Evolution of the Angular Momentum of Molecular Cloud Cores in Magnetized Molecular Filaments

Figure 1 displays the fragmentation of a filament in our simulation. The cores are formed along the filament due to the growth of initial turbulent fluctuations. The magnetic field lines are perpendicular to the filament axis at the initial state and are deformed by the turbulence and the core formation motions as time progresses. These deformations also impact the overall shape of the filament, which is straight at the initial stage. Since the density profile rapidly drops in the outer region of the filament (ρ ∝ r⁻⁴) and plasma beta is so small, the magnetic field lines are bent only close to the ridge of the filament. The perpendicular magnetic fields with the free boundary condition cannot halt the fragmentation of the filaments (Hanawa et al. 2017).

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Here we study the internal rotational structure of the cores averaging the results of the 40 simulations for both sets of simulations with weak and strong magnetic fields. Figure 2(a) displays the averaged j-M profile in the cores at the final state of our simulations, where j and M are the specific angular momentum and the mass of the spherical enclosed region around the density peak, respectively. The black dashed line represents j ∝ M, which corresponds to the self-similar profile (Basu 1997; Saigo & Hanawa 1998; Misugi et al. 2023). Figure 2(a) indicates that the profile of the angular momentum converges to the self-similar solution in the weak magnetic field case similar to the hydrodynamic case, as shown in Paper I. On the other hand, in the strong magnetic field case, the angular momentum of the inner region is smaller, and the slope of the profile is steeper than that of the weak magnetic field case. This result suggests that the transfer of the angular momentum is more efficient in the strong magnetic field case. Note that the dependence of the angular momentum on the strength of the initial magnetic field can be seen only in the inner region M ≲ 3.0 M_⊙, which corresponds to the radii r ≲ 0.1 pc. In the outer part of the core, for enclosed mass M ≳ 3.0 M_⊙, the angular momentum profiles are similar for both weak and strong magnetic field cases. The black dotted line in Figure 2(a) represents the angular momentum profile due to the turbulence. Since the initial turbulent velocity field follows the Kolmogorov turbulence, the relation between the velocity dispersion σ and the length scale l is σ ∝ l^1/3. Therefore, the specific angular momentum can be described as j ∼ σ l ∝ l^4/3. If the length scale is larger than the width of the filament (l ≫ 2H₀), the mass contained in the spherical regions follows M ∼ M_line l. Using these relations, we can derive j ∼ σ l ∝ M^4/3. The angular momentum profile of the outer region shown in Figure 2(a) is consistent with j ∝ M^4/3 both in the weak and the strong magnetic field cases. This indicates that the angular momentum transfer due to the magnetic field is inefficient in the outer region (M ≳ 3.0 M_⊙). When the radius of the spherical region is larger than the width of the filament, the derived physical properties of the region are affected not only by the core but also by the filament. These results suggest that, although magnetic breaking is not important at the filament scale, magnetic breaking is important for the angular momentum transfer in the core.

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The averaged j-r profile for the 40 cores at the final state is shown in Figure 2(b). Here r denotes the radial distance from the density peak of the core. The resultant angular momentum in the outer region (≳10⁴ au) does not depend on the strength of the initial magnetic field similar to the j–M relation, and follows j ∝ r^4/3. In the inner region (r ≲ 10³ au), the resultant profile with the weak magnetic field at the initial state follows the self-similar profile (j ∝ r). In the weak magnetic field case, a shallower slope appears for 10³ au ≲ r ≲ 10⁴ au. This is due to the contraction during the core formation phase in the filaments. The displacement of the fluid elements due to the contraction at r ∼ 10³ au is larger than that at r ∼ 10⁴ au, consequently, a shallower slope appears at 10³ au ≲ r ≲ 10⁴ au (compare to the j ∝ r^4/3 at r ≳ 10⁴ au) before the transition to the j ∝ r in the inner part of the core at r ≲ 10³ au. In the strong magnetic field case, the region with a shallower slope is not clearly seen since the angular momentum transfer is efficient due to the strong magnetic field. We will analyze the angular momentum transfer in more detail in the following subsections.

In Section 3.2, we discussed the internal angular momentum profile at the final state. In this subsection, we investigate the evolution of the core angular momentum by tracing the trajectory of the SPH particles. To do so, first, we identify the core around the density peak along the filament at the final state of the simulation. That core corresponds to the fastest collapsing core along the filament. Then, we determine the mass scale M_ana which we analyze. We choose a density contour threshold ρ_ana to satisfy the following equation:

$$\begin{eqnarray}&&{M}_{\mathrm{ana}}({\rho }_{\mathrm{ana}})={\int }_{\rho \gt {\rho }_{\mathrm{ana}}}\rho {d}^{3}x.\end{eqnarray} \tag{ 5 }$$

Note that the enclosed region is not spherical since the region is defined by the density contour in contrast with Section 3.2. In this analysis, M_ana is in the range of 0.01 M_⊙–3.0 M_⊙. We take the periodic boundary conditions into account. Then, the trajectories of the SPH particles are traced from the final state to the initial state of the simulation. Figure 3 displays the evolution of the SPH particles that form the core at the final state for M_ana = 3.0 M_⊙, 1.0 M_⊙, and 0.1 M_⊙. At the initial state, the core is elongated along the z-axis, which is parallel to the longitudinal axis of the filament. The aspect ratio decreases with time due to the contraction along the filament axis (Figure 3).

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We calculate the angular momentum of the core around the center of mass J at each mass scale as follows:

$$\begin{eqnarray}&&{\boldsymbol{J}}=\displaystyle \sum _{i}{m}_{i}({{\boldsymbol{x}}}_{i}-{{\boldsymbol{x}}}_{{\rm{c}}})\times ({{\boldsymbol{v}}}_{i}-{{\boldsymbol{v}}}_{{\rm{c}}}),\end{eqnarray} \tag{ 6 }$$

where x _i and v _i denote the position and velocity vectors of the SPH particles, respectively. x _c and v _c are the position and velocity of the center of mass of the core, respectively. The summation includes the SPH particles that belong to the core in the final state. The specific angular momentum of a core is defined as follows:

$$\begin{eqnarray}&&{\boldsymbol{j}}=\displaystyle \frac{| {\boldsymbol{J}}| }{{M}_{\mathrm{ana}}}.\end{eqnarray} \tag{ 7 }$$

We note that, if there is no interaction with the ambient gas around the core, the angular momentum should be conserved. In our simulations, since the cores are formed and embedded in the filament, they interact with the surrounding medium. The histograms of the specific angular momentum of the cores at the initial and at the final state are displayed in Figure 4. The left panel of Figure 4 shows that the distribution of the initial angular momentum of the cores does not depend on the strength of the magnetic field. The mean of the distribution at the final state with B₀ = 2 μG and B₀ = 10 μG are j = 5.0 × 10²⁰ cm² s⁻¹ and j = 3.6 × 10²⁰ cm² s⁻¹, respectively (Figure 4 right), suggesting a more efficient transport of the angular momentum in the strong magnetic field case. Figure 5 shows the dependence of the ratio of the angular momentum at the final state to that at the initial state on the mass scale. Although the angular momentum transfer is efficient in small mass scales (M_ana ≲ 0.1 M_⊙). The average ratio J_fin/J_ini at M_ana = 1.0 M_⊙ in the case of weak and strong magnetic field cases are 0.78 and 0.46, respectively. These results indicate that, although the resultant angular momentum of the cores with the strong magnetic field is smaller than that obtained in the weak magnetic field case, the difference in the angular momentum between both cases is relatively small at M_ana ≳ 1.0 M_⊙. To investigate which force dominates the angular momentum transfer at each mass scale, we will analyze the evolution of the torques in Section 3.4.

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To investigate the mechanism of angular momentum transfer in the cores, we calculate the torque that is acting on the core. The torque T can be written formally as follows:

$$\begin{eqnarray}&&\begin{array}{l}{\boldsymbol{T}}=\displaystyle \int \rho \left({\boldsymbol{x}}-{{\boldsymbol{x}}}_{{\rm{c}}}\right)\times {\boldsymbol{F}}{d}^{3}x,\end{array}\end{eqnarray} \tag{ 8 }$$

where F is a force per unit of mass. We discretize Equation (8) and calculate the magnetic tension torque ( T _bt), magnetic pressure torque ( T _bp), gas pressure torque ( T _f), and gravity torque ( T _g) as follows:

$$\begin{eqnarray}&&\begin{array}{l}{{\boldsymbol{T}}}_{\mathrm{bt}}=\displaystyle \sum _{i}{m}_{i}\left({{\boldsymbol{x}}}_{i}-{{\boldsymbol{x}}}_{{\rm{c}}}\right)\times {{\boldsymbol{F}}}_{\mathrm{bt},i},\end{array}\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&\begin{array}{l}{{\boldsymbol{T}}}_{\mathrm{bp}}=\displaystyle \sum _{i}{m}_{i}\left({{\boldsymbol{x}}}_{i}-{{\boldsymbol{x}}}_{{\rm{c}}}\right)\times {{\boldsymbol{F}}}_{\mathrm{bp},i},\end{array}\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&\begin{array}{l}{{\boldsymbol{T}}}_{{\rm{f}}}=\displaystyle \sum _{i}{m}_{i}\left({{\boldsymbol{x}}}_{i}-{{\boldsymbol{x}}}_{{\rm{c}}}\right)\times {{\boldsymbol{F}}}_{{\rm{f}},i},\end{array}\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&\begin{array}{l}{{\boldsymbol{T}}}_{{\rm{g}}}=\displaystyle \sum _{i}{m}_{i}\left({{\boldsymbol{x}}}_{i}-{{\boldsymbol{x}}}_{{\rm{c}}}\right)\times {{\boldsymbol{F}}}_{{\rm{g}},i},\end{array}\end{eqnarray} \tag{ 12 }$$

where F _bt,i , F _bp,i , F _f,i , and F _g,i are the magnetic tension force, the magnetic pressure force, the gas pressure force, and the gravity per unit of mass of the ith particle. ⁵ In the case of the weak magnetic field, the mechanism of angular momentum transfer is the same as that without the magnetic field, as shown in Paper I. In the hydrodynamic case, the angular momentum of the core is transferred by the negative pressure torque during the core formation phase from the filament fragmentation (ρ_c < 10⁻¹⁷ g cm⁻³). This can be understood as follows. First, most of the cores rotate perpendicularly to the filament axis, and the major axis of the core is inclined with respect to the filament longitudinal axis at the initial state. Then, as time progresses, the major axis of the cores aligns with the filament longitudinal axis with the rotation axis of the cores perpendicular to the filament axis. This rotational direction makes it easier for the cores to gather their mass. Since, at the initial state, the filament follows the hydrostatic equilibrium profile, the pressure torque and the gravitational torque are negative and positive, respectively, with respect to the core rotation axis, which is perpendicular to the filament axis. However, in the runaway collapse phase, the effect of the initial filament geometry decreases. Since the core has a nonaxisymmetric structure and its inner region rotates faster than the outer region, both torques become negative.

Figure 6 shows the evolution of the torques exerted on the cores in the strong magnetic field case. The orange, magenta, and olive solid lines represent the magnetic tension, magnetic pressure, and the sum of the gas pressure and gravity torques, respectively. Figure 6 shows that the magnetic pressure term does not contribute to the angular momentum transfer for all mass scales. As shown in Figure 6(a), magnetic braking is the dominant mechanism to transfer the angular momentum at 0.1 M_⊙. Since the magnetic field lines are twisted due to the spin-up of the inner region of the core, the magnetic tension increases as time progresses. The magnetic pressure and the sum of the gas pressure and gravity torques do not play an important role in the angular momentum transfer compared to the magnetic braking at the mass scale of 0.1 M_⊙. At 1.0 M_⊙ (Figure 6 (b)), although the difference between the gas pressure and gravity is responsible for the angular momentum transfer until the central density reaches ρ_c = 10⁻¹⁷ g cm⁻³, the magnetic braking becomes more important for densities larger than ρ_c = 10⁻¹⁷ g cm⁻³. On the other hand, the sum of the gas pressure and gravity torques is always larger than the magnetic tension torque at 3.0 M_⊙. For the small mass scales (M_crit ≲ 1.0 M_⊙), the magnetic braking is efficient since the surrounding gas is massive enough to extract the angular momentum from the inner region. On the other hand, the mass of the surrounding gas is relatively small for the large mass scales (M_crit ≳ 3.0 M_⊙), since the filament has a steep density profile in the outer region (ρ ∝ r⁻⁴). This is the reason why the magnetic braking is not efficient at 3.0 M_⊙. All the torques remain constant ρ_c ≳ 10⁻¹⁶ g cm⁻³ because the timescale of the gravitational collapse becomes shorter and shorter as the gravitational collapse proceeds. Figure 6 indicates that we can define the critical mass, which determines whether the angular momentum transfer is efficient or not. If the mass scale is smaller than the critical mass, the magnetic field plays a role in the angular momentum transfer. On the other hand, if the mass scale is larger than the critical mass, the magnetic tension does not affect the evolution of the angular momentum. Our results suggest that the critical mass M_crit is 1.0 M_⊙ ≲ M_crit ≲ 3.0 M_⊙ in the case of the strong magnetic field because we see a change of behavior of the importance of the torques between these two mass scales (Figure 6). This behavior of torques is consistent with the results shown in Figure 2. Since the magnetic tension is not dominant in the outer region (≳3.0 M_⊙), the angular momentum profile of the outer region is almost the same irrespective of the strength of the initial magnetic field.

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To study the anisotropy of the angular momentum transfer, we divide all cores into two samples at ρ_c = 10⁻¹⁸ g cm⁻³. The first sample is composed of the cores with the angle between their angular momentum and the local magnetic field smaller than 30°. The cores of the second sample have an angle between their angular momentum and the local magnetic field larger than 60°. The former and the latter samples are referred to as the parallel sample and the perpendicular sample, respectively. The parallel sample contains 8, 9, and 10 cores at M_ana = 0.1, 1.0, 3.0 M_⊙, respectively. The perpendicular sample consists of 10, 9, and 11 cores at M_ana = 0.1, 1.0, 3.0 M_⊙, respectively. The local magnetic field is defined as the volume-averaged magnetic field in the enclosed region at each time step. Note that the magnetic field orientation can change due to the initial turbulent velocity field. Figure 7 displays the evolution of the angular momentum of the parallel and the perpendicular samples in the strong magnetic field case. At M_ana = 0.1 M_⊙, the angular momentum of the perpendicular sample is smaller than that of the parallel sample at the final state, while both have almost the same angular momentum at ρ_c = 10⁻¹⁸ g cm⁻³. The same tendency can be seen in the case of M_ana = 1.0 M_⊙, although the difference in the angular momentum between the perpendicular and parallel sample is a bit smaller compared with the results of M_ana = 0.1 M_⊙. On the other hand, at M_ana = 3.0 M_⊙, the evolution of the angular momentum of the perpendicular sample is almost the same as that of the parallel sample. This dependence on the mass scale can be understood from the results shown in Figure 6. In Figure 6, we show that the magnetic tension is the dominant process of the angular momentum transfer only for M_ana ≲ 3.0 M_⊙. Since only the magnetic field causes the anisotropy in the angular momentum transfer, the difference in the angular momentum between samples can be seen only at M_ana = 0.1, 1.0 M_⊙.

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In Section 3.5, we show that the anisotropy of the angular momentum transfer due to the magnetic field can be found in the mass scale M_ana ≲ 1.0 M_⊙. Since the angular momentum is efficiently removed in the perpendicular case compared to the parallel case (Mouschovias 1985), it is expected that the core rotation axis tends to be parallel to the local magnetic field direction at the final state. To confirm whether this is true or not, in this subsection, we investigate the distribution and evolution of the 3D angle between the angular momentum and the magnetic field in more detail.

First, in Figure 8, we show the distribution of the angle between the angular momentum of the core and the filament axis at the final state of our simulations. Here, the z-axis (the initial filament axis) is used as the reference axis for measuring the rotation axis. Figure 8 shows that the rotation directions of the cores are mostly perpendicular to the filament axes, irrespective of the initial strength of the magnetic field. The reason for this is the same in the case of nonmagnetized filaments shown in Paper I: this is because the cores are elongated along the z-axis (filament longitudinal direction) at the initial state (Figure 3). In the case of M_ana = 0.1 M_⊙, the tendency of the perpendicular rotation is weak compared with the case of M_ana = 3.0 M_⊙ because the effect of the filament geometry is less effective for smaller mass scales.

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Figure 9 displays the distribution of the angle between the angular momentum and the local magnetic field, θ_j,B at the final state. The black dotted line in Figure 9 represents the random distribution in the 3D space. The distribution increases toward larger θ because of the solid angle. We refer to the black dotted line as the 3D random distribution. On the other hand, if both rotation and magnetic field vectors are random but confined in the 2D plane perpendicular to the filament axis, the distribution is flat. This flat distribution is described as the black dashed line in Figure 9. Since, at the initial state, the magnetic field lines are parallel to the x-axis and the rotation axis of the core is randomly oriented in the x-y plane due to the elongation of the cores along the filament longitudinal axis (Figure 3), the angle between the rotation axis and the magnetic field line follows the flat distribution at the initial state. We note that the direction of the magnetic field within the core also changes with time due to the turbulent velocity field and the gravitational fragmentation of the filament. Figure 9 indicates that the distribution of θ_j,B is flatter than the 3D random distribution. This means that the magnetic field lines tend to be confined in the x-y plane compared with the 3D random distribution. Even at M_ana = 0.1 M_⊙, 11 cores have θ_j,B > 60°. As shown in Figure 8, most cores rotate perpendicular to the filament axis. In the strong magnetic field case, the magnetic field direction of half of the cores in the sample remains perpendicular even just before the first core formation (Section 4.1). Therefore, in the strong magnetic field case, both the rotation vector and the magnetic field direction of half of the cores are roughly on the x–y plane, which is perpendicular to the filament axis. On the other hand, in the weak magnetic field case, since the magnetic field direction significantly changes during the core formation phase (Section 4.1), the magnetic field lines are not confined in the x-y plane. In the weak magnetic field case, although the rotation axis of the cores tends to be perpendicular to the filament axis, the magnetic field lines show a parallel component along the filament axis at the core scale. This means that the distribution of θ_j,B tends to have a 3D random distribution rather than on the x-y plane compared with the strong magnetic field case.

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These results of a strong magnetic field case indicate that the rotation axis of the cores is not always aligned with the local magnetic field direction. To reveal what causes this diversity, we focus on the two cores in the perpendicular sample that have an angle between the angular momentum and the local magnetic field larger than 60° at ρ_c = 10⁻¹⁸ g cm⁻³. At the final state, one is aligned to the magnetic field; the other has a large inclination between the rotation axis and the magnetic field direction, even though both cores are in the perpendicular sample. The former and latter cores are referred to as aligned core and misaligned core, respectively.

The time evolution of θ_j,B at M_ana = 0.1 M_⊙ is shown in the top panels of Figure 10. The rotation axis of the misaligned core remains perpendicular to the local magnetic field direction. However, the aligned core experiences a strong alignment when 10⁻¹⁸ g cm⁻³ < ρ_c < 10⁻¹⁷ g cm⁻³. The middle panels of Figure 10 display the evolution of the torques exerted on each core at M_ana = 0.1 M_⊙. As shown in panel (c) of Figure 10, the magnetic tension torque exerted on the misaligned core monotonically increases with time. On the other hand, in the case of an aligned core, the magnetic tension torque rapidly increases until the central density reaches ρ_c ≃ 10⁻¹⁷ g cm⁻³. Since the timescale becomes shorter as the gravitational collapse proceeds, the torque in the early evolutionary stage transfers the angular momentum more efficiently compared to that at the later evolutionary stage. Therefore, the strong magnetic tension torque acting on the aligned core during 10⁻¹⁸ g cm⁻³ < ρ_c < 10⁻¹⁷ g cm⁻³ leads to strong alignment between the rotation axis and the local magnetic field direction shown in panel (b) of Figure 10. The magnetic field structure is shown in the bottom panels of Figure 10. The magnetic field lines are relatively straight in the case of the misaligned core, while the mean direction is inclined with the initial magnetic field direction (x-axis) due to the large eddy motion. On the other hand, the magnetic field lines are strongly pinched in the case of the aligned core. Our results indicate that the pinched magnetic field configuration causes strong magnetic braking. This is consistent with the analytical discussion in Mouschovias (1985). The pinched geometry has a longer lever arm compared to the case of the misaligned core. These results imply that the cores formed in the filament with the same line mass and the same initial magnetic field have a diversity of the magnetic field structure just before the first core formation due to the difference of the initial seed of the turbulence, even in the case of transonic initial turbulence.

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At the initial state of our simulations, the filaments are perpendicularly threaded by the magnetic field. However, the turbulent motion of the gas can deform the magnetic field lines, and hourglass magnetic field structures are created by the longitudinal contraction motions due to gravity. Observations show that the magnetic field directions toward the cores formed along the same filament show a variety of orientations compared to the filament axis (Eswaraiah et al. 2021). Therefore, it is important to study whether our simulation results can explain the diversity of the configuration of magnetic field lines observed in molecular cloud cores along a given filament or not. To make synthetic observational-like maps of the plane of the sky component of the magnetic field as traced by observations of the dust polarized emission, we integrate the magnetic field along the y-direction weighted by density as follows:

$$\begin{eqnarray}&&{{\boldsymbol{B}}}_{\mathrm{obs}}=\displaystyle \frac{\int \rho {\boldsymbol{B}}{dy}}{\int \rho {dy}}.\end{eqnarray} \tag{ 13 }$$

Figure 11 displays the synthetic observations of our simulations toward a few examples of cores. In Figure 11(a), one can clearly see the hourglass structure formed by the filament fragmentation and the contraction of the core due to gravity. Figures 11(b) and (c) show that, although the magnetic field lines are relatively straight in the outer low-density region, the magnetic field lines in the central high-density region of the cores are bent and not aligned with the initial magnetic field direction due to the core rotation motion inherited from the initial turbulent velocity field. Figure 12 displays the histogram of the angle between the initial magnetic field direction and the local magnetic field direction in the high-density region enclosed by the contour of 50% of the peak column density. The histogram shown in Figure 12 suggests that the magnetic field direction in the cores has diversity, and this result is compatible with the observations that report the variety of the magnetic field directions inside the core (Eswaraiah et al. 2021). Figure 12 also suggests that the histogram of the weak magnetic field case shows a flat distribution compared with that of the strong magnetic field case. On the other hand, the local magnetic field direction of half of the cores in the strong magnetic field case is aligned with the initial magnetic field direction (0° < θ_b < 15°). Our result indicates that the statistical measurement of the angle between the filament axis and the local magnetic field direction will give us a hint on how to determine the strength of the magnetic field.

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In Section 3.5, we show that the distribution of the angle between the local magnetic field and the rotation axis is random within the x-y plane in the strong magnetic field case. However, if the magnetic field is strong enough, the gas falls onto the core along the initial magnetic field direction (x-axis). This means that the strong magnetic field could cause the anisotropy of the angular momentum of the core even at the initial state of our simulations. To investigate the effect of selective infall, we derive the initial shape of the cores, as shown in Figure 13(a). The size of the cores along each direction is defined as, for example ${x}_{\max }-{x}_{\min }$, where ${x}_{\max }$ and ${x}_{\min }$ are the maximum and minimum value of x of the SPH particles contained in the core, respectively. The size of the y and z-directions are defined using the same method. Figure 13(a) suggests that the effect of the magnetic field on the shape of the core can be clearly seen in the inner region. This means that the selective infall due to the magnetic field occurs in the case of a strong magnetic field. The anisotropy of the shape can be seen only in the inner region since the contraction due to gravity occurs only in the inner region. The outer region, which traces the filament, is mainly supported by the thermal pressure. Tsukamoto et al. (2018) showed that selective accretion affects the evolution of the angular momentum of the central high-density region. If the angle between the rotation axis and the magnetic field lines is parallel, the fluid elements with smaller angular momentum fall onto the central region. On the other hand, if the misalignment angle is large, the gas elements with larger angular momentum selectively accrete onto the central region. Figure 13(b) displays the initial specific angular momentum at each mass scale. The green and magenta solid lines represent the specific angular momentum of parallel and perpendicular samples at the initial state, respectively. The difference between both samples is relatively small in all mass scales. This is because the cores are elongated along the z-axis even on the small mass scale. The elongation along the z-axis hides the effect of the selective accretion guided by the strong magnetic field due to the flux-freezing in our ideal MHD simulations. The evolution of the angular momentum due to selective accretion should be tested in the future with nonideal MHD simulations.

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In Section 3.4, we define M_crit below which the angular momentum transfer by the magnetic field is not important, and our results show 1.0 M_⊙ ≲ M_crit ≲ 3.0 M_⊙. To study the origin of M_crit, we calculate the timescales of magnetic braking based on the concept of the analytical model shown in Mouschovias & Paleologou (1979, 1980) and Mouschovias (1985). The timescale of magnetic braking of the core is estimated as the time needed for the Alfvén wave to sweep a region that has the same inertia as the core. To calculate this timescale, we consider a simple model as shown in Figure 14. In this model, the core is a concentric cylindrical region in the filament whose length and radius are written as $2{l}_{\mathrm{core}}$ and ${R}_{\mathrm{core}}$, respectively. The magnetic field is uniform and perpendicular to the filament axis. The filament axis and magnetic field direction are along the z- and x- axes, respectively. We adopt the same hydrostatic density profile of the initial condition of our simulations (Equation (3)). The Alfvén wave propagates from the boundary between the core and the surrounding medium along the magnetic field described as the light blue region in Figure 14. l_A(t) represents the distance propagated by the Alfvén wave during the time t. In the model, we consider two configurations. First, the perpendicular configuration, where the rotation direction of the core is perpendicular to the magnetic field direction and to the filament axis. Second, the parallel configuration is where the rotation vector is parallel to the magnetic field and perpendicular to the filament axis. In Figure 14, the rotation motions of the perpendicular and parallel configurations are described as the red and yellow arrows, respectively. The equations relative to the moment of inertia of the core and the external medium in our model are shown in the Appendix. Figure 15 displays the timescale of magnetic braking derived from our semianalytical model. Since we adopt the density profile of hydrostatic equilibrium, the moment of inertia has a finite value in both perpendicular and parallel cases, even when l_A → ∞ . Moreover, the Alfvén wave reaches infinity in a finite time which is described as follows:

$$\begin{eqnarray}\begin{array}{rcl}{t}_{{\rm{A}}} & = & {\displaystyle \int }_{0}^{\infty }\displaystyle \frac{{ds}}{{v}_{A}(s)}\\ & = & \displaystyle \frac{\pi {H}_{0}}{2{v}_{{\rm{A}}}(0)},\end{array}\end{eqnarray} \tag{ 14 }$$

where ${v}_{{\rm{A}}}(s)\equiv {B}_{0}/\sqrt{4\pi \rho (s)}$ is the Alfvén wave. The region in which the magnetic braking is ineffective is described by the shaded region in Figure 15. The critical mass for the cores in the parallel case is smaller than that of the perpendicular cores since the moment of inertia in the perpendicular case is larger than that of the parallel case. This is consistent with the results of Mouschovias (1985). The average elapsed time from the initial to the final state of our simulation in the strong magnetic filed case is t_fin = 5.54 t_ff, which includes the timescale of the filament fragmentation and that of the collapse of the core. Using the elapsed time t_fin as the timescale of the magnetic braking and adopting the initial strength of the magnetic field of 10 μG, we can determine the critical mass scale from Figure 15. The critical masses in the case of perpendicular and parallel configuration are M_crit = 2.10 and 0.79 M_⊙. ⁶ These critical masses are plotted as the blue stars in Figure 15. These results are consistent with the conclusions from our simulations in which the magnetic braking is not important for M ≳ 3.0 M_⊙. In summary, the magnetic field does not make a significant impact on the angular momentum transfer at the core scale (∼0.1 pc).

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Recent observational results using thermal dust emission at (sub-)millimeter wavelengths suggest that the circumstellar disks around young stellar objects, such as Class 0 and Class I objects, have a large diversity in size and mass of protoplanetary disks (e.g., Cox et al. 2017; Maury et al. 2019; Tobin et al. 2020). For a recent detailed review, see Tsukamoto et al. (2022). Tobin et al. (2020) show that the mean disk radii of Class 0, Class I, and flat spectrum protostars measured from dust continuum emission in the Orion molecular clouds are 44.9, 14.9, and 28.5 au, respectively. They also reported that the dispersion of the distribution of disk radii of Class 0, Class I, and flat spectrum protostars are $\sigma \left({\mathrm{log}}_{10}\left({R}_{\mathrm{disk}}/(1\,\mathrm{au})\right)\right)\,=\,0.38,0.42,\mathrm{and}\ 0.38$, respectively. To investigate the origin of this diversity, Kuffmeier & Haugbølle (2017) and Bate (2018) performed simulations starting from the molecular cloud scale with an initial turbulent velocity field. Although the distribution of disk radius shown in Bate (2018) seems to be consistent with the observations, the simulation of Bate (2018) does not include the effect of a magnetic field. Kuffmeier & Haugbølle (2017) performed zoom-in simulations with the maximum resolution reaching 2 au. Although the resolution of their simulations is relatively high, the sample size of the disk is not large enough.

We define the "equivalent angular momentum radius," ${\tilde{R}}_{\mathrm{disk}}$, by the following equation:

$$\begin{eqnarray}&&\begin{array}{l}{M}_{\mathrm{disk}}\sqrt{{{GM}}_{\mathrm{star}}{\tilde{R}}_{\mathrm{disk}}}={\varepsilon }_{\mathrm{AM}}{J}_{\mathrm{tot}},\end{array}\end{eqnarray} \tag{ 15 }$$

where M_star and M_disk are the mass of the central star and disk, respectively. ε_AM is the parameter to control the angular momentum transfer from the first core formation epoch to the disk formation epoch. ε_AM is defined as the ratio of the specific angular momentum of the core at the final state of our simulations to that of the disk at the disk formation epoch. At the final state of our simulations, which is just before the first core formation, we measure the total angular momentum, J_tot, of the spherical region with the enclosed mass of 1.0 M_⊙. Then, we calculate ${\tilde{R}}_{\mathrm{disk}}$ using Equation (15). To calculate ${\tilde{R}}_{\mathrm{disk}}$ from Equation (15), M_star and M_disk should be determined. Roughly speaking, one-third of the core mass is ejected by the outflow, and another one-third of the mass in the outer region of the core tends to escape from the core because it becomes gravitationally unbound by the loss of the system mass due to the outflow (e.g., Machida et al. 2009; Machida & Matsumoto 2012; Machida & Hosokawa 2013). Then, we define the disk-to-star mass ratio, ε_disk = M_disk/M_star. Here, we adopt ε_disk = 0.1 (e.g., Gammie 2001). The correct value of ε_AM(< 1) may depend on the properties of each core, such as the magnetic field strength, magnetic field orientation, and gravitational energy. Here we simply adopt ε_AM = 1/3 as a typical value found in protostellar collapse calculations (e.g., Tomisaka 2000; Tsukamoto et al. 2018). Figure 16 plots the distributions of ${\tilde{R}}_{\mathrm{disk}}$, where we have adopted ε_AM = 1/3 and ε_disk = 0.1. The disk size is estimated from Equation (15) at the time when the central maximum density of the core reaches 10⁻¹⁴ g cm⁻³. In Figure 16, the average of the distribution in the case of B₀ = 10 μG and B₀ = 2 μG are 11 au and 27 au, respectively. The standard deviations of the distributions in the case of B₀ = 10 μG and B₀ = 2 μG are $\sigma \left({\mathrm{log}}_{10}\left({\tilde{R}}_{\mathrm{disk}}/(1\,\mathrm{au})\right)\right)=0.51$ and 0.54, respectively (Figure 16). Note that these dispersions are independent of ε_AM because the ε_AM is constant.

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In addition, it is worth estimating the disk size formed from a part of the gas of the envelope by assuming the conservation of the angular momentum. Therefore, we also define the outer radius of the disk at its birth by the following equation:

$$\begin{eqnarray}&&{R}_{\mathrm{disk}}=\displaystyle \frac{{j}_{\mathrm{shell}}^{2}}{{{GM}}_{\mathrm{star}}}.\end{eqnarray} \tag{ 16 }$$

We measure the specific angular momentum, j_shell, of the most outer concentric shell region with mass of M_disk. In Figure 17, the average of the distribution of R_disk in the case of B₀ = 10 μG and B₀ = 2 μG are 444 au and 842 au, respectively. The standard deviations of the distributions of R_disk in the case of B₀ = 10 μG and B₀ = 2 μG are $\sigma \left({\mathrm{log}}_{10}\left({R}_{\mathrm{disk}}/(1\,\mathrm{au})\right)\right)\,=0.50$ and 0.62, respectively. Our results indicate that the initial diversity of the angular momentum inherited from the turbulent velocity field could explain the observed variety of the disk radii. Since we adopt the angular momentum at the final state of our simulations as J_tot, the distributions in Figures 16 and 17 include the effect of the magnetic field on the angular momentum transfer during the core formation and gravitational collapse phases. The fact that the dispersion of the distribution does not strongly depend on the strength of the magnetic field indicates that the angular momentum transfer due to the magnetic field during the core formation and collapse stages does not have a large impact on the diversity of the disk size. Note that, since in our model ε_AM does not depend on the strength of the magnetic field, our model does not include the effect of the magnetic field on the angular momentum transfer during the disk formation phase.

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Note that the present estimation is based on the assumption that each core produces only one protostar. If a binary or a higher-order multiple system is created, the radius of the protoplanetary disk around each star should be smaller than our estimate. Since the specific angular momentum is a slowly increasing function of the radius in the rotationally supported disk (see Equations (15) and (16)), the estimated disk size is significantly affected by the value of the specific angular momentum. In addition, as shown in Equation (15), the disk size is proportional to the square of ε_AM. This means that variations of ε_AM will also create larger scatters in ${\tilde{R}}_{\mathrm{disk}}$. The results shown in Hirano et al. (2020) indicate that the resultant disk size from the gravitational collapse of molecular cloud cores changes by a factor of 3 or 4 depending on the angle between the initial magnetic field line and rotation axis even if the magnitude of the angular momentum is the same at the initial state. Moreover, Tsukamoto et al. (2015b) conducted simulations of cloud core collapse with the Hall effect and showed that the Hall effect also has a large impact on the disk size. When the angular momentum vector and the initial magnetic field are in the antiparallel configuration, the resultant circumstellar disk radius is ∼20 au at the protostar formation epoch. On the other hand, the disk with a size of ∼1 au is formed in the case of the parallel configuration. These results in Tsukamoto et al. (2015b) suggest that the Hall effect can change the disk size by an order of magnitude at the protostar formation epoch. Although these effects may tend to increase the diversity of disk radius, the dispersion of the distribution of disk radius from the initial turbulent velocity field in the filaments alone is large enough to reproduce the observed diversity of disk size. To understand which mechanism is dominant for the diversity of disk properties, we need to solve the evolution from the filament to the circumstellar disk, including nonideal MHD effects, which is one of our future works and beyond the scope of this paper.