An early transition to magnetic supercriticality in star formation

Magnetic fields have an important role in the evolution of interstellar medium and star formation1,2. As the only direct probe of interstellar field strength, credible Zeeman measurements remain sparse owing to the lack of suitable Zeeman probes, particularly for cold, molecular gas3. Here we report the detection of a magnetic field of +3.8 ± 0.3 microgauss through the H I narrow self-absorption (HINSA)4,5 towards L15446,7—a well-studied prototypical prestellar core in an early transition between starless and protostellar phases8–10 characterized by a high central number density11 and a low central temperature12. A combined analysis of the Zeeman measurements of quasar H I absorption, H I emission, OH emission and HINSA reveals a coherent magnetic field from the atomic cold neutral medium (CNM) to the molecular envelope. The molecular envelope traced by the HINSA is found to be magnetically supercritical, with a field strength comparable to that of the surrounding diffuse, magnetically subcritical CNM despite a large increase in density. The reduction of the magnetic flux relative to the mass, which is necessary for star formation, thus seems to have already happened during the transition from the diffuse CNM to the molecular gas traced by the HINSA. This is earlier than envisioned in the classical picture where magnetically supercritical cores capable of collapsing into stars form out of magnetically subcritical envelopes13,14.

velope.The molecular envelope traced by HINSA is found to be magnetically supercritical, with a field strength comparable to that of the surrounding diffuse, magnetically subcritical CNM despite a large increase in density.The reduction of the magnetic flux relative to the mass, necessary for star formation, thus seems to have already happened during the transition from the diffuse CNM to the molecular gas traced by HINSA , earlier than envisioned in the classical picture where magnetically supercritical cores capable of collapsing into stars form out of magnetically subcritical envelopes 13,14 .
In non-masing interstellar medium, only HI , OH, and CN have successfully produced systematic Zeeman measurements.The comprehensive Zeeman surveys 15 indicate that the magnetic fields in diffuse CNM probed by HI do not scale significantly with density, whereas above a critical break-point density ∼ 300 cm −3 , the magnetic fields in dense cores probed by OH tend to increase with density.However, owing to the gap in densities between the HI (∼ 40 cm −3 ) and OH ( 10 3 cm −3 ) Zeeman measurements, the field transition around the critical density of ∼ 300 cm −3 (where the dependence of the field strength on the density changes behavior) remains a controversial topic and could have crucial implication on star formation [16][17][18][19] .Recently, a CCS Zeeman detection 20 sheds light into regions denser than that probed by OH.A Zeeman probe sensitive for a wide range of densities, particularly the low-density molecular envelope, is highly desirable and could help to distinguish different core formation scenarios.
We developed the so-called HINSA technique to provide a probe of the transition from HI to H 2 4, 5 .HINSA traces cold atomic hydrogen well mixed with H 2 , which provides the necessary cool-ing, not available in the CNM, of HI through collision.Close to the steady state between H 2 formation and destruction, the HINSA strength is independent of the gas density 5 , and thus capable of probing the transition around the critical density.Although the Zeeman effect of HI self-absorption feature has been reported 21,22 , the broad line widths of the absorption components are mostly associated with diffuse atomic gas rather than dense molecular gas.Considering that HINSA typically has much higher brightness temperatures than most molecular lines, impervious to depletion 23 , and can be detected in a wide range of H 2 densities, HINSA is a promising Zeeman probe for molecular gas.
The HINSA feature in L1544 has a strong absorption dip and a nearly thermalized narrow line width at a temperature lower than 15 K 4 .The non-thermal line width and centroid velocity of the HINSA are very close to those of the emission lines of OH, 13 CO, and C 18 O molecules, and their column densities are well correlated, suggesting that a significant fraction of the atomic hydrogen is located in the cold, well-shielded portions of L1544 5 .We thus assume that the column density sampled by the HINSA can be approximated by that obtained from dust, despite the substantially larger apparent area covered by HINSA (Fig. 1a).The previous OH Zeeman detection with Arecibo 24 toward the L1544 center resulted in a field strength of B los = +10.8± 1.7 µG, where B los is the magnetic field component along the line of sight with positive sign representing field pointing away from the observer.In contrast, the OH Zeeman observations of the Green Bank Telescope (GBT) toward four envelope locations 6.0' (0.24 pc) from the center yielded a marginal detection B los = +2 ± 3 µG 16 , leaving the structure of envelope field undetermined.
With the Five-hundred-meter Aperture Spherical radio Telescope (FAST) 25 , we detected Zeeman splittings in a 2.9' beam (0.12 pc) toward the HINSA column density peak, 3.6' (0.15 pc) away from the L1544 center (Fig. 1).The spectra of the Stokes I(v) and V(v) parameters (where v denotes velocity) are shown in Fig. 2. The I(v) spectrum contains HI emission of CNM and warm neutral medium (WNM) clouds in the direction toward the Taurus complex and a HINSA feature at the centroid velocity of L1544.Fig. 2a shows our decomposition of I(v) into a foreground HINSA component, a background WNM component, and three CNM components between the HINSA and WNM.Our fitted parameters of the HINSA component are in good agreement with the previous HINSA observations 4,5 , and our parameters of the CNM and WNM components are similar to the Arecibo results toward quasars around L1544 26 .
The V(v) spectrum shows features of classic 'S curve' patterns proportional to the first derivatives of I(v) for the HINSA , CNM, and WNM components, as expected for Zeeman splittings.The Zeeman splitting profile of HINSA has a maximum at high velocity and a minimum at low velocity, opposite to the Zeeman splitting profile of CNM1, the closest CNM component at a velocity similar to L1544, that shows positive V at low velocity and negative V at high velocity.From our least-squares fits to V(v), Fig. 2b shows the Zeeman splitting of the HINSA and the total Zeeman profile of the five components, and Fig. 3  Comparing the Zeeman observations of HINSA , OH, and HI tracing the CNM1 and the molecular envelope of L1544, it is clear that the magnetic fields at distances of 0.15, 0.24, 0.72, and 7.1 pc from the center all have the same direction of B los and consistent strengths roughly within the 1σ.This finding is in agreement with the conclusion of a median value of 6 µG in absolute total strength in HI clouds inferred from comprehensive Zeeman surveys 15 .HINSA Zeeman effect thus provides a connection between the magnetic fields from HI clouds to molecular clouds.
The HI emission components (CNM1, CNM2, CNM3) and the HI absorption toward 3C133 and 3C132 trace CNM with a kinematic temperature of about 100 K 27 and a number density of about 40 cm −3 15 , whereas the HINSA and OH observations trace the envelope of about 10-15 K 4 and ∼ 10 3 cm −3 5, 15 .Despite the 1-2 orders of magnitude change in both temperature and density in the phase transition from the atomic CNM to the molecular envelope, the Zeeman observations reveal a magnetic field that is coherent in both direction and strength across multi-scales and multi-phases of the interstellar medium.To constrain the uniformity of the coherent magnetic field, our likelihood analysis of the HINSA , OH, and HI Zeeman measurements suggests a gaussian distribution of the B los with a mean strength of B 0 = +4.1 ± 1.6 µG and an intrinsic spread of σ 0 = 1.2 +1.2 −0.6 µG, a significantly better constraint than the previous estimation of B 0 = +4 +10 −8 µG based only on the OH results 18 .
It is well known that the progenitor of molecular gas, the atomic CNM, is strongly magnetized, as measured by the dimensionless mass-to-magnetic flux ratio λ in units of the critical value where N H 2 is the column density of H 2 gas and B tot is the total magnetic field strength 1 ), which is well below unity (i.e., magnetically subcritical) 28 .
On the other hand, the immediate progenitors of stars, the prestellar cores of molecular clouds such as the L1554 core, are observed to be magnetically supercritical (λ > 1) 24 , which is required for the self-gravity to overwhelm the magnetic support and form stars through gravitational collapse.
When and how the transition from the magnetically subcritical CNM that is incapable of forming stars through direct gravitational collapse to the supercritical star-forming cores occurs is a central unresolved question in star formation.
Our HINSA Zeeman observations can be used to address this question.Using the physical parameters of the clouds (Table 1) and the statistically most probable value of B tot = 2B los , the λ of CNM1 is about 0.10-0.18,consistent with previous results 28 .The λ of the envelope and core of L1544 core is 2.5-3.5, which is well above unity, indicating that the transition to magnetic supercriticality has already occurred.We further consider the relative values of λ between CNM1 and L1544 to avoid the geometrical correction from B los to B tot 16 , assuming that the inclination angles of the magnetic fields in the L1544 core and envelope are similar.Therefore, the molecular envelope of the L1544 core traced by HINSA is at least 13 times less magnetized relative to its mass compared to its ambient CNM.This is different from the "classic" theory of low-mass star formation, which envisions the transition from magnetic subcriticality to supercriticality occurring as the supercritical core forms out of the magnetically supported (subcritical) envelope 13,14 .Our results suggest that the transition from magnetic subcriticality to supercriticality occurs earlier, during the formation of the molecular envelope, favoring the more rapidly evolving scenario of core formation and evolution for L1544 8 over the slower, magnetically retarded scenario 9 .In other words, by the time that the molecular envelope is formed, the problem of excessive magnetic flux as a fundamental obstacle to gravitational collapse and star formation is already resolved.This early reduction of flux relative to mass is unlikely due to the "classical" scenario where gravity drives neutrals through ions (and the magnetic field tied to them) in a process called "ambipolar diffusion" because the CNM is not self-gravitating.The coherent magnetic fields reviewed here provide a new specific question on how to create supercritical dense cores such as L1544 from subcritical clouds.
Plausible scenarios include mass accumulation along field lines 29 and (turbulence-enhanced) magnetic reconnection 30 , although whether such scenarios can reproduce the distributions of gas and magnetic field observed in the L1544 region remains to be seen.In any case, the already magnetically supercritical envelope can in principle go on to form dense cores and stars without having to further reduce its magnetic flux relative to the mass.The CNM and WNM Zeeman profiles include the absorption from the CNM components that lie in front but not include the absorption from HINSA .The sum of the red profiles of the five components is the black dashed profile in Fig. 2b.

Data reduction
The FAST Zeeman observations toward the HINSA column density peak in L1544 were carried out on five days between August and November of 2019 with a total integration time of 7.6 hours.The HINSA spectra were obtained with the central beam of the L-band 19-beam receiver 31 .The central beam has an average system temperature of 24 K, a main beam efficiency of 0.63, and a main beam diameter at the half-power point of 2.9' with a pointing accuracy of 7.9".The 19-beam receiver had orthogonal linear polarization feeds followed by a temperature stabilized noise injection system and low noise amplifiers to produce the X and Y signals of the two polarization paths.The XX, YY, XY, and YX correlations of the signals then were simultaneously recorded using the ROACH backend with 65536 spectral channels in each polarization.The spectral bandwidth was 32.75 MHz centered at the frequency of the HI 21-cm line for a channel spacing of 500 Hz, and the V(v) spectrum presented in this work was Hanning-smoothed, which produced a spectral resolution of 0.21 km s −1 .
The data reduction including gain and phase calibrations of the two polarization paths, bandpass calibrations of the four correlated spectra, and polarization calibrations to generate Stokes I, Q, U, and V spectra was made with the IDL RHSTK package written by C. Heiles and T. Robishaw that is widely used for Arecibo and GBT polarization data.The 19-beam receiver is rotatable from -80 • to +80 • with respect to the line of equatorial latitude.The polarization calibrations used drifting scans of the continuum source 3C286 at rotation angles of -60 • , -30 • , 0 • , 30 • , and 60 • over 1.5 hours surrounding its transit.The details of the polarization calibration procedure were provided in 32 .We performed polarization calibrations once a month during the observations.The calibrated polarization of 3C286 of the three epochs were 8.9% ± 0.1%, 8.7% ± 0.2%, and 9.0% ± 0.1% for polarization degrees and 30.4 • ± 0.3 • , 33.8 • ± 0.5 • , and 29.4 • ± 0.3 • for polarization angles.
Considering that the ionosphere can generate a faraday rotation of 1 • -3 • in polarization angle at L-band 33 , our results were consistent with the intrinsic polarization degree of 9.5% and polarization angle of 33 • of 3C286 at 1450 MHz 34 .In addition to the polarization observations of L1544 and 3C286, we observed the circularly polarized OH maser source IRAS02524+2046 35 in order to verify that our procedures produced consistent B los , including the sign or direction of the magnetic field, as had been obtained previously.
The convolutions of the sidelobes of the Stokes V beam with the spatial gradient of the Stokes I emission may generate a false 'S curve' in the V spectrum 27 .In order to check the credibility of our Zeeman detections, we measured the Stokes V beam of FAST and convolved the beam with the GALFA Stokes I cube 36 of L1544.The convolved V spectrum showed a profile with a shape similar to the I spectrum and a strength less than 0.03% of the I spectrum, different from the 'S curve' patterns in the observed V spectrum.Meanwhile, the 19-beam receiver was rotated to -45 • , 0 • , and 45 • in the three epochs of the L1544 observations, and all of the three epochs showed 'S curve' patterns in the V spectra, indicating that our Zeeman results were true detections.
Although the data of the 19 beams of the FAST L-band receiver were simultaneously taken in our observations, only the polarization of the central beam were commissioned at the time of writing.The results represented in this work were made with only the central beam pointing toward the HINSA column density peak in Fig. 1.The Zeeman results of the 18 off-central beams will be published in the future.
Multiple gaussians and radiative transfer fitting to I(v) and V(v) We adopts the least-squares fits of multiple gaussians with radiative transfer 26 to decompose the I(v) into HINSA , CNM, and WNM components.The expected profile of I(v) consists of multiple CNM components providing opacity and also brightness temperature and a WNM component providing only brightness temperature: (1) The I CN M (v) is an assembly of N CNM components where the subscript m with its associated optical depth profile τ m (v) represents each of the M CNM clouds that lie in front of cloud n.The optical depth of the ith component is in which τ 0 represents the HINSA providing only opacity and no brightness temperature.For the WNM in the background, The fitting of I(v) thus yields values for I peak , τ, v 0 , and σ v of the components.
We consider the radiative transfer of V(v) in terms of right circular polarization (RCP) and left circular polarization (LCP).The Zeeman effect states that with the existence of B los , the frequency of RCP shifts from its original frequency ν 0 to ν 0 + ν z and the frequency of LCP shifts to ν 0 − ν z with ν z = (Z/2) × B los , where Z is the Zeeman splitting factor (2.8 Hz µG −1 for HI 21cm line).Since the RCP and LCP are orthogonal components of radiation, the radiative transfer fits of the 250/350/500 µm spectral energy distributions weighted by the squares of the measured noise levels to derive the pixel-to-pixel distributions of dust temperature T d and dust optical depth for the calibration uncertainty in SPIRE 40 .We then estimated the uncertainty in each pixel with 1000 fittings of the H 2 column density.The N H 2 and its uncertainty in Table 1 were obtained from the convolutions of the H 2 column density map and uncertainty map with the FAST and Arecibo beams.
Note that the equivalent H 2 column density N H 2 of the CNM1 are derived from HI data toward 3C132 and 3C133, a method different to the N H 2 of the L1544 envelope and core that are derived from dust emission.Therefore, in addition to the statistical errors listed in Table 1, there is a systematic difference between the N H 2 derived from the two methods.Considering that the regime traced by dust emission can be different from those traced by HINSA or OH, which is particularly noticeable from the different spacial extents of dust and HINSA in Fig. 1a, we expect that the systematic difference could be as large as a factor of a few.Since the values of λ between lihoods of the observations (L = N J=1 l j ), the B 0 and σ 0 can be estimated by maximizing the likelihood L. After performing the integration in Equation ( 7) and some algebraic manipulations, Extended Data Figure 1 shows the distribution of L as functions of B 0 and σ 0 and the probability distributions of B 0 and σ 0 by integrating L along the B 0 axis and the σ 0 axis, respectively.The probability distribution of B 0 is similar to a normal distribution with a mean value of +4.1 µG and a standard deviation of 1.6 µG.The probability distribution of σ 0 is highly asymmetric since the values of σ 0 cannot be negative.The first, second, and third quartiles of the σ 0 distribution are 0.6, 1.2, and 2.4 µG.We therefore suspect that the Zeeman measurements in the L1544 envelope can be explain by a magnetic field with B 0 = +4.1 ± 1.6 µG and σ 0 = 1.2 +1.2 −0.6 µG.
Inclination angle of magnetic field Given the uniformity of magnetic fields in the envelope of L1544 and CNM1 is well constrained by the maximum likelihood analysis, the coherent B los suggests that the inclination angles of magnetic fields in the CNM1 and L1544 envelope are likely to be similar, or a special geometry of magnetic field structure across multi-scales and multi-phases of the interstellar medium is needed.In contrast, the B los of the L1544 envelope and core differ by a factor of 2.6.There are two physical explanations for the 2.6-time difference between the HINSA and OH Zeeman measurements.First, the OH measurement likely samples a denser gas than the HINSA measurement, since the column density along the OH sightline is twice that along the HINSA sightline (Table 1).Since the magnetic field strength in molecular clouds tends to increase with number density 15 , the stronger field is naturally expected in the denser core.Alternatively, the inclination angle of the L1544 core magnetic field could differ substantially from CCS Zeeman Measurements Ref. 20 reported a CCS Zeeman detection of 117 ± 21 µG in a dense core of TMC-1 that has an estimated H 2 column density of 3 × 10 22 cm −2 , which is 4 times higher than that probed by OH Zeeman measurements in L1544 and nearly one order of magnitude higher than that probed by our HINSA measurements.It appears to provide further support to the evolutionary scenario suggested by our HINSA measurements: namely, once the gas loses its magnetic support during the transition from CNM to the molecular envelope (or ridge) and becomes magnetically supercritical, there is no longer any need to lose magnetic flux further (relative to the mass) in order for a piece of the envelope/ridge to condense into a (magnetically supercritical) core (e.g., the L1544 core probed by OH) and for the core to evolve further by increasing its column density (e.g., the TMC-1 core probed by CCS).
Technically, we note that one potential source of significant uncertainty in frequency shift, namely the uncertainty of beam squint, was not included in the CCS result, which may affect the level of significance.In comparison, the HINSA measurement is robust with a > 10σ significance with the beam squint and velocity gradient been taken into account by convolving the FAST Stokes V beam with the Stokes I cube of L1544 (see the third paragraph in the section of data reduction in Methods).
https://github.com/taochung/HINSAzeeman. || The order of the component along the line of sight.Order begins with 0, and increasing numbers mean increasing distance along the line of sight.We fix order = 0 for HINSA since Taurus cloud is one of the closest cloud to us.
The orders of the other components are free parameters in the fitting.
Extended Data Figure 1: Likelihood L for the coherent magnetic field to have mean (B 0 ) and spread (σ 0 ) values.a, contours of L as functions of B 0 and σ 0 plotted at 10%, 30%, 50%, 70%, and 90% of the peak value.b, the probability distribution of B 0 while allowing all possible values of σ 0 .c, the probability distribution of σ 0 while allowing all possible values of B 0 .
shows the individual Zeeman splittings and B los of the components.The HINSA Zeeman effect gives B los = +3.8± 0.3 µG, and the HI Zeeman effect of CNM1 gives B los = +4.0 ± 1.1 µG.The magnetic field strengths of HINSA and CNM1 are consistent with the results of B los = +5.8± 1.1 µG and +4.2 ± 1.0 µG obtained from the Zeeman observations toward quasars 3C133 and 3C132, probing the magnetic fields of CNM1 at distances of 17.7' (0.72 pc) and 174.5' (7.1 pc) from L1544, respectively 27 .For the second and third CNM components (CNM2 and CNM3) along the line of sight, our results of B los,CN M2 = −7.6 ± 1.0 µG and B los,CN M3 = +2.9± 0.4 µG are also consistent with the results of B los,CN M2 = −9.6 ± 6.3 µG and B los,CN M3 = −0.3± 1.7 µG toward quasar 3C133 27 .

Figure 1 :
Figure 1: L1544 core and illustration of the structure of interstellar medium from CNM to

Figure 3 :
Figure 3: The individual V(v) profiles for the HINSA , CNM, and WNM components.In each where S ν is the flux density at frequency ν, Ω m is the is the solid angle of the pixel, B µ (T d ) is the Planck function at T d , and τ ν = τ 230 (ν[GHz]/230) β with dust opacity index β of 1.8.Next, we obtained the H 2 column density with N H 2 = gτ 230 /(κ 230 µ m m H ), where g = 100 is the gas-to-dust mass ratio, κ 230 = 0.09 cm 2 g −2 39 is the dust opacity at 230 GHz, µ m = 2.8 is the the mean molecular weight, and m H is the atomic mass of hydrogen.To estimate the uncertainties in the H 2 column density, we used a Monte-Carlo technique.For each pixel, we created artificial 250/350/500 µm flux densities by adding the original flux densities with normal-distributed errors taking account the uncertainty in the measured flux and a 10% correlation
1. McKee, C. F. & Ostriker, E. C. Theory of Star Formation.Annu.Rev. Astron.Astrophys.45, Formation.Front.Astron.Space Sci. 6, 66 (2019).28 Extended Data Table 1: Gaussian Fit ParametersComponent I peak [K] * τ † v LS R [km s −1 ] ‡ σ v [km s −1 ] § I peakis the intrinsic peak Stokes I emission.We do not fit I peak for HINSA because it is an absorption component.† τ is the central opacity.We do not fit τ for WNM because it is a background component.‡ v LS R is the central LSR velocity.§ σ v is the gaussian dispersion. *