Free Access
Issue
A&A
Volume 615, July 2018
Article Number A58
Number of page(s) 11
Section Interstellar and circumstellar matter
DOI https://doi.org/10.1051/0004-6361/201732195
Published online 13 July 2018

© ESO 2018

1 Introduction

With recent interferometric observations at (sub-)millimeter wavelengths, Keplerian disks with radii of tens to hundreds of au have been detected around several embedded Class 0 and I protostars (Lommen et al. 2008; Takakuwa et al. 2012; Tobin et al. 2012; Brinch & Jørgensen 2013; Murillo et al. 2013; Ohashi et al. 2014; Chou et al. 2014, 2016; Harsono et al. 2014; Lindberg et al. 2014; Lee et al. 2014, 2016; Yen et al. 2014, 2017; Aso et al. 2015). On the other hand, there is a group of Class 0 protostars with envelope rotations on a 1000 au scale that are one order of magnitude slower than other Class 0 protostars (Brinch et al. 2009; Yen et al. 2010, 2015a; Maret et al. 2014), and they do not exhibit Keplerian disks with sizes larger than 10–20 au (Maury et al. 2010; Oya et al. 2014; Yen et al. 2017). The origin of the discrepancy in the gas kinematics and the disk sizes between these young protostars and others is not clear. It can be due to difference in their ages, initial rotation of their parental cores, and/or effects of the magnetic field (Yen et al. 2015a,b, 2017).

Molecular clouds are magnetized (Crutcher 2012). The magnetic field can slow down the gas motion in collapsing dense cores and transfer the angular momentum of the collapsing material outward, and consequently suppress the formation and growth of Keplerian disks around protostars (Allen et al. 2003). Ideal magnetohydrodynamic (MHD) simulations of collapsing dense cores where the rotational axis is aligned with the magnetic field show that no Keplerian disk with a size larger than 10 au can form as a result of efficient magnetic braking (e.g., Galli et al. 2006; Mellon & Li 2008; Joos et al. 2012). This is consistent with the observations of some Class 0 protostars exhibiting slow envelope rotation and Keplerian disks smaller than 10–20 au, but it contradicts other observations showing an increasing number of Keplerian disks with radii larger than tens of au. In addition, 100 au scale Keplerian disks are often observed around T Tauri and Herbig Ae/Be stars (Williams & Cieza 2011). Simulations have demonstrated that when non-ideal MHD effects, dissipation of protostellar envelopes, initially misaligned rotational axis and magnetic field, more realistic treatment of ionization degrees, or turbulence are considered, the efficiency of magnetic braking can be reduced to enable the formation of Keplerian disks larger than 10 au (Hennebelle & Ciardi 2009; Krasnopolsky et al. 2011; Li et al. 2011, 2013; Machida & Matsumoto 2011; Machida et al. 2011, 2014; Dapp et al. 2012; Joos et al. 2012, 2013; Santos-Lima et al. 2012; Seifried et al. 2012, 2013; Padovani et al. 2013, 2014; Tomida et al. 2015; Tsukamoto et al. 2015a,b; Hennebelle et al. 2016; Zhao et al. 2016, 2018). Nevertheless, these mechanisms have not yet been observationally confirmed or constrained.

Ambipolar diffusion is one of the non-ideal MHD effects proposed to enable the formation of large-scale Keplerian disks. With ambipolar diffusion, ions and neutrals can have a relative drift, and the magnetic field, which is tied to the ions, is not draggedtoward the center during the collapse as quickly as in the ideal MHD case (Li et al. 2014). Thus, the magnetic flux is redistributed and is partially left in an outer region, and the mass-to-flux ratio increases toward the center (e.g., Fig. 7 in Dapp et al. 2012 and Fig. 6 in Tomida et al. 2015). As a result, the efficiency of magnetic braking can be reduced to form 10 au or larger Keplerian disks depending on the magnetic diffusivity of ambipolar diffusion (e.g., Dapp et al. 2012; Tomida et al. 2015; Tsukamoto et al. 2015a; Masson et al. 2016; Zhao et al. 2016, 2018). Thus, observational studies comparing the motions of ionized and neutral gas are essential to understand ambipolar diffusion in protostellar sources and its effects on the formation and growth of Keplerian disks (e.g., Caselli et al. 2002a,b).

The isolated Bok globule B335 has an embedded Class 0 protostar (IRAS 19347 + 0727) at a distance of ~100 pc (Keene et al. 1980, 1983; Stutz et al. 2008; Olofsson & Olofsson 2009). It is associated with a large-scale CO outflow (e.g., Hirano et al. 1988; Yen et al. 2010; Hull et al. 2014) as well as Herbig–Haro objects (Reipurth et al. 1992; Gâlfalk & Olofsson 2007). The signatures of the infalling motion in B335 have been observed with single-dish telescopes and interferometers on scales from hundreds to thousands of au (Zhou et al. 1993; Zhou 1995; Choi et al. 1995; Evans et al. 2005, 2015; Saito et al. 1999; Yen et al. 2010; Kurono et al. 2013). B335 is slowly rotating with a decreasing specific angular momentum on scales from 0.1 pc to 1000 au (Saito et al. 1999; Yen et al. 2011; Kurono et al. 2013), and the radial profile of its specific angular momentum flattens at an inner radius of 1000 au (Yen et al. 2015b). No Keplerian disk with a size larger than 10 au is detected in B335 with the Atacama Large Millimeter/submillimeter Array (ALMA) observations (Yen et al. 2015b). Near-infrared polarimetric observations show that the magnetic field on a 0.2 pc scale in B335 is tilted from the outflow axis by 35°–60° (Bertrang et al. 2014). The structures of the magnetic field become more disordered and are misaligned with the outflow axis on a scale of a few thousand au, as revealed by single-dish and interferometric polarimetric observations at (sub-)millimeter wavelengths (Wolf et al. 2003; Davidson et al. 2011; Chapman et al. 2013; Hull et al. 2014). Because of its slowly rotating inner envelope and absence of a large disk, B335 is a promising candidate with efficient magnetic braking (Yen et al. 2015b). Therefore, B335 is an excellent target for probing the absence or existence of ambipolar diffusion and for studying the effects of the magnetic field in the process of star formation.

We here report our observational results of the H13 CO+ (3–2; 260.255342 GHz) emission in B335 obtained with ALMA, and we compare the gas kinematics traced by the H13 CO+ (3–2) line with the C18O (2–1) results presented in Yen et al. (2015b). C18O and H13 CO+ are chemically similar (Lee et al. 2004; Aikawa et al. 2008), and both are expected to be abundant in the inner envelope on a scale of hundreds of au in B335 (Evans et al. 2005). The C18 O (2–1) line has an upper energy level of 16 K and a critical density of ~104 cm−3, and the H13 CO+ (3–2) line has an upper energy level of 25 K and a critical density of ~105 cm−3. The upper energy levels and critical densities of both lines are below the temperature and density in the protostellar envelope on a scale within a few hundred au in B335 (Harvey et al. 2003; Shirley et al. 2011), and both lines are expected to be optically thin on this scale based on the typical column densities of C18 O and H13 CO+ in protostellar envelopes on a scale of hundreds of au (Hogerheijde et al. 1998). Therefore we here adopt these two lines to trace the kinematics of the ionized and neutral gas in the protostellar envelope in B335, and we compare the measured infalling velocities from these two lines and discuss the implication of our results on the effects of ambipolar diffusion in B335.

2 Observations

The data of B335 presented in this paper were obtained with the ALMA observations with 40 antennas during the cycle-3 observing period on May 23 and June 3, 2016. The array configuration was C36-4 with the shortest baseline length of ~15 m (~13 kλ). Because of different array configurations, the shortest baseline length in the H13 CO+ observations is half of that in the ALMA C18 O observations with the C34-6 configuration (Yen et al. 2015b). With this shortest baseline length, our H13 CO+ observations have a largest recoverable angular scale of 7′′ at a 50% level (Wilner & Welch 1994). The pointing center was α(J2000) = 19h 37m00. s89, δ(J2000) = 7°34′9. ′′6. The on-source integration time on B335 was ~80 min. The correlator was configured in the frequency division mode, and a spectral window with a bandwidth of 117.2 MHz was assigned to the H13 CO+ emission with 960 channels, resulting in a channel width of 122.1 kHz. The 1.2 mm continuum was observed simultaneously with a total bandwidth of 2 GHz. We have confirmed that the peak position of the 1.2 mm continuum emission is consistent with the 1.3 mm continuum emission in Yen et al. (2015b), so that there is no relative positional offset between the two data sets. The position of the continuum peak, which is the same as the pointing center, was adopted as the protostellar position. J1935 + 2031 (0.75 Jy at 260.2 GHz) and J2035 + 1056 (0.31 Jy at 260.2 GHz) were observed as gain calibrators in the first and second observations, respectively. J2148 + 0657 was observed as a bandpass and flux calibrator. Calibration of the raw visibility data was performed with the standard reduction script for the cycle-3 data, which uses tasks in Common Astronomy Software Applications (CASA; McMullin et al. 2007) of version 4.5.3. The image of the H13 CO+ emission was generated with the Briggs robust parameter of 0 from the calibrated visibility data and CLEANed with the CASA task “clean” at a velocity resolution of 0.17 km s−1. This velocity resolution is the same as that in the C18 O image obtained from Yen et al. (2015b). The achieved synthesized beam is 0. ′′ 5 × 0. ′′ 4 with a position angle (PA) of 111°, which is a factor of 1.5 larger than that of the C18O image. The achieved rms noise is 3.3 mJy beam−1 per channel, comparable to that of the C18O observations (3.8 mJy beam−1 per channel).

thumbnail Fig. 1

Moment 0 (contours) overlaid on moment 1 (color) maps of the (a) H13 CO+ (3–2) and (b) C18 O (2–1) emission in B335 obtained with the ALMA observations. Color scales are in units of km s−1 in the LSR frame. The color scales are the same in (a) and (b). Crosses denote the protostellar position. Red and blue arrows show the direction of the blue- and redshifted outflows, respectively. Contour levels in (a) are 3σ, 6σ, 10σ, 15σ, 20σ, 30σ, and 45σ, and in (b) are 3σ, 6σ, 10σ, 15σ, 25σ, and 40σ, where 1σ is 3.3 and 3.4 mJy beam−1 km s−1, respectively.

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3 Results

Figure 1 presents the total integrated-intensity (moment 0) and mean intensity-weighted velocity (moment 1) maps of the H13 CO+ and C18O emission. The maps of the C18O emission have been presented in Yen et al. (2015b) and are also shown here for a direct comparison with the H13 CO+ maps. The distributions of both emission lines are centrally peaked at the protostellar position, and the H13 CO+ emission exhibits a blueshifted extension on a scale of 5′′ (500 au) toward the east. A similar blueshifted extension toward the east is also observed in the C18 O emission, but on a smaller scale. The observed H13 CO+ emission is more extended than the C18O emission because the shortest baseline length in the H13 CO+ observationsis almost half of that in the C18O observations. The central parts of both emission lines are more elongated along the north–south direction. By fitting a two-dimensional Gaussian distribution to the central part of these emission lines, where the integrated intensity is >20σ in the H13 CO+ emission and >10σ in the C18O emission, the PA of the elongation of the H13 CO+ and C18O emission around the protostar is measured to be − 12°±19° and 17°±11°, respectively.These elongations are perpendicular to the outflow direction with a PA of 99° (Hull et al. 2014) and are more aligned with the flattened envelope with a PA of 16° on a 100 au scale traced by the 1.3 mm continuum (Yen et al. 2015b). The H13 CO+ and C18O emission lines both exhibit a clear velocity gradient along the east–west direction, where the eastern side is more blueshifted and the western side is more redshifted. The direction of this velocity gradient is consistent with the associated outflow. As discussed in Yen et al. (2015b), the blueshifted extended emission toward the east is most likely related to the blueshifted outflow, and the velocity gradient in the central part, where the emission is elongated along the direction of the flattened envelope, can be due to the infalling motion in the envelope.

Figure 2 presents the spectra of the C18 O and H13 CO+ emission at the protostellar position. To compare the spectra of the two lines obtained with different observations, both data were first convolved with the same circular beam with a size of 0. ′′ 5, and the intensity is converted into brightness temperature in units of K. Both emission lines show intensity peaks at similar velocities, VLSR of 8 and 9 km s−1 as well as a dip close to the systemic velocity of VLSR = 8.34 km s−1. The H13 CO+ emission additionally shows a redshifted line wing at VLSR > 10.5 km s−1. We integrated the H13 CO+ emission over the three different velocity regimes, low velocity (VLSR = 7.9–8.3 and 8.3– 9.1 km s−1), high velocity (VLSR = 6–7.9 and 9.1–10.8 km s−1), and very high velocity (VLSR = 10.8–12 km s−1), shown in Fig. 3. Similar maps of the C18O emission have been presented in Yen et al. (2015b). The extended emission is primarily observed at the low velocities, and the emission is elongated along the north-south direction with a blueshifted extension toward the east. A separated redshifted component is seen at ~7′′ southwest to the protostar, possibly associated with the wall of the outflow cavity. Thus, at the low velocities, the H13 CO+ emission likely traces the flattened envelope with a possible contamination from the outflow. On the other hand, at high velocities, blue- and redshifted compact components with a size of ~2′′ (200 au) are clearly seen and are not surrounded by any extended structures, suggesting that the emission becomes less contaminated by the outflow. There is an additional component at even higher velocities with a relative velocity (Δ V) of 2.5–4 km s−1 in the H13 CO+ emission. This component is not observed in the C18O emission, and neither does it have a blueshifted counterpart. Because this very high velocity component is not resolved with our observations, its origin is not clear. If it originates from the infalling motion in the inner envelope, its distance to the protostar is estimated to be ~5–10 au on the assumption of a protostellar mass of 0.04 M (Yen et al. 2015b). However, in this case, a blueshifted counterpart is expected, which is not observed. The other possibility is that this very high velocity component in the H13 CO+ emission is related to the jets in B335. Emission at an extremely high velocity up to ~20 km s−1 tracing the jets has been observed in the 12CO (2–1) line with the SMA (Yen et al. 2010).

Figure 4 presents the position–velocity (PV) diagrams along the major and minor axes of the flattened envelope passing through the protostellar position. The PA of the major axis of the flattened envelope has been measured to be 16° from the 1.3 mm continuum at an angular resolution of 0. ′′ 3 (Yen et al. 2015b). The PV diagrams of the H13 CO+ emission are morphologically similar to those in the C18O emission (Fig. 4 in Yen et al. 2015b). The PV diagrams show that the velocity width increases as radii decrease, and in the central 2′′ region alongthe major axis, there are both blue- and redshifted components at the same offsets. These features are consistent with the signature of an edge-on infalling envelope (e.g., Fig. 10 in Ohashi et al. 1997) and can be explained with accelerating infalling gas behind and in front of the protostar along the line of sight. On the other hand, there is no clear positional offset between the blue- and redshifted emission in the central H13 CO+ component (|offset| < 1′′) in both PV diagrams, different from the C18O observations, which show clear velocity gradients along both axes (Yen et al. 2015b). The reason might be the lower angular resolution of the H13 CO+ observations. The linear velocity gradient along the major axis of the envelope in the C18 O emission is measured to be 40.5 km s−1 arcsec−1 from the high-velocity (|Δ V | > 1.2 km s−1) channels in the C18O PV diagram (Yen et al. 2015b). From this linear velocity gradient, the peak offset between blueshifted and redshifted emission at Δ V of ±1.2 km s−1 is expected to be 0. ′′06. At ΔV of ±1.2 km s−1, the signal-to-noise ratios (S/N) of the H13 CO+ emission in our observations is ~10 at the peaks. The relative positional accuracy of our observations at these velocities is approximately the angular resolution of 0. ′′ 5 divided by the S/N of 10, yielding 0. ′′05, which is comparable to the expected positional offset from the velocity gradient. Therefore, the angular resolution of our H13 CO+ observations is not sufficient to resolve the velocity gradient observed in the C18 O emission. We are thus not able to detect the slow envelope rotation on a 100 au scale, which we did observe in the C18 O and SO emission, with our H13 CO+ observations.

thumbnail Fig. 2

Spectra of the C18O and H13CO+ emission at the protostellar position obtained with our ALMA observations, shown in blue and green histograms. The data were first convolved with the same circular beam with a size of 0. ′′ 5 to extract the spectra. A vertical dotted line denotes the systemic velocity of 8.34 km s−1.

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thumbnail Fig. 3

Moment 0 maps of the H13CO+ emission integrated over different velocity regimes, (a) low velocity VLSR = 7.9–8.3 and 8.3–9.1 km s−1, (b) high velocity VLSR = 6–7.9 and 9.1–10.8 km s−1, and (c) very high velocity VLSR = 10.8–12 km s−1. Blue and red contours present the blue- and redshifted emission, respectively. Crosses denote the protostellar position. Red and blue arrows show the direction of the blue- and redshifted outflows, respectively. A green box in (a) presents the image size of (b) and (c). Filled blue ellipses show the size of the synthesized beam of 0. ′′ 5 × 0. ′′ 4. Contour levels in (a) are 5σ, 10σ, 15σ, and 20σ and then in steps of 10σ, in (b) are 3σ, 6σ, 10σ, 15σ, 20σ, and 25σ and then in steps of 10σ, and in (c) are from 3σ in steps of 3σ, where 1σ is 1.1, 1.8, and 1.6 mJy beam−1 km s−1, respectively.

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4 Kinematical models of infalling and rotational motions

The central component with a radius of 2′′ in the H13 CO+ emission is elongated along the major axis of the flattened envelope and is perpendicular to the outflow direction (Fig. 1). Its velocity features can be explained with infalling motion in an edge-on envelope (Fig. 4). Thus, the central H13 CO+ emission likely traces the infalling flattened envelope around B335, and no envelope rotation is detected in the H13 CO+ line. To measure the infalling velocity in the H13 CO+ line, we constructed kinematical models of an infalling flattened envelope to fit the observed velocity structures in the PV diagrams of the H13 CO+ emission. In order to compare the infalling velocities measured in the H13 CO+ and C18O lines, we also reanalyzed the C18O data with the same kinematical models adopted in this work.

The model envelope has three dimensions and is assumed to have power-law density and temperature profiles. Its number density within 60° from the polar axis is artificially set to zero to mimic the outflow cavity and the flattened envelope, meaning that the thickness of the model envelope h(r) = rsin30°, identical to Yen et al. (2015b). This radial profile of the envelope thickness was determined based on the aspect ratios of the 1.3 mm continuum emission observed with the SMA and ALMA because B335 is almost edge on. We fixed the temperature profile in the model envelope because the temperature cannot be well constrained with only one transition in the H13 CO+ and C18O lines. The power-law index of the temperature profile is adopted to be − 0.4 (Shirley et al. 2000, 2011), and the temperature is adopted to be 38 K at a radius of 100 au (Evans et al. 2015). As discussed in Yen et al. (2017), because the emission is optically thin, the fitted velocity profile is not sensitive to the adopted temperature profile, and adopting a different temperature profile results in a different fitted density profile, which compensates for the change in the line intensity due to the different temperature profile. Thus, the density and temperature profiles of the model envelope are described as (1)

and (2)

where n is the number density of C18O or H13 CO+ and R0 is the characteristic radius adopted to be 100 au. The outer radius is adopted to be 7′′ (700 au), so the diameter of the model envelope is twice larger than the maximum recoverable angular scale of the H13 CO+ observations. The model envelope is assumed to be free-falling with a constant specific angular momentum. The radial profiles of the infalling and rotational velocity (Vin and Vrot) are described as (3)

and (4)

where j is the specific angular momentum and θ is the anglebetween the radius and the polar axis. As described below, the H13 CO+ and C18O data are fitted separately. The best-fit n(R0), p, Vin (R0), and j can be different for the H13 CO+ and C18O emission. We adopt the latest estimates of the distance of 100 pc (Olofsson & Olofsson 2009) and an inclination angle of 87° (Stutz et al. 2008) for B335 to compute the model images.

We first performed the χ2 fitting on the C18 O PV diagrams along the major and minor axes. The model images in the C18 O emission were computed on the assumption of the local thermal equilibrium (LTE) condition because the C18 O (2–1) line is expected to be thermalized with the typical physical condition (nH2 > 104 cm−3) in protostellar envelopes on a scale of a few hundred au (e.g., Shirley et al. 2000). Then we simulated ALMA observations of the model images with the same array configuration and coverage of hour angle as our real observations. Therefore, the effects of different uv sampling and angular resolutions between the H13 CO+ and C18 O observations are included in our analysis. We generated synthetic images from the simulated visibility data with the same imaging parameters as Yen et al. (2015b) and extracted PV diagrams from the synthetic images. The channels at the low velocities of VLSR = 7.7–8.9 km s−1 (vertical dashed lines in Fig. 5) are excluded in the fitting to have minimal contamination from the extended structures and the outflows. As demonstrated in Yen et al. (2015b), the envelope rotation is so slow that it does not affect the measurements of the infalling velocity. In addition, the signature of the envelope rotation is not detected in the H13 CO+ emission. Thus, the fitting was performed with two fixed j, 3 × 10−5 km s−1 pc, as derived from Yen et al. (2015b), with the correction of the different adopted distances, and 0, meaning no rotation. We confirmed that the best-fit Vin(R0) is the same with these two j because the gas kinematics is dominated by the infalling motion.

The best-fit results of the C18O data are shown in Fig. 5 and are listed in Table 1. The χ2 of our best-fit model is 2.6. The uncertainties of our best-fit parameters are estimated from the parameter ranges of the models having χ2 less than the minimum χ2 + 1 (Table 1). Figure 6 presents χ2 as a function of Vin(100 au) from our fitting results. In Appendix A, we present PV diagrams extracted from the synthetic images of our kinematical models with different Vin(100 au) to demonstrate the dependence of the velocity structures on the infalling velocity. These comparisons show that the infalling velocity at a radius of 100 au is most likely within the range of 0.85 ± 0.2 km s−1.

The best-fit C18 O number density is 2.7 cm−3. On the assumption of a typical C18 O abundance of 3 × 10−7 relative to H2 (Frerking et al. 1982), the H2 number density is estimated to be 9 × 106 cm−3. In addition, the C18 O abundance in the protostellar envelope in B335 is possibly lower than the typical value in the interstellar medium (ISM) by a factor of ten (Evans et al. 2005; Yen et al. 2010), and this leads to an even higher H2 number density. The H13 CO+ (3–2) emission is also expected to be thermalized at this H2 number density and the temperature of 38 K, as calculated with the non-LTE radiative transfer code RADEX (van der Tak et al. 2007). Thus, the model images in the H13 CO+ emission were also computed with the LTE condition. We generated synthetic PV diagrams following the same process as described above, and performed the χ2 fitting on the H13 CO+ PV diagrams along the major and minor axes. The best-fit results of the H13 CO+ emission are also shown in Fig. 5 and are listed in Table 1. The χ2 of our best-fit model for H13 CO+ is 3, and χ2 as a function of Vin (100 au) from our fitting results is shown in Fig. 6. For comparison with the best-fit model, PV diagrams extracted from the synthetic images of the kinematical models with different Vin (100 au) for H13 CO+ are presented in Appendix A.

In addition, to examine the influence of the uncertainty in the PA of the major and minor axes on our fitting results, we have changed the PA by ±10° and performed the same χ2 fitting. The resulting best-fit infalling velocities remain unchanged. Thus, our results are not sensitive to the uncertainty in the PA of the major and minor axes. Furthermore, we note that in the non-ideal MHD simulations (e.g., Li et al. 2011; Zhao et al. 2018), the radial profile of infalling velocity can be shallower than our assumed profile, r−0.5, and the velocity profile of ions can be almost flat or even decrease with decreasing radii when the matter and the magnetic field are decoupled. Nevertheless, our data show that the line widths of the C18 O and H13 CO+ emission increase with decreasing radii (Fig. 5), suggesting that the velocity profiles of both neutral and ionized gas increase with decreasing radii on a scale of 100 au. To test the influence of our assumption of the velocity profile, we have also adopted a shallower radial profile of infalling velocity by changing the power-law index by a factor of two to be r−0.25 and performed the same χ2 fitting. The difference between the best-fit Vin(100 au) with the two different velocities profiles is 0.1 km s−1, less than our uncertainty. Therefore, our results are also not sensitive to the assumption of the velocity profile.

The best-fit infalling velocities in the C18O and H13 CO+ emission are identical within the uncertainty. The difference in the infalling velocities in the C18 O and H13 CO+ emission is estimated to be less than 0.3 km s−1 at a radius of 100 au from the error propagation of their uncertainties in Vin (100 au). This measured infalling velocity corresponds to a protostellar mass of ~0.04 M on the assumption that the infalling motion is free fall. This is consistent with Yen et al. (2015b) but is lower than the estimate from fitting the inverse P-Cygni profile in the HCN and HCO+ emission with the inside-out collapse model, 0.2 M, by Evans et al. (2015). In Evans et al. (2015), there are high-velocity wings in the spectra of the HCN and HCO+ lines, which need to be explained with the infalling velocity caused by the higher protostellar mass. The lines analyzed in Evans et al. (2015) are optically thicker than the lines analyzed in this work, and these high-velocity wings are not detected in the C18 O line. There is a hint of a redshifted high-velocity wing in the H13 CO+ emission, but there is no blueshifted counterpart. Observations at higher angular resolutions to reveal the origins of these high-velocity wing and to resolve the Keplerian disk in B335 are needed to clarify the difference in the estimated protostellar mass. Nevertheless, our results showing that the infalling velocities are identical within the uncertainty are not affected by the discrepancy in the estimated protostellar mass because the observed velocity structures in the C18 O and H13 CO+ emission can be explained with the same kinematical models.

thumbnail Fig. 4

Position–velocity diagram of the H13CO+ emission along the (a) major and (b) minor axes of the flattened envelope passing through the protostellar position. Velocity axes are in units of km s−1 in the LSR frame, and offset axes are in arcsecond with respect to the protostellar position. Vertical and horizontal green lines denote the systemic velocity of 8.34 km s−1 and the protostellar position, respectively. Contour levels are 3σ, 6σ, 10σ, 15σ, and 20σ and then in steps of 10σ, where 1σ is 3.3 mJy beam−1.

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thumbnail Fig. 5

Observed (black) and best-fit model (red; with parameters in Table 1) position–velocity diagrams of the C18 O (upper panels) and H13CO+ (lower panels) emission along the major (left panels) and minor (right panels) axes of the flattened envelope passing through the protostellar position. Vertical and horizontal dotted lines denote the systemic velocity of 8.34 km s−1 and the protostellar position, respectively. Vertical green dashed lines show the velocity range that was not included in our model fitting, VLSR of 7.7–8.9 km s−1, because of a possible contamination from the outflow. Contour levels are from 3σ in steps of 5σ, where 1σ is 3.8 mJy beam−1 in the C18O emission andis 3.3 mJy beam−1 in the H13CO+ emission.

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Table 1

Best-fit parameters of kinematical models.

thumbnail Fig. 6

χ2 as a function of infalling velocity at a radius of 100 au of our kinematical models for C18 O (blue curve) and H13CO+ (green curve). For each infalling velocity, χ2 presented here is the smallest χ2 achieved by varying all the other parameters. The minimum χ2 for the neutral and ionized lines are at 0.85 km s−1.

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5 Discussion

By fitting our kinematical models to the observed velocity structures in the PV diagrams of the C18 O and H13 CO+ emission, wehave found that there is no detectable difference in the infalling velocities traced by the two emission lines, and the velocity difference between the C18O and H13 CO+ gas is estimated to be at most 0.3 km s−1 at a radius of 100 au in B335.

If H13 CO+ is attached to the magnetic field on a 100 au scale in B335, the velocity difference between H13 CO+ and C18O can trace the ambipolar drift velocity, which is the relative velocity between the magnetic field and the neutrals. Whether H13 CO+ is attached to the magnetic field depends on the ratio between the Lorenz force and the drag force on H13 CO+, and it can be evaluated with the Hall parameter (βi,H2; Zhao et al. 2016), (5)

where mi and Zi e are the mass and the charge of H13 CO+, B is the magnetic field strength, c is the speed of light, μ is the mean molecular weight of 2.36, mH and mH2 are the masses of atomic and molecular hydrogen, n(H2) is the number density of H2, and is the momentum transfer rate coefficient. When βi,H2 ≫ 1, meaning that the Lorenz force dominates over the drag force, H13 CO+ is attached to the magnetic field. At a radius of 100 au in B335, n(H2) is estimated to be 9 × 106 cm−3 with the ISM C18O abundance in Sect. 4. is a function of temperature and velocity difference between ions and neutrals (vd), and for HCO+ (Pinto & Galli 2008). Our results suggest that vd is less than 0.3 km s−1. With the formulae in Pinto & Galli (2008), is estimated to be 1.2 × 109 cm3 s−1 at a temperature of 38 K and vd of 0.3 km s−1. With the method described by Chandrasekhar & Fermi (1953), B has been estimated to be 10–40 μG on a 0.1 pc scale from the infrared polarimetric observations (Bertrang et al. 2014) and to be 134 μG on a 4000 au scale from the polarized thermal dust emission at the submillimeter wavelengths (Wolf et al. 2003) in B335. On the other hand, if the simple power-law relation between the magnetic field strength and the density in molecular clouds, μG, is valid in B335 (Nakano et al. 2002), B is estimated to be 400 μG on a 100 au scalewith our estimated n(H2). In other protostellar sources, B on a scale of a few hundred au has been estimated to be even higher with ~5 mG from the polarized thermal dust emission (Girart et al. 2006; Hull et al. 2017). It thus seems very plausible to have B of a few hundred μG in B335 on a 100 au scale. Still, with a conservative value of B > 100 μG, βi,H2 is estimated to be >20 on a 100 au scale. Therefore, on the assumptions of the typical magnetic field strength and C18 O abundance in protostellar sources, H13 CO+ is expected to be attached to the magnetic field on a 100 au scale in B335.

In non-ideal MHD theoretical calculations and simulations incorporating ambipolar diffusion, the ambipolar drift velocity starts to increase on a scale of tens to hundreds of au in infalling protostellar envelopes, where the magnetic field starts to decouple from the neutral gas (e.g., Fig. 7 in Krasnopolsky & Königl 2002, Fig. 3 in Mellon & Li 2009, Fig. 5 in Li et al. 2011, and Fig. 1 in Zhao et al. 2018). The radius, where the magnetic field starts to decouple from the neutral gas, is proportional to the magnetic diffusivity of ambipolar diffusion (Krasnopolsky & Königl 2002; Zhao et al. 2016, 2018). In the simulations with a standard Mathis–Rumpl–Nordsieck (MRN) grain size distribution (Mathis et al. 1977) and a typical cosmic-ray ionization rate of 10−17 s−1, the magnetic field is well coupled with the neutral gas and is dragged to the inner few hundred au or even a smaller scale in infalling envelopes (Li et al. 2011; Zhao et al. 2018). Then the magnetic field gradually decouples from the neutral gas in the inner regions and diffuses outward to form an ambipolar diffusion shock, which efficiently decelerates the infalling and rotational motions. (e.g., Krasnopolsky & Königl 2002; Li et al. 2011). In this case, the formation of a Keplerian disk with a size larger than 10 au is suppressed because a large amount of magnetic flux is accumulated in the inner envelope and efficiently transports away the angular momentum of the infalling material (Mellon & Li 2009; Li et al. 2011). Several simulations have shown that incorporating ambipolar diffusion with a typical diffusivity does not enable large-scale Keplerian disks to form (e.g., Mellon & Li 2009; Li et al. 2011; Zhao et al. 2016, 2018). In these simulations, the ambipolar drift velocity is lower than 0.2 km s−1 at a radius of 100 au, increases to 0.3 km s−1 at a a radius of 70 au, and is higherer than 1 km s−1 at radii smaller than 30 au, while the infalling velocities of the neutrals are 0.3, 0.5, and 1.5 km s−1 at radii of 100, 70, and 30 au, respectively, when the central protostellar mass is 0.57 M (Fig. 5 in Li et al. 2011). If the ions are attached to the magnetic field, such an ambipolar drift velocity implies that the ion velocity is lower than 30%–40% of the neutral velocity.

On the other hand, the ambipolar diffusivity is enhanced by one to two orders of magnitude when small dust grains with sizes of a few to tens of nanometers are removed from the grain size distribution (Dapp et al. 2012; Padovani et al. 2014; Zhao et al. 2016). As a consequence, the magnetic field starts to decouple from the neutrals on a larger scale of several hundred or even few thousand au. In the simulations, the ambipolar drift velocity can be larger than 0.5 km s−1 at radii of 100–300 au and increase to 1 km s−1 at a radius of 100 au when the protostellar mass is 0.07 M (Zhao et al. 2018). This effect enables the formation of Keplerian disks with sizes of tens of au in the simulations (Masson et al. 2016; Zhao et al. 2016, 2018). We note that in such a paradigm with an enhanced ambipolar diffusivity, the formation of large-scale Keplerian disks with a typical mass-to-flux ratio of a few can still be suppressed when the initial rotation1 of parental cores is slow with an angular velocity lower than (5– 8) × 10−14 s−1, corresponding to a ratio of rotational to gravitational energy of 0.5–1.6% (Dapp et al. 2012; Tomida et al. 2015; Zhao et al. 2018).

With our observations, we set an upper limit to the velocity difference between the ionized and neutral gas at a radius of 100 au to be 0.3 km s−1 in B335. Our results are consistent with the non-ideal MHD simulations with the ambipolar diffusivity computed with the standard MRN grain size distribution, where the ambipolar drift velocity is expected to be <0.2 km s−1 at a radius of 100 au. Since our observations are not able to resolve the inner region of a few tens of au, we are not able to further unambiguously detect any increase in drift velocity toward the inner regions, which is predicted inthe simulations. On the contrary, our observational results of B335 are inconsistent with the non-ideal MHD simulations with the enhanced ambipolar diffusivity and the removal of the small dust grains2, where the minimum grain size is 0.1 μm. The ambipolar drift velocity of 0.5–1 km s−1 on a scale of 100 au predicted in these simulations with the enhanced ambipolar diffusivity is not detected in B335 with our observations.

Therefore, based on the fact that H13 CO+ is likely well coupled to the magnetic field and that there is no measurable difference in the infall velocities between H13 CO+ and C18O on a 100 au scale probed by our observations, our results suggest that in B335 any significant magnetic decoupling from the bulk neutral matter, if present, likely occurs on a scale smaller than 100 au, as in the non-ideal MHD simulations with the typical ambipolar diffusivity. Therefore, the magnetic field is dragged and accumulated on the small scale, and magnetic braking can efficiently remove the angular momentum from the material infalling to the small scale and suppress the disk formation in B335. This can result in the absence of a Keplerian disk with a size larger than 10 au in B335, as found in the observations (Yen et al. 2015b).

There are also other non-ideal MHD simulations that only incorporate Ohmic dissipation but no ambipolar diffusion (e.g., Dapp & Basu 2010; Tomida et al. 2013). These simulations, which resolve the formation of first cores, show that small Keplerian disks with a size of few au can form during the phase of first cores or during the formation of second cores. These simulations were unable to follow the evolution of the Keplerian disks further. Although it is not clear whether the size of these small disks will grow to a scale of 100 au as expected (Tomida et al. 2015) or remains on a scale of 10 au (Dapp et al. 2012), in these simulations the magnetic field is decoupled from the neutral gas in the innermost regions. B335 can also be an observational analog to these simulations. In addition, the non-detection of the velocity difference between the ionized and neutral gas is also consistent with the ideal MHD case. Nevertheless, no Keplerian disk is expected to form in the ideal MHD case unless the magnetic field is weak and largely misaligned from the rotational axis of parental cores (Mellon & Li 2008; Joos et al. 2012; Li et al. 2013), while the presence of the outflow and jets in B335 suggests that there is at least a small rotating disk around the protostar (e.g., Blandford & Payne 1982; Pudritz & Norman 1983, 1986). Thus, B335 is less likely the ideal MHD case. Future observations with higher angular resolutions to measure the velocity difference between the ionized and neutral gas on a scale of tens of au in B335 are needed to distinguish these simulations and the ones with the typical ambipolar diffusivity.

We also note that the velocity drift between ionized and neutral gas is expected to be most significant in the midplane of infalling protostellar envelopes, where the magnetic field is highly pinched, and there is almost no velocity drift in the upper layers of envelopes, as discussed in Krasnopolsky & Königl (2002) and Zhao et al. (2018). Although the inclination of B335 is edge on, the most suitable case to probe the gas motions in the midplane, our observations may not be able to distinguish the gas motions in the midplane and upper layers in the protostellar envelope in B335 because of the limited angular resolution, and the signature of the velocity drift (if present) is possibly diluted and smoothed out. Further observations with an angular resolution of ~0. ′′1 to resolve the vertical velocity structures within 10–20 au from the midplane at radii of 100–200 au are required to examine this possibility.

Finally, if the magnetic field strength on a 100 au scale in B335 is only a few tens of μG rather than >100 μG, specific ionspecies such as H13 CO+ are not the ideal indicator of the kinematics of the magnetic field because they are detached from the magnetic field (βi,H2 < 1). In this case, ions and neutrals in the inner 100 au region can move together, but the magnetic field can still be left behind the bulk infall motion. This picture of ions detaching from the magnetic field is also consistent with our non-detection or upper limit of the velocity difference between ions and neutrals at a radius of 100 au. In this case, the magnetic field is not dynamically important in the infalling protostellar envelope (Mellon & Li 2009). The Keplerian disk is expected to grow in size with the proceeding collapse (Terebey et al. 1984; Basu 1998), and the absence of a Keplerian disk larger than 10 au in B335 might simply be due to its young age (Yen et al. 2015b).

6 Summary

We analyzed the data of the H13 CO+ (3–2) and C18O (2–1) emission in the Class 0 protostar B335 obtained with our ALMA observations. The goal is to investigate ambipolar diffusion in protostellar envelopes by comparing the gas motions traced by the ionized and neutral molecular lines. Our main results are summarized below.

  • The H13CO+ and C18O emission lines both show a central compact component with a size of 1′′–2′′ (100–200 au) elongated perpendicular to the outflow direction, and there are additional blueshifted extensions toward the east. Clear velocity gradients along the outflow direction are observed in both lines. The extensions toward the east are likely associated with the outflow. The elongated central components in the H13CO+ and C18O emission likely trace the flattened envelope around the protostar, and their velocity structures can be explained with an edge-on infalling envelope.

  • We constructed kinematical models of an infalling and rotating envelope and fitted the observed velocity structures in the PV diagrams along the major and minor axes of the flattened envelope in the H13CO+ and C18O emission. The infalling velocities traced by the H13CO+ and C18O emission are both measured to be 0.85±0.2 km s−1 at a radius of 100 au, suggesting that the velocity difference between the ionized and neutral gas is at most 0.3 km s−1 at a radius of 100 au.

  • The Hall parameter of H13CO+ is estimated to be ≫1 on a 100 au scale in B335 on the assumption of a typical magnetic field strength of >100 μG on a 100 au scale in protostellar envelopes. Thus, H13CO+ is expected to be attached to the magnetic field, and the difference in the infalling velocities measured in the H13CO+ and C18O lines cantrace the ambipolar drift velocity.

  • Our non-detection or upper limit of the ambipolar drift velocity suggests that the magnetic field and the bulk neutral matter remain well coupled on a 100 au scale, and any significant decoupling, if present, likely occurs on smaller scales than probed by our observations. Consequently, the magnetic field isdragged and accumulated on the small scale, and efficient magnetic braking can suppress the disk formation in B335, as expected in non-ideal MHD simulations with a standard MRN grain size distribution and a typical cosmic-ray ionization rate. On the other hand, the high ambipolar drift velocity of 0.5–1 km s−1 predicted in the non-ideal MHD simulations with the enhanced ambipolar diffusivity by removing the small dust grains, where the minimum grain size is 0.1 μm, is not detected in our observations. However, we cannot rule out that there is a significant ambipolar drift only in the midplane of the infalling envelope. Such signatures could be smoothed out in our observations due to the limited angular resolution. While the non-detection of a velocity difference is still also consistent with ideal MHD simulations, the existence of a small disk in B335 makes this scenario less probable. Future observations with higher angular resolutions are needed to establish the definite presence of ambipolar diffusion in B335.

  • Although it is less likely, if the magnetic field strength is only tens of μG on a 100 au scale in B335, H13CO+ can be detached from the magnetic field and is no longer an indicator of the kinematics of the magnetic field. Thus, H13CO+ and the neutrals in the inner 100 au region can move together, and there would be no difference in the infalling velocities traced by the H13CO+ and C18O emission. In this case, the magnetic field is not dynamically important in the infalling protostellar envelope.

Acknowledgements

This paper makes use of the following ALMA data: ADS/JAO.ALMA#2013.1.00879.S, ADS/JAO.ALMA#2015.1.01188.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. We thank all the ALMA staff supporting this work. ZYL is supported in part by NASA NNX14AB38G and NSF AST-1313083 and 1716259. PMK acknowledges support from the Ministry of Science and Technology through grant 103-2119-M-001-009 and from an Academia Sinica Career Development Award. ST is supported by NAOJ ALMA Scientific Research Grant Numbers 2017-04A. ST acknowledges a grant from the Ministry of Science and Technology (MOST) of Taiwan (MOST 102-2119-M-001-012-MY3), and JSPS KAKENHI Grant Number JP16H07086, in support of this work.

Appendix A Position–velocity diagrams of kinematical models

Figure A.1 compares the observed PV diagrams with models with different Vin (100 au) from 0.4 km s−1 to 1.2 km s−1. For these models, all the other parameters, n(R0) and p, were varied to minimize χ2 with the given Vin (100 au). This comparison shows that when Vin(100 au) is beyond the range of 0.85 ± 0.2 km s−1, the line widths in the models become wider or narrower than the observations for both C18 O and H13 CO+, especially at offsets from 0. ′′5 to 1′′, and the velocities of the intensity peaks in the model PV diagrams are offset from the observations. These results suggest that the infalling velocities of the C18O and H13 CO+ gas at a radius of 100 au are most likely within the range of 0.85 ± 0.2 km s−1.

thumbnail Fig. A.1

Same as Fig. 5, but for comparison with models with different Vin (100 au) for C18 O (left columns) and H13CO+ (right columns), and Vin(100 au) of each model is labeled at the upper left corner in the panel.

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1

The initial core rotation is assumed to be rigid-body rotation in the simulations.

2

In non-ideal MHD simulations incorporating ambipolar diffusion, the ambipolar drift velocity is expected to lower when the minimum size in the grain size distribution is smaller (Zhao et al. 2018).

All Tables

Table 1

Best-fit parameters of kinematical models.

All Figures

thumbnail Fig. 1

Moment 0 (contours) overlaid on moment 1 (color) maps of the (a) H13 CO+ (3–2) and (b) C18 O (2–1) emission in B335 obtained with the ALMA observations. Color scales are in units of km s−1 in the LSR frame. The color scales are the same in (a) and (b). Crosses denote the protostellar position. Red and blue arrows show the direction of the blue- and redshifted outflows, respectively. Contour levels in (a) are 3σ, 6σ, 10σ, 15σ, 20σ, 30σ, and 45σ, and in (b) are 3σ, 6σ, 10σ, 15σ, 25σ, and 40σ, where 1σ is 3.3 and 3.4 mJy beam−1 km s−1, respectively.

Open with DEXTER
In the text
thumbnail Fig. 2

Spectra of the C18O and H13CO+ emission at the protostellar position obtained with our ALMA observations, shown in blue and green histograms. The data were first convolved with the same circular beam with a size of 0. ′′ 5 to extract the spectra. A vertical dotted line denotes the systemic velocity of 8.34 km s−1.

Open with DEXTER
In the text
thumbnail Fig. 3

Moment 0 maps of the H13CO+ emission integrated over different velocity regimes, (a) low velocity VLSR = 7.9–8.3 and 8.3–9.1 km s−1, (b) high velocity VLSR = 6–7.9 and 9.1–10.8 km s−1, and (c) very high velocity VLSR = 10.8–12 km s−1. Blue and red contours present the blue- and redshifted emission, respectively. Crosses denote the protostellar position. Red and blue arrows show the direction of the blue- and redshifted outflows, respectively. A green box in (a) presents the image size of (b) and (c). Filled blue ellipses show the size of the synthesized beam of 0. ′′ 5 × 0. ′′ 4. Contour levels in (a) are 5σ, 10σ, 15σ, and 20σ and then in steps of 10σ, in (b) are 3σ, 6σ, 10σ, 15σ, 20σ, and 25σ and then in steps of 10σ, and in (c) are from 3σ in steps of 3σ, where 1σ is 1.1, 1.8, and 1.6 mJy beam−1 km s−1, respectively.

Open with DEXTER
In the text
thumbnail Fig. 4

Position–velocity diagram of the H13CO+ emission along the (a) major and (b) minor axes of the flattened envelope passing through the protostellar position. Velocity axes are in units of km s−1 in the LSR frame, and offset axes are in arcsecond with respect to the protostellar position. Vertical and horizontal green lines denote the systemic velocity of 8.34 km s−1 and the protostellar position, respectively. Contour levels are 3σ, 6σ, 10σ, 15σ, and 20σ and then in steps of 10σ, where 1σ is 3.3 mJy beam−1.

Open with DEXTER
In the text
thumbnail Fig. 5

Observed (black) and best-fit model (red; with parameters in Table 1) position–velocity diagrams of the C18 O (upper panels) and H13CO+ (lower panels) emission along the major (left panels) and minor (right panels) axes of the flattened envelope passing through the protostellar position. Vertical and horizontal dotted lines denote the systemic velocity of 8.34 km s−1 and the protostellar position, respectively. Vertical green dashed lines show the velocity range that was not included in our model fitting, VLSR of 7.7–8.9 km s−1, because of a possible contamination from the outflow. Contour levels are from 3σ in steps of 5σ, where 1σ is 3.8 mJy beam−1 in the C18O emission andis 3.3 mJy beam−1 in the H13CO+ emission.

Open with DEXTER
In the text
thumbnail Fig. 6

χ2 as a function of infalling velocity at a radius of 100 au of our kinematical models for C18 O (blue curve) and H13CO+ (green curve). For each infalling velocity, χ2 presented here is the smallest χ2 achieved by varying all the other parameters. The minimum χ2 for the neutral and ionized lines are at 0.85 km s−1.

Open with DEXTER
In the text
thumbnail Fig. A.1

Same as Fig. 5, but for comparison with models with different Vin (100 au) for C18 O (left columns) and H13CO+ (right columns), and Vin(100 au) of each model is labeled at the upper left corner in the panel.

Open with DEXTER
In the text

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