by dividing the panel (b) relationship by the 2D array of correction factors (Figure 19), so that panel (c) corresponds to panel (d) in Figure 4 of Paper III. In the revision, the g cos 2 mag factor with the angle of 35°was used in the calculations for panel (c). This does not change the linearity of the plot but changes the steepness in the slope. The slope, -P E H H K s , for each panel is 2.43%±0.05% mag −1 , 4.76%±0.33% mag −1 , and 6.60%±0.41% mag −1 for panels (a), (b), and (c), respectively. Figure 4(b) of Paper V was revised in the same way, and the corrected relationships are shown in Figures 20 and 21. The dotted line in Figures 20 Figure 15. c 2 distribution of the polarization angle (c q 2 ). The best magnetic curvature parameter (C) was determined for each inclination angle (g mag ). g =  0 mag and 90°correspond to the edge-on and pole-on geometries with respect to the magnetic axis. Figure 16. c 2 distribution of the polarization degree (c P 2 ). The calculations of c 2 in polarization degree were performed after determining the best magnetic curvature parameter (C) that minimized c 2 in the polarization angle. This calculation was carried out for each inclination angle (g mag ). g =  0 mag and 90°c orrespond to the edge-on and pole-on geometries with respect to the magnetic axis.   Figure 15, but the background image was made using the LIC technique (Cabral & Leedom 1993). The direction of the LIC "texture" is parallel to the magnetic field direction, and the background image is based on the polarization degree of the model core.  Figure 20 shows the linear fitting to the data, resulting in a slope of 0.002%±0.002% mag −1 . The dotteddashed lines in Figures 20 and 21 show the observational upper limit as determined by Jones (1989). The relation was calculated based on the equation , and the parameter η is set to 0.875 (Jones 1989). t K denotes the optical depth in the K band, and » P A 0.62 Note that although the above revisions are minor in terms of the shape/linearity of the   plots, the steepness of the slope is important when we discuss the efficiency of dust grain alignment.
The correlation coefficients for the Figure 19 relationship are 0.68, 0.76, and 0.85 for panels (a), (b), and (c), respectively. It is evident that the corrections (subtraction of ambient off-core polarization components, depolarization correction, and inclination correction) improve the tightness in the polarizationextinction relationship. Finally, we explain Figure 22, showing the depolarization and inclination correction factor. To obtain the factor, we divided the g =  35 mag model by the g =  0 mag model with the same magnetic curvature. In Figure 22, the factors in the regions around the equatorial plane are less than unity, showing that the depolarization effect applies. This is due to the crossing of the polarization vectors at the front and back sides of the core along the line of sight (see the explanatory illustration of Figure 7 of Kataoka et al. 2012 for the case of a uniform field, for the parabolic field case, most of the magnetic field lines around the poles are inclined with respect to the magnetic axis, reducing the polarization degree in the regions in the g =  0 mag model and consequently increasing the correction factors from 0.67.

A.2. Power-law Index of Submillimeter Polarimetry Data
As shown in Figures 20 and 21, the polarization efficiency at NIR wavelengths is nearly constant against A V , indicating that the observations trace the dust alignment, i.e., the magnetic field structure, in FeSt 1-457. However, the probing depth in our polarimetry is limited toÃ 25 V mag. To investigate the magnetic field structure deep inside the core, polarimetric observations at longer wavelengths are important. In Paper V, using the data of Alves et al. (2014Alves et al. ( , 2015, obtained with the APEX 12 m telescope and PolKa polarimeter at 870 μm (for the instrument see Siringo et al. 2004Siringo et al. , 2012Wiesemeyer et al. 2014), we showed that the magnetic field orientations obtained from submillimeter polarimetry (132°.1±22°.0) and NIR polarimetry (2°.7±16°.2) differ significantly. This may indicate a change of magnetic field orientation inside the core. However, the polarization fraction at submillimeter wavelengths P submm has an a submm index of 0.92±0.17 for the µ a -P I submm submm submm relationship ). An a submm index close to unity indicates that the alignment of dust inside the core should be lost (e.g., Andersson et al. 2015).
The polarization fraction data points obtained with dust emission polarimetry are usually debiased (e.g., Wardle & Kronberg 1974), and points having a signal-to-noise ratio (S/N) larger than a certain value are selected for the power-law fitting. Recently, Pattle et al. (2019) reported that the usual method for obtaining the α power-law index can lead to an overestimation of α, and they demonstrated that the Riceanmean model fitting to the whole data (without debias) can provide a better estimation of the α index. We followed this method to revise/improve the a submm index. The P submm versus I submm data were fitted using the following equation: , which they refer to as the Ricean-mean model. In the equation, s QU is the rms noise in the Stokes Q and U measurements, s P QU is a parameter to be fitted simultaneously with a submm , and  1 2 is a Laguerre polynomial of order 1 2 . We fitted the observations using this function, and the results are shown in Figure 23 as a solid line. The dotted line shows the relationship for the Figure 22. Distribution of the depolarization and inclination correction factor. The field of view is the same as the diameter of the core, 288″. Figure 23. Relationship between the polarization fraction P submm and intensity I submm at submillimeter wavelengths. The solid line shows the best-fitting Ricean-mean model. The dotted line shows the relationship of the low-S/N limit. low-S/N limit defined by Equation (12) of Pattle et al. (2019). Note that the P submm values greater than unity are physically meaningless. The best-fit parameters are a =  0.41 submm 0.10 and =  s P 0.30 0.10 QU . We obtained a significantly low value of a submm compared with the fitting based on the ordinary method . Thus, we conclude that the alignment of dust grains is better than previously thought.

A.3. List of Physical Parameters
In Table 1, we summarize the physical parameters for FeSt 1-457. This parameter list does not contain all the values reported so far but shows the physical parameters mainly used in our studies related to this core (Kandori et al. 2005; Papers I, II, V, IV, V; this work). For example, the parameters for the chemical properties reported by Juárez et al. (2017) or the dust grain (growth) properties reported by Forbrich et al. (2015) are not included.

A.4. Modified Davis-Chandrasekhar-Fermi Method
Cho & Yoo (2016) and Yoon & Cho (2019) studied the reduction of variation in polarization angle dq due to the averaging effect along the line of sight. If there is more than one independent turbulent eddy along the line of sight, the measured value of dq will be reduced. They suggested to use dV c , the standard deviation of centroid velocity of the optically thin molecular line, instead of s turb , the turbulent velocity dispersion, in the original Davis-Chandrasekhar-Fermi formulation. The conventional form of the Davis-Chandrasekhar-Fermi method is¯( wherer is the mean density and = C 0.5 corr is a correction factor suggested by theoretical studies (Ostriker et al. 2001; see also Heitsch et al. 2001;Padoan et al. 2001;Heitsch 2005;Matsumoto et al. 2006 where ξ is a constant of order unity that can be determined by numerical simulations. The standard deviation of centroid velocity is given by where N eddy is the number of independent turbulent eddies along the line of sight. We obtained s =  0.0573 0.006 turb km s −1 based on the N 2 H + ( = -J 1 0) molecular line observations using the Nobeyama 45 m radio telescope (Kandori et al. 2005). Using the same data, we obtained d » V 0.023 c km s −1 . Note that the standard deviation of V c was calculated after subtracting the rigid rotation component estimated by plane fitting. Comparing this value with´= C 0.0573 0.029 corr km s −1 , the difference is about 20%, indicating that the applications of the Davis-Chandrasekhar-Fermi method and its modified version to FeSt 1-457 yield consistent results. The expected number of independent turbulent eddies is »6.2. The relatively small N eddy enables the use of the classic Davis-Chandrasekhar-Fermi formula for FeSt 1-457, and such situations might be common for other low-mass dense cores.