QUANTIFYING THE SIGNIFICANCE OF THE MAGNETIC FIELD FROM LARGE-SCALE CLOUD TO COLLAPSING CORE: SELF-SIMILARITY, MASS-TO-FLUX RATIO, AND STAR FORMATION EFFICIENCY

The dust continuum Stokes I maps of JCMT, BIMA, and the SMA are shown in the left column panels in Figure 1. In this sequence of subsequently higher resolution maps, the JCMT main core (∼40'' in size) is resolved into two cores (∼8'') in the BIMA observation, where the northern core is further resolved into W51 e2 (∼2'') in the SMA map. Overlaid on the dust continuum maps are the magnetic field segments (red) and the intensity gradient segments (blue). For completeness, the SMA result for W51 e2 is reproduced from Koch et al. (2012). A tight correlation between magnetic field P.A. and the corresponding intensity gradient P.A. (angle δ < π/2 in Figure 3 in Koch et al. 2012) is obvious for many of the pairs. The correlation coefficients—in the definition of Pearson's linear correlation coefficient—are 0.71, 0.72, and 0.95 for the JCMT, BIMA, and the SMA data, respectively. Thus, there is seemingly a trend for an increasing alignment between magnetic field and intensity gradient orientations with smaller scales, with the tightest correlation found for the collapsing core. This correlation is being investigated in a separate work on a larger data sample (P. M. Koch et al. 2012, in preparation). It suffices to mention here that the above correlation can possibly serve as an indicator for the role of the magnetic field through the evolutionary stages of a molecular cloud.

Maps of the magnetic field to gravitational force ratio, Σ_B = sin ψ/sin α, are displayed in the right column panels in Figure 1. The pressure gradient force, ∇P, typically being small compared to gravity, is omitted here. For the JCMT and BIMA data, two gravity centers at the two emission peaks in each map are assumed in order to calculate ψ. The angle ψ is then simply measured in between the intensity gradient direction (left column panels in Figure 1) and the gravity center direction. For W51 e2 (SMA data), a single gravity center at the emission peak in the SMA map is adopted. The angle α = π/2 − δ, where δ is the difference in between the intensity gradient and the magnetic field orientations, is deduced from the left column panels in Figure 1. All three maps—originating from three different instruments, one single dish and two interferometers—show clear differences in the ratios between core regions and areas in between the cores or further away. With the largest mapping area, the JCMT observation reveals very distinct features. With the exception of only two segments in the eastern core, both core regions show ratios below 0.5. In between the two cores the ratios reveal peaks in between 1.5 and ∼5, with large stretches in the northeast–southwest direction larger than 1. The BIMA observation is rather restricted to the cores, with an area in the northern core without polarization detection. Nevertheless, both cores clearly show ratios below 1, except one segment in the southern core. The four segments in the east–west direction (around y-offset ∼−4) reveal ratios larger or around one. Though limited to a few segments only, this still points toward a trend of increased ratios outside the immediate core areas. The detection by the SMA mostly resolves polarization features in the e2 core. The ratio averaged over the core is ∼0.2. The northwest extension shows two segments with a ratio larger than 1 and an average of about 0.8. If assuming an additional new core being formed here, the ratios get reduced to about 0.5. In any case, the main collapsing core very clearly shows ratios below 1, whereas the more distant northwest extension shows larger values. We remark that the total outflow mass in e2 (∼1.4 M_☉; Shi et al. 2010) is negligibly small compared to the total core mass of about 220 M_☉ (Tang et al. 2009b). Additionally, with a dust-to-gas ratio of about 1–100, the outflow is unlikely to be detected in the dust continuum with the SMA sensitivity. Consequently, we do not expect the magnetic field and the dust continuum Stokes I morphologies to be significantly affected by outflows. Our analysis, therefore, still quantifies the field to gravitational force ratio. The SMA e8 core is not further analyzed here because only a few polarization segments are detected (Tang et al. 2009b).

In summary, the cores in the JCMT, BIMA, and SMA data have average ratios of 0.33, 0.45 (east, west), 0.38, 0.49 (north, south), and ∼0.2 (main core), respectively. The other areas reveal significantly and systematically larger ratios (≳ 1). Despite probing three very different physical scales, the three data sets equally reveal a minor role of the magnetic field, F_B < F_G, in the center regions, and an increasingly more significant field, F_B ≳ F_G, at larger distances.

The distinct features of the field-to-gravity force ratio in the right panels in Figure 1 are further analyzed in the following. Azimuthally averaged radial profiles, binned at half of the beam resolution, are displayed in Figure 2. The profiles are centered at the dust continuum peaks. From the JCMT data, the main (eastern) peak covering the BIMA observation and the SMA e2 core is selected. On the next smaller scale, the BIMA southern core is chosen because its force ratio in the center has a smaller uncertainty than the northern core. As already manifest in the right panels in Figure 1, a clear difference between center and outer regions is seen. Both the JCMT and SMA profiles show ratios across the center (∼0.3 for JCMT and ∼0.2 for SMA) with relatively little variations. This is followed by a rather abrupt increase over a short distance (one bin) by a factor of ∼2 (JCMT) and ∼4 (SMA). A plateau, with ratios ∼0.8 for both JCMT and SMA, then extends over distances comparable to or larger than the center regions. The JCMT data, with the largest mapped area, then show another abrupt change to a ratio beyond 1. No polarized emission is detected for W51 e2 at larger distances. The case of BIMA is less clear, but still shows values in the center of ∼0.1 and ∼0.6, followed by a plateau around 0.6 and an increase to larger than 1 at the largest distances. For comparison, azimuthally averaged emission intensity profiles are also shown.

Zoom In Zoom Out Reset image size

For completeness, profiles for the JCMT western and the BIMA northern core are shown in Figure 3. The result for the very central region in the BIMA northern core is less conclusive because only a single value around 0.7 with a ±50% error is available for the force ratio. Nevertheless, values are clearly below one in the core region with a trend to grow at larger distances up to about 0.9. Similarly to the eastern core, the western core in the JCMT data shows ratios around 0.2–0.6 up to about 20'' in distance. Further away from the peak, ratios grow up to about 5 with some oscillatory behavior remaining larger than 1. Thus, despite being less pronounced, the two additional cores here also reveal a transition in their force ratios between central and more distant ratios. In particular, in the overlapping region between the BIMA northern core and the SMA e2 core, the ratios show similar trends regardless of the different field morphologies (hourglass-like for SMA, more uniform for BIMA).

Zoom In Zoom Out Reset image size

Errors in the field-to-gravity force ratio are calculated by propagating the measurement uncertainties Δψ and Δα through Equation (2). The measured magnetic field P.A. uncertainty, in the range of a few degrees to ∼10°, is assumed for Δα. A typical uncertainty of 3°–5° in Δψ results after interpolation when calculating the intensity gradients. This leads to errors for individual ratios in Figure 2 of ∼ ± 4% to ±19%, ∼ ± 4% to ±40%, and ∼ ± 2% to ±20% (with a single outlier at ±60%) for the SMA, BIMA, and the JCMT data, respectively. After averaging in each bin, the errors are typically reduced due to the sample variance factor, resulting in average errors of ∼ ± 10% or less. Average errors of ∼ ± 20% remain for two bins in the JCMT data. A 10% measurement uncertainty is conservatively estimated for the emission intensity when calculating errors for its radial profiles. Binned errors are then typically at the percent level (Figure 2). Except for the single central value for the force ratio in the BIMA northern core (∼ ± 50% error), similar errors are present in Figure 3.

Finally, it is important to keep in mind that the correlation presented in Section 4.1 (based on one angle between two orientations) can be affected by projection effects. Generally, all values are integrated along the line of sight. The field significance Σ_B presented here is much less or not at all affected by projection, because it is based on the ratio of two angles (Koch et al. 2012). It nevertheless still deals with quantities averaged along the line of sight.

Given the common features in the radial profiles of the magnetic to gravitational force ratio (Figures 2 and 3), we address here the question of self-similarity. We focus on the JCMT main core where the BIMA and SMA data provide higher resolution follow-up observations. The BIMA southern core is adopted due to its better statistics. All three data sets in Figure 2 show a rather constant ratio across their cores (plateau-like or a single binned radius), followed by a plateau with a larger ratio before eventually the ratio grows to values larger than or around one. With the cores in the maps of Figure 1 being clearly identified, we choose to normalize the distances from the peaks to units of core sizes, i.e., all profiles are aligned to one normalized core size. The normalization radii, chosen to be those bins where the emission intensity profiles in Figure 2 flatten out, are 2185, 342, and 106 for the JCMT, the BIMA, and the SMA data, respectively. At these radii, the emissions are ∼15%–20% of the peak emissions. Figure 4 shows the result.

Zoom In Zoom Out Reset image size

Even with some uncertainty left in the normalization, it is obvious that the aligned profiles show a close resemblance. This self-similarity suggests that the interplay between magnetic and gravitational force is independent of the three different scales probed here. Thus, the analysis here quantifies the relative magnetic field significance from being generally dominant or comparable to gravity at distances twice the core size to minor or negligible inside the core. We stress that this result seems to hold generally for all of the different field morphologies analyzed here, including even the more irregular cases shown in Figure 3.

In this section, we investigate the connection between the force ratio in Equation (2) and the mass-to-flux ratio for a molecular cloud. The mass scale associated with the amount of magnetic flux Φ threaded by a self-gravitating cloud was introduced in Mestel & Spitzer (1956). The magnetic critical mass M_Φ = Φ/2πG^1/2, with Φ = πR²B where R is the cloud radius, defines the maximum mass that can be supported by the magnetic field against gravitational collapse if no other forces are present (e.g., Shu et al. 1999). This leads to the notions of magnetically supercritical clouds when M > M_Φ, and magnetically subcritical clouds when M < M_Φ. Contrary to the mass-to-flux ratio—which is typically applied to the entire (global) molecular cloud—the force ratio in Equation (2) is a local criterion comparing the field significance with gravity at a specific location in a cloud. Neglecting pressure gradients and explicitly writing out the local field force and gravity force terms, it can be expressed as a ratio of local flux over local mass:

$$\begin{equation} \frac{\sin \psi _{\ell }}{\sin \alpha _{\ell }}=\frac{\Phi ^2_{\ell }}{m^2_{\ell }}\cdot \frac{m_{\ell }}{M_{\ell }} \cdot \frac{1}{R_{B,\ell }} \cdot \frac{1}{R_{\ell }^4} \cdot R_{G,\ell }^2 \cdot \frac{1}{4\pi ^3}\frac{\eta }{G}, \end{equation} \tag{ 3 }$$

where the lower index ℓ refers to a local quantity. M_ℓ is the gravitating mass leading to the local gravitational force F_{G, ℓ} = G(m_ℓM_ℓ/R²_{G, ℓ}) acting upon a local mass element m_ℓ at a distance R_{G, ℓ}. Φ_ℓ is the flux associated with the local mass within a flux tube of radius R_ℓ (Figure 5). R_{B, ℓ} is the local field radius. G and η are the gravitational constant and a numerical unit conversion factor, respectively. Equation (3) is generally valid. However, in order to make further use of it, the local mass element m_ℓ and M_ℓ need to be specified. We therefore make the basic assumption of spherical symmetry. The enclosed mass within a distance r from the center is then $M_{\ell }(r)=\int ^r_0 4\pi r^{\prime 2} \rho (r^{\prime })\, dr^{\prime }\equiv N_r \bar{m_{\ell }}(r)$, where N_r is the number of local flux tubes with an average local mass $\bar{m_{\ell }}(r)$ where both depend on the integration radius r. A local mass-to-flux ratio then results from Equation (3):

$$\begin{eqnarray} \frac{m_{\ell }}{\phi _{\ell }}&=& \left(\frac{\sin \psi _{\ell }}{\sin \alpha _{\ell }}\right)^{-1/2} \cdot \left(\frac{m_{\ell }(r)}{\bar{m_{\ell }}(r)}\right)^{1/2}\nonumber\\ && \cdot R_{B,\ell }^{-1/2} \cdot R_{\ell }^{-1}\cdot (4\pi ^3)^{-1/2}\cdot \left(\frac{\eta }{G}\right)^{1/2}, \end{eqnarray} \tag{ 4 }$$

where we have used $M_{\ell }(r)=N_r \bar{m_{\ell }}(r)$ with N_r = R²_{G, ℓ}/R²_ℓ. We note that the above expression is still fairly general and valid for any density profile ρ(r). In particular, since the ratio depends on the ratio of the mass profile over a mean mass, only the functional form of the density profile is relevant, but not its absolute value. Introducing a mean local mass, $\bar{m_{\ell }}(r)$, allows us to write the local mass-to-flux ratio with the explicit spherical radius dependence only in the second term on the right-hand side of Equation (4). Besides the spherical symmetry assumption for the mass distribution, Equation (4) is basically valid at any position in a molecular cloud.⁹ The average inverse of the force ratio decreases with larger radius (Figure 4), and $({m_{\ell }(r)}/{\bar{m_{\ell }}(r)})^{1/2}\le 1$ is monotonically decreasing for centrally peaked density profiles, with unity in the center. The local mass-to-flux ratio in Equation (4) is thus decreasing with larger radius from the center. Azimuthally averaging Equation (4) and normalizing it to the critical mass-to-flux ratio, we get the local normalized mass-to-flux ratio—for individual subvolumes of fixed size R_ℓ in a cloud—as a function of radius:

$$\begin{eqnarray} \left(\!\frac{\Delta M}{\Delta \Phi }(r)\!\right)_{{\rm norm}}&=& \left<\left(\!\frac{\sin \psi _{\ell }}{\sin \alpha _{\ell }}\!\right)^{-1/2} \cdot R_{B,\ell }^{-1/2}\right>_r \cdot \left(\frac{m_{\ell }(r)}{\bar{m_{\ell }}(r)}\right)^{1/2}\nonumber\\ && \cdot \frac{R_{\ell }^{-1}}{R_0^{-3/2}} \cdot \pi ^{-1/2} \le \left<\left(\!\frac{\sin \psi _{\ell }}{\sin \alpha _{\ell }}\!\right)^{-1/2}\right>_r \cdot \pi ^{-1/2}, \nonumber\\ \end{eqnarray} \tag{ 5 }$$

where 〈⋅⋅⋅〉_r denotes azimuthal averaging at radius r. When normalizing to a local flux tube, we can simply set R₀ ≡ R_ℓ. Furthermore, as found in Koch et al. (2012), R_{B, ℓ} is roughly constant over an observed map, and similar to R_ℓ, which we identify with the beam resolution of an observation. With this simplification, R^−1/2_{B, ℓ}R_ℓ⁻¹/R^−3/2₀ ∼ 1. Assuming the cloud gravitating mass to be proportional to the dust emission, m_ℓ/M_ℓ is derived from the observed emission intensity profiles in Figure 2. As already indicated with the notation, Equation (5) quantifies the differential mass-to-flux ratio as a function of radius.

Zoom In Zoom Out Reset image size

In a next step, we derive an integrated mass-to-flux ratio, i.e., we are asking whether the entire cloud inside a certain radius r is magnetically supercritical. We then have to evaluate

$$\begin{equation} \frac{M}{\Phi }(\le r)=\frac{\sum ^{N_r}m_{\ell }}{\sum ^{N_r}\phi _{\ell }}=M_{\ell }(r) \cdot \left(\sum ^{N_r}\phi _{\ell }\right)^{-1}. \end{equation} \tag{ 6 }$$

ϕ_ℓ can be expressed with Equation (4). The summation is over all local mass and flux elements within the radius r, where the summation limit N_r depends on r. After some algebra and after normalizing with the critical mass-to-flux ratio, we find

$$\begin{eqnarray} \left(\!\frac{M}{\Phi }(\le r)\!\right)_{{\rm norm}}&=&\Bigg(\!\sum _{i=1}^{N_b}\left<\!\left(\!\frac{\sin \psi _{\ell }}{\sin \alpha _{\ell }}\!\right)^{1/2}\cdot R_{B,\ell }^{1/2}\!\right>_{r_i} \nonumber\\ && \!\!\!\! \cdot \frac{\bar{m}^{1/2}(r_i)\cdot m^{1/2}(r_i)}{\bar{m}(r)}\cdot N_{r_i}\!\Bigg)^{-1} \cdot \frac{R_l^{-3}}{r^{-7/2}} \cdot \pi ^{-1/2},\nonumber\\ \end{eqnarray} \tag{ 7 }$$

where the averaging $\langle \cdots \rangle _{r_i}$ is for each bin r_i, with $N_{r_i}$ being the number of subvolumes within r_i and r_{i + 1}. N_b is the number of bins within r (Figure 5). Equation (7) states that the integrated mass-to-flux ratio is calculated by adding bin-averaged force ratios which are weighted with a mass function and $N_{r_i}$, where the latter one results from the integrated volume growing with r. The normalization is with respect to r, i.e., the increasingly larger cloud (with growing radius r) is normalized with its corresponding critical mass-to-flux ratio. R_ℓ is constant and again set by the beam resolution. R_{B, ℓ} is also roughly constant. When omitting the summation, setting $N_{r_i}\equiv 1$ and r ≡ R_ℓ and replacing $\bar{m}(r)$ with $\bar{m}(r_i)$, Equation (7) reduces to the differential mass-to-flux ratio in Equation (5).

Finally, a global mass-to-flux ratio for the entire cloud as a single entity can be derived from Equation (4) in the limiting case where all (small) local quantities grow to cloud-size: m_ℓ → M and 〈R_{B, ℓ}〉 ≈ 〈R_ℓ〉 ≈ 〈R_G〉 ≡ R. The force ratio needs to be averaged over the entire cloud: (sin ψ_ℓ/sin α_ℓ) → 〈sin ψ_ℓ/sin α_ℓ〉. This then leads to the global mass-to-flux ratio, normalized to the critical mass-to-flux ratio with R₀:

$$\begin{equation} \left(\frac{M}{\Phi }\right)_{{\rm norm}}=\left<\left(\frac{\sin \psi _{\ell }}{\sin \alpha _{\ell }}\right)^{-1/2}\right> \cdot \left(\frac{R}{R_0}\right)^{-3/2}\cdot \pi ^{-1/2}, \end{equation} \tag{ 8 }$$

where R₀ ≡ R is the cloud radius.

The set of Equations (5), (7), and (8) defines a differential, integrated, and global mass-to-flux ratio, respectively, which can readily be calculated from observations. These equations provide the criteria to evaluate whether a subvolume (local or integrated up to a certain radius) or an entire cloud is magnetically supercritical. In all cases, the inverse square root of the force ratio is the key measure leading to an estimate independent of any mass or field strength input. Except for the global mass-to-flux ratio, the local ratios need the functional form of the mass profile as an additional input.

A schematic illustration of the different subvolumes analyzed by the various ratios is shown in Figure 5. Figure 6 illustrates these mass-to-flux ratios for the SMA observation. The differential mass-to-flux ratio shows a clear transition from a magnetically subcritical state at larger radii to a supercritical state at smaller radii. This might indicate that a fixed subvolume can only collapse closer to the center, but not in the outer parts of a cloud/core. This finding possibly also provides an explanation for why fragmentation and clustering of stars preferentially occur at the center but not at larger radii. Such a result is further consistent with the theoretical work by Shu & Li (1997), where the dilution of self-gravitational forces by the magnetic tension force (see Section 5.3 and Figure 4) is found to reduce the tendency of a disk to fragment. As further utilized in Section 5.3, this dilution is more significant at larger radii. Consequently, the fragmentation process is more significantly suppressed here. The number of individual collapsing subvolumes in the center region (with smaller or non-existent dilution) might then be linked to the multiplicity of stars. Additionally, the finding here might give support to the suggestion by Mouschovias (1976) that magnetic tension might be able to hold back the envelope and, therefore, prevent it from following the cloud core collapse. In a related later work, Shu et al. (2004) investigated whether magnetic tension can define the masses of forming stars by holding up the subcritical envelope of a molecular cloud while its supercritical core is collapsing. A split monopole—as it is possibly seen in the field morphology in the e2 core (Tang et al. 2009b)—leads to magnetic levitation in their work. However, they find that magnetic suspension alone cannot keep the subcritical envelope from falling onto the center, but additional dynamic levitation by winds and jets is needed for a halt of infall. The results here are also in qualitative agreement with the recent numerical simulation of giant molecular clouds with ambipolar diffusion by Vázquez-Semadeni et al. (2011). They find that the mass-to-flux ratio inside a cloud exhibits large spatial fluctuations, spanning an order of magnitude or more. Thus, local clumps can become magnetically supercritical within a globally subcritical system. Estimating $\mathcal {R}$ (the ratio between the mass-to-flux ratio in the core and in the envelope as introduced in Crutcher et al. 2009) from the values at the largest distances and the values close to the center (Figure 6), we find $\mathcal {R}>1$. This would be in favor of ambipolar diffusion driven models. However, the recent statistical analysis of $\mathcal {R}$ based on numerical simulations with supersonic MHD turbulence (Bertram et al. 2011) yields a large scatter in $\mathcal {R}$ with values both smaller and larger then unity. It thus has to be acknowledged that the interpretation of $\mathcal {R}$ as a measure to distinguish between ambipolar diffusion and turbulence driven star formation is possibly not unambiguous.

Zoom In Zoom Out Reset image size

The integrated mass-to-flux ratio in Figure 6 is driven by the increasing volume. It therefore grows with radius. As expected, this ratio is generally larger than the differential ratio of a subvolume because mass scales with volume whereas flux scales with area. The slight mismatch in the innermost bin results from different weightings for the two ratios. Finally, the global mass-to-flux ratio is close to the integrated ratio at large radii. The difference here results from the incomplete sampling of $N_{r_i}$ in azimuth for those bins which contain some depolarization zones. It is important to remark that both the integrated and global ratios present average values for the entire cloud. Therefore, they are likely biased toward some subvolumes. Whereas these ratios provide a description as to whether a cloud as an entity is in a subcritical or a supercritical state, they cannot properly assess the state of individual subvolumes. This question can only be further addressed with the differential mass-to-flux ratio. The finding here clearly demonstrates the differences and the need for a measurement of the local field significance.

In this section, we aim at investigating further implications of the local force ratio sin ψ/sin α (Figure 4) and the differential mass-to-flux ratio (Figure 6). We limit the discussion here to the collapsing core W51 e2. Global and/or average cloud properties are typically used to characterize its state, e.g., in order to determine whether a system is gravitationally bound or whether it can collapse or not. With the local criteria—the magnetic field to gravity force ratio in Equation (2) and the differential mass-to-flux ratio in Equation (4)—we have tools to go one step further. We cannot only address the question whether a cloud collapses or not, but we can even ask: Where does a cloud collapse? How does it collapse? Does all of the gas take part in the collapse? And finally, depending on the answers to these questions, what are the consequences for the inferred star formation efficiency?

The change with radius in the magnetic field to gravity force ratio (Figure 4) clearly demonstrates that gravity is not everywhere equally efficient to initiate and to keep driving a collapse. The result in Figure 4 suggests and quantifies an effective gravitational force which is reduced by the presence of the magnetic field. Indeed, on the theory side, a concept of diluted gravity (by the magnetic field) was put forward by Shu & Li (1997). Thus, we can define an effectively acting gravitational force F_* as

$$\begin{equation} F_{\ast }(r)=\left(1-\left<\frac{\sin \psi }{\sin \alpha }\right>_r\right)\cdot F_{{\rm max}}(r) \equiv \langle \epsilon _G\rangle _r \cdot F_{{\rm max}}(r), \end{equation} \tag{ 9 }$$

where 〈⋅⋅⋅〉_r denotes azimuthal averaging at radius r and F_max(r) is the maximum non-diluted gravitational force resulting from an enclosed spherically symmetric mass distribution within r. F_* is only defined where sin ψ(r)/sin α(r) ⩽ 1, which possibly holds only for a limited area. In Equation (9), we have introduced the gravity efficiency factor _G ∈ [0, 1]¹⁰ .

Besides the diluted gravity which slows down the accretion and collapse process, the fraction of volume of a cloud (or core) that actually can collapse is relevant, i.e., where is the cloud/core magnetically supercritical? The differential mass-to-flux ratio (Figure 6) provides a local criterion that answers this question. The increasing ratio toward the center indicates that only the central part of the core has accumulated enough mass to overcome the magnetic flux. By reading off the radius where the core state changes from magnetically sub- to supercritical, the potentially collapsing volume and its associated mass can be estimated. This defines a volume efficiency, _V ∈ [0, 1], compared to the maximum efficiency _V ≡ 1 where the entire core can collapse.

Based on the two findings described above, we proceed to estimate a star formation efficiency with the assumptions: (1) the available effective gravitational force is diluted by the presence of the magnetic field and (2) only the regions where the differential mass-to-flux ratio is larger than one will eventually collapse and form stars. We choose to reference our estimate to the well-known pressureless free-fall collapse of a spherical gas sphere. The dynamics in this case are governed by the momentum equation (Dv/Dt) = −G(M_r/r²), where the convective derivative is (Dv/Dt) = (∂v/∂t) + v(∂v/∂r) for the gas infall velocity v. The enclosed mass within radius r is M_r = ∫^r_o4πr'²ρ(r') dr' with the gas density ρ. With the diluted gravity, the momentum equation reads

$$\begin{equation} \frac{Dv}{Dt}=-\langle \epsilon _G\rangle _r \cdot \frac{G M_r}{r^2}. \end{equation} \tag{ 10 }$$

Technically, the gravity efficiency factor 〈_G〉_r could be absorbed into an effective density ρ_*(r). Therefore, the general solution (for any M_r) of Equation (10) for a fluid element starting at a distance r₀ and reaching the center at r = 0 at the time t is still valid. The time t_* in the presence of the magnetic field is then¹¹

$$\begin{equation} t_{\ast }=\frac{\pi }{2}\sqrt{\frac{r_0^3}{2 G M_r(\rho (r),\langle \epsilon _G\rangle _r)}} \approx \langle \epsilon _G\rangle ^{-1/2}\cdot t . \end{equation} \tag{ 11 }$$

Strictly speaking, 〈_G〉_r in the above equation is still a function of radius. In writing the right-hand side we have assumed it to be averaged within an appropriate radius. The average accretion rate $\langle \dot{M}\rangle$, assuming the entire mass M within a cloud/core to be accreted and to collapse, is $\langle \dot{M}\rangle =M/t$. Adding the volume efficiency factor _V together with Equation (11), the effective accretion rate $\langle \dot{M}_{\ast }\rangle$, corrected for our observed magnetic field features, becomes

$$\begin{equation} \langle \dot{M}_{\ast }\rangle =\frac{M_{\ast }}{t_{\ast }} \approx \frac{\epsilon _V\cdot M}{\langle \epsilon _G\rangle ^{-1/2}\cdot t}. \end{equation} \tag{ 12 }$$

The resulting star formation efficiency relative to the standard free fall one is then expressed as

$$\begin{equation} \frac{\langle \dot{M}_{\ast }\rangle }{\langle \dot{M}\rangle }\approx \epsilon _V \cdot \langle \epsilon _G\rangle ^{1/2}. \end{equation} \tag{ 13 }$$

We note that the above phenomenological modeling is still valid in the presence of additional force terms in the momentum equation with any density profiles. The pressureless free-fall collapse simply allows us to analytically express the results. Figure 7 shows the reduced star formation efficiency taking into account the effects of the magnetic field. The red dots indicate efficiencies estimated from the gravity dilution and the reduced collapse volume based on the Figures 4 and 6. Since 1 − (sin ψ(r)/sin α(r)) ∼ 0 at large distances, an arbitrarily low star formation efficiency would result from adopting these values for _G. Instead, we adopt values averaged over the entire core (force ratio ∼0.33), the central core only (∼0.2), and the outer plateau (∼0.85). We consider these values adequate to estimate and sample the gravity dilution for W51 e2. Similarly, for the volume efficiency factor we read a radius of about 1'' within which the core is magnetically supercritical compared to an overall size of about 2''. For comparison, a larger volume of about 15 is considered as well. The resulting efficiencies are reduced at least to a conservative value of ∼0.35 or less when still assuming the larger volume to collapse. The efficiencies drop to ∼0.1 with the smaller volume, with a likely limit of about ∼0.05 in combination with the largest observed gravity dilution. These values are close to the observationally inferred efficiencies of a few percent (e.g., Krumholz & Tan 2007). For comparison, recent numerical simulations by Nakamura & Li (2011) find a star formation efficiency per global free-fall time of ∼20% to ∼30% depending on the magnetic field strength. When they further take into account the feedback from outflows, their efficiencies can be reduced by another order of magnitude.

Zoom In Zoom Out Reset image size

We finally remark that the discussion here is limited to the core e2. A more complete picture, estimating the star formation efficiency starting from the largest scales of the cloud envelop, will need to dynamically link the observed self-similar profiles from Figure 4. Caution is needed here because linking structures of different size scales is non-trivial due to additional structures possibly generated by, e.g., fragmentation and turbulence. Nevertheless, this first estimate further solidifies the significance and impact of the magnetic field. Moreover, it demonstrates that the magnetic field is capable of reducing the star formation efficiency by at least one order of magnitude or even more.