Formation of Magnetized Prestellar Cores with Ambipolar Diffusion and Turbulence
Abstract
We investigate the roles of magnetic fields and ambipolar diffusion during prestellar core formation in turbulent giant molecular clouds (GMCs), using threedimensional numerical simulations. Our simulations focus on the shocked layer produced by a converging largescale flow, and survey varying ionization and angle between the upstream flow and magnetic field. We also include ideal magnetohydrodynamic (MHD) and hydrodynamic models. From our simulations, we identify hundreds of selfgravitating cores that form within 1 Myr, with masses M\sim 0.042.5 M{}_{\odot} and sizes L\sim 0.0150.07 pc, consistent with observations of the peak of the core mass function (CMF). Median values are M=0.47~{}\mathrm{M}_{\odot} and L=0.03 pc. Core masses and sizes do not depend on either the ionization or upstream magnetic field direction. In contrast, the masstoflux ratio does increase with lower ionization, from twice to four times the critical value. The higher masstoflux ratio for low ionization is the result of enhanced transient ambipolar diffusion when the shocked layer first forms. However, ambipolar diffusion is not necessary to form lowmass supercritical cores. For ideal MHD, we find similar masses to other cases. These masses are 12 orders of magnitude lower than the value M_{\mathrm{mag,sph}}=0.007~{}B^{3}/(G^{3/2}\rho^{2}) that defines a magnetically supercritical sphere under postshock ambient conditions. This discrepancy is the result of anisotropic contraction along field lines, which is clearly evident in both ideal MHD and diffusive simulations. We interpret our numerical findings using a simple scaling argument which suggests that gravitationally critical core masses will depend on the sound speed and mean turbulent pressure in a cloud, regardless of magnetic effects.
1 Introduction
The formation of stars begins with dense molecular cores (McKee & Ostriker, 2007; André et al., 2009). These cores form through the concentration of overdense regions within turbulent, filamentary GMCs; subsequent core collapse leads to protostellar (or protobinary)/disk systems. Magnetic fields are important at all scales during this process (McKee & Ostriker, 2007; Crutcher, 2012): the cloudscale magnetic field can limit compression in interstellar shocks that create dense clumps and filaments in which cores form, while the local magnetic field within individual cores can prevent collapse if it is large enough (Mestel & Spitzer, 1956; Strittmatter, 1966; Mouschovias & Spitzer, 1976), and can help to remove angular momentum during the disk formation process if cores are successful in collapsing (Mestel, 1985; Mouschovias, 1991; Allen et al., 2003; Li et al., 2013a). The significance of magnetic fields in selfgravitating cores can be quantified by the ratio of mass to magnetic flux; only if the masstoflux ratio exceeds a critical value is gravitational collapse possible. How the masstoflux ratio increases from the stronglymagnetized interstellar medium to weaklymagnetized stars is a fundamental problem of star formation (Shu et al., 1987; McKee & Ostriker, 2007). Here, as suggested in Chen & Ostriker (2012, hereafter CO12), we consider core formation in GMCs with highly supersonic turbulence and nonideal MHD.
Magnetic fields are coupled only to charged particles, while the gas in GMCs and their substructures is mostly neutral. The ability of magnetic fields to affect core and star formation thus depends on the collisional coupling between neutrals and ions. Ambipolar diffusion is the nonideal MHD process that allows charged particles to drift relative to the neutrals, with a drag force proportional to the collision rate (Spitzer, 1956). Ambipolar drift modifies the dynamical effect of magnetic fields on the gas, and may play a key role in the star formation.
In classical theory, quasistatic ambipolar diffusion is the main mechanism for prestellar cores to lose magnetic support and reach supercritical masstoflux ratios. Through ambipolar drift, the mass within dense cores can be redistributed, with the neutrals diffusing inward while the magnetic field threading the outer region is left behind (Mouschovias, 1979). However, the quasistatic evolution model (e.g. Mouschovias & Ciolek, 1999; Ciolek & Basu, 2001) gives a prestellar core lifetime considerably longer (up to a factor of 10) than the gravitational freefall timescale, t_{\mathrm{ff}}, while several observational studies have shown that cores only live for (25) t_{\mathrm{ff}} (e.g. WardThompson et al., 2007; Evans et al., 2009).
The failure of the traditional picture to predict core lifetimes indicates that supercritical cores may not have formed quasistatically through ambipolar diffusion. Indeed, it is now generally recognized that, due to pervasive supersonic flows in GMCs, core formation is not likely to be quasistatic. Realistic star formation models should take both ambipolar diffusion and largescale supersonic turbulence into consideration. This turbulence may accelerate the ambipolar diffusion process (Heitsch et al., 2004; Li & Nakamura, 2004), with an analytic estimate of the enhanced diffusion rate by a factor of 23 for typical conditions in GMCs (Fatuzzo & Adams, 2002).
In our previous work (CO12), we investigated the physical mechanism driving enhanced ambipolar diffusion in onedimensional Ctype shocks. These shocks pervade GMCs, and are responsible for the initial compression of gas above ambient densities. We obtained a formula for the Cshock thickness as a function of density, magnetic field, shock velocity, and ionization fraction, and explored the dependence of shockenhanced ambipolar diffusion on environment through a parameter study. Most importantly, we identified and characterized a transient stage of rapid ambipolar diffusion at the onset of shock compression, for onedimensional converging flows. For an interval comparable to the neutralion collision time and before the neutralion drift reaches equilibrium, the neutrals do not experience drag forces from the ions. As a consequence, the initial shock in the neutrals is essentially unmagnetized, and the neutrals can be very strongly compressed. This transient stage, with timescale t_{\mathrm{transient}}\sim 1 Myr (but depending on ionization), can create dense structures with much higher \rho/B than upstream gas. CO12 suggested this could help enable supercritical core formation. CO12 also found that (1) the perpendicular component of the magnetic field is the main determinant of the shock compression, and (2) the perpendicular component of the magnetic field B_{\perp} must be weak (\lesssim 5~{}\muG) for transient ambipolar diffusion in shocks to significantly enhance \rho/B_{\perp}.
Observations of nearby clouds provide direct constraints on the role of magnetic fields, as well as other properties of prestellar cores. The typical mean masstoflux ratio of dark cloud cores is \Gamma\sim 2 (in units of critical value; see Equation (16)) from Zeeman studies (Falgarone et al., 2008; Troland & Crutcher, 2008). Due to the instrumental limitations, magnetic field observations in solarmass and smaller scale regions are relatively lacking compared with observations of larger scales (see review in Crutcher, 2012), however. Surveys in nearby clouds have found that prestellar cores have masses between \sim 0.110 M{}_{\odot} and sizes \sim 0.011 pc (e.g. Motte et al., 2001; Ikeda et al., 2009; Rathborne et al., 2009; Kirk et al., 2013). In addition, a masssize relation has been proposed as a power law M\propto R^{k}, with k=1.22.4 dependent on various molecule tracers (e.g. Elmegreen & Falgarone, 1996; Curtis & Richer, 2010; RomanDuval et al., 2010; Kirk et al., 2013).
The magnetic field strength within prestellar cores is important for late evolution during core collapse, since disk formation may be suppressed by magnetic braking (for recent simulations see Allen et al. 2003; Hennebelle & Fromang 2008; Mellon & Li 2008; Hennebelle et al. 2011; or see review in Li et al. 2013a). However, many circumstellar disks and planetary systems have been detected (e.g. Haisch et al., 2001; Maury et al., 2010), suggesting that the magnetic braking “catastrophe” seen in many simulations does not occur in nature. The proposed solutions include the misalignment between the magnetic and rotation axes (e.g. Hennebelle & Ciardi, 2009; Ciardi & Hennebelle, 2010; Joos et al., 2012; Krumholz et al., 2013), turbulent reconnection and other turbulent processes during the rotating collapse (e.g. SantosLima et al., 2012; Seifried et al., 2012, 2013), and nonideal MHD effects including ambipolar diffusion, Hall effect, and Ohmic dissipation (e.g. Krasnopolsky et al., 2010; Li et al., 2011; Machida et al., 2011; Dapp et al., 2012; Tomida et al., 2013). If prestellar cores have sufficiently weak magnetic fields, however, braking would not be a problem during disk formation (e.g. Mellon & Li, 2008; Li et al., 2013a, b). Therefore, the magnetic field (and masstoflux ratio) within a prestellar core is important not just for the ability of the core to collapse, but also of a disk to form.
Fragmentation of sheetlike magnetized clouds induced by smallamplitude perturbation and regulated by ambipolar diffusion has been widely studied (e.g. Indebetouw & Zweibel, 2000; Basu & Ciolek, 2004; Boss, 2005; Ciolek & Basu, 2006; Basu et al., 2009a). Analogous fully threedimensional simulations have also been conducted (e.g. Kudoh et al., 2007). Supercritical cores formed in the flattened layer have masses \sim 0.110 M{}_{\odot} (e.g. Indebetouw & Zweibel, 2000; Basu et al., 2009a), at timescales \sim 110 Myr dependent on the initial masstoflux ratio of the cloud (e.g. Indebetouw & Zweibel, 2000; Kudoh et al., 2007; Basu et al., 2009a). The above cited simulations start from relatively high densities (\sim 10^{4} cm{}^{3}; e.g. Kudoh et al. 2007) and included only the lowamplitude perturbations. Alternatively, Li & Nakamura (2004) and Nakamura & Li (2005) took the formation of these overdense regions into consideration by including a direct treatment of the largescale supersonic turbulence. They demonstrated that ambipolar diffusion can be sped up locally by the supersonic turbulence, forming cores with masses \sim 0.5 M{}_{\odot} and sizes \sim 0.1 pc within \sim 2 Myr, while the strong magnetic field keeps the star formation efficiency low (110\%). Similarly, Basu et al. (2009b) found that turbulenceaccelerated, magneticallyregulated core formation timescales are \sim 1 Myr in twodimensional simulations of magnetized sheetlike clouds, with corresponding threedimensional simulations showing comparable results (Kudoh & Basu, 2008, 2011). In addition, Nakamura & Li (2008) measured the core properties in their threedimensional simulations to find L_{\mathrm{core}}\sim 0.040.14 pc, \Gamma_{\mathrm{core}}\sim 0.31.5, and M_{\mathrm{core}}\sim 0.1512.5 M{}_{\odot}, while Basu et al. (2009b) found a broader core mass distribution M_{\mathrm{core}}\sim 0.0425 M{}_{\odot} in their parameter study using thinsheet approximation.
Supersonic turbulence within GMCs extends over a wide range of spatial scales (Mac Low & Klessen, 2004; BallesterosParedes et al., 2007). Although turbulence contains sheared, diverging, and converging regions in all combinations, regions in which there is a largescale convergence in the velocity field will strongly compress gas, creating favorable conditions for the birth of prestellar cores. Gong & Ostriker (2011, hereafter GO11) investigated core formation in an idealized model containing both a largescale converging flow and multiscale turbulence. These simulations showed that the time until the first core collapses depends on inflow Mach number {\cal M} as t_{\mathrm{collapse}}\propto{\cal M}^{1/2}. With a parameter range {\cal M}=1.1 to 9, cores formed in the GO11 simulations had masses 0.0550 M{}_{\odot}. Following similar velocity power spectrum but including ideal MHD effects, Myers et al. (2014) performed simulations with sink particle, radiative transfer, and protostellar outflows to follow the protostar formation in turbulent massive clump. They demonstrated that the median stellar mass in the simulated star cluster can be doubled by the magnetic field, from 0.05 M{}_{\odot} (unmagnetized case) to 0.12 M{}_{\odot} (star cluster with initial masstoflux ratio \Gamma=2). This is qualitatively consistent with the conclusion in Inoue & Fukui (2013), that the mass of the cores formed in the postshock regions created by cloudcloud collision is positively related to (and dominated by) the strong magnetic field in the shocked layer. Note that, though the main focus of Inoue & Fukui (2013) is the cloud’s ability to form massive cores (\sim 20200 M{}_{\odot} in their simulations), the idea of cloudcloud collision is very similar to the converging flows setup adopted in GO11 and this study.
In this paper, we combine the methods of CO12 for modeling ambipolar diffusion with the methods of GO11 for studying selfgravitating structure formation in turbulent converging flows. Our numerical parameter study focuses on the level of ambipolar diffusion (controlled by the ionization fraction of the cloud) and the obliquity of the shock (controlled by the angle between the magnetic field and the upstream flow). We show that filamentary structures similar to those seen in observations (see review in André et al., 2013) develop within shocked gas layers, and that cores form within these filaments. We measure core properties to test their dependence on these parameters. As we shall show, our models demonstrate that lowmass supercritical cores can form for all magnetic obliquities and all levels of ionization, including ideal MHD. However, our models also show that ambipolar diffusion affects the magnetization of dynamicallyformed cores.
The outline of this paper is as follows. We provide a theoretical analysis of oblique MHD shocks in Section 2, pointing out that a quasihydrodynamic compression ratio (which is \sim 5 times stronger than in fast MHD shocks for the parameters we study) can exist when the converging flow is nearly parallel to the magnetic field. We also show that shock compression cannot increase the masstoflux ratio except in the nearlyparallel case or with ambipolar diffusion. Section 3 describes methods used in our numerical simulations and data analysis, including our model parameter set and method for measuring magnetic flux within cores. The evolution of gas structure (including development of filaments) and magnetic fields for varying parameters is compared in Section 4. In Section 5 we provide quantitative results for masses, sizes, magnetizations, and other physical properties of the bound cores identified from our simulations. Implications of these results for core formation is discussed in Section 6, where we argue that the similarity of core masses and sizes among models with different magnetizations and ionizations can be explained by anisotropic condensation preferentially along the magnetic field. Section 7 summarizes our conclusions.
2 Theoretical Analysis
2.1 Oblique MHD Shock
CO12 describe a onedimensional simplified MHD shock system with velocity and magnetic field perpendicular to each other, including a short discussion of oblique shocks. Here we review the oblique shock equations and write them in a more general form to give detailed jump conditions.
We shall consider a planeparallel shock with uniform preshock neutral density \rho_{0} and ionizationrecombination equilibrium everywhere. The shock front is in the xy plane, the upstream flow is along the zdirection (\mathbf{v}_{0}=v_{0}\hat{\mathbf{z}}), and the upstream magnetic field is in the xz plane, at an angle \theta to the inflow (\mathbf{B}_{0}=B_{0}\sin\theta\hat{\mathbf{x}}+B_{0}\cos\theta\hat{\mathbf{z}}) such that B_{x}=B_{0}\sin\theta is the upstream component perpendicular to the flow. The parameters {\cal M} and \beta (upstream value of the Mach number and plasma parameter) defined in CO12 therefore become
{\cal M}\equiv{\cal M}_{z}=\frac{v_{0}}{c_{s}},\ \ \ \frac{1}{\beta_{0}}\equiv% \frac{B_{0}^{2}}{8\pi\rho_{0}c_{s}^{2}}=\frac{1}{\beta_{x}}\frac{1}{\sin^{2}% \theta}.  (1) 
The jump conditions of MHD shocks are described by compression ratios of density and magnetic field:
r_{f}\equiv\frac{\rho_{n,\ \mathrm{downstream}}}{\rho_{n,\ \mathrm{upstream}}}% ,\ \ \ r_{B_{\perp}}\equiv\frac{B_{\perp,\ \mathrm{downstream}}}{B_{\perp,\ % \mathrm{upstream}}}.  (2) 
From Equations (A10) and (A14) in CO12, we have
\frac{\sin^{2}\theta{r_{f}}^{2}}{\beta_{0}}\left(1\frac{2\cos^{2}\theta}{% \beta_{0}{\cal M}^{2}}\right)^{2}=\left({\cal M}^{2}+1+\frac{\sin^{2}\theta}{% \beta_{0}}\frac{{\cal M}^{2}}{r_{f}}r_{f}\right)\left(1\frac{2r_{f}\cos^{2}% \theta}{\beta_{0}{\cal M}^{2}}\right)^{2},  (3) 
which can be solved numerically to obtain explicit solution(s) r_{f,\mathrm{exp}}(\theta). The compression ratio for the magnetic field perpendicular to the inflow is
r_{B_{\perp}}(\theta)=r_{f}(\theta)\frac{1\frac{2\cos^{2}\theta}{\beta_{0}{% \cal M}^{2}}}{1\frac{2r_{f}(\theta)\cos^{2}\theta}{\beta_{0}{\cal M}^{2}}}.  (4) 
Equation (A17) of CO12 gives an analytical approximation to r_{f}(\theta):
r_{f,\mathrm{app}}(\theta)=\frac{\sqrt{\beta_{0}}{\cal M}}{\sin\theta}\left[% \frac{2\sin\theta}{\sqrt{\beta_{0}}{\cal M}\tan^{2}\theta}+\frac{\sqrt{\beta_{% 0}}}{2{\cal M}\sin\theta}+1\right]^{1}.  (5) 
Since Equation (3) is a quartic function of \theta, there are four possible roots of r_{f} for each angle, and r_{f}(\theta)=const.=1 (noshock solution) is always a solution. When \theta is large, Equation (3) has one simple root (r_{f}=1) and a multiple root with multiplicity =3. When \theta drops below a critical value, \theta_{\mathrm{crit}}, Equation (3) has four simple roots, which give us four different values of r_{B_{\perp}}. Figure 1 shows the three explicit solutions for r_{f} and r_{B_{\perp}} (r_{f,\mathrm{exp}}(\theta) and r_{B,\mathrm{exp}}(\theta)) as well as the approximations (r_{f,\mathrm{app}}(\theta) and r_{B,\mathrm{app}}(\theta)) that employ Equation (5).
The fact that there are multiple solutions for postshock properties is the consequence of the nonunique Riemann problem in ideal MHD (see discussions in e.g. Torrilhon, 2003; Delmont & Keppens, 2011; Takahashi & Yamada, 2013), and whether all solutions are physically real is still controversial. The first set of solutions r_{f,\mathrm{exp}}(\theta),1 and r_{B,\mathrm{exp}}(\theta),1 shown in Figure 1 gives positive r_{f} and r_{B_{\perp}}, classified as fast MHD shocks (Shu, 1992; Draine & McKee, 1993), and is the principal oblique shock solution referred to in this contribution^{1}^{1}1We use Equation (5) as analytical approximation for r_{f}(\theta), if necessary.. The other two solutions for postshock magnetic field, r_{B,\mathrm{exp}}(\theta),2 and r_{B,\mathrm{exp}}(\theta),3, both become negative when \theta<\theta_{\mathrm{crit}}, indicating that the tangential component of the magnetic field to the shock plane is reversed in the postshock region. These two solutions are commonly specified as intermediate shocks (e.g. Wu, 1987; Karimabadi, 1995; Inoue & Inutsuka, 2007). Among these two fieldreversal solutions, we notice that r_{f,\mathrm{exp}}(\theta),2 approaches the hydrodynamic jump condition (r_{f,\mathrm{hydro}}={\cal M}^{2}) when \theta\rightarrow 0, and r_{B,\mathrm{exp}}(\theta),2 is smaller in magnitude than other solutions when \theta<\theta_{\mathrm{crit}}. Thus, we classify this set of solutions r_{f,\mathrm{exp}}(\theta),2 and r_{B,\mathrm{exp}}(\theta),2 as the quasihydrodynamic shock. This quasihydrodynamic solution can create gas compression much stronger than the regularlyapplied fast shock condition, and may be the reason that when \theta<\theta_{\mathrm{crit}}, even ideal MHD simulations can generate shocked layers with relatively high masstoflux ratio (see Sections 4 and 5 for more details).
The definition of \theta_{\mathrm{crit}} can be derived from Equation (4), which turns negative when 1\frac{2\cos^{2}\theta}{\beta_{0}{\cal M}^{2}}>0 and 1\frac{2r_{f}(\theta)\cos^{2}\theta}{\beta_{0}{\cal M}^{2}}<0:
\cos^{2}\theta>\cos^{2}\theta_{\mathrm{crit}}=\frac{\beta_{0}{\cal M}^{2}}{2r_% {f}(\theta_{\mathrm{crit}})}.  (6) 
Using Equation (5) and considering only the terms \sim{\cal M}, this becomes
\frac{\cos^{2}\theta_{\mathrm{crit}}}{\sin\theta_{\mathrm{crit}}}\approx\frac{% \sqrt{\beta_{0}}{\cal M}}{2},  (7) 
or
\sin^{2}\theta_{\mathrm{crit}}+\frac{\sqrt{\beta_{0}}{\cal M}}{2}\sin\theta_{% \mathrm{crit}}1=0.  (8) 
Assuming \theta_{\mathrm{crit}}\ll 1, this gives
\theta_{\mathrm{crit}}\sim\frac{2}{\sqrt{\beta_{0}}{\cal M}}=\sqrt{2}\frac{v_{% \mathrm{A,0}}}{v_{0}},  (9) 
where v_{\mathrm{A,0}}\equiv B_{0}/\sqrt{4\pi\rho_{0}} is the Alfvén speed in the cloud. Therefore, the criterion to have multiple solutions, \theta<\theta_{\mathrm{crit}}, is approximately equivalent to
v_{\perp}=v_{0}\sin\theta\lesssim v_{0}\cdot\sqrt{2}\frac{v_{\mathrm{A,0}}}{v_% {0}}\sim v_{\mathrm{A,0}}  (10) 
where v_{\perp} is the component of the inflow perpendicular to the magnetic field. Though Equation (9) only provides a qualitative approximation^{2}^{2}2For parameters used in Figure 1, Equation (9) gives \theta_{\mathrm{crit}}=18^{\circ}, approximately 2 times larger than the exact solution. for \theta_{\mathrm{crit}}, Equation (10) suggests that when v_{\perp}/v_{\mathrm{A,0}} is sufficiently small, highcompression quasihydrodynamic shocks are possible.
2.2 Gravitational Critical Scales in Spherical Symmetry
For a core to collapse gravitationally, its selfgravity must overcome both the thermal and magnetic energy. For a given ambient density \rho\equiv\mu_{n}n and assuming spherical symmetry, the mass necessary for gravity to exceed the thermal pressure support (with edge pressure \rho{c_{s}}^{2}) is the mass of the critical BonnorEbert sphere (see e.g. Gong & Ostriker, 2009):
M_{\mathrm{th,sph}}=4.18\frac{{c_{s}}^{3}}{\sqrt{4\pi G^{3}\rho}}=4.4~{}% \mathrm{M}_{\odot}\left(\frac{T}{10~{}\mathrm{K}}\right)^{3/2}\left(\frac{n}{1% 000~{}\mathrm{cm}^{3}}\right)^{1/2}  (11) 
(see Section 3.2 for discussion about the value of \mu_{n}). The corresponding length scale at the original ambient density is
R_{\mathrm{th,sph}}\equiv\left(\frac{3M_{\mathrm{th,sph}}}{4\pi\rho}\right)^{1% /3}=2.3\frac{c_{s}}{\sqrt{4\pi G\rho}}=0.26~{}\mathrm{pc}\left(\frac{T}{10~{}% \mathrm{K}}\right)^{1/2}\left(\frac{n}{1000~{}\mathrm{cm}^{3}}\right)^{1/2},  (12) 
although the radius of a BonnorEbert sphere with mass given by Equation (11) would be smaller than Equation (12) by 25\%, due to internal stratification.
In a magnetized medium with magnetic field B, the ratio of mass to magnetic flux for a region to be magnetically supercritical^{3}^{3}3See Section 3.3 for more detailed discussion about the critical value of M/\Phi_{B}. can be written as
\frac{M}{\Phi_{B}}\bigg{}_{\mathrm{mag,crit}}\equiv\frac{1}{2\pi\sqrt{G}}.  (13) 
With M=4\pi R^{3}\rho/3 and \Phi_{B}=\pi R^{2}B for a spherical volume at ambient density \rho, this gives
M_{\mathrm{mag,sph}}=\frac{9}{128\pi^{2}G^{3/2}}\frac{B^{3}}{\rho^{2}}=14~{}% \mathrm{M}_{\odot}\left(\frac{B}{10~{}\mu\mathrm{G}}\right)^{3}\left(\frac{n}{% 1000~{}\mathrm{cm}^{3}}\right)^{2}.  (14) 
and
R_{\mathrm{mag,sph}}=\frac{3}{8\pi\sqrt{G}}\frac{B}{\rho}=0.4~{}\mathrm{pc}% \left(\frac{B}{10~{}\mu\mathrm{G}}\right)\left(\frac{n}{1000~{}\mathrm{cm}^{3% }}\right)^{1},  (15) 
A spherical region must have M>M_{\mathrm{th,sph}} as well as M>M_{\mathrm{mag,sph}} to be able to collapse. In the cloud environment (the preshock region), B\sim 10~{}\muG and n\sim 1000 cm{}^{3} are typical. Comparing Equation (11) and (14), the magnetic condition is more strict than the thermal condition; if cores formed from a spherical volume, the mass would have to exceed \sim 10 M{}_{\odot} in order to collapse. This value is much larger than the typical core mass (\sim 1 M{}_{\odot}) identified in observations. This discrepancy is the reason why traditionally ambipolar diffusion is invoked to explain how lowmass cores become supercritical.
We can examine the ability for magnetically supercritical cores to form isotropically in a postshock layer. The normalized masstoflux ratio
\Gamma\equiv\frac{M}{\Phi_{B}}\cdot 2\pi\sqrt{G}  (16) 
of a spherical volume with density \rho, magnetic field B, and mass M is
\displaystyle\Gamma_{\mathrm{sph}}  \displaystyle=\frac{8\pi\sqrt{G}}{3}\left(\frac{3}{4\pi}\right)^{1/3}M^{1/3}% \rho^{2/3}B^{1}  
\displaystyle=0.4\left(\frac{M}{\mathrm{M}_{\odot}}\right)^{1/3}\left(\frac{n}% {1000~{}\mathrm{cm}^{3}}\right)^{2/3}\left(\frac{B}{10~{}\mu\mathrm{G}}\right% )^{1}.  (17) 
Or, with \Sigma=4R\rho/3\equiv\mu_{n}N_{n} for a sphere, we have
\Gamma_{\mathrm{sph}}=2\pi\sqrt{G}\cdot\frac{\Sigma}{B}=0.6\left(\frac{N_{n}}{% 10^{21}~{}\mathrm{cm}^{2}}\right)\left(\frac{B}{10~{}\mu\mathrm{G}}\right)^{% 1}.  (18) 
Considering the cloud parameters from Figure 1 ({\cal M}=10, B_{0}=10 \muG, n_{0}=1000 cm{}^{3}), the postshock density and magnetic field are approximately n_{\mathrm{ps}}\sim 10^{4} cm{}^{3} and B_{\mathrm{ps}}\sim 50 \muG when \theta>\theta_{\mathrm{crit}}. A solarmass spherical region in this shocked layer will have \Gamma_{\mathrm{ps,sph}}\approx 0.37; spherical contraction induced by gravity would be suppressed by magnetic fields. Thus, typical postshock conditions are unfavorable for forming lowmass cores by spherical contraction in ideal MHD.
Furthermore, using r_{f} and r_{B_{\perp}} defined in Section 2.1, we can compare \Gamma_{\mathrm{ps,sph}} and the preshock value \Gamma_{\mathrm{pre,sph}} for spherical postshock and preshock regions:
\frac{\Gamma_{\mathrm{ps,sph}}}{\Gamma_{\mathrm{pre,sph}}}=\left(\frac{M_{% \mathrm{ps}}}{M_{\mathrm{pre}}}\right)^{1/3}\left(\frac{\rho_{\mathrm{ps}}}{% \rho_{\mathrm{pre}}}\right)^{2/3}\left(\frac{B_{\mathrm{ps}}}{B_{\mathrm{pre}}% }\right)^{1}\approx\left(\frac{M_{\mathrm{ps}}}{M_{\mathrm{pre}}}\right)^{1/3% }{r_{f}}^{2/3}{r_{B_{\perp}}}^{1}.  (19) 
Considering volumes containing similar mass, M_{\mathrm{ps}}\sim M_{\mathrm{pre}}, the ratio between the postshock and preshock \Gamma_{\mathrm{sph}} is smaller than unity when \theta>\theta_{\mathrm{crit}}, because Equation (4) shows that r_{B_{\perp}} is larger than r_{f}. Thus, provided \theta>\theta_{\mathrm{crit}}, the postshock layer will actually have stronger magnetic support than the preshock region for a given spherical mass.
Based on the above considerations, formation of lowmass supercritical cores appears difficult in ideal MHD. Adapting classical ideas, one might imagine that lowmass subcritical cores form quasistatically within the postshock layer, then gradually lose magnetic support via ambipolar diffusion to become magnetically supercritical in a timescale \sim 110~{}Myr. A process of this kind would, however, give prestellar core lifetimes longer than observed, and most cores would have \Gamma<1 (inconsistent with observations).
Two alternative scenarios could lead to supercritical core formation in a turbulent magnetized medium. First, the dynamic effects during a turbulenceinduced shock (including rapid, transient ambipolar diffusion and the quasihydrodynamic compression when \theta<\theta_{\mathrm{crit}}) may increase the compression ratio of neutrals, creating r_{f}\gg r_{B_{\perp}} and \Gamma_{\mathrm{ps,sph}}>1, enabling lowmass supercritical cores to form. Second, even if the postshock region is strongly magnetized, mass can accumulate through anisotropic condensation along the magnetic field until both the thermal and magnetic criteria are simultaneously satisfied. In this study, we carefully investigate these two scenarios, showing that both effects contribute to the formation of lowmass supercritical cores within timescale \lesssim 0.6 Myr, regardless of ionization or magnetic obliquity.
3 Numerical Methods and Models
3.1 Simulation Setup and Equations
To examine core formation in shocked layers of partiallyionized gas, we employ a threedimensional convergent flow model with ambipolar diffusion, selfgravity, and a perturbed turbulent velocity field. We conducted our numerical simulations using the Athena MHD code (Stone et al., 2008) with Roe’s Riemann solver. To avoid negative densities if the secondorder solution fails, we instead use firstorder fluxes for bad zones. The selfgravity of the domain, with an open boundary in one direction and periodic boundaries in the other two, is calculated using the fast Fourier transformation (FFT) method developed by Koyama & Ostriker (2009). Ambipolar diffusion is treated in the strong coupling approximation, as described in Bai & Stone (2011), with super timestepping (Choi et al., 2009) to accelerate the evolution.
The equations we solve are:
\displaystyle\frac{\partial\rho_{n}}{\partial t}  \displaystyle+\mathbf{\nabla}\cdot\left(\rho_{n}\mathbf{v}\right)=0,  (20a)  
\displaystyle\frac{\partial\rho_{n}\mathbf{v}}{\partial t}  \displaystyle+\mathbf{\nabla}\cdot\left(\rho_{n}\mathbf{v}\mathbf{v}\frac{% \mathbf{B}\mathbf{B}}{4\pi}\right)+\mathbf{\nabla}P^{*}=0,  (20b)  
\displaystyle\frac{\partial\mathbf{B}}{\partial t}  \displaystyle+\mathbf{\nabla}\times\left(\mathbf{B}\times\mathbf{v}\right)=% \mathbf{\nabla}\times\left[\frac{\left(\left(\mathbf{\nabla}\times\mathbf{B}% \right)\times\mathbf{B}\right)\times\mathbf{B}}{4\pi\rho_{i}\rho_{n}\alpha}% \right],  (20c) 
where P^{*}=P+B^{2}/(8\pi). For simplicity, we adopt an isothermal equation of state P=\rho{c_{s}}^{2}. The numerical setup for inflow and turbulence is similar to that adopted by GO11. For both the whole simulation box initially and the inflowing gas subsequently, we apply perturbations following a Gaussian random distribution with a Fourier power spectrum as described in GO11. The scaling law for supersonic turbulence in GMCs obeys the relation
\frac{\delta v_{\mathrm{1D}}(\ell)}{\sigma_{v,\mathrm{cloud}}}=\left(\frac{% \ell}{2R_{\mathrm{cloud}}}\right)^{1/2},  (21) 
where \delta v_{\mathrm{1D}}(\ell) represents the onedimensional velocity dispersion at scale \ell, and \sigma_{v,\mathrm{cloud}} is the cloudscale onedimensional velocity dispersion. In terms of the virial parameter \alpha_{\mathrm{vir}}\equiv 5{\sigma_{v}}^{2}R_{\mathrm{cloud}}/(GM_{\mathrm{% cloud}}) with M_{\mathrm{cloud}}\equiv 4\pi\rho_{0}{R_{\mathrm{cloud}}}^{3}/3, and for the inflow Mach number {\cal M} comparable to \sigma_{v}/c_{s} of the whole cloud, the threedimensional velocity dispersion \delta v=\sqrt{3}\cdot\delta v_{\mathrm{1D}} at the scale of the simulation box would be
\delta v(L_{\mathrm{box}})=\sqrt{3}\left(\frac{\pi G\alpha_{\mathrm{vir}}}{15}% \right)^{1/4}{\cal M}^{1/2}{c_{s}}^{1/2}{\rho_{0}}^{1/4}{L_{\mathrm{box}}}^{1/% 2}.  (22) 
To emphasize the influence of the cloud magnetization instead of the perturbation field, our simulations are conducted with 10\% of the value \delta v(L_{\mathrm{box}}), or \delta v=0.14 km/s with \alpha_{\mathrm{vir}}=2. With larger \delta v(L_{\mathrm{box}}), simulations can still form cores, but because nonselfgravitating clumps can easily be destroyed by strong velocity perturbations and no core can form before the turbulent energy dissipates, it takes much longer, with corresponding higher computational expense.
3.2 Model Parameters
A schematic showing our model setup is shown in Figure 2. Our simulation box is 1 pc on each side and represents a region within a GMC where a largescale supersonic converging flow with velocity \mathbf{v}_{0} and \mathbf{v}_{0} (i.e. in the centerofmomentum frame) collides. The zdirection is the largescale inflow direction, and we adopt periodic boundary conditions in the x and ydirections. We initialize the background magnetic field in the cloud, \mathbf{B}_{0}, in the xz plane, with an angle \theta with respect to the convergent flow. For simplicity, we treat the gas as isothermal at temperature T=10 K, such that the sound speed is c_{s}=0.2 km/s. The neutral density within the cloud, \rho_{0}, is set to be uniform in the initial conditions and in the upstream converging flow.
It has been shown that ionizationrecombination equilibrium generally provides a good approximation to the ionization fraction within GMCs for the regime under investigation (CO12). Thus, the number density of ions in our model can be written as
n_{i}=\frac{\rho_{i}}{\mu_{i}}=10^{6}\chi_{i0}\left(\frac{\rho_{n}}{\mu_{n}}% \right)^{1/2},  (23) 
with
\chi_{i0}\equiv 10^{6}\times\sqrt{\frac{\zeta_{\mathrm{CR}}}{\alpha_{\mathrm{% gas}}}}  (24) 
determined by the cosmicray ionization rate (\zeta_{\mathrm{CR}}) and the gasphase recombination rate (\alpha_{\mathrm{gas}}). The ionization coefficient, \chi_{i0}, has values \sim 120 (McKee et al., 2010), and is the model parameter that controls ambipolar diffusion effects in our simulations, following CO12. We use typical values of the mean neutral and ion molecular weight \mu_{n} and \mu_{i} of 2.3m_{\mathrm{H}} and 30m_{\mathrm{H}}, respectively, which give the collision coefficient (see Equation (20c)) between neutrals and ions \alpha=3.7\times 10^{13} cm{}^{3}s{}^{1}g{}^{1}.
The physical parameters defining each model are \rho_{0}, v_{0}=\mathbf{v}_{0}, B_{0}=\mathbf{B}_{0}, \theta, and \chi_{i0}. We set the upstream neutral number density to be n_{0}=\rho_{0}/\mu_{n}=1000 cm{}^{3} in all simulations, consistent with typical mean molecular densities within GMCs^{4}^{4}4Note that the upstream neutral number density we adopted here is n_{0}=n_{\mathrm{neutral},0}\equiv n_{\mathrm{H_{2}}}+n_{\mathrm{He}}=0.6n_{% \mathrm{H}}=1.2n_{\mathrm{H_{2}}}, with GMC observations giving n_{\mathrm{H_{2}}}\sim 10^{2}10^{3} cm{}^{3}. Also note that \mu_{n}\equiv\rho_{n}/n_{n}=(\rho_{\mathrm{H_{2}}}+\rho_{\mathrm{He}})/(n_{% \mathrm{H_{2}}}+n_{\mathrm{He}})=(0.5n_{\mathrm{H}}\times 2m_{\mathrm{H}}+0.1n% _{\mathrm{H}}\times 4m_{\mathrm{H}})/(0.5n_{\mathrm{H}}+0.1n_{\mathrm{H}})=2.3% m_{\mathrm{H}}. (e.g. Larson, 1981; Williams et al., 2000; Bot et al., 2007; Bolatto et al., 2008). We choose the upstream B_{0}=10 \muG as typical of GMC values (Goodman et al., 1989; Crutcher et al., 1993; Heiles & Crutcher, 2005; Heiles & Troland, 2005) for all our simulations. To keep the total number of simulations practical, we set the largescale inflow Mach number to {\cal M}=10 for all models. Exploration of the dependence on Mach number of ambipolar diffusion and of core formation has been studied in previous simulations (CO12 and GO11, respectively). For our parameter survey, we choose \theta=5, 20, and 45 degrees to represent small (\theta<\theta_{\mathrm{crit}}), intermediate (\theta>\theta_{\mathrm{crit}}), and large (\theta\gg\theta_{\mathrm{crit}}) angles between the inflow velocity and cloud magnetic field. For each \theta, we conduct simulations with \chi_{i0}=3, 10, and ideal MHD to cover situations with strong, weak, and no ambipolar diffusion. We also run corresponding hydrodynamic simulations with same \rho_{0} and v_{0} for comparison.
A full list of models is contained in Table 1. Table 1 also lists the steadystate postshock properties, as described in Section 2.1. Solutions for all three types of shocks are listed for the \theta=5^{\circ} (A5) case. For the \theta=20^{\circ} and \theta=45^{\circ} cases, there is only one shock solution. Also included in Table 1 are the nominal values of critical mass and radius for spherically symmetric volumes to be selfgravitating under these steadystate postshock condition, as discussed in Section 2.2 (see Equations (11), (12), (14) and (15)). Both “thermal” and “magnetic” critical masses are listed. In most models, M_{\mathrm{mag,sph}}>M_{\mathrm{th,sph}} and M_{\mathrm{mag,sph}}\gg\mathrm{M}_{\odot}, indicating the postshock regions are dominated by magnetic support, and either ambipolar diffusion or anisotropic condensation would be needed to form lowmass supercritical cores, as discussed in Section 2.2. On the other hand, the quasihydrodynamic shock solution for models with \theta<\theta_{\mathrm{crit}} (i.e. A5 cases) has M_{\mathrm{mag,sph}}<M_{\mathrm{th,sph}}<\mathrm{M}_{\odot} downstream. If this shock solution could be sustained, then in principle lowmass supercritical cores could form by spherical condensation of postshock gas.
Model  model settings^{∧}^{∧}footnotemark:  steadystate postshock solutions  gravitational critical scales^{§}^{§}footnotemark:  
\theta  \chi_{i0}  B_{\perp}  n_{\mathrm{ps}}  B_{\perp}  B_{\mathrm{tot}}  M_{\mathrm{th,sph}}  R_{\mathrm{th,sph}}  M_{\mathrm{mag,sph}}  R_{\mathrm{mag,sph}}  
(deg)  (\muG)  (10^{4} cm{}^{3})  (\muG)  (\muG)  (M{}_{\odot})  (pc)  (M{}_{\odot})  (pc)  
HD^{¶}^{¶}footnotemark:        10.0      0.44  0.03     
A5X3  5  3  0.87  
A5X10  5  10  0.87  
A5ID^{*}^{*}footnotemark: ^{‡}^{‡}footnotemark:  5  ^{*}^{*}footnotemark:  0.87  1.51  55.3  56.2  1.14  0.07  11  0.15 
8.93^{†}^{†}footnotemark:  20.2^{†}^{†}footnotemark:  22.5^{†}^{†}footnotemark:  0.47^{†}^{†}footnotemark:  0.03^{†}^{†}footnotemark:  0.01^{†}^{†}footnotemark:  0.01^{†}^{†}footnotemark:  
2.79  51.8  52.7  0.84  0.05  2.6  0.07  
A20X3  20  3  3.42  
A20X10  20  10  3.42  
A20ID^{*}^{*}footnotemark:  20  ^{*}^{*}footnotemark:  3.42  0.96  56.0  56.7  1.43  0.08  28  0.23 
A45X3  45  3  7.07  
A45X10  45  10  7.07  
A45ID^{*}^{*}footnotemark:  45  ^{*}^{*}footnotemark:  7.07  0.69  57.9  58.3  1.68  0.10  59  0.33 

{}^{\wedge}In the model settings, \theta is the angle between inflow velocity and the magnetic field, and B_{\perp} is the upstream magnetic field perpendicular to the shock front.

{}^{\lx@paragraphsign}Hydrodynamics; no magnetic field, or \chi_{i0}=0.

{}^{\ast}Ideal MHD; neutrals and ions are perfectly coupled.

{}^{\ddagger}The A5 model satisfies \theta<\theta_{\mathrm{crit}} and has three shock solutions (see Section 2.1). We list all three. The postshock conditions in simulations may be a combination of these possible solutions.

{}^{\dagger}The quasihydrodynamic solution.
In order to collect sufficient statistical information on the core properties from simulations, we repeat each parameter set 6 times with different random realizations of the same perturbation power spectrum for the turbulence. The resolution is 256^{3} for all simulations such that \Delta x\approx 0.004 pc, or \sim 800 AU. We tested this setup with two times of this resolution (\Delta x\approx 0.002 pc), and the resulting dense structures are highly similar. Though the individual core properties vary around \pm 50\%, the median values (which are more important in our statistical study) only change within \pm 1030\%. Thus, our simulations with \Delta x\approx 0.004 pc are wellresolved for investigations of core properties.
3.3 Analysis of Core Properties
To measure the physical properties of the cores formed in our simulations, we apply the GRID corefinding method developed by GO11, which uses gravitational potential isosurfaces to identify cores. In this approach, the largest closed potential contour around a single local minimum of the gravitational potential defines the material eligible to be part of a core. We define the bound core region as all the material within the largest closed contour that has the sum of gravitational, magnetic, and thermal energy negative.^{5}^{5}5The gravitational, thermal, and magnetic energy density in each zone are u_{g}=\rho\Delta\Phi_{g}, u_{\mathrm{th}}=3nkT/2, and u_{B}=B^{2}/8\pi, respectively, where \Delta\Phi_{g} is the difference in gravitational potential relative to the largest closed contour, and n is the neutral number density defined as n=\rho/\mu_{n}. The selfgravitating core consists of all zones with u_{g}+u_{\mathrm{th}}+u_{B}<0. All of our cores are, by definition, selfgravitating.
The essential quantity to measure the significance of magnetic fields in selfgravitating cores is the ratio of mass to magnetic flux (Mestel & Spitzer, 1956; Mouschovias & Spitzer, 1976). From Gauss’s law the net flux of the magnetic field through a closed surface is always zero. As a result, to measure the magnetic flux within a core, we need firstly to define a crosssection of the core, and then measure the net magnetic flux through the surface of the core defined by this crosssection (which is the same as the flux through the crosssection itself).
To define the crosssection through a core, we use the plane perpendicular to the average magnetic field that also includes the minimum of the core’s gravitational potential. This choice ensures that we measure the magnetic flux through the part of the core with strongest gravity. After defining this plane, we separate the core into an upper half and a lower half, and measure the magnetic flux \Phi_{B} through one of the halves. In practice, we compute this by firstly finding all zones that contain at least one face which is on the core surface, and assign normal vectors \mathbf{{\hat{n}}} (pointing outwards) to those faces. From these, we select only those in the upper “hemisphere” of the core. After we have a complete set of those gridfaces that are on the upper half of the core surface, we sum up their \mathbf{B}\cdot\mathbf{{\hat{n}}} to get the net magnetic flux of the core. This method is tested in spherical and rectangular ‘cores’ with magnetic fields in arbitrary directions. Note that this method works best when the core is approximately spherical (without corners).
After we have the measurement of magnetic flux \Phi_{B}, we can calculate the masstoflux ratio of the core, M/\Phi_{B}. This determines whether the magnetic field can support a cloud against its own selfgravity. The critical value of M/\Phi_{B} differs with the geometry of the cloud, but the value varies only within \sim 10\% (e.g. Mouschovias & Spitzer, 1976; Nakano & Nakamura, 1978; Tomisaka et al., 1988, or see review in McKee & Ostriker 2007). We therefore choose the commonly used value (2\pi\sqrt{G})^{1} (e.g. Kudoh & Basu 2011; VázquezSemadeni et al. 2011; CO12) as a reference value, and define the normalized masstoflux ratio as \Gamma\equiv 2\pi\sqrt{G}\cdot M/\Phi_{B} (see Equation (16)). For a prestellar core with \Gamma>1, the gravitational force exceeds the magnetic support and the core is magnetically supercritical. A subcritical core has \Gamma<1 and is ineligible for wholesale collapse unless magnetic fields diffuse out.
4 Sample Evolution of Structure
Figure 3 shows typical evolution of column density and magnetic field^{6}^{6}6The magnetic field lines shown in left panels of Figure 3, 4, and 5 are contours of the absolute value of the magnetic vector potential \mathbf{\Psi} in the direction perpendicular to the plane plotted. By definition, \mathbf{B}=\nabla\times\mathbf{\Psi}, and therefore B_{x}=d\Psi_{z}/dy, B_{y}=d\Psi_{z}/dx. If we start with \Psi_{z}=0 in the lowerleft corner (x=y=0), we can compute \Psi_{z}(0,y)=\int_{0}^{y}B_{x}(0,y^{\prime})dy^{\prime}, and \Psi_{z}(x,y)=\Psi_{z}(0,y)\int_{0}^{x}B_{y}(x^{\prime},y)dx^{\prime}. After we have \Psi_{z} everywhere, we make contours to show the magnetic field structures, with fixed spacing so \delta\Psi=constant. in our numerical simulations. The simulations start with uniform density and constant magnetic field. When compressed by the supersonic converging flows, the magnetic fields perpendicular to the converging flows are amplified in the postshock dense region. Seeded by turbulent velocity perturbations, dense structures form within the compressed layer.
The postshock structure can be very different for different model parameters. Figure 4 and 5 provide examples with weak (small \theta and/or small \chi_{i0}) and strong (large \theta and/or large \chi_{i0}) magnetic effects in the shocked gas. The thickness of the postshock layer is very different for these two extreme cases. Especially at early time (0.3 Myr), structure is also different in these two cases, with stronger magnetic effects producing filaments perpendicular to the magnetic field. The timescale at which compressed layers become gravitationally unstable and start to form cores also differ. Note that in the cases with ambipolar diffusion (Figure 3 and 4), a highlycompressed layer forms in the center of the postshock region. Quantitatively, we measured the average density within the z=0.5~{}\mathrm{pc}\pm\Delta x layer at t=0.3~{}\mathrm{Myr} for model A5X3, and found this overdense layer has \overline{n}\approx 1.4\times 10^{5} \mathrm{cm}^{3}, which exceeds the steadyMHD shock jump condition predicted in Table 1 even for the quasihydrodynamic solution. This is a direct evidence of the existence of transient stage of ambipolar diffusion (CO12).
Table 2 lists the physical properties of the postshock layers measured at t=0.2 Myr as well as the corresponding values of the critical mass and size of a spherical region under these ambient conditions. Generally, models with upstream magnetic field almost parallel to the inflow (A5 models) have weaker postshock magnetic field than that for a fast shock (see Table 1) even with ideal MHD (A5ID), indicating that the quasihydrodynamic shock mode discussed in Section 2 plays a role. Also, models with stronger transient ambipolar diffusion effect (smaller \chi_{i0}) have higher density and weaker magnetic field in the postshock layer, and thus it would be easier to form selfgravitating cores promptly (small M_{\mathrm{th,sph}} and M_{\mathrm{mag,sph}} values).
The difference in postshock magnetic field among models with same upstream magnetic obliquity but various ionization levels can be explained by varying transient ambipolar diffusion. From Equation (61) in CO12, the timescale before the shock profile transitions to that of a steady Cshock is
t_{\mathrm{transient}}\approx\frac{2{r_{f}}^{1/2}}{\alpha\rho_{i,0}}=0.34~{}% \mathrm{Myr}\left(\frac{r_{f}}{10}\right)^{1/2}\left(\frac{\chi_{i0}}{10}% \right)^{1}\left(\frac{n_{0}}{1000~{}\mathrm{cm}^{3}}\right)^{1/2}.  (25) 
Therefore, while the latetime (ideal MHD) value of r_{f} is the same for models with same \theta value, it will take 3.33 times longer for the X3 models to reach steadystate postshock values than the X10 models. Correspondingly, the compression rate of the magnetic field in X3 models is 0.3 times slower than in X10 models, and thus the magnetic field within the postshock layer is weaker in X3 models than in X10 or ideal MHD models at a given time. This tendency is clearly shown in Table 2; note that since r_{f} might be larger because of the transient ambipolar diffusion effect, the difference in postshock magnetic field is further enhanced (smaller \chi_{i0} causes higher r_{f}, resulting in longer t_{\mathrm{transient}} and weaker B_{\mathrm{ps}}).
Model  postshock properties^{§}^{§}footnotemark:  gravitational critical scales  
\overline{n}_{\mathrm{ps}}  \overline{B}_{\mathrm{ps}}  \overline{\beta}_{\mathrm{ps}}  M_{\mathrm{th,sph}}  R_{\mathrm{th,sph}}  M_{\mathrm{mag,sph}}  R_{\mathrm{mag,sph}}  
(10^{4} cm{}^{3})  (\muG)  (M{}_{\odot})  (pc)  (M{}_{\odot})  (pc)  
HD  5.5      0.60  0.04     
A5X3  5.3  26  3.0  0.61  0.04  0.09  0.02 
A5X10  5.3  40  1.3  0.60  0.04  0.31  0.03 
A5ID  2.4  47  0.43  0.90  0.05  2.4  0.08 
A20X3  5.3  45  1.02  0.61  0.04  0.45  0.03 
A20X10  3.6  68  0.30  0.74  0.04  3.4  0.07 
A20ID  1.4  78  0.09  1.2  0.07  33  0.22 
A45X3  4.2  60  0.45  0.69  0.04  1.7  0.06 
A45X10  2.7  86  0.14  0.85  0.05  12  0.13 
A45ID  0.91  96  0.04  1.5  0.09  151  0.41 

{}^{\lx@sectionsign}Postshock properties are measured at t=0.2 Myr in each model, averaged over the whole postshock layer. The timescale is chosen so the downstream properties are measured before the postshock layer becomes strongly selfgravitating.
Figure 6 compares the density structures formed under different physical conditions, at the timescale when n_{\mathrm{max}}\geq 10^{7} cm{}^{3} in each simulation. With low ionization (strong ambipolar diffusion), the clumps are relatively more isolated and randomly distributed, following the initial perturbation pattern. Models with high ionization (weak or no ambipolar diffusion) show wellordered largescale filament structures. Structures are also at larger scales for models with larger magnetic field parallel to the shock front (large \theta). The filaments are around 0.05 pc wide, consistent with the observed characteristic width of filaments (\sim 0.1 pc, Arzoumanian et al., 2011; or see review in André et al., 2013). Note that the filaments are not necessary perpendicular to the magnetic field as indicated in Inoue & Fukui (2013) because the initial velocity field in our simulations is not homogeneous.
In addition, models with moderately strong magnetization have a network of small subfilaments aligned parallel to the magnetic field (A20X10, A20ID, A45X10, and A45ID models in Figure 6). These features are very similar to the striations identified in {}^{12}CO emission map of the Taurus molecular cloud (Goldsmith et al., 2008), subsequently observed in other clouds (Sugitani et al., 2011; Hennemann et al., 2012; Palmeirim et al., 2013; or see review in André et al., 2013). This filament pattern is likely due to the anisotropy of turbulence at small scales in a magnetized medium (Goldreich & Sridhar, 1995), which tends to have more power for wavenumbers \hat{k}\perp\mathbf{B}. This leads to the formation of threads/striations/subfilaments with small separations aligned parallel to the magnetic field in molecular clouds if the magnetic field is sufficiently strong. Vestuto et al. (2003) and Heyer et al. (2008) found that in order to have significant turbulent anisotropy, the plasma \beta must satisfy \beta\lesssim 0.2, which agrees with our results for when these striations are seen (see \overline{\beta}_{\mathrm{ps}} values listed in Table 2).
5 Survey of Core Properties
We define the timescale used in Figure 6 (at which n_{\mathrm{max}}\geq 10^{7} cm{}^{3}) as the moment t_{\mathrm{collapse}} when the most evolved core starts to collapse, and measure the physical properties of all cores formed at this time. We identified hundreds of gravitationally bound cores from our 60 simulations (6 runs for each parameter set), with examples illustrated in Figure 7. The simulation results are summarized in Table 3, including the following core properties: mean density \overline{n}, size L, mass M, mean magnetic field \overline{B}, and normalized masstoflux ratio \Gamma. To ensure the measured core properties are only for resolved structures, we omit cores with less than 27 zones, or L_{\mathrm{core}} smaller than \sim 0.015 pc. Table 3 also shows for each parameter set the mean value of time t_{\mathrm{collapse}} (at which the core properties are measured). These cores have masses, sizes, and masstoflux ratios similar to observed values (e.g. Falgarone et al., 2008; Troland & Crutcher, 2008; Rathborne et al., 2009; Kirk et al., 2013).
Our results show that lowmass supercritical cores form at t<1 Myr in all models: with converging velocity either nearly aligned with the magnetic field (small \theta) or highly oblique (large \theta), and for all levels of ambipolar diffusion. We also calculated the core formation efficiency (CFE) from our simulations:
\mathrm{CFE}\equiv\frac{\mathrm{mass\ in\ cores}}{\mathrm{mass\ of\ the\ % shocked\ layer}}=\frac{\sum\limits_{i}M_{\mathrm{core},i}}{2\rho_{0}v_{0}t_{% \mathrm{collapse}}\cdot L_{x}L_{y}}\approx 3.1\%.  (26) 
This is similar to the observed star formation efficiency (SFE), which is around 110\% (e.g Myers et al., 1986; Evans et al., 2009; Lada et al., 2010). Note that, though the core formation timescale is slightly different from model to model (see Figure 6), the CFE does not vary significantly between models; the variance in CFE among all models is only \sim 10\%.
Model  # Cores  CFE^{¶}^{¶}footnotemark:  t_{\mathrm{collapse}}^{§}^{§}footnotemark:  \overline{n}_{\mathrm{core}}  L_{\mathrm{core}}^{‡}^{‡}footnotemark:  M_{\mathrm{core}}  \overline{B}_{\mathrm{core}}  \Gamma_{\mathrm{core}} 

Identified^{⋆}^{⋆}footnotemark:  (%)  (Myr)  (10^{5} cm{}^{3})  (pc)  (M{}_{\odot})  (\muG)  (normalized)  
HD  32  3.1  0.56  5.8  0.036  0.75     
A5X3  40  3.1  0.58  5.5  0.032  0.63  42  4.4 
A5X10  49  3.7  0.61  6.6  0.031  0.65  64  3.7 
A5ID  51  3.9  0.54  5.6  0.030  0.58  67  2.6 
A20X3  34  3.0  0.59  5.6  0.032  0.72  60  3.9 
A20X10  54  3.1  0.60  9.7  0.025  0.47  79  3.3 
A20ID  36  3.3  0.62  9.5  0.031  0.78  90  2.7 
A45X3  42  3.7  0.60  8.9  0.031  0.73  83  3.7 
A45X10  38  3.2  0.60  9.2  0.030  0.70  82  3.0 
A45ID  21  1.9  0.90  11  0.035  1.12  137  2.1 

{}^{\star}We only consider gravitationally bound cores with E_{\mathrm{grav}}+E_{\mathrm{thermal}}+E_{\mathrm{B}}<0.

{}^{\lx@paragraphsign}CFE is the ratio of the total mass in cores to the total mass in the shocked layer at t_{\mathrm{collapse}}.

{}^{\dagger}Columns (5)(9) are averaged over all cores for each parameter set (6 simulation runs).

{}^{\lx@sectionsign}Collapse is defined as the time when n_{\mathrm{max}}=10^{7} cm{}^{3} in each simulation. The t_{\mathrm{collapse}} shown here is the mean value over all 6 runs for each parameter set.

{}^{\ddagger}L_{\mathrm{core}} is calculated from the total number of zones N within a core, for an equivalent spherical volume: L_{\mathrm{core}}=2\times(3N/(4\pi))^{1/3}\Delta x, where \Delta x=1/256 pc is the grid size.
5.1 Mass and Size
Figure 8 shows the distribution of mass and size of cores for all model parameters. The masses range between 0.04 to 2.5 M_{\odot}, with peak around \sim 0.6 M_{\odot}; the core sizes are between 0.0150.07 pc, with peak around \sim 0.03 pc. These are consistent with observational results (e.g. Motte et al., 2001; Ikeda et al., 2009; Rathborne et al., 2009; Kirk et al., 2013). Also, the distribution of the core mass shows a similar shape to the observed core mass function (CMF) (e.g. Simpson et al., 2008; Rathborne et al., 2009; Könyves et al., 2010). Interestingly, the peak in the distribution is close to value given by Equation (7) from GO11:
M_{\mathrm{BE,\ ps}}=1.2\frac{{c_{s}}^{4}}{\sqrt{G^{3}P_{\mathrm{ps}}}}=1.2% \frac{{c_{s}}^{3}}{\sqrt{G^{3}\rho_{0}}}\frac{1}{{\cal M}}\rightarrow 0.45\ % \mathrm{M}_{\odot}.  (27) 
This mass is characteristic of what is expected for collapse of a thermallysupported core that is confined by an ambient medium with pressure equal to the postshock value^{7}^{7}7The postshock total pressure (whether for an unmagnetized medium, as considered by GO11, or for a magnetized medium as considered here) will be comparable to the momentum flux of the converging flow, P_{\mathrm{ps}}\approx\rho_{0}{v_{0}}^{2}=\rho_{0}c_{s}^{2}{\cal M}^{2}. , where the numerical figure uses values for the mean cloud density and largescale Mach number equal to those of the converging flow in our simulations, n_{0}=1000 cm{}^{3} and {\cal M}=10. Correspondingly, since the critical ratio of mass and radius is M_{\mathrm{BE}}/R_{\mathrm{BE}}=2.4{c_{s}}^{2}/G (Bonnor, 1956), the characteristic size expected for a collapsing core formed in a postshock region when the Mach number of the largescale converging flow is {\cal M} and the mean cloud density is \rho_{0}, is
L_{\mathrm{BE}}=2R_{\mathrm{BE}}=\frac{c_{s}}{\sqrt{G\rho_{0}}}\frac{1}{{\cal M% }}\rightarrow 0.04\ \mathrm{pc}.  (28) 
This is again comparable to the peak value of the core size distribution in Figure 8.
We also separately explore the dependence of core mass, size, magnetic field strength, and masstoflux ratio on model parameters, as shown in Figure 9. Our results show that the core mass is relatively insensitive to both the ionization (i.e. ambipolar diffusion effect) and obliquity of the upstream magnetic field (Figure 9, top left). The median masses are within a factor 2.4 of the mean of the whole distribution, 0.68 M{}_{\odot}, or a factor 2 of the median of all core masses (0.47 M{}_{\odot}). Similarly, median core sizes vary only between 0.022 pc and 0.034 pc for the various parameter sets, with a median of 0.03 pc. Note that we chose to compare median values between different parameter sets in Figure 9 instead of mean values used in Table 3, because an average can be affected by any single value being high or low relative to the other samples. The median value, on the other hand, represents the central tendency better, and with the \pm 25\% values we can have a better understanding of the sample distribution.
We note in particular that for the \theta=20^{\circ} and \theta=45^{\circ} ideal MHD cases, the masses in Figure 8 and 9 are more than an order of magnitude lower than the limits for a spherical region at postshock conditions to be magnetically supercritical, as listed in Table 1 and 2. This implies that the lowmass bound cores found in the simulations did not form isotropically. We discuss this further in Section 6.
To further investigate the relationship between core masses and sizes, we binned the data set by \log L_{\mathrm{core}} and calculate the average core mass and mean density for different model parameters. The results are shown in Figure 10, where we chose four models with different magnetization and ionization levels to compare: HD (hydrodynamics; no magnetization), A5X3 (low ionization, weak upstream magnetic field parallel to the shock), A20X10 (moderate ionization and magnetic field), and A45ID (ideal MHD, strong magnetic field). In both the masssize and densitysize plots, the differences among models are small, and all four curves have similar shape. In fact, from all resolved cores identified in our simulations, we found a powerlaw relationship between the core mass and size, M\propto L^{k}, with bestfitted value k=2.28. This is consistent with many coreproperty surveys towards different molecular clouds (e.g. Elmegreen & Falgarone, 1996; Curtis & Richer, 2010; RomanDuval et al., 2010; Kirk et al., 2013), in which k=1.22.4 with various molecule tracers (for more details, see Figure 7 and corresponding discussions in Kirk et al., 2013).
5.2 Magnetization
Figure 11 shows the distribution of core masstoflux ratio, a roughly normal distribution with peak at \Gamma\sim 3. This range of \Gamma is quite similar to observational results (\Gamma\sim 14; Falgarone et al., 2008; Troland & Crutcher, 2008). In addition, the colorcoded histogram in Figure 11 shows how the masstoflux ratio depends on magnetization: the highend region (\Gamma\gtrsim 5) is comprised of bluegreen pieces (which represent models with lower ionization), while the lowend tail is mostly red and orange (highly ionized models). Note that essentially all of the cores in our simulations are magnetically supercritical (\Gamma>1), which is selfconsistent with our corefinding criterion (gravitationally bound; E_{g}+E_{\mathrm{th}}+E_{B}<0).
The tendency of models with lower ionization to form cores with higher masstoflux ratio is very clearly seen in Figure 9 (bottom right). The median value of the core masstoflux ratio also decreases with increasing \theta as the value of the upstream B_{x}=B_{0}\sin\theta increases. Also from Figure 9 (top right), the average core magnetic fields show a similar tendency as in postshock magnetic field (see Table 2), which decrease at lower ionization fractions for models with same preshock magnetic field structure (same \theta). The larger and more systematic variation of \overline{B} than M with model parameters suggests that the core masstoflux ratio is not decided by the core mass, but by the core magnetic field. This is also shown in Figure 12, where we binned the data by M_{\mathrm{core}} and calculated the average core masstoflux ratio in each bin for different models. For cores with similar mass, the masstoflux ratios of cores formed in environments with low ionization and magnetization are much higher than those with stronger and better coupled magnetic fields.
The fact that the median value of magnetic field strength within the core depends on preshock magnetic obliquity and ionization is consistent with our discussions in Section 4 that magnetic fields are lower in shocked regions that have longer transient timescales. Since lower ionization fraction leads to stronger ambipolar diffusion and a longer transient stage^{8}^{8}8From CO12 and Equation (25), the predicted duration of the transient stage is 0.31.4 Myr for \chi_{i0}=3 to 10 and our range of model parameters, assuming r_{f}=r_{f,\ \mathrm{ideal\ MHD}}., it is logical to expect the cores formed in weaklyionized clouds have lower magnetic field than those formed with higher ionization fraction (or stronglycoupled ions and neutrals).
In addition, Figure 9 (top right) shows that cores formed in models with small \theta (A5 cases) have weaker magnetic fields inside even with higher ionization fraction or ideal MHD, which indicates that the magnetic field is less compressed by the shock when the inflow is almost parallel to the upstream magnetic field. This is consistent with the discussion in Section 2: when \theta<\theta_{\mathrm{crit}}, the MHD shock becomes a composite compounded of the regular (fast) mode and the quasihydrodynamic mode, which has relatively small magnetic field compression ratio. Thus, the magnetic obliquity relative to the shock has a similar effect to the cloud ionization fraction in determining field strengths in prestellar cores.
Based on the results shown in Section 5.1 and 5.2, we conclude that magnetic effects do not appear to control core mass and size. This suggests that once a core becomes strongly gravitationally bound, magnetic effects are relatively unimportant to its internal structure. However, the formation process of gravitationally bound cores is highly dependent on magnetic effects. As noted above, Figure 6 shows clear differences in the largescale structures from which cores condense; we discuss core formation further in Section 6. Also, cores are born with either lower or higher magnetic field, depending on the magnetic field structure and the ambipolar diffusion in their surrounding environment.
6 Anisotropic Core Formation
6.1 Examples of Simulation Evolution
The fact that gravitationally supercritical lowmass cores (with M\ll M_{\mathrm{mag,sph}}) can form in the highly magnetized postshock medium even without ambipolar diffusion suggests that these cores did not contract isotropically. Figure 13 provides a closeup view of the core forming process in highly magnetized environment with ideal MHD, from model A20ID. At stages earlier than shown, the directions of the perturbed magnetic field and gas velocity are determined randomly by the local turbulence. The magnetic field is compressed by the shock (similar to Figures 35), such that in the postshock layer it is nearly parallel to the shock front (along \hat{\mathbf{x}}). When the magnetic field strength increases, the velocity is forced to become increasingly aligned parallel to the flow, as shown in Figure 13. By the time t=0.65 Myr, a very dense core has formed by gathering material preferentially along the magnetic field lines. After the core becomes sufficiently massive, its selfgravity will distort the magnetic field and drag material inward even in the direction perpendicular to the magnetic field lines (t=0.77 Myr, Figure 13). This collapsing process with a preferential direction is similar to the postshock focusing flows found in previous studies (e.g. Inoue & Fukui, 2013; Vaidya et al., 2013) where the gas is confined by the strong magnetic field in the shockcompressed region.
In fact, anisotropic condensation is key to core formation not only with ideal MHD, but for all cases. Figure 14 shows spacetime diagrams of all three velocity components (v_{x}, v_{y}, v_{z}) around collapsing cores in different parameter sets. Though models with stronger postshock magnetic fields (larger \theta, larger \chi_{i0}, or ideal MHD) have more dominant v_{x} (bluer/redder in the colormap), there is a general preference to condense preferentially along the magnetic field lines (in the xdirection) among all models, regardless of upstream magnetic obliquity and the ambipolar diffusion level. Figure 14 also shows that there are flows perpendicular to the mean magnetic field (along \mathbf{\hat{y}} or \mathbf{\hat{z}}) in the final \sim 0.1 Myr of the simulations, indicating the stage of core collapse. The prominent gas movement along \mathbf{\hat{x}} long before each core starts to collapse shows that cores acquire masses anisotropically along the magnetic field lines, and thus anisotropic condensation is important for all models.
6.2 Theoretical Scalings
We have shown in Section 2.2 quantitatively that isotropic formation of lowmass supercritical cores is not possible for oblique shocks with ideal MHD, because the minimum mass for a spherical volume to become magnetically supercritical is \geq 10 M{}_{\odot} (see Equation (14) and the M_{\mathrm{mag,sph}} entries in Table 1 and 2 for cases A20ID and A45ID), much larger than the typical core mass (\sim 1 M{}_{\odot}). However, nonspherical regions may have smaller critical mass. Consider, for example, a core that originates as a prolate spheroid with semimajor axis R_{1} along the magnetic field and semiminor axis R_{2} perpendicular. The masstoflux ratio is then
\frac{M}{\Phi_{B}}\bigg{}_{\mathrm{prolate}}=\frac{4\pi R_{1}{R_{2}}^{2}\rho/% 3}{\pi{R_{2}}^{2}B}=\frac{4}{3}\frac{R_{1}\rho}{B}.  (29) 
The critical value for R_{1} would be the same as given in Equation (15), but the critical mass would be lower than that in Equation (14) by a factor (R_{2}/R_{1})^{2}. For R_{1}/R_{2}\sim 34, the critical mass will be similar to that found in the simulations.
Here we provide a physical picture for core formation via initial flow along the magnetic field, as illustrated in Figure 13 and 14. Consider a postshock layer with density \rho_{\mathrm{ps}} and magnetic field B_{\mathrm{ps}}. For a cylinder with length L along the magnetic field and radius R, the normalized masstoflux ratio is
\Gamma_{\mathrm{cyl}}=\frac{\pi R^{2}L\rho_{\mathrm{ps}}}{\pi R^{2}B_{\mathrm{% ps}}}\cdot 2\pi\sqrt{G},  (30) 
and the critical length along the magnetic field for it to be supercritical is
L_{\mathrm{mag,cyl}}=\frac{B_{\mathrm{ps}}}{\rho_{\mathrm{ps}}}\frac{1}{2\pi% \sqrt{G}}  (31) 
(note that up to a factor 3/4, this is the same as Equation (15)). The critical mass M_{\mathrm{mag,cyl}}=\pi R^{2}L_{\mathrm{mag,cyl}}\rho_{\mathrm{ps}} can then be written as
M_{\mathrm{mag,cyl}}=\frac{R^{2}B_{\mathrm{ps}}}{2\sqrt{G}}=1.2~{}\mathrm{M}_{% \odot}\left(\frac{R}{0.05~{}\mathrm{pc}}\right)^{2}\left(\frac{B_{\mathrm{ps}}% }{50~{}\mu\mathrm{G}}\right).  (32) 
This cylinder is gravitationally stable to transverse contraction unless L\lesssim 2R (Mestel & Spitzer, 1956). However, contraction along the length of the cylinder is unimpeded by the magnetic field, and will be able to overcome pressure forces provided L_{\mathrm{mag,cyl}} exceeds the thermal Jeans length, which is true in general for oblique shocks in typical conditions under consideration. The longitudinal contraction will produce an approximately isotropic core of radius R when the density has increased by a factor
\frac{\rho^{\prime}}{\rho_{\mathrm{ps}}}=\frac{L_{\mathrm{mag,cyl}}}{2R},  (33) 
and at this point transverse contraction would no longer be magnetically impeded. For the core to have sufficient selfgravity to overcome thermal pressure at this point, the radius would have to be comparable to R_{\mathrm{th,sph}} (see Equation (12)):
R\sim R_{\mathrm{th,sph}}=2.3\frac{c_{s}}{\sqrt{4\pi G\rho^{\prime}}}.  (34) 
Combining Equations (31), (33), and (34) yields
\rho^{\prime}=0.19\frac{{B_{\mathrm{ps}}}^{2}}{4\pi{c_{s}}^{2}},  (35) 
and
R=5.3\frac{{c_{s}}^{2}}{\sqrt{G}B_{\mathrm{ps}}}=0.05~{}\mathrm{pc}\left(\frac% {B_{\mathrm{ps}}}{50~{}\mu\mathrm{G}}\right)^{1}\left(\frac{T}{10~{}\mathrm{K% }}\right).  (36) 
Substituting Equation (36) in Equation (32), the minimum mass that will be both magnetically and thermally supercritical, allowing for anisotropic condensation along \mathbf{B}, will be
M_{\mathrm{crit}}=14\frac{{c_{s}}^{4}}{G^{3/2}B_{\mathrm{ps}}}=1.3\ \mathrm{M}% _{\odot}\left(\frac{B_{\mathrm{ps}}}{50\mu\mathrm{G}}\right)^{1}\left(\frac{T% }{10~{}\mathrm{K}}\right)^{2}.  (37) 
Thus, anisotropic contraction can lead to lowmass supercritical cores, with values comparable to those formed in our simulations.^{9}^{9}9Note that up to factors of order unity, Equations (35) to (37) can equivalently be obtained by taking B=B_{\mathrm{ps}} and requiring that the density \rho\rightarrow\rho^{\prime} in Equations (11)(12) and (14)(15) is such that R_{\mathrm{th,sph}}\sim R_{\mathrm{mag,sph}} and M_{\mathrm{th,sph}}\sim M_{\mathrm{mag,sph}}.
In addition, anisotropic condensation also helps explain why the core masses are quite similar for HD and MHD models, and independent of the angle between upstream magnetic field and converging flow. Note that Equation (37) only depends on the postshock magnetic field strength. For a magnetized shock, the postshock magnetic pressure must balance the preshock momentum flux: {B_{\mathrm{ps}}}^{2}/8\pi\sim\rho_{0}{v_{0}}^{2}. Therefore, Equation (37) can be expressed as
M_{\mathrm{crit}}=2.8\frac{{c_{s}}^{4}}{\sqrt{G^{3}\rho_{0}{v_{0}}^{2}}}=2.1~{% }\mathrm{M}_{\odot}\left(\frac{n_{0}}{1000~{}\mathrm{cm}^{3}}\right)^{1/2}% \left(\frac{v_{0}}{1~{}\mathrm{km/s}}\right)^{1}\left(\frac{T}{10~{}\mathrm{K% }}\right)^{2}.  (38) 
This is equivalent to Equation (24) of GO11 with \psi=2.8. GO11 also pointed out that \rho_{0}{v_{0}}^{2} will be proportional to G{\Sigma_{\mathrm{GMC}}}^{2} for a gravitationallybound turbulencesupported GMC. Thus, using Equation (28) of GO11 in a cloud with virial parameter \alpha_{\mathrm{vir}}, Equation (38) would become
M_{\mathrm{crit}}=2.8~{}\mathrm{M}_{\odot}\left(\frac{T}{10~{}\mathrm{K}}% \right)^{2}\left(\frac{\Sigma_{\mathrm{GMC}}}{100~{}\mathrm{M}_{\odot}~{}% \mathrm{pc}^{2}}\right)^{1}{\alpha_{\mathrm{vir}}}^{1/2}.  (39) 
Equations (38) and (39) suggest that M_{\mathrm{crit}} is not just independent of magnetic field direction upstream, it is also independent of magnetic field strength upstream. That is, when cores form in postshock regions (assuming the GMC is magnetically supercritical at large scales), the critical mass is determined by the dynamical pressure in the cloud, independent of the cloud’s magnetization. The models studied here all have the same dynamical pressure \rho_{0}{v_{0}}^{2}, and same upstream B_{0}. It will be very interesting to test whether for varying B_{0} the core masses remain the same, and whether the scaling proposed in Equation (38) holds for varying total dynamic pressure.
7 Summary
In this work, we have used numerical simulations to study core formation in magnetized, highly dynamic environments, including the effect of ambipolar diffusion. Our simulations are fully threedimensional, including a largescale convergent flow, local turbulence, and selfgravity, and allow for varying ambipolar diffusion levels (parameterized by the ionization fraction coefficient \chi_{i0}) and shock obliquity (parameterized by the angle \theta between the converging inflow and the global magnetic field). Filaments and then cores form in postshock dense layers, with dense structures very similar to those found in observations.
In all of our models (with or without ambipolar diffusion), magnetically supercritical cores form with physical properties similar to those found in observations. However, our parameter survey suggests that the transient ambipolar diffusion timescale and quasihydrodynamic shocks are crucial in setting the magnetization of cores formed in postshock regions. In addition, we demonstrate and quantitatively explain how lowmass supercritical cores form in stronglymagnetized regions, via anisotropic condensation along the magnetic field.
Our main conclusions are as follows:

Under typical GMC conditions, isotropic formation of lowmass supercritical cores is forbidden under ideal MHD by the relatively strong magnetic support (Equation (14)). This is true even downstream from strong MHD shocks where gas density is enhanced, because the magnetic field is compressed as well. In fact, for a spherical volume of given mass, the masstoflux ratio is generally larger for preshock conditions than postshock conditions (Equation (19); except for the special case described in #2 below). For typical conditions, the minimum postshock critical mass for a spherical volume exceeds 10 M{}_{\odot} when ideal MHD applies (Table 1, 2). This suggests that either transient ambipolar diffusion in shocks must be taken into consideration, or that core formation is not spherically symmetric.

When the incoming flows are almost parallel to the background magnetic field, MHD shocks will have compound postshock conditions, including the regular fast mode (Shu, 1992) and the quasihydrodynamic mode in which gas is compressed more strongly (Figure 1). This happens when the angle \theta between the inflow and the magnetic field is smaller than a critical value, \theta_{\mathrm{crit}} (Equation (8)). For small \theta, the postshock layer will have relatively high gas density and weak magnetic field compared to fastmode MHD shocks (Table 2).

Our threedimensional simulations demonstrate the effect of transient ambipolar diffusion, as earlier identified and explained by CO12. During the earliest stage of shock formation (t\lesssim 0.3 Myr), a thin but extremely dense layer appears in the middle of the shocked region in models with ambipolar diffusion (Figure 3 and 4), just like the central dense peak in the onedimensional shocks analyzed by CO12. Consequently, postshock densities are generally higher in models with lower ionizations (smaller \chi_{i0}; see Table 2), which correspond to stronger ambipolar diffusion as predicted in CO12.

The ionization fraction is the main parameter controlling the transient ambipolar diffusion timescale needed for the gas to reach steady postshock conditions (t_{\mathrm{transient}}). Models with smaller \chi_{i0} have longer transient timescales (Equation (25)), indicating lower growth rate of the postshock magnetic field and more weakly magnetized postshock layers (Table 2). Therefore, transient ambipolar diffusion is crucial in reducing the magnetic support in the postshock regions (see M_{\mathrm{mag,sph}} and R_{\mathrm{mag,sph}} in Table 2).

The filament network in more strongly magnetized postshock cases is similar to those found in observations: in addition to largescale main filaments, there are many thinner, lessprominent subfilaments parallel to the magnetic field (Goldsmith et al., 2008; Sugitani et al., 2011; Hennemann et al., 2012; Palmeirim et al., 2013; André et al., 2013). Dense cores form within the largescale main filaments for all models.

In our simulations, magnetically supercritical cores are able to form in the shockcompressed dense layers in all models, and the first collapse occurs at t\lesssim 0.6 Myr in most cases. Cores formed in our simulations have masses \sim 0.042.5 M{}_{\odot} and sizes \sim 0.0150.07 pc (Table 3 and Figure 8), similar to the values obtained in observations (e.g. Motte et al., 2001; Ikeda et al., 2009; Rathborne et al., 2009; Kirk et al., 2013). The medians from the distributions are 0.47 M{}_{\odot} and 0.03 pc. The masssize relationship derived from our cores, M\propto L^{2.3}, also agrees with observations (e.g. Elmegreen & Falgarone, 1996; Curtis & Richer, 2010; RomanDuval et al., 2010; Kirk et al., 2013).

Our results show that the core mass and size are relatively independent of both the ambipolar diffusion and the upstream magnetic obliquity (Figure 9). Hydrodynamic and ideal MHD models also have very similar core masses and sizes. The core masses for ideal MHD cases with oblique shocks are more than an order of magnitude lower than the magnetic critical mass for a spherical region in the postshock environment. Thus, simple estimates of the form in Equation (14) should not be used in predicting magnetically supercritical core masses from ambient environmental conditions in a GMC.

The magnetic field of cores follows the same trends as the postshock magnetization, in terms of variation with the upstream magnetic obliquity and ionization (Table 2, 3). This indicates that further ambipolar diffusion is limited during the core building phase, and instead cores form by anisotropic selfgravitating contraction as described in Section 6. The masstoflux ratio in cores secularly increases with decreasing ionization (Figure 9), ranging from \Gamma\sim 0.5 to 7.5 (Figure 11). From all models combined, the median masstoflux ratio within cores is \Gamma\sim 3 (Figure 11), agreeing with the observed range of \Gamma (\Gamma\sim 14; Falgarone et al., 2008; Troland & Crutcher, 2008).

Anisotropic selfgravitating condensation is likely the dominant mechanism for supercritical core formation in magnetized environments, regardless the magnetization strength and ionization fraction. Figures 13 and 14 clearly show how gas preferentially flows along the magnetic field lines in all models, creating dense cores that are both magnetically and thermally supercritical. The theoretical analysis of Section 6.2 shows that the characteristic mass expected from anisotropic contraction (Equation (37)) is similar to the median core mass obtained from our simulations (Figure 8). For anisotropic core formation in a postshock region, the critical mass is expected to depend only on the momentum flux entering the shock. We believe this explains why core masses in our simulations are similar regardless of the ionization level, whether the converging flow is nearly parallel to or highly oblique to the upstream magnetic field, or indeed whether the medium is even magnetized at all.
References
 Allen et al. (2003) Allen, A., Li, Z.Y., & Shu, F. H. 2003, ApJ, 599, 363
 André et al. (2009) André, P., Basu, S., & Inutsuka, S. 2009, Structure Formation in Astrophysics, 254
 André et al. (2013) André, P., Di Francesco, J., WardThompson, D., et al. 2013, arXiv:1312.6232
 Arzoumanian et al. (2011) Arzoumanian, D., André, P., Didelon, P., et al. 2011, A&A, 529, L6
 Bai & Stone (2011) Bai, X.N., & Stone, J. M. 2011, ApJ, 736, 144
 BallesterosParedes et al. (2007) BallesterosParedes, J., Klessen, R. S., Mac Low, M.M., & VazquezSemadeni, E. 2007, Protostars and Planets V, 63
 Basu & Ciolek (2004) Basu, S., & Ciolek, G. E. 2004, ApJ, 607, L39
 Basu et al. (2009a) Basu, S., Ciolek, G. E., & Wurster, J. 2009b, New A, 14, 221
 Basu et al. (2009b) Basu, S., Ciolek, G. E., Dapp, W. B., & Wurster, J. 2009a, New A, 14, 483
 Bolatto et al. (2008) Bolatto, A. D., Leroy, A. K., Rosolowsky, E., Walter, F., & Blitz, L. 2008, ApJ, 686, 948
 Bonnor (1956) Bonnor, W. B. 1956, MNRAS, 116, 351
 Boss (1997) Boss, A. P. 1997, ApJ, 483, 309
 Boss (2005) Boss, A. P. 2005, ApJ, 622, 393
 Bot et al. (2007) Bot, C., Boulanger, F., Rubio, M., & Rantakyro, F. 2007, A&A, 471, 103
 Chen & Ostriker (2012) Chen, C.Y., & Ostriker, E. C. 2012, ApJ, 744, 124
 Choi et al. (2009) Choi, E., Kim, J., & Wiita, P. J. 2009, ApJS, 181, 413
 Ciardi & Hennebelle (2010) Ciardi, A., & Hennebelle, P. 2010, MNRAS, 409, L39
 Ciolek & Basu (2001) Ciolek, G. E., & Basu, S. 2001, ApJ, 547, 272
 Ciolek & Basu (2006) Ciolek, G. E., & Basu, S. 2006, ApJ, 652, 442
 Crutcher et al. (1993) Crutcher, R. M., Troland, T. H., Goodman, A. A., et al. 1993, ApJ, 407, 175
 Crutcher (2012) Crutcher, R. M. 2012, ARA&A, 50, 29
 Curtis & Richer (2010) Curtis, E. I., & Richer, J. S. 2010, MNRAS, 402, 603
 Dapp et al. (2012) Dapp, W. B., Basu, S., & Kunz, M. W. 2012, A&A, 541, A35
 Delmont & Keppens (2011) Delmont, P., & Keppens, R. 2011, Journal of Plasma Physics, 77, 207
 Draine & McKee (1993) Draine, B. T., & McKee, C. F. 1993, ARA&A, 31, 373
 Elmegreen & Falgarone (1996) Elmegreen, B. G., & Falgarone, E. 1996, ApJ, 471, 816
 Evans et al. (2009) Evans, N. J., II, Dunham, M. M., Jørgensen, J. K., et al. 2009, ApJS, 181, 321
 Falgarone et al. (2008) Falgarone, E., Troland, T. H., Crutcher, R. M., & Paubert, G. 2008, A&A, 487, 247
 Fatuzzo & Adams (2002) Fatuzzo, M., & Adams, F. C. 2002, ApJ, 570, 210
 Goldreich & Sridhar (1995) Goldreich, P., & Sridhar, S. 1995, ApJ, 438, 763
 Goldsmith et al. (2008) Goldsmith, P. F., Heyer, M., Narayanan, G., et al. 2008, ApJ, 680, 428
 Gong & Ostriker (2009) Gong, H., & Ostriker, E. C. 2009, ApJ, 699, 230
 Gong & Ostriker (2011) Gong, H., & Ostriker, E. C. 2011, ApJ, 729, 120
 Goodman et al. (1989) Goodman, A. A., Crutcher, R. M., Heiles, C., Myers, P. C., & Troland, T. H. 1989, ApJ, 338, L61
 Haisch et al. (2001) Haisch, K. E., Jr., Lada, E. A., & Lada, C. J. 2001, ApJ, 553, L153
 Heiles & Crutcher (2005) Heiles, C., & Crutcher, R. 2005, Cosmic Magnetic Fields, 664, 137
 Heiles & Troland (2005) Heiles, C., & Troland, T. H. 2005, ApJ, 624, 773
 Heitsch et al. (2004) Heitsch, F., Zweibel, E. G., Slyz, A. D., & Devriendt, J. E. G. 2004, ApJ, 603, 165
 Hennebelle & Fromang (2008) Hennebelle, P., & Fromang, S. 2008, A&A, 477, 9
 Hennebelle & Ciardi (2009) Hennebelle, P., & Ciardi, A. 2009, A&A, 506, L29
 Hennebelle et al. (2011) Hennebelle, P., Commerçon, B., Joos, M., et al. 2011, A&A, 528, A72
 Hennemann et al. (2012) Hennemann, M., Motte, F., Schneider, N., et al. 2012, A&A, 543, L3
 Heyer et al. (2008) Heyer, M., Gong, H., Ostriker, E., & Brunt, C. 2008, ApJ, 680, 420
 Hosking & Whitworth (2004) Hosking, J. G., & Whitworth, A. P. 2004, MNRAS, 347, 1001
 Ikeda et al. (2009) Ikeda, N., Kitamura, Y., & Sunada, K. 2009, ApJ, 691, 1560
 Indebetouw & Zweibel (2000) Indebetouw, R., & Zweibel, E. G. 2000, ApJ, 532, 361
 Inoue & Fukui (2013) Inoue, T., & Fukui, Y. 2013, ApJ, 774, L31
 Inoue & Inutsuka (2007) Inoue, T., & Inutsuka, S. 2007, Progress of Theoretical Physics, 118, 47
 Joos et al. (2012) Joos, M., Hennebelle, P., & Ciardi, A. 2012, A&A, 543, A128
 Karimabadi (1995) Karimabadi, H. 1995, Advances in Space Research, 15, 507
 Kirk et al. (2013) Kirk, J. M., WardThompson, D., Palmeirim, P., et al. 2013, MNRAS, 432, 1424
 Könyves et al. (2010) Könyves, V., André, P., Men’shchikov, A., et al. 2010, A&A, 518, L106
 Koyama & Ostriker (2009) Koyama, H., & Ostriker, E. C. 2009, ApJ, 693, 1316
 Krasnopolsky et al. (2010) Krasnopolsky, R., Li, Z.Y., & Shang, H. 2010, ApJ, 716, 1541
 Krumholz et al. (2013) Krumholz, M. R., Crutcher, R. M., & Hull, C. L. H. 2013, ApJ, 767, L11
 Kudoh et al. (2007) Kudoh, T., Basu, S., Ogata, Y., & Yabe, T. 2007, MNRAS, 380, 499
 Kudoh & Basu (2008) Kudoh, T., & Basu, S. 2008, ApJ, 679, L97
 Kudoh & Basu (2011) Kudoh, T., & Basu, S. 2011, ApJ, 728, 123
 Lada et al. (2010) Lada, C. J., Lombardi, M., & Alves, J. F. 2010, ApJ, 724, 687
 Larson (1981) Larson, R. B. 1981, MNRAS, 194, 809
 Li et al. (2013a) Li, Z.Y., Banerjee, R., Pudritz, R., Joergensen, J., Shang, H., Krasnopolsky, R., & Maury, A. 2013a, Protostars & Planets VI
 Li et al. (2011) Li, Z.Y., Krasnopolsky, R., & Shang, H. 2011, ApJ, 738, 180
 Li et al. (2013b) Li, Z.Y., Krasnopolsky, R., & Shang, H. 2013b, ApJ, 774, 82
 Li & Nakamura (2004) Li, Z.Y., & Nakamura, F. 2004, ApJ, 609, L83
 Mac Low & Klessen (2004) Mac Low, M.M., & Klessen, R. S. 2004, Reviews of Modern Physics, 76, 125
 Machida et al. (2011) Machida, M. N., Inutsuka, S.I., & Matsumoto, T. 2011, PASJ, 63, 555
 Maury et al. (2010) Maury, A. J., André, P., Hennebelle, P., et al. 2010, A&A, 512, A40
 McKee et al. (2010) McKee, C. F., Li, P. S., & Klein, R. I. 2010, ApJ, 720, 1612
 McKee & Ostriker (2007) McKee, C. F., & Ostriker, E. C. 2007, ARA&A, 45, 565
 Mellon & Li (2008) Mellon, R. R., & Li, Z.Y. 2008, ApJ, 681, 1356
 Mestel (1985) Mestel, L. 1985, Protostars and Planets II, 320
 Mestel & Spitzer (1956) Mestel, L., & Spitzer, L., Jr. 1956, MNRAS, 116, 503
 Motte et al. (2001) Motte, F., André, P., WardThompson, D., & Bontemps, S. 2001, A&A, 372, L41
 Mouschovias & Spitzer (1976) Mouschovias, T. C., & Spitzer, L., Jr. 1976, ApJ, 210, 326
 Mouschovias (1979) Mouschovias, T. C. 1979, ApJ, 228, 475
 Mouschovias (1991) Mouschovias, T. C. 1991, ApJ, 373, 169
 Mouschovias & Ciolek (1999) Mouschovias, T. C., & Ciolek, G. E. 1999, NATO ASIC Proc. 540: The Origin of Stars and Planetary Systems, 305
 Myers et al. (1986) Myers, P. C., Dame, T. M., Thaddeus, P., et al. 1986, ApJ, 301, 398
 Myers et al. (2014) Myers, A., Klein, R., Krumholz, M., & McKee, C. 2014, arXiv:1401.6096
 Nakamura & Li (2005) Nakamura, F., & Li, Z.Y. 2005, ApJ, 631, 411
 Nakamura & Li (2008) Nakamura, F., & Li, Z.Y. 2008, ApJ, 687, 354
 Nakano & Nakamura (1978) Nakano, T., & Nakamura, T. 1978, PASJ, 30, 671
 Palmeirim et al. (2013) Palmeirim, P., André, P., Kirk, J., et al. 2013, A&A, 550, A38
 Rathborne et al. (2009) Rathborne, J. M., Lada, C. J., Muench, A. A., et al. 2009, ApJ, 699, 742
 RomanDuval et al. (2010) RomanDuval, J., Jackson, J. M., Heyer, M., Rathborne, J., & Simon, R. 2010, ApJ, 723, 492
 SantosLima et al. (2012) SantosLima, R., de Gouveia Dal Pino, E. M., & Lazarian, A. 2012, ApJ, 747, 21
 Seifried et al. (2012) Seifried, D., Banerjee, R., Pudritz, R. E., & Klessen, R. S. 2012, MNRAS, 423, L40
 Seifried et al. (2013) Seifried, D., Banerjee, R., Pudritz, R. E., & Klessen, R. S. 2013, MNRAS, 432, 3320
 Shu et al. (1987) Shu, F. H., Adams, F. C., & Lizano, S. 1987, ARA&A, 25, 23
 Shu (1992) Shu, F. H. 1992, Physics of Astrophysics, Vol. II, by Frank H. Shu. Published by University Science Books, ISBN 0935702652, 476pp, 1992.,
 Simpson et al. (2008) Simpson, R. J., Nutter, D., & WardThompson, D. 2008, MNRAS, 391, 205
 Spitzer (1956) Spitzer, L. 1956, Physics of Fully Ionized Gases, New York: Interscience Publishers, 1956,
 Stone et al. (2008) Stone, J. M., Gardiner, T. A., Teuben, P., Hawley, J. F., & Simon, J. B. 2008, ApJS, 178, 137
 Strittmatter (1966) Strittmatter, P. A. 1966, MNRAS, 132, 359
 Sugitani et al. (2011) Sugitani, K., Nakamura, F., Watanabe, M., et al. 2011, ApJ, 734, 63
 Takahashi & Yamada (2013) Takahashi, K., & Yamada, S. 2013, Journal of Plasma Physics, 79, 335
 Tomida et al. (2013) Tomida, K., Tomisaka, K., Matsumoto, T., et al. 2013, ApJ, 763, 6
 Tomisaka et al. (1988) Tomisaka, K., Ikeuchi, S., & Nakamura, T. 1988, ApJ, 335, 239
 Torrilhon (2003) Torrilhon, M. 2003, Journal of Plasma Physics, 69, 253
 Troland & Crutcher (2008) Troland, T. H., & Crutcher, R. M. 2008, ApJ, 680, 457
 Vaidya et al. (2013) Vaidya, B., Hartquist, T. W., & Falle, S. A. E. G. 2013, MNRAS, 433, 1258
 VázquezSemadeni et al. (2011) VázquezSemadeni, E., Banerjee, R., Gómez, G. C., Hennebelle, P., Duffin, D., & Klessen, R. S. 2011, MNRAS, 414, 2511
 Vestuto et al. (2003) Vestuto, J. G., Ostriker, E. C., & Stone, J. M. 2003, ApJ, 590, 858
 WardThompson et al. (2007) WardThompson, D., André, P., Crutcher, R., Johnstone, D., Onishi, T., & Wilson, C. 2007, Protostars and Planets V, 33
 Williams et al. (2000) Williams, J. P., Blitz, L., & McKee, C. F. 2000, Protostars and Planets IV, 97
 Wu (1987) Wu, C. C. 1987, Geophys. Res. Lett., 14, 668