# The role of magnetic fields in the formation of protostars, disks, and
outflows

Yusuke Tsukamoto  Kaghoshima University,Department of Physics and Astronomy
Graduate School of Science and Engineering, Korimoto, Kagoshima, Japan
Anaëlle Maury CEA/DRF/IRFU Astrophysics department, Université Paris-Saclay,
UMR AIM, F-91191 Gif-sur-Yvette, France Benoît Commerçon Univ Lyon, Ens de
Lyon, Univ Lyon1, CNRS, Centre de Recherche Astrophysique de Lyon UMR5574,
F-69007, Lyon, France Felipe O. Alves Max-Planck-Institut für
extraterrestrische Physik, Gießenbachstraße 1, Garching, 85748, Germany Erin
G. Cox Center for Interdisciplinary Exploration and Research in Astrophysics
(CIERA), 1800 Sherman Avenue, Evanston, IL 60201, USA Nami Sakai RIKEN Cluster
for Pioneering Research, 2-1 Hirosawa, Wako, Saitama, Japan Tom Ray Astronomy
and Astrophysics Section, School of Cosmic Physics, Dublin Institute for
Advanced Studies, Ireland Bo Zhao Max-Planck-Institut für extraterrestrische
Physik, Gießenbachstraße 1, Garching, 85748, Germany Masahiro N. Machida
Kyushu University, Department of Earth and Planetary Sciences, Motoka,
Fukuoka, Japan

###### Abstract

Abstract We present our current understanding of the formation and early
evolution of protostars, protoplanetary disks, and the driving of outflows as
dictated by the interplay of magnetic fields and partially ionized gas in
molecular cloud cores. In recent years, the field has witnessed enormous
development through sub-millimeter observations which in turn have constrained
models of protostar formation. As a result of these observations the state-of-
the-art theoretical understanding of the formation and evolution of young
stellar objects is described. In particular, we emphasize the importance of
the coupling, decoupling, and re-coupling between weakly ionized gas and the
magnetic field on appropriate scales. This highlights the complex and intimate
relationship between gravitational collapse and magnetic fields in young
protostars.

##  1 Introduction

The evolution of solar-type protostars is classified in stages following an
empirical sequence. It starts from the most embedded objects, Class 0
protostars, where the envelope dominates the mass: this main accretion stage
is believed to last less than 0.1 Myr (e.g. _Evans et al._ , 2009; _Maury et
al._ , 2011). Once the envelope mass and the mass accumulated in the central
embryo are roughly equal, protostars enter the Class I phase which sees most
of the second half of the final stellar mass accreted, over a fraction of a
Myr. Finally, the young stellar object is no longer enbedded and becomes
visible during the T-Tauri phase that precedes its arrival onto the Main
Sequence. In this review, we focus on the physics at work during the early
stages of star formation, in embedded protostars.

Recent years have witnessed the major development of realistic magneto-
hydrodynamics simulations modelling the physics of protostellar formation, as
well as a cascade of detailed, sensitive observations of large samples of
protostellar objects. These advances have transformed our understanding and
highlighted the paramount role of the magnetic field during the early
evolution of young stellar objects. Indeed, radio interferometers sensitive to
the cold and dense material typical of embedded protostars, e.g., the Atacama
Large Millimeter Array (ALMA), Northern Extended Millimeter Aray (NOEMA), and
the Very Large Array (VLA), have significantly extended their capabilities
since Protostars and Planets VI (henceforth PP VI). Exquisite maps probing the
gas, dust, and magnetic field properties on the very small scales of
protostellar cores, where material is accreted into a stellar embryo, stored
in a disk, and partially ejected via outflows and jets, have placed
unprecedented constraints on theoretical models.

In parallel, state-of-the art computations now provide a comprehensive
theoretical landscape thanks to recent progress in 3-D magneto-hydrodynamic
simulations which are able to follow the time evolution of the physics from
the large spatial scales typical of dense cores ($\sim 10^{4}$ au) to the
protostar-disk system ($\sim 10^{2}$ au), over long timescales from the onset
of prestellar collapse to the end of the Class I phase ($\sim 10^{5}$ years
after protostar formation). They revealed the essential physics in the complex
processes involved in the formation of protostar, disk, and outflow; i.e., the
coupling, decoupling, and re-coupling between weakly ionized gas and magnetic
field on appropriate scales. in these processes, microscopic ion chemistry,
especially adsorption of charged particles by dust, plays an essential role.

In this chapter, we aim to present a synthesis of the observational and
theoretical progresses made in the past decade to enrich our understanding of
the formation of protostars, their protostellar disks, outflows, and jets.

This chapter is organized as follows: Section 2 presents the state-of-the-art
observational characterization of magnetic fields, gas and dust properties in
protostellar environments. Section 3 describes our current understanding of
the formation and early evolution of protostars, protoplanetary disks,
outflows, and jets based on recent theoretical studies. We present theories
that are in the main successful in reproducing current observations, but also
highlight some remaining discrepancies and key challenges for both
observations and modelling to tackle in the near future. In Section 4, we
describe a promising avenue for future research regarding the evolution of
dust in magnetized protostellar environments, and its relevance to
understanding not only the formation of planetesimals but also the pristine
properties of the disks and stars that host them.

##  2 Observations of embedded protostars: envelopes, young disks, outflows,
and magnetic fields

###  2.1 Observing magnetic fields: techniques

Since Protostar and Planets VI (see chapter by _Li et al._ 2014a), our view of
interstellar magnetic fields as a key ingredient in threading the interstellar
medium from clouds, to filaments and their embedded cores has been firmly
established by multi-wavelength analysis of polarized light. Several
observational techniques have been developed or improved. We summarise them
below and the constraints they allow us to put on the strength and topology of
magnetic fields in star-forming structures.

The linear polarization of dust thermal emission and absorption is widely used
to trace the magnetic field topology from the plane-of-the-sky component
$B_{\mathrm{pos}}$ of the magnetic vector B, in dense ISM structures. The
polarized dust thermal emission largely maps the field topology at mm/sub-mm
wavelengths: assuming dust grains populating the star-forming structures are
elongated, they are expected to produce polarized dust emission if they are
aligned along B-field lines in an anisotropic radiation field (radiative
torques alignment, _Lazarian and Hoang_ , 2007). Optical and near-IR
observations of the dichroic polarization of background stars are also used to
probe magnetic fields along highly extincted lines-of-sight, such as those
typical of prestellar cores ($A_{\mathrm{v}}>10$). The magnetic field strength
can be estimated from these polarimetric observations, by performing a
statistical analysis of the dispersion of polarization angles compared to the
velocity dispersion of the gas, such as in the Davis-Chandrasekhar-Fermi
method (DCF, _Davis_ , 1951; _Chandrasekhar and Fermi_ , 1953). Although this
method relies heavily on the assumed physical properties of the medium and how
the latter couples to the magnetic field (_Falceta-Gonçalves et al._ , 2008;
_Liu et al._ , 2021, 2022), it has been widely used in studies from clouds to
cores scales.

The only direct technique to estimate the strength of the magnetic field in
star-forming structures is through utilizing the Zeeman effect via spectro-
polarimetry. Observations of molecular species with significant magnetic
susceptibility, i.e., those with an unpaired electron, can be used to trace
the magnitude of the line-of-sight component $B_{\mathrm{los}}$ of the
magnetic vector B in star-forming clouds, dense cores, and disks (_Crutcher_ ,
2012; _Crutcher and Kemball_ , 2019). Spectral-line polarization can also
arise from the so-called Goldreich-Kylafis (GK) effect (_Goldreich and
Kylafis_ , 1982), because individual molecular rotational levels, in the
presence of a magnetic field, split into sub-levels and the resulting line
emission can be polarized either parallel or perpendicular to the magnetic
field depending on the sign of $\rm{n}_{u}-\rm{n}_{\pm}$, where $\rm{n}_{u}$
and $\rm{n}_{\pm}$ are the populations of an upper energy level and its
magnetic sub-levels, respectively. It is worth pointing out however that the
GK effect requires optical depths $\tau\approx 1$ to be observed, i.e. the
radiation field has to be anisotropic. It is not usable as a means of
measuring the magnetic field if the optical depth is large.

When magnetic field strengths can be estimated, they are typically used to
compute two physical quantities that tell about the importance of magnetic
field against other key ingredients, turbulence and gravity. The Alfvénic Mach
number captures the relative importance of magnetic and turbulent energies: it
is the velocity dispersion in the gas flow divided by the Alfvén waves
velocity, which is directly computed from the B-field strength. In the densest
star-forming structures, observations are used to estimate the ratio of
gravitational to magnetic energies, a.k.a. the mass-to-flux ratio $\mu$
expressed in units of the critical value when the magnetic energy equals the
gravitational energy (_Nakano and Nakamura_ , 1978). Gravitational collapse to
form a stellar system requires $\mu>1$, otherwise the magnetic forces will
dominate over gravity and stabilize the structure. In the latter case
evolution out of equilibrium can only happen over long timescales by diffusive
processes that allow to remove the magnetic flux from the central region of
the core, for example by ambipolar diffusion (see, e.g. _Shu_ , 1983).

While many interesting constraints have been obtained regarding the magnetic
field in star-forming clouds (see for example the reviews by _Pattle and
Fissel_ , 2019; _Hennebelle and Inutsuka_ , 2019), in the following we focus
on observations that probe the magnetic properties of dense cores, protostars,
their disks and outflows.

###  2.2 Magnetic fields in prestellar cores and protostars

In prestellar cores, both near-IR starlight polarization and sub-mm polarized
dust emission have revealed a large fraction of cores with highly organized
magnetic field topologies, at the typical $\sim 0.1$ pc core scales (see e.g.
_Kirk et al._ , 2006; _Jones et al._ , 2015; _Kandori et al._ , 2018, and
references therein). In the few cores where a magnetic field strength can be
estimated, values are typically between 10 and 100 $\mu$G. A comparison of
gravitational and magnetic energies then suggests a dynamical state in which
the mass is close to the magnetic critical mass, and this could significantly
delay gravitational collapse (_Stephens et al._ , 2013; _Girart et al._ ,
2006; _Kandori et al._ , 2017, 2020b, 2020a; _Myers and Basu_ , 2021). The
orientations of B-field lines mapped with optical/near-IR polarimetry are
sometimes smoothly connected to the ones recovered with sub-mm single-dish
observations on core scales (_Soler et al._ , 2016; _Soam et al._ , 2019), but
are sometimes significantly different (_Alves et al._ , 2014; _Clemens et al._
, 2016). Detailed analyses using statistics from large single-dish surveys,
such as the JCMT BISTRO survey (_Ward-Thompson et al._ , 2017; _Liu et al._ ,
2019), and starlight polarimetry, may give more insights into the role and
structure of magnetic fields in prestellar cores.

Thanks to the Zeeman effect, the OH molecule is used to probe the strength of
magnetic fields in molecular clouds (_Troland and Crutcher_ , 2008). However,
OH cannot be used in the same way to investigate magnetic field strength in
cores, due to their high gas densities. Here other tracers, such as CN, must
be employed. Only the most massive protostellar cores have detections of the
Zeeman effect in CN single-dish surveys: the values found for the mean total
magnetic field, $B_{\mathrm{tot}}$, are typically around a few hundred $\mu$G
(_Falgarone et al._ , 2008). They seem in broad agreement with statistical
estimates derived from polarized dust emission in the mm, which suggest values
up to 1 mG towards massive cores. Observed mass-to-magnetic flux ratios,
derived from Zeeman measurements are found to be super-critical with typical
values $\mu\sim 2-3$. The average Alfvénic Mach number in star-forming gas at
densities $\sim 10^{3}-10^{6}$ cm-3 inferred from these observations is then
found to be $M_{\mathrm{A}}\sim 1.5$ (_Falgarone et al._ , 2008; _Girart et
al._ , 2009). This suggests that from clouds to cores scales, the turbulent
and magnetic energies may be in approximate equipartition. The magnetic field
strengths observed are systematically higher in the CN cores than in the lower
density clouds (typical magnetic field strengths of $\sim$ 10 to 30 $\mu$G for
densities of a few $10^{3}$ cm-3, and on $\sim$0.1 pc scales, see e.g., Figure
1 in _Crutcher et al._ 2010). This suggests intrinsically stronger magnetic
fields in denser gas, but there could also be a bias through the specific
conditions necessary to form the most massive cores. Note that it is still
unclear however whether the switch in Zeeman molecular tracer is responsible
for the steep increase in the observed $B_{\mathrm{tot}}$ between clouds and
protostellar cores. Finally, the total magnetic pressure of a core with a
complex magnetic field could be higher than what is inferred from the Zeeman
effect, due to the cancelling effect of field vectors with opposite directions
along the line of sight. With all these caveats in mind, nevertheless these
measurements suggest magnetic fields are relatively strong in most massive
star-forming cores.

Single-dish observations of the polarized dust emission have allowed us to map
the magnetic field line spatial distribution in several low-mass protostellar
cores, particularly in recent times as a result of the extensive JCMT BISTRO
survey (see, e.g. _Eswaraiah et al._ , 2021). In the inner envelope, on
typical scales $\sim 100-5000$ au, polarized dust emission can be probed using
(sub-) millimeter interferometry and has been detected at the few percent
level (see e.g. _Galametz et al._ , 2018; _Cox et al._ , 2018). The wealth of
recent interferometric observations of magnetic field lines have suggested
that the early detections of the “hourglass” pattern (e.g. _Girart et al._ ,
2006; _Rao et al._ , 2009) may actually be a common feature in low-mass cores
(_Maury et al._ , 2018; _Kwon et al._ , 2019; _Yen et al._ , 2019; _Kandori et
al._ , 2020b). An example can be found in the left panel of Figure 1 showing
the B-field topology reconstructed from ALMA observations of the millimeter
polarized dust emission in a Class 0 protostar. Multi-scale observations and
comparison of the magnetic field topology with the gas kinematics in the inner
envelope suggest this hourglass shape may be a natural result from an
initially quasi-poloidal configuration being affected by the collapse of the
inner envelope: magnetic field lines efficiently coupled to the infalling gas
are pulled towards the core center as the collapse proceeds. However, this
morphology is not seen as ubiquitously as one might expect from a simple
strongly magnetized scenario for low-mass star formation (_Hull and Zhang_ ,
2019) and this may stem from several causes. First, identifying hourglass
fields is not simple due to uncertainties caused by projection effects and
limited instrument sensitivity; thus a robust survey probing the magnetic
field in star-forming cores with high spatial dynamic range still needs to be
carried out. Second, the simple view that protostellar accretion proceeds
largely in a purely symmetric fashion, with gas flowing down to the central
embryo along the equatorial plane and creating a distinguishable hourglass
pattern, has received much criticism, and contradicts some recent observations
(see the PPVII review chapter by Pineda et al., and references in $\S$2.5 of
the present chapter). The early development of non-isotropic infall and
differences in gas ionization levels affecting the coupling at different
locations in the protostellar cores may lead to the complex field geometries
observed. Finally, local irradiation conditions may enhance polarized dust
emission towards specific locations, making it difficult to assess the global
3-D B-field topology. Observations of the magnetic field on core scales have
suggested in a few cases that the magnetic field lines could correlate with
density and/or kinematic structures such as (gravo-)turbulence (e. g., _Hull
et al._ , 2017b), accretion streamers (e. g., _Alves et al._ , 2018; _Le
Gouellec et al._ , 2019), irradiated layers (_Le Gouellec et al._ , 2020) and
protostellar outflows (e. g. _Hull et al._ , 2017a).

Protostars are actively accreting envelope material onto the central stellar
embryo: as such they are expected to be primarily super-critical ($\mu>1$).
Unfortunately magnetic field strengths in solar-type cores have not been
widely explored using the Zeeman effect, because it is a subtle effect in
relatively weak spectral lines. Only recent sensitive observations have
allowed us to detect large numbers of independent B-field orientations within
single low-mass cores, and use the DCF method to estimate B-field strengths.
Measured values suffer from large uncertainties due to the gravitational
energies associated with these collapsing objects, but typical field strengths
are around a few tens of $\mu$G (_Hull and Zhang_ , 2019). Finally, since the
polarized flux is a cumulative quantity that is prone to cancellation if the
polarization angle is highly disorganized along the line of sight,
observations of a few percent of polarization suggests the magnetic field may
be strong enough to remain at least partly organized inside star-forming cores
(_Le Gouellec et al._ , 2020).

Figure 1: Illustrative examples of protostellar magnetic fields mapped with
millimeter/sub-millimeter polarized dust continuum emission on similar
envelope/disk scales in the L1448-2A Class 0 protostar (left panel, from _Kwon
et al._ , 2019) and the [BHB2007]11 Class 0/I protostar (right panel, data
published in _Alves et al._ , 2017, 2018). Both panels show the ALMA dust
continuum emission at sub-millimeter wavelength as a background image, and
contours of integrated CO ($2\rightarrow 1$) emission tracing the outflow at
blue/red velocities. The B-field lines, obtained from the polarized dust
emission, are shown towards both objects as green line segments. The
synthesized beam of the observations are shown in the lower right corners.

###  2.3 Magnetic fields observed in protostellar disks and outflows

Polarization observations at high angular resolution should be able to
differentiate the properties (i.e., polarization fraction and position angle)
between components from the disk and the envelope (e.g. _Cox et al._ , 2018;
_Sadavoy et al._ , 2019). However, although a small number of embedded disks
show a few detectable polarization measurements, most observations at these
small scales are still limited by sensitivity (e. g., Oph-emb-1, _Sadavoy et
al._ , 2019). Moreover, in relevant surveys the disk emission is often
optically thick and B-field aligned grains are not the only plausible origin
for the polarized emission, complicating the interpretation. Indeed, the
scattering opacity of large grains (a${}_{\mathrm{max}}>100~{}\mu$m) is
predicted to prevail over their absorption opacity (_Kataoka et al._ , 2015),
in most conditions typical of disks. This results in polarized emission due to
self-scattering of the dust grains, up to a few percent at millimeter
wavelengths in the most extreme cases. In edge-on disks such as HH 212 or
L1527, the impact of large optical depth is clear on the polarization maps,
where polarization angles are predominantly parallel to the disk minor axis
(_Segura-Cox et al._ , 2015; _Harris et al._ , 2018; _Lee et al._ , 2021b).
Hence, only very few measurements of dust polarization from disks in young
protostars can be interpreted as magnetically aligned grains rather than by
the polarized emission from dust self-scattering (this is usually probed by
large polarization fractions and the combined analysis of the polarization
properties at varying wavelengths). For example, ALMA observations of the
Class I protostar TMC-1A shows polarization components from the disk poles
that can be interpreted as arising from toroidal magnetic fields associated
the disk or the outflow or magnetic accretion flows within the disk (_Aso et
al._ , 2021). Another example is the disk around the Class 0/I [BHB2007] 11
protostar, the magnetic field of which is modelled as a combination of
poloidal and toroidal components produced by disk rotation and infalling
material from the envelope, respectively (see Fig. 1 and _Alves et al._ ,
2018). However, if the dust population in this source is dominated by mm-size
grains, the polarization observed at mm-wavelengths (i.e., in the Mie regime)
is expected to be negative, which in practical terms means a 90-degree flip in
the polarization direction (_Guillet et al._ , 2020b). In this scenario, the
disk polarization is parallel to the magnetic fields, potentially following
the accretion streamers seen within the circumbinary disk (_Alves et al._ ,
2019). A poloidal hour-glass magnetic field was also interpreted as causing
the polarized dust emission around the disk of VLA 1623A, at scales of 200 au
(_Sadavoy et al._ , 2018). The non-detection of a toroidal component in the
dust polarization signal in this protostar could be due to the predominance of
scattering polarization from large grains at disk scales, non-ideal MHD
effects decoupling the magnetic field from the rotating gas in the disk at
small scales, or a combination of an intrinsically weak magnetic field and a
weakly ionized medium on disk scales (_Vaytet et al._ , 2018; _Hennebelle et
al._ , 2020).

Spectro-polarimetry may become a powerful tool to measure the magnetic
strength in embedded disks. The high abundance of the Zeeman-sensitive CN
molecule in their midplane (_Chapillon et al._ , 2012), for example, make this
molecule a potential tracer of magnetic fields in disks as well (_Brauer et
al._ , 2017). Several attempts to carry out Zeeman measurements in mm lines
with ALMA have provided stringent upper limits of a few mG for the line-of-
sight magnetic field intensity in a couple of evolved disks (_Vlemmings et
al._ , 2019; _Harrison et al._ , 2021), while polarized emission from spectral
lines has been tentatively linked to the magnetic field morphology in the TW-
Hya disk (_Teague et al._ , 2021). In the future, it may be possible to extend
this technique to embedded disks, and hopefully lift the current challenge to
measuring their B-fields. Maser emission has also proved to be a powerful tool
to study magnetic fields in the densest portions of star-forming regions
(_Lankhaar et al._ , 2018). As masers are often very bright, polarimetric
observations can detect the line-of-sight component of the magnetic field,
reliably determining its strength (and direction, if VLBI techniques are
employed) in high-mass (e.g., using 6.7 GHz methanol masers, _Surcis et al._ ,
2013) and low-mass (e.g., using 22 GHz water masers, _Alves et al._ , 2012)
disks. Despite these observations being sensitive to very specific conditions
such as shocks, where magnetic fields may be amplified to several mG,
comparison to models may allow for a finer characterization of maser emission
over a wide range of pumping and excitation conditions, using sources such as
SiO and H2O masers, in the near future (_Lankhaar and Vlemmings_ , 2019).

###  2.4 The physical and chemical conditions in inner envelopes:
observational characterization

The rotating/infalling envelopes of embedded protostars are the cradles where
the most pristine disks form and evolve, at the same time as the accretion of
envelope material onto the protostellar embryo. Both the gas kinematics and
the chemical processes at work on disk-forming scales can directly affect the
disk properties. We refer the reader to the dedicated PPVII chapters devoted
to the gas kinematics in star-forming structures (Pineda et al.) and the
chemical conditions in protostellar environments (Ceccarelli et al.) for
detailed reviews. We briefly describe here the gas properties in the inner
envelopes which have important consequences for how efficiently magnetic field
couple to the protostellar gas, the transport and redistribution of angular
momentum, and ultimately the properties of disk-forming material.

The recent development of observational capabilities (increased bandwidth of
the Sub-millimeter Array (SMA), the upgrade of the NOEMA interferometer, and
development of the ALMA observatory) have allowed us to observe a wide range
of molecular lines as probes, not only to study the pristine chemistry in
star-forming cores, but also to selectively study the physical processes at
work in protostellar interiors, down to scales where disks form (_Jørgensen et
al._ , 2020; _Öberg and Bergin_ , 2021). The warm inner envelope, apart from
showing emission from complex organic molecules (COMs), also presents compact
emission from small molecules like H2S, SO, OCS and H13CN, most likely related
to ice sublimation and high-temperature chemistry (_Tychoniec et al._ , 2021).

Molecular line observations are also used to characterize the angular momentum
contained in the star-forming gas, ultimately responsible for the formation of
the protostellar disks. Recent observations of Class 0 protostellar envelopes
at large scales ($>$1000 au) suggest the specific angular momentum scales with
envelope radii, following a power-law relation $j\propto r^{1.8}$ between 1000
and 10000 au, with values ranging from $10^{-5}$ to a few $10^{-3}$
$\mathrm{km}\,{{\rm{s}}}^{-1}\,\mathrm{pc}$ at 1000 au (_Yen et al._ , 2015b;
_Pineda et al._ , 2019; _Heimsoth et al._ , 2021). In the CALYPSO survey,
molecular line observations at sub-arcsecond resolution have allowed
characterization of protostellar gas motions down to $\sim 50$ au scales.
_Gaudel et al._ (2020) finds a break in the angular momentum evolution with
radius, from a steep profile $j\propto r^{1.6}$ at radii $r>1600$ au, to a
quasi-flat profile at radii 50–1600 au. These profiles are shown in the right
panel of Figure 2, together with the SMA MASSES measurements from _Heimsoth et
al._ (2021) in the left panel.

Figure 2: Specific angular momentum of the gas measured at clouds and core
scales (left panel, from _Heimsoth et al._ 2021) and inside protostellar
envelopes (right panel, from _Gaudel et al._ 2020).

Envelope mass infall rates are estimated by modelling molecular lines showing
inverse P-Cygni profiles (_Pineda et al._ , 2012; _Mottram et al._ , 2013;
_Evans et al._ , 2015) but these are rare due to he intrinsic complexity of
the velocity field of the gas in the inner envelopes. Typical values measured
on scales of a few hundred au suggest a gas mass accretion rate $\sim 10^{-5}$
$\rm{M}_{\odot}$ per year in Class 0 protostars. In the Class I protostars of
the ALMA Ophiuchus survey, typical mass accretion rates are a few $10^{-7}$
$\rm{M}_{\odot}$ per year (_Artur de la Villarmois et al._ , 2019). The rather
high rates found in Class 0 protostars at first sight seem inconsistent with
their observed protostellar luminosities because one would expect typical
accretion luminosities of a few tens of $\rm{L}_{\odot}$ while the median
luminosity is observed to be around a few $\rm{L}_{\odot}$ (with large
variations from object to object). This conundrum, known as the luminosity
problem (_Dunham and Vorobyov_ , 2012; _Dunham et al._ , 2014), may be solved
by by invoking episodic accretion, an idea that is also supported by
observations of molecular species that are found at radii larger than expected
based on their gas temperature. Assuming the chemical timescale is
considerably longer than the cooling timescale of the gas, sudden temperature
changes due to short but vigorous accretion bursts would explain why CO is
observed where the current envelope temperature predicts it should be frozen
onto dust grains ($<30$K) for example (_Anderl et al._ , 2016; _Cieza et al._
, 2016; _Frimann et al._ , 2017). A recent ALMA survey of N2H+ and HCO+ toward
39 Class 0 and I protostars in Perseus suggest almost all sources in the
sample show evidence for post-burst signatures in N2H+ while the frequency of
the bursts may decrease with protostellar evolution, from a burst every 2400
yr in the Class 0 stage to 8000 yr in the Class I stage (_Hsieh et al._ ,
2019). Moreover, observations show that the velocity field of the gas at radii
$r<5000$ au is not organized as expected from the collapse of axisymmetric
rotating cores, with many envelopes exhibiting reversal of their velocity
fields or multiple velocity components (_Gaudel et al._ , 2020; _Maureira et
al._ , 2017), and well-developed asymmetric features which may trace preferred
pathways that funnel accretion, such as streamers (_Pineda et al._ , 2020) or
supersonic infall along outflow cavity walls (_Cabedo et al._ , 2021).

The study of the molecular line emission at the disk surface and at the
interface between the rotationally-supported disk and the infalling-rotating
envelope may reveal the physical conditions of the accreted gas, at scales
where the kinetic energy of the infalling gas is converted into thermal energy
and rotational motions. The gas from the infalling envelope landing onto the
disk can create shocks for example by friction between the rotating material
and the infalling material, raising the temperatures of the gas and dust at
the disk-envelope interface to values much higher than from heating by stellar
photons alone. These accretion shocks may therefore be crucial in setting the
pristine chemical composition of young disks(_Sakai et al._ , 2014b).
Observational signatures of accretion shocks have been reported, on scales
between the centrifugal barrier (where most of the gas kinetic energy
contained in infalling motion is converted to rotational motion, producing an
azimuthal velocity, $v_{\theta}$, larger than the Keplerian velocity) and the
centrifugal radii (inside which the circular gas motions are nearly
Keplerian), in a handful of young protostars. For instance, SO and other
sulfur-containing species were found to be enhanced at a radius $\sim 100$ au
around the L1527 disk as a result of sublimation of grain mantles due to a
weak accretion shock (_Sakai et al._ , 2014a; _Miura et al._ , 2017). Gas
temperatures are also found to be enhanced in the shocked region from $\sim$30
K pre-shock to $>$60 K inside the shocked region (_Sakai et al._ , 2017). In
IRAS16293-2422A, for comparison, it is enhanced from 80-120 K to 130-160 K
(_Oya et al._ , 2016). Equivalently, warm SO2 emission, possibly related to an
accretion shock, was observed toward the B335 Class 0 protostar (Bjerkeli et
al. 2019) and a few Class I sources (_Oya et al._ , 2019; _Artur de la
Villarmois et al._ , 2019). The observed abundances of SO can be reproduced
either with low-density shocks ($<10^{6}$ cm-3) in a weak UV field, if the SO
ice is not thermally sublimated, or high density shocks in strong UV fields
(_van Gelder et al._ , 2021). This second hypothesis seem to be supported by
tentative evidence of a correlation between the amount of warm SO2 and the
bolometric luminosity, as well as observed SO2/SO column density ratios in the
Class I source Elias 29 (_Artur de la Villarmois et al._ , 2019). However,
high-angular resolution observations ($\sim 30$ au) of the Class I source
TMC1-A show narrow SO emission lines coming from a ring-shaped morphology
which may be linked to the warm inner envelope (_Harsono et al._ , 2021) and
SO is also found to trace molecular outflows and jets in some objects (_Podio
et al._ , 2016; _Tabone et al._ , 2017; _Lee_ , 2020; _Podio et al._ , 2021):
emission from warm SO and SO2 is thus not an unambiguous tracer of accretion
shocks. Observations of mid-infrared shock tracers such as H2O, high-J CO, [S
I], and [O I] with the James Webb Space Telescope may allow a finer
characterization of shock conditions and put constraints on both the chemical
composition of the gas incorporated into the protostellar disks and the
magnetic field coupling at the disk/envelope interface.

###  2.5 Protostellar disks: observed properties

In the standard paradigm of star and planet formation, circumstellar disks
have long since been described as a natural, almost unavoidable, outcome of
the envelope’s angular momentum conservation during its collapse, ultimately
leading to the formation of a young star (see e.g. the reviews by
_Bodenheimer_ , 1991; _Larson_ , 2010). As such, they were a sound and
convenient solution to the so-called “angular momentum problem” in star
formation (see e.g., _Mestel_ , 1965; _Prentice and Ter Haar_ , 1971, for
early reviews), as buffer structures preventing the transfer of all the
envelope’s angular momentum to the forming star. In the traditional analytical
description of the “inside-out” collapse of a singular isothermal sphere (SIS)
in solid-body rotation (_Terebey et al._ , 1984), the centrifugal radius grows
very quickly with time with $t^{3}$ as a result of incoming material with
increasingly larger specific angular momentum, producing disk radii of a few
hundreds of au in a few thousands years (_Stahler et al._ , 1994). In a
magnetized scenario, however, disk growth is mitigated by magnetic braking and
diffusive processes, such as ambipolar diffusion, regulating magnetic flux
evolution, and the centrifugal radius grows more slowly, as $t$ (_Basu_ ,
1997, 1998). Although different initial conditions may better reproduce
observations of prestellar and protostellar cores (such as, e.g., Bonnort-
Ebert spheres, see for example _Banerjee et al._ , 2004; _Keto et al._ ,
2015), and affect the details of disk formation during the main accretion
phase, it is a common feature of all protostellar formation models that the
introduction of magnetic fields will produce somewhat smaller rotationally-
supported disks. While models and their main features are described in the
following section 3, we outline here how observations of the youngest embedded
protostars have been used to discriminate between them, and ultimately shed
light on the role of key physical ingredients in disk formation, and in
determining pristine properties of planet-forming disks.

####  2.5.1 Disk radii

Disentangling the disk from the envelope contribution on scales where
protostellar disks are expected to form has long been an observational
challenge (_Looney et al._ , 2000; _Jørgensen et al._ , 2009). Since PP VI,
many sensitive high-resolution interferometer surveys of protostellar
populations have been carried out. Observations of the dust continuum emission
at small envelope radii, and comparison with protostellar models of dust
emission, has allowed us to determine which protostars likely host a disk, and
to estimate dusty disk radii down to the observational limit (typically a few
tens of au). We review here of some of the largest recent surveys, and their
conclusions. Figure 3 presents all dust disk sizes reported in the literature
at short wavelengths $\lambda<2.7$mm, associated with embedded protostars.

The CALYPSO survey (_Maury et al._ , 2010, 2019) used the Plateau de Bure
Interferometer, at 1.3 mm and 2.7 mm, to characterize disk properties of 26
Class 0 and Class I protostars. Modeling the visibility profiles of the
millimeter dust continuum emission with a combination of envelope and disk
contributions, they found an average disk size of $<$ 50 au $\pm$ 10 au in the
Class 0 objects, and 115 $\pm$ 15 au in the Class I objects. While this survey
had the advantage of samples from several different star forming regions, it
is likely biased towards the more luminous sources that were easily observed
in the pre-ALMA era.

The VANDAM survey produced VLA observations of the 8 mm dust continuum
emission which were used to measure the radii of all observed Class 0 and
Class I disks in Perseus using a power-law intensity model. _Segura-Cox et
al._ (2018) find 14/43 ($33\%$) Class 0 and 4/37 ($11\%$) Class I sources have
resolved disks with r $>$ 12 au, and 62/80 ($78\%$) of Class 0 and I
protostars are not associated with resolved disk emission within r $>$ 8 au.
_Cox et al._ (2017) observed a sample consisting largely of Class II YSOs,
Flat and Class I objects in Ophiuchus with ALMA in the 870 $\mu$m dust
continuum emission and found a median radius of 12.6 au for the Class I/Flat
Sources by fitting a 2D Gaussian to the images. A complementary study of
Ophiuchus done by _Encalada et al._ (2021) found a median radius of 23.5 au
for Class I protostars. The VANDAM survey, also (_Tobin et al._ , 2020),
looked at Orion protostars at 870 $\mu$m using ALMA: from the Gaussian fitting
of the dust continuum emission, they find the median dust disk radii for the
69 Orion Class 0 protostars $\sim 48$ au and $\sim 38.0$ au for the 110 Class
I sources. While these results confirm Class 0 disks are typically compact,
they reveal median dust disk sizes smaller than the median radii computed by
using all the literature available (see Figure 3). This difference may stem
from the statistical completeness of the VANDAM survey in one single region,
while other Class I disk sizes estimates such as the ones reported in _Maury
et al._ (2019) may be biased with mostly bright Class I protostars in the
sample. Note however that Orion is a unique cloud due to its very active
environment as well as the number of massive stars forming therein: this could
be the cause of the peculiar decrease in disk sizes from Class 0 to Class I
protostars, which do not seem to be routinely observed in other star-forming
regions.

We stress that characterising disk size depends markedly on the choice of
observations performed, and the analysis carried out. For example in some
cases, disk radii are estimated from the additional thermal component which
cannot be accounted by small-radii envelope emission. Alternatively, some
studies assume the envelope emission is completely filtered out on sub-
arcsecond scales and attribute the entire emission obtained from long-baseline
maps as exclusively due to disk emission. While simple Gaussian models measure
the spatial extent of the dust thermal emission, they ignore the potential
contribution from envelope emission, which may differ, statistically, from
Class 0 to Class I with the envelope dispersal/accretion. This method may
overlook complex structures (streamers, outflow cavities, etc.) which are
often reported when carefully examining the spatial distribution in sensitive
dust emission maps. Finally, the choice of wavelength where to observe the
thermal emission of the dust may also influence the dusty disk sizes observed.
Using long wavelengths may be more sensitive to the spatial distribution of
large grains, especially at more advanced protostellar stages when dust
settling may have had time to operate. This was noted by _Tobin et al._ (2020)
who finds systematically more extended 0.8mm emission than 8mm emission in
their sample of Orion disks. The use of a shorter wavelength may bias the
sizes of more evolved sources because optically-thick emission is better
modeled by flatter Gaussian sources with larger FWHMs.

Figure 3: Compilation of Class 0 (in blue) and Class I (in orange)
protostellar disk radii, from the literature, observed from the dust continuum
emission at (sub-)millimeter wavelengths ($\lambda<2.7$mm). From _Oya et al._
(2014); _Lee et al._ (2018b); _Yen et al._ (2015a); _Yen et al._ (2017); _Aso
et al._ (2015); _Ohashi et al._ (2014); _Brinch and Jørgensen_ (2013);
_Harsono et al._ (2014); _Miotello et al._ (2014); _Tobin et al._ (2020);
_Sheehan and Eisner_ (2017); _Maury et al._ (2019). The two horizontal lines
show the median disk radii of 61 au and 81 au for each Class (0 in blue, I in
orange) respectively, from these distributions. We stress that upper-limit
radii from (yet?) unresolved disks are not included in those statistics so
these median values should certainly be interpreted as upper-bounds.

These aforementioned studies rely mainly on the analysis of the thermal dust
emission, but protostellar disks are gaseous structures, which sizes should be
evaluated also using the gas emission. This seems even more important as
recent observations towards Class II YSOs have revealed a large population of
compact dusty disks but suggest as well that the gaseous disks may be on
average a factor of two larger than the dust extent (see, e.g., _Barenfeld et
al._ , 2016; _Sanchis et al._ , 2021; _Ansdell et al._ , 2016, 2018, , as well
as the PPVII review chapter by Miotello et al.). This could suggest the
existence of a potential bias in the youngest protostellar disk size
distribution, which are mostly obtained from dust emission. The dusty disk
sizes could be different from gaseous disk sizes, although at the high
densities typical of Class 0 inner envelopes, no significant gas/dust
decoupling can be expected if the grain sizes remain relatively small
(_Lebreuilly et al._ , 2020). While the small sample of protostellar disks
characterized in dust and molecular line emission appears to show gaseous
radii similar to dust radii, only a larger sample can confirm this.

Embedded disks are expected to be largely rotationally supported and as such
their gaseous sizes could be measured with the detection of (nearly) Keplerian
gas motions around the central stellar embryo. A few studies have been able to
detect Keplerian rotation from protostellar dusty disks, finding gaseous radii
close to the dust radii in Class 0 protostars (_Ohashi et al._ , 2014; _Yen et
al._ , 2017; _Bjerkeli et al._ , 2019; _Maret et al._ , 2020) and Class I
protostars (_Harsono et al._ , 2014; _Chou et al._ , 2014; _Yen et al._ ,
2014; _Aso et al._ , 2015). Observations may also be able to trace the
location of the centrifugal radius and centrifugal barrier of the gas, thanks
to shock tracers, as detailed in the previous section 2.4 (_Sakai et al._ ,
2014a; _Alves et al._ , 2017; _Lee et al._ , 2017b; _Imai et al._ , 2019).

The small numbers are due to the complexity of the velocity field in embedded
protostars on the smallest scales. Deciphering the kinematic signature of
disks in embedded protostars is hampered by the intrinsic complexity of the
protostellar environment which naturally produces entangled contributions from
e.g., envelope infall, rotation, and outflowing gas, on overlapping scales and
mixed along the line of sights. For example in the HH 212 Class 0/I protostar
the sequential analysis of ALMA observations providing increased spatial
resolution and sensitivity have led to a revision downwards of the disk radius
from $\sim$100 au (_Codella et al._ , 2014; _Lee et al._ , 2014) to a smaller
$<44$ au radius (_Lee et al._ , 2017b), thanks to the characterization of
complex kinematic structures contributing at intermediate radii 50-200 au
(_Podio et al._ , 2015). Similarly in L483, where _Oya et al._ (2017) argued
that the centrifugal barrier in this source is at radii of 30–200 au while
_Jacobsen et al._ (2019) suggest molecular lines trace infalling gas down to
radii of 10–15 au, rather than Keplerian rotation.

Moreover, the kinematic signature of embedded and heavily accreting disk
structures may present significant deviations from purely axisymetric
Keplerian rotation curves (e.g. because of the generation of relatively
massive tidal streams and spiral arms, _Takakuwa et al._ 2014; _Pérez et al._
2016; _Tobin et al._ 2016; _Takakuwa et al._ 2017; _Alves et al._ 2019),
making it difficult to characterize young embedded disks as easily as their
older counterparts. An example is found in the L1527 Class 0/I protostar where
the $\sim 60$ au disk, although one of the first resolved kinematically and
seen in a favorable edge-on configuration, was recently proposed to be warped
because of either anisotropic accretion of gas with different rotational axes,
or misalignment of the rotation axis of the disk with the magnetic field
direction (_Sakai et al._ , 2019).

Confirming the rotationally-supported nature of candidate protostellar disks
observed in the dust continuum emission is also observationally expensive as
it requires sensitive observations of faint molecular emission, from species
which remain optically thin at high densities. Future observational studies
should not only rely on observations of multiple molecular species, but also
model them jointly to characterize the gaseous disks in the embedded phases.

####  2.5.2 Disk masses

Several observational studies have measured disk masses in embedded
protostars, using the emission due to dust which is then corrected to a gas
mass using an assumed gas-to-dust ratio. In Orion, _Tobin et al._ (2020) use
the 870 $\mu$m ALMA dust continuum fluxes to measure the disk masses, finding
a median dust mass $52.5{\rm\,M_{\oplus}}$ for the 69 Class 0 single
protostars, and $15.2{\rm\,M_{\oplus}}$ for the 110 Class I protostars. The
choice of wavelength(s) for the dust observations is crucial not only for
estimating disk sizes but also their masses. For example in Perseus,
_Tychoniec et al._ (2020) measure median dust masses of the embedded disks
$\sim 158{\rm\,M_{\oplus}}$ for Class 0 and $52{\rm\,M_{\oplus}}$ for Class I
from the VLA dust continuum emission at centimeter wavelengths. In comparison,
the lower limits on the median dust disk masses extrapolated from sub-
millimeter dust emission in ALMA bands are in better agreement with the masses
found in Orion, $47{\rm\,M_{\oplus}}$ and $12{\rm\,M_{\oplus}}$ for 38 Class 0
and 39 Class I respectively in Perseus.

These disk dust masses are however subject to several caveats due to the
possible presence of optically thick dust emission and what is assumed as
regards the dust opacity and temperatures in these yet largely unconstrained
environments. First, most studies are based on a single sub-millimeter flux
measurement, the mass estimates assume optically thin dust emission: if the
dust emission is not optically thin (as shown in several cases, see _Ko et
al._ 2020 and discussions therein), then the dust masses will be lower limits.
We stress that this could cause a bias leading to an underestimate of the
masses of the most evolved protostellar disks, which could be optically-thick
as collapse proceeds (_Galván-Madrid et al._ , 2018; _Zhu et al._ , 2019).
Regarding mass estimates based on dust emission at longer (cm) wavelengths,
the contribution of large dust grains to the observed emission could be
enormously variable from object to object, and tracing mostly the compact
emission from the inner disk. Obtaining a dust mass from cm dust emission is
thus also highly uncertain. (Sub-)millimeter dust emission is also a uncertain
probe of dust masses, unfortunately, as it suffers from uncertainties on dust
emissivity properties which have been found to be very different from those of
ISM dust, see section 4. Finally, the considerable unknowns regarding the
typical gas and dust temperature of these young disks, actively accreting and
bombarded by envelope material, also impair the dust continuum observations in
providing accurate disk masses. That said a few clues have been recently come
to light suggesting these disks are warmer than their older counterparts,
although these conclusions still suffer from poor statistics (_van’t Hoff et
al._ , 2020).

Demographics of older, Class II, disks appear to imply that most do not
contain enough material to form the known census of exoplanets (_Ansdell et
al._ , 2016; _Manara et al._ , 2018). Both dust and gas mass estimates
(usually from CO emission) suffer from large uncertainties and their
reliability is still widely debated by the community (e.g., _Aikawa et al._ ,
1996; _Yoneda et al._ , 2016; _Thi et al._ , 2001; _Dodson-Robinson et al._ ,
2018; _Powell et al._ , 2019; _Bergin et al._ , 2013; _Favre et al._ , 2013;
_Kama et al._ , 2016; _McClure et al._ , 2016). Details are beyond the scope
of this review as we focus on the youngest disks. Nevertheless, the resultant
uncertainties highlight the importance of determining disk masses from the
earliest phases, as such masses determine how early planets might form. Recent
observations largely suggest the dust masses of young Class 0 and I disks are
larger, by at least a factor of a few, compared to more evolved Class II
disks. In Ophiuchus, for example, the mean Class I disk dust mass is found to
be significantly lower than in Perseus and Orion, $\sim
2.8-3.8{\rm\,M_{\oplus}}$ (_Williams et al._ , 2019; _Encalada et al._ ,
2021). Nevertheless it is still about 5 times greater than the mean Class II
disk mass in the same region, although the dispersion in each class is so high
that there is a large overlap between the two distributions. Interestingly,
some annular structures in the dust continuum emission of Class I protostars
have recently been observed (_Harsono et al._ , 2018; _Segura-Cox et al._ ,
2020; _Sheehan et al._ , 2020). While their origin is still debated, it is
possible they are carved by the early planetary seeds. Observations in the
coming years of the gas kinematics in young disks should help constrain
whether this is the case. Moreover, these preliminary findings, regarding both
dust masses and their spatial distributions, still need to be validated
through the use of large, high spatial resolution, multi-wavelength samples
that allow one to remove current degeneracies. If confirmed, they would make
embedded protostars the likely cradles of planetary formation.

###  2.6 Outflows and jets in embedded protostars

Jets from young stars have been observed from the X-ray regime right down in
frequency to the radio band. Historically the emphasis has been on studying
them at optical and near-infrared wavelengths but with the advent of high
spatial and spectral resolution millimeter arrays such as ALMA, increasingly
the focus has shifted to molecular emission lines and more embedded sources.
Whether we are looking at the ionized, neutral atomic or molecular component,
the presence of line emission ensures we can determine fundamental outflow
parameters such as velocity, density, temperature, etc. Of course these
various species need not be co-located, nor are they expected to be, and so no
single set of waveband limited derived parameters can characterise an outflow.
Outflow emission arises primarily from the cooling zones of shocks, shocks
that are generated by higher velocity gas catching up with slower material
ahead of it. This means the strength of the shock is determined by velocity
differences rather than the actual outflow velocity with respect to the
source.

There is insufficient space here to provide an in-depth review of the
properties of outflows generated by the wide variety of YSOs, or even proto-
brown dwarfs, from which they are observed, the reader instead is referred to
_Frank et al._ (2014), _Bally_ (2016) and _Ray and Ferreira_ (2021). Here we
will concentrate on a few salient properties of those from the most embedded
(Class 0/I) sources and describe in general terms how observations can give us
clues to their launching mechanism. Note that as Class 0/I sources are highly
embedded, observations have largely been confined to millimeter wavelengths,
i.e. their molecular emission, although not exclusively.

Of course molecular outflows from young stars have been known for several
decades (e.g., _Arce et al._ , 2007) but the realisation that such outflows
can appear (e.g. in SiO, SO or even CO) just as jet-like as their
ionized/neutral atomic counterparts was slower in coming (e.g., _Plunkett et
al._ , 2015; _Lee et al._ , 2017a; _Jhan and Lee_ , 2021). Studies using mm
interferometers of Class 0/I sources show their outflows can be traced right
back to the protostar with in some cases very narrow opening angles (_Bjerkeli
et al._ , 2019) and velocities comparable to the young star’s escape velocity
(_Hirano et al._ , 2010). All of this suggests such outflows, or at least
their most highly collimated component i.e. the jet, arise from a region no
greater than $\approx$1 au centred on the YSO. A classic example, and
something of a poster child is HH 212 (e.g., _Lee_ , 2020, and see Fig. 4).
While extended emission in H2 is observed on pc scales, SiO can be traced back
to at least 10 au from the source. Certainly the presence of a disk is a
necessary, although not necessarily sufficient, condition for jet launching.

Figure 4: ALMA SiO and continuum observations of the rotating outflow from the
Class 0 protostar HH 212. While the outflow itself extends to much larger
scales, this shows the region within $\approx$ 120 au of the central source,
at a resolution of $\approx$ 8 au on top of the continuum map of the disk. The
maps show the intensity (in unit of K km s-1) integrated over the outflow
velocity range. (a) A chain of SiO knots trace the primary jet emanating from
the disk testifying to the episodic nature of the outflow. (b) Blueshifted and
redshifted SiO emission of the jet plotted with the continuum (disk) emission.
The direction of rotation of the disk is shown with blue and red arrows and
this is the same as the direction of rotation of the red and blueshifted jets.
From _Lee et al._ (2017a) reproduced with permission ⓒ Springer Nature.

The other ingredient needed, as in almost all astrophysical models for the
generation of jets is a magnetic field (e.g., _Livio_ , 2011). In 2.7.2, we
will examine the evidence for magnetic fields in outflows but here we mention
some plausibility arguments as to why we think they are present and why they
are thought to be dynamically important as they transfer angular momentum from
infalling/rotating motions to outflowing motions, at least close to a YSO.
While there is very little data on magnetic fields threading the base of the
jets and protostellar disk surfaces where disk winds are launched in embedded
sources, we know that the surface magnetic fields of Class II/III young stars
have strengths of several kG (e.g., _Donati et al._ , 2020) as derived from
Zeeman splitting measurements. Data on field strengths in the innermost region
of protostellar disks (i.e. $\lessapprox$ a few stellar radii) is sparse but
observations obtained from spectro-polarimetry in the disk of FU Ori (_Donati
et al._ , 2005) lead to similar values as derived for the parent star. In
addition studies of paleomagnetism in chondrules from the early period in the
formation of the Solar System (_Weiss et al._ , 2021) suggest field strengths
of $\approx$ 1 G at 1 au. Such values, in combination with the expected
density, temperature and velocity of a stellar/disk wind, are clearly capable
of collimating a wind into a jet, at least very close to the star. The most
efficient means of achieving this is for the ionised gas (dragging through
collisional coupling associated atomic/neutral material with it) to be
centrifugally ejected along the magnetic field lines. These field lines are
either anchored to the star (stellar winds), at the star-disk interface (so-
called X-winds), or alternatively in the disk itself (Disk or D-winds). For a
review of these various options the reader is referred to _Pudritz and Ray_
(2019) but also see 3.1.5. In each case the magnetic field associated with the
wind/outflow, while starting off largely poloidal, increasingly, particularly
in the case of D-winds, becomes toroidal with distance as a result of the
material carrying more and more angular momentum (see below). Tension in these
wound-up toroidal fields then generate forces towards the axis, known as
$``$hoop stresses” that focus the outflow further into a jet (_Spruit et al._
, 1997).

As explained elsewhere in this chapter (3.1.5) in order for accretion to occur
in a protostellar disk, angular momentum must either be removed poloidally,
for example through a magnetic tower flow (3.1.5, through a disk wind or
outflow, or alternatively redistributed radially outwards within the disk (for
example through gravitational torques at large disk radii, _Vorobyov and Basu_
(2006, 2010, 2015). In any event, in recent years it has become increasingly
clear that angular momentum in a disk is removed through the former mechanisms
as possible sources of disk viscosity, e.g. MRI driven turbulence, seem
unlikely to work due to non-ideal MHD effects (e.g. for Class 0 disks see
_Kawasaki et al._ (2021)). While winds/outflows from the star, star-disk
interface and the disk itself are all likely to play a role in angular
momentum transport (_Pudritz and Ray_ , 2019), it is clear that a significant
fraction of the angular momentum must be removed through a disk wind/outflow
in any event for material to reach the inner disk radius and be accreted onto
the star along magnetic field lines as suggested by observations (_Gravity
Collaboration et al._ , 2020). Thus depending on where the wind/outflow is
launched, it is expected to carry varying amounts of angular momentum per unit
mass. Thus rotation is expected to be a key signature of disk generated
outflows (_Pudritz and Ray_ , 2019). The search for rotation in
ionized/neutral atomic jets has been challenging from an observational
perspective as it is difficult to spatially resolve such jets transversely,
the maximum rotation (toroidal) velocity is only a small fraction of the
poloidal velocity and the signal to noise ratio is often not high (e.g. _Frank
et al._ , 2014). All of these issues are less of a problem with mm-
interferometers such as ALMA, and even radio interferometers such as the
Jansky VLA, that not only have high spatial resolution, to resolve the outflow
transversely, but also the necessary high spectral resolution. Thus, for
example, at the start of the ALMA era, _Hirota et al._ (2017) observed
rotation in the outflow from Orion Source I. In turn, using basic magneto-
centrifugal wind theory, including angular momentum conservation (for details,
see _Anderson et al._ 2003), they could estimate from where the outflow was
launched in the disk and discovered it to be far ($\gtrsim 10$ au) from the
protostar (see also _Bjerkeli et al._ , 2016). That the magnetic field has the
configuration and strength expected to collimate this outflow has also
recently been found (_Hirota et al._ , 2020). Rotation has now been detected
in many molecular outflows including TMC1A (_Bjerkeli et al._ , 2016), HH 211
(_Lee et al._ , 2018a), HH 212 (_Lee et al._ , 2017a), for several outflow in
NGC1333 (_Choi et al._ , 2011; _Chen et al._ , 2016b; _Zhang et al._ , 2018),
HH 30 (_Louvet et al._ , 2018) and OMC2/FIR6b (_Matsushita et al._ , 2021). In
the latter it was found that the high-velocity jet axis is significantly
inclined from the low-velocity outflow axis, indicating that the launching
radii of the high- and low-velocity components differ in an inclined disk. In
all cases a disk origin for the jet/outflow is indicated and, where observed,
the direction of disk rotation matches that of the outflow (see Fig. 4) as
would be expected. It would also appear that all of these observations support
the direct disk wind expected from core collapse simulations.

###  2.7 The role of magnetic fields in shaping protostellar disks and
outflows: observational keys

####  2.7.1 Observations of magnetic fields and disks

Here we discuss the main clues that observations provide regarding the role
played by magnetic fields in setting disk sizes and masses during the embedded
phase.

In a simple scenario for the collapse of a rotating core, even a small
difference in initial specific angular momentum on large scales is expected to
result in a large change in the radius of the resultant rotationally-supported
disk. Observations of the angular momentum on scales where the gas should
contribute to the building of the disk (_Sai et al._ , 2022) suggest most
protostars should give rise to disks that are larger, by a factor of a few,
than those observed (_Gaudel et al._ , 2020). Here it is assumed that the
angular momentum contained in rotation is entirely transmitted to the disk
scale. Naively such observations potentially support models with efficient
magnetic braking. Note however, that several theoretical studies have shown
that the observations of angular momentum in protostellar envelopes can be
satisfactorily reproduced without large scale core rotation, questioning
whether observed angular momentum values really stem from rotational motion
(_Verliat et al._ , 2020; _Xu and Kunz_ , 2021; _Lee et al._ , 2021c).

From an observational perspective, the relationship between the outflow axis
and the mean magnetic field direction has received a lot of recent attention.
This is because it is thought that the efficiency of magnetic braking in
redistributing angular momentum (and hence preventing the growth of disks to
large radii) when a core collapses depends on the configuration of the core’s
magnetic field with respect to its rotation axis (_Joos et al._ , 2012;
_Hirano et al._ , 2020). Starlight polarization observations show large-scale
magnetic fields that are misaligned with outflows powered by evolved (Class
II) young stellar objects (e. g., _Ménard and Duchêne_ , 2004), but the fields
are better aligned with outflows from younger protostars (e. g. _Vrba et al._
, 1986; _Targon et al._ , 2011; _Soam et al._ , 2015). Single-dish
observations examining the relative orientation of the B-field on core-scales
with the outflow axis are inconclusive: some studies suggest cores exhibit a
mean B-field that is not randomly distributed with respect to the large-scale
outflow axis (_Soam et al._ , 2015; _Yen et al._ , 2021b) while others suggest
no correlation (_Doi et al._ , 2020). Finally, some sub-mm polarization
observations of dense cores have found the direction of the core’s minor axis
correlates with their magnetic field orientation (_Chapman et al._ , 2013). A
flattening in this preferential direction could be explained by the
development of large magnetic pseudo-disks (equatorial flattened gas
structures which are not supported by rotation _Galli and Shu_ , 1993).

Likewise, observations using interferometers have not reached any firm
conclusions regarding correlations between outflow, disk and magnetic field
axes. On very small envelope scales, the mean B-field direction is observed to
be randomly aligned with respect to the outflow axis (_Hull et al._ , 2013).
However, as reported by _Hull et al._ (2014), there is evidence that cores
with lower fractional polarization tend to have their outflows perpendicular
to the mean B-field. This suggests the existence of a non-negligible toroidal
field (caused by the core/disk rotation) in addition to the poloidal one
coming from envelope-disk accretion. An SMA survey of 20 low-mass protostars
(_Galametz et al._ , 2018, 2020) has found that protostellar envelopes tend to
have a higher angular momentum on 1000 au scales if the mean envelope magnetic
field measured on similar scales is misaligned with respect to the rotational
axis of the core (assumed to coincide with the outflow axis). In contrast,
observations analyzed in _Yen et al._ (2021a) have shown no correlation of
dust continuum protostellar disk radii with misalignment of the magnetic
fields and outflow axes in Orion A cores. Both results may be consistent with
the magnetic field being efficient at reducing the amount of angular momentum
transmitted to the inner envelope scales ($<1000$ au), inhibiting the
formation of large hydro-like disks. However, they may also support the
hypothesis that the angular momentum responsible for the formation and growth
in size of protostellar disks has a more local origin, on a few hundreds of au
scales, as discussed previously.

Figure 5: Observations and best model to match the properties of the B335
protostar, from _Maury et al._ (2018). These authors have demonstrated that
the observed B field geometry shown in the left panel, the small disk size and
the kinematics of the B335 inner envelope can only be explained by a family of
MHD models where the initially poloidal field is pulled in the dominant
direction of the collapse, but yet is strong enough ($\mu\sim 6$) to partially
counteract the transfer of angular momentum inwards and set the disk size in
this Class 0 protostar: the best model is shown in the right panel.

Comparison of observed properties with models of disk formation in both
specific objects and statistical samples may also help our understanding.
_Maury et al._ (2018) has shown that the geometry of the B field, small disk
size and the kinematics of the B335 inner envelope can only be explained by a
family of MHD models where the initially poloidal field is pulled in the
direction of collapse, while at the same time being strong enough ($\mu\sim
6$) to partially counteract the transfer of angular momentum inwards. In B335,
they show the magnetic field likely regulates the formation of the disk,
constraining its size to $<20$ au (see Figure 5). The role of the magnetic
field in determining the disk size in B335 has been also proposed in follow on
studies (_Yen et al._ , 2019; _Bjerkeli et al._ , 2019; _Imai et al._ , 2019).
Detailed analyses of the gas kinematics, disk sizes, and B field configuration
in other objects are needed to ascertain whether the magnetically-regulated
disk growth in B335 is common, or a special case.

Since Protostars and Planets VI, and despite their complexity, observations
have successfully explored the properties of Class 0 disks in large part
thanks to the increased spatial resolution and sensitivity of new millimeter
surveys. As shown in Figure 3, the vast majority of observations now point
towards a population of young protostellar disks that are primarily compact.
From a statistical point of view this, and the confirmed presence of magnetic
fields on the scales where disks form, may naively point to magnetic fields
having a key dynamical role in regulating the early growth of stellar embryos
and their surrounding disks. Indeed, such disk sizes are difficult to
reconcile with purely hydrodynamical models of disk formation and, if the
B-field is well coupled, magnetic braking naturally produces such small disks
(see, e.g. _Lebreuilly et al._ , 2021, for a statistical comparison of
observed disk sizes and protostellar disk sizes in magnetized models).
However, observations are still largely unable to characterize how effective
magnetic fields are at coupling on the smallest scales where protostellar
disks form. This coupling is mostly due to small grains and the ionized
species in the gas (see Section 3). Measuring the amount of these two key
ingredients in and around embedded disks is thus fundamental to setting
constraints to the role of the magnetic field, and the importance of diffusive
processes, such as ambipolar diffusion, in counteracting the outward transport
of angular momentum from magnetic braking. In embedded sources, the ionization
rate from cosmic rays is inferred using various chemical signatures, often
HCO+, and N2H+ (_Bron et al._ , 2021). Measurements suggest typical cosmic ray
ionization rates $\zeta\sim 10^{-17}-10^{-15}\,\rm{s}^{-1}$ with large
uncertainties (_Ceccarelli et al._ , 2014; _Podio et al._ , 2014; _Favre et
al._ , 2017, 2018). Such measurements however remain rare, not only because
they are difficult to make from an observational perspective but also because
of chemical model degeneracy. In any event such high values seem inconsistent
with ionization from external radiation only: galactic cosmic rays, mostly
relativistic protons, are the dominant source of ionization in dense molecular
gas where ultraviolet radiation cannot penetrate but their flux should be
severely attenuated inside dense cores (_Grenier et al._ , 2015; _Padovani et
al._ , 2018). Several recent studies (_Padovani et al._ , 2015; _Silsbee et
al._ , 2018; _Padovani et al._ , 2021; _Fitz Axen et al._ , 2021; _Gaches and
Offner_ , 2018) have investigated the role of shocks at the protostellar
surface and magnetic acceleration within jets as efficient forges to locally
accelerate low-energy cosmic rays, and their role in increasing the ionization
rate of the shielded protostellar material. While the ionization can be
increased locally, for example at the accretion shock, the typical gas
densities in these regions should lead to a significant decrease in the
ionization fraction. Observational studies of the chemical composition of the
dense gas, and other properties could shed light on the small scale conditions
that determine the coupling of the magnetic field to the disk gas. Note that,
in a recent study of the TMC1-A Class I protostar, _Harsono et al._ (2021)
estimate an ionization from cosmic rays $\zeta\sim 10^{-17}s^{-1}$ at the disk
surface. This low ionization suggests the B-field is weakly coupled and
accretion through the disk may not be driven by the magneto-rotational
instability.

Future radio observations will undoubtedly bring more constraints on the
dynamical role of magnetic fields in regulating protostellar collapse and in
setting pristine disk properties. Future large surveys will allow us to
statistically investigate the dynamical role magnetic fields play by careful
examination of their relationship with the angular momentum and mass transport
processes from envelopes to disks (_Galametz et al._ , 2020; _Yen et al._ ,
2021a). The true spatial distribution of magnetic fields on disk-forming
scales may become routinely observed thanks to the development of polarimetric
capabilities and modeling tools, e.g. in connection with polarized dust
emission. Finally, observations with, for example, the next generation of mid-
infrared facilities will help characterize the small scale structures of these
young disks, still largely unresolved spatially. These studies may allow us to
put observational constraints on the models of magnetic field properties and
coupling efficiency inside protostellar disks (_Lee et al._ , 2021d), and test
how mass accretion onto a protostar might be mediated through its disk.

####  2.7.2 Observations of Magnetic Fields in Outflows Versus Prediction

Perhaps the most challenging physical quantity to determine in an outflow from
a young star is the strength and direction of its magnetic field. Measuring
the Zeeman effect for example using optical/near-infrared atomic lines is not
practical as any splitting is so weak. Interestingly, however, in the near
future it may be possible to use molecular lines, e.g., those of sulphur
monoxide SO (_Cazzoli et al._ , 2017). This of course might be particularly
pertinent in the case of molecular outflows from less evolved sources. In
general, however, indirect methods have to be employed to assess field
strengths. For example, as pointed out many years ago, magnetic fields can
cushion an outflow shock, altering the degree of compression in the post-shock
radiative cooling zone and changing observed emission line ratios (_Hartigan
et al._ , 1994). Just measuring line ratios however is not sufficient as the
shock solutions are degenerate. Instead additional parameters, such as the
spatial extent of the cooling zone are needed, to break the degeneracy and
this requires very high spatial resolution, for example as afforded by HST
(_Hartigan and Wright_ , 2015).

An alternative approach, that at least can be applied in some outflows, is to
measure their non-thermal radio emission. Of course most of the radio flux
from outflows is thermal in origin and is simply the free-free emission from
the collimated ionized jet. That said a number of outflows show pockets of
non-thermal emission, usually well away from the parent YSO (_Anglada et al._
, 2018). A classic example is HH 80/81 which has been shown through VLA
observations to display polarized synchrotron emission (_Carrasco-González et
al._ , 2010). This indicates, perhaps somewhat surprisingly, the presence of
relativistic particles which presumably have been energised through diffusion
shock acceleration even though the shock velocities are low (_Rodríguez-
Kamenetzky et al._ , 2016). In any event minimum energy requirements including
a equipartition field, in conjunction with the know non-thermal flux, can lead
us to estimates of the magnetic field (for example following _Longair_
(2011)). These values, typically 20–200 $\mu$G, appear consistent with what we
might expect in the regions observed far away from the collimation zone. More
recently it has emerged that low frequency radio observations of non-thermal
emission can offer us a novel method of deriving at least the field strength,
if not its direction. This technique relies on the fact that in an outflow
there can be a mixture of relativistic particles and thermal electrons. The
latter, depending on the electron density, de-collimates the forward-beamed
radiation from the non-thermal electrons, giving rise to a decrease in flux,
or effectively a low frequency turnover in the synchrotron spectrum. If the
thermal electron density is known independently, for example from optical
emission line diagnostics, then the magnetic field can be derived. This, so-
called Razin Effect, may have been seen in an outflow for the first time
(_Feeney-Johansson et al._ , 2019). The derived magnetic field strengths are
in agreement with what might be expected based on the expansion of the outflow
while allowing for some amplification due to shock compression (_Feeney-
Johansson et al._ , 2019).

Finally, it is worth noting that the GK effect (see 2.1) has been used to
probe the magnetic field in protostellar outflows. In particular _Lee et al._
(2018c) have detected this effect in SiO observations of the HH 211 outflow
using ALMA. Despite the inherent uncertainty of 900 in the magnetic field
orientation, the implied magnetic field geometry may well be largely toroidal
as expected for a collimating field.

##  3 A unified scenario for the formation of protostars, protoplanetary
disks, jets, and outflows

As described in §2, the rapid progress of observational research since PP VI
has provided strong constraints on theoretical models. In §3, we describe the
theoretical progress and present a comprehensive scenario for the formation
and early evolution of protostars, disks, and outflows that is consistent with
observational constraints. The key mechanism of this scenario is the coupling,
decoupling, and recoupling of the gas and magnetic field at appropriate
scales.

###  3.1 From cloud cores to protostars: formation and evolution of outflows,
jets, and protoplanetary disks

####  3.1.1 Essential microscopic physics for the formation of protostars,
disks, and outflows

In this section, as a basis for the ones that follow, we will introduce the
physical processes that are crucial for the formation and evolution of
protostars and protoplanetary disks.

Molecular cloud cores, from which protostars, disks, and outflows are formed,
are generally weakly ionized (with typical electron abundances on the order of
10-7; _Bergin and Tafalla_ , 2007). Applying the Ohm’s law, the electric field
in the neutral frame $(\mathbf{E^{\prime}})$ is tied to the current density
($\mathbf{j}$) as,

$\mathbf{j}=\sigma_{\parallel}\mathbf{E^{\prime}_{\parallel}}+\sigma_{\rm
H}\frac{\mathbf{B}}{B}\times\mathbf{E^{\prime}_{\perp}}+\sigma_{\rm
P}\mathbf{E^{\prime}_{\perp}}~{},$ (1)

where $\sigma_{\parallel}$, $\sigma_{\rm P}$, and $\sigma_{\rm H}$ are
parallel, Pedersen, and Hall conductivities, respectively, and
$\mathbf{E}^{\prime}_{\parallel}$ and$\mathbf{E}^{\prime}_{\perp}$ are the
components of $\mathbf{E^{\prime}}$ parallel and perpendicular to the magnetic
field $\mathbf{B}$.

The three components of the conductivity tensor can be expressed as (_Norman
and Heyvaerts_ , 1985; _Wardle and Ng_ , 1999; _Wardle_ , 2007),

$\displaystyle\sigma_{\parallel}$ $\displaystyle=$ $\displaystyle{{ec}\over
B}\sum_{i}Z_{i}n_{i}\beta_{i,\rm H_{2}}~{},$ (2) $\displaystyle\sigma_{\rm P}$
$\displaystyle=$ $\displaystyle{{ec}\over B}\sum_{i}{{Z_{i}n_{i}\beta_{i,\rm
H_{2}}}\over{1+\beta_{i,\rm H_{2}}^{2}}}~{},$ (3) $\displaystyle\sigma_{\rm
H}$ $\displaystyle=$ $\displaystyle-{{ec}\over
B}\sum_{i}{{Z_{i}n_{i}}\beta_{i,\rm H_{2}}^{2}\over{1+\beta_{i,\rm
H_{2}}^{2}}}~{};$ (4)

here, $e$ and $c$ are elementary charge and speed of light,
$Z_{i},n_{i},\beta_{i,\rm{H_{2}}}$ are the charge number, number density, and
Hall parameter of the charged species $i$, respectively. The Hall parameter
$\beta_{i,\rm H_{2}}$, which is the ratio of the gyro-frequency and neutral
collision frequency, determines the relative importance of the Lorentz and
drag forces in balancing the electric force. By solving equation (1) for the
electric field and substituting it into Faraday’s law. The induction equation
can be formulated as (e.g., _Kunz and Mouschovias_ , 2009),

$\begin{split}{\partial\mathbf{B}\over\partial
t}&=\nabla\times(\mathbf{v}\times\mathbf{B})-\nabla\times\biggl{\\{}\eta_{\rm
Ohm}\nabla\times\mathbf{B}\\\ &+~{}\eta_{\rm
Hall}(\nabla\times\mathbf{B})\times{\mathbf{B}\over B}+~{}\eta_{\rm
AD}{\mathbf{B}\over
B}\times\left[(\nabla\times\mathbf{B})\times{\mathbf{B}\over
B}\right]\biggl{\\}}~{},\end{split}$ (5)

which departs from the ideal MHD limit by the introduction of the three non-
ideal MHD terms, characterized by the Ohmic ($\eta_{\rm Ohm}$), Hall
($\eta_{\rm Hall}$), and ambipolar ($\eta_{\rm AD}$) resistivities. The
resistivities are related to the conductivities as

$\displaystyle\eta_{\rm AD}$ $\displaystyle=$ $\displaystyle{c^{2}\over
4\pi}({\sigma_{\rm P}\over\sigma_{\rm P}^{2}+\sigma_{\rm
H}^{2}}-{1\over\sigma_{\parallel}})~{},$ (6) $\displaystyle\eta_{\rm Ohm}$
$\displaystyle=$ $\displaystyle{c^{2}\over{4\pi\sigma_{\parallel}}}~{},$ (7)
$\displaystyle\eta_{\rm Hall}$ $\displaystyle=$ $\displaystyle{c^{2}\over
4\pi}({\sigma_{\rm H}\over\sigma_{\rm P}^{2}+\sigma_{\rm H}^{2}})~{}.$ (8)

The microscopic origin of these three non-ideal MHD effects is related to how
the particles interact in each scenario. Ohmic dissipation is the collision
between neutral and charged particles whereas ambipolar diffusion is charged-
particles slipping through neutral particles. The Hall effect is the velocity
difference between positively and negatively charged particles.

In the cloud core and disk, $\eta_{\rm Hall}$ (or equivalently $\sigma_{\rm
H}$) is negative over almost the entire density range (see, Figure 6). This
may be counter-intuitive. Therefore, the following will discuss the origin of
the negative $\sigma_{\rm H}$. The key is the total dust charge. The total
dust charge is negative in low density regions due to the large thermal
velocity of the electrons. In this case, due to charge neutrality, total
charge of ion is slightly larger than the charge of electron. Consider the
regime in which the Hall parameter of ions ($\beta_{\rm i}$), electrons
($\beta_{\rm e}$), and dust ($\beta_{\rm d}$) obeys, $\beta_{\rm i}\gg 1$,
$\beta_{\rm e}\gg 1$, $\beta_{\rm d}\ll 1$ i.e., ions and electrons are tied
with magnetic field but dust is not. In this case, $\sigma_{\rm H}\to
ec/B(n_{\rm e}-n_{\rm i})$ (we can neglect the contribution of dust because
$\beta_{\rm d}\ll 1$). Therefore, $\sigma_{\rm H}$ is negative because $n_{\rm
e}<n_{\rm i}$. Intuitively, it can be understood as follows. In a situation
where negatively charged dust is coupled with neutral gas and the ions and
electrons are coupled to the magnetic field, a relative motion of the ion
electron mixture (which is slightly positively charged) and the negatively
charged dust creates the electric current. In this case, the role of positive
and negative charges is reversed compared to ordinary Hall effect, and it
causes the negative $\sigma_{H}$.

In general, an ionization chemistry network (_Oppenheimer and Dalgarno_ ,
1974; _Umebayashi and Nakano_ , 1990; _Nakano et al._ , 2002; _Marchand et
al._ , 2016; _Wurster_ , 2016; _Zhao et al._ , 2018b; _Guillet et al._ ,
2020a) should be properly constructed in order to obtain the resistivities in
the induction equation (5). Cosmic-rays (CRs) are the main source of
ionization that triggers the chain of ionization chemistry as most of the
interstellar UV-radiation is attenuated in dense molecular cores (visual
extinction Av$>$4 mag; _McKee_ , 1989). On the other hand, dusts act as a
charge absorber and can also be a major contributor to conductivity. As an
example, Figure 6 shows the resistivities values for various dust
distributions. Note that the Ohmic resistivity is many orders of magnitude
smaller than the ambipolar resistivity in $\rho_{\rm g}\lesssim
10^{-10}~{}{\rm g~{}cm}^{-3}$ (or in $n_{\rm g}\lesssim 10^{13}~{}{\rm
cm}^{-3}$).

In this section, we use the following approximated formulae for Ohmic and
ambipolar resisitivities for a quantitative discussion. Ohmic dissipation,
$\eta_{\rm Ohm}$, is approximated as (_Nakano et al._ , 2002; _Machida et al._
, 2007, see also Figure 6)

$\displaystyle\eta_{\rm Ohm}\sim 1.6\times 10^{13}\rho_{\rm g,10^{-16}~{}{\rm
g~{}cm}^{-3}}\sqrt{T_{10\rm K}}~{}{\rm cm^{2}~{}s^{-1}},$ (9)

where $\rho_{\rm g}$ and $T$ denote the gas density and temperature, and
$f_{X}$ means $f_{X}=(\frac{f}{X})$. For ambipolar diffusion, the diffusion
rate, $\eta_{\rm AD}$, is approximated as (_Shu_ , 1983; _Zhao et al._ ,
2018b, see also Figure 6)

$\begin{split}\eta_{\rm AD}&\sim 2\times 10^{18}~{}{\rm cm^{2}~{}s^{-1}}\\\
\times&\begin{cases}\rho^{-1/2}_{\rm g,10^{-16}~{}{\rm
g~{}cm}^{-3}}&(\rho_{g,~{}{\rm g~{}cm}^{-3}}<10^{-16})\\\
1&(10^{-16}<\rho_{\rm g,~{}{\rm g~{}cm}^{-3}}<10^{-13})\\\ \rho_{\rm
g,10^{-13}~{}{\rm g~{}cm}^{-3}}&(10^{-13}<\rho_{\rm g,~{}{\rm
g~{}cm}^{-3}}<10^{-9}).\end{cases}\end{split}$ (10)

Here, we assume $B=0.2n_{\rm H}^{1/2}\mu G$ (_Nakano et al._ , 2002) and flux
freezing during the evolution. This is valid for a supercritical core and its
isothermal collapse phase ($\rho_{\rm g}\lesssim 10^{-13}~{}{\rm g~{}cm}^{-3}$
or $n_{\rm g}\lesssim 10^{10}~{}{\rm cm}^{-3}$) as we will discuss in
subsequent section.

Another potentially important magnetic diffusion process is turbulent (or
reconnection) diffusion. On small scales, reconnection from turbulent vortices
causes effective magnetic field diffusion. The effective resistivity of this
process is given as (_Lazarian_ , 2005; _Santos-Lima et al._ , 2021),

$\displaystyle\eta_{\rm turb}=L_{\rm turb}\delta v_{\rm turb}{\rm
min}(1,M_{A}^{3}).$ (11)

where $L_{\rm turb}$ and $\delta v_{\rm turb}$ are the injection scale and
turbulent velocity at the scale. $M_{A}$ is the Alfven Mach number
$M_{A}=\delta v_{\rm turb}/v_{A}$ and is typically $M_{A}\lesssim 1$ in cloud
cores.

Figure 6: The magnetic resistivities computed for different dust-grain size
distributions (top left: $\eta_{\rm Ohm}$; top right: $\eta_{\rm AD}$; bottom
left: $\eta_{\rm Hall}$; bottom right: $\eta_{\rm Ohm}$/$\eta_{\rm AD}$) with
$a_{\rm min}$=0.005 $\mu$m and $a_{\rm max}$=0.25 $\mu$m (MRN: black), $a_{\rm
min}$=0.03 $\mu$m and $a_{\rm max}$=0.25 $\mu$m (blue), $a_{\rm min}$=0.1
$\mu$m and $a_{\rm max}$=0.25 $\mu$m (red), $a$=1.0 $\mu$m (cyan) from (_Zhao
et al._ , 2018b). The reference level of ambipolar resistivity of Shu (1991)
is shown in purple dashed line (top right).

####  3.1.2 Isothermal collapse phase: magnetic diffusion timescale vs free-
fall timescale

The gravitational collapse of molecular cloud core initially proceeds
isothermally. The compressional heating by gravitational contraction is
immediately radiated away via dust thermal emission and the gas keeps its
temperature approximately constant at $T=10~{}$K. In this phase, the effective
polytropic index ($\Gamma_{\rm eff}$) is $\Gamma_{\rm eff}=1$ where $P_{\rm
gas}=K_{\rm s}\rho^{\Gamma{\rm eff}}$ ($K_{\rm s}$ is a constant) is smaller
than the critical value of $4/3$ for spherical gravitational collapse, and the
pressure gradient force never overtakes the gravity. As observations have
shown (_Crutcher_ , 2012), most clouds cores are supercritical. Therefore,
magnetic pressure is also not enough to support the core against gravity. As a
result, the gas contracts with free-fall timescale.

In general, the evolution of the magnetic field is determined by the balance
between the gas dynamical timescale and the magnetic diffusion timescale
$t_{\rm diff}=L^{2}/\eta$ where $L$ is the length scale of the system and
$\eta$ is the magnetic resistivity. In the case of isothermal collapse phase,
the gas dynamical timescale is free-fall timescale, because 3D simulations
have shown that even in the presence of a magnetic field, once gravitational
collapse begins, evolution proceeds in free-fall timescale (e.g., _Bate et
al._ , 2014), and observationally, the lifetime of a cores with
$n_{g}>10^{6}~{}{\rm cm}^{-3}$ is about free-fall time (_Könyves et al._ ,
2015). Therefore, by comparing the diffusion timescales of ambipolar
diffusion, Ohmic dissipation, and turbulent diffusion with the free-fall time,
we can understand the behavior of the magnetic field in the isothermal
collapse phase.

As shown in the equation (9) and (10), ambipolar diffusion dominates the Ohmic
dissipation during the isothermal collapse phase ($\rho_{\rm g}\lesssim
10^{-13}~{}{\rm g~{}cm}^{-3}$ or $n_{\rm g}\lesssim 10^{10}~{}{\rm cm}^{-3}$).
Thus, let us compare the free-fall timescale and the ambipolar diffusion
timescale using the equation (10),

$\begin{split}&\frac{t_{\rm ff}}{t_{\rm AD}}=\frac{\eta_{\rm AD}t_{\rm
ff}}{\lambda_{\rm J}^{2}}=\\\ &\begin{cases}2.2\times 10^{-3}&(\rho_{\rm
g,~{}{\rm g~{}cm}^{-3}}<10^{-16})\\\ 7.1\times 10^{-2}\rho^{1/2}_{\rm
g,10^{-13}~{}{\rm g~{}cm}^{-3}}&(10^{-16}<\rho_{\rm g,~{}{\rm
g~{}cm}^{-3}}<10^{-13})\end{cases}\end{split}$ (12)

where $t_{\rm ff}=\sqrt{3\pi/(32G\rho_{\rm g})}$ and we assume the length
scale to be $L=\lambda_{\rm J}=\sqrt{\pi c_{\rm s}^{2}/(G\rho_{\rm g})}$,
where $\lambda_{\rm J}$ is the Jeans length. This estimate shows $t_{\rm
ff}<0.1t_{\rm AD}~{}$ even at the density where ambipolar diffusion is most
efficient in isothermal collapse phase, and that the free-fall timescale is
much shorter than the ambipolar diffusion timescale. More quantitatively, to
diffuse the magnetic field over a spatial scale of Jeans length $\lambda_{\rm
J}=8.7\times 10^{2}~{}\rho_{\rm g,10^{-16}~{}{\rm g~{}cm}^{-3}}$ au, it takes
about $3.4\times 10^{11}~{}\rho_{\rm g,10^{-16}~{}{\rm g~{}cm}^{-3}}^{-2}$ yr
with Ohmic dissipation and $2.7\times 10^{6}~{}\rho_{\rm g,10^{-16}~{}{\rm
g~{}cm}^{-3}}^{-1}$ yr with ambipolar diffusion (we assume $\rho_{\rm
g}>10^{-16}~{}{\rm g~{}cm}^{-3}$ here). This estimate is consistent with
previous studies with more realistic $\eta_{O}$ and $\eta_{A}$ (_Nakano and
Umebayashi_ , 1986; _Umebayashi and Nakano_ , 1990). Hence, this estimate
shows that ambipolar diffusion (as well as Ohmic dissipation) may not play a
role during the isothermal collapse phase.

The estimate is also consistent with three-dimensional simulations. _Tsukamoto
et al._ (2015b) and _Masson et al._ (2016) compared the magnetic field
evolution between ideal and non-ideal MHD simulations and showed that, in the
isothermal collapse phase, the magnetic field evolution is identical between
simulations confirmed that, even with Ohmic dissipation and ambipolar
diffusion, the gas and magnetic field are completely coupled during the
isothermal collapse phase.

Note that equation (10) and hence (12) assumes the flux freezing. This
assumption is justified and our discussion is self-consistent because we
conclude that ambipolar diffusion does not work during isothermal collapse
phase.

Then, can turbulent (or reconnection) diffusion play the role during the
isothermal collapse phase? The following inequality holds for the turbulent
resistivity,

$\displaystyle\eta_{\rm turb}=L_{\rm turb}\delta v_{\rm turb}{\rm
min}(1,M_{A}^{3})\leq L_{\rm turb}\delta v_{\rm turb}.$ (13)

Here, we assume that the injection length is of the order of Jeans length
$L_{\rm turb}\sim\lambda_{\rm J}$.

As shown in _Ward-Thompson et al._ (2007), the turbulent velocity is subsonic
in the cloud core. Thus, we have the inequality $\delta v_{\rm turb}<c_{\rm
s}$ and we can show that the ratio of the free-fall timescale to the magnetic
diffusion timescale due to the turbulent reconnection diffusion as

$\displaystyle\frac{t_{\rm ff}}{t_{\rm turb}}=\frac{\eta_{\rm turb}t_{\rm
ff}}{\lambda_{\rm J}^{2}}\leq\frac{\delta v_{\rm turb}t_{\rm ff}}{\lambda_{\rm
J}}\leq\frac{c_{\rm s}t_{\rm ff}}{\lambda_{\rm J}}\sim\frac{t_{\rm ff}}{t_{\rm
sound}}<1,$ (14)

where $t_{\rm sound}=\lambda_{\rm J}/c_{\rm s}$ is the sound-crossing
timescale. The last inequality follows from the fact that the sound-crossing
time is longer than the free-fall time due to the graviationally collapsing
core. Hence, the turbulent reconnection diffusion also may not play a role in
the isothermal collapse phase. Note, however, that since supersonic turbulence
exists in massive cores, turbulent diffusion could be an important effect
there.

In summary, any magnetic diffusion may not play a role during the isothermal
collapse phase, which is consistent with previous studies (_Nakano and
Umebayashi_ , 1986; _Umebayashi and Nakano_ , 1990; _Tomida et al._ , 2015;
_Masson et al._ , 2016). The estimates made in this subsection show that the
magnetic field structure in the cloud core and envelope reflects the dynamics
of the gas. Therefore, it should be possible to study the dynamics of the gas
from the structure of the magnetic field, and vice versa.

####  3.1.3 Formation of hydrostatic first-core and magnetic diffusion

As gravitational collapse proceeds and the central density increases, drastic
changes in gas thermal evolution and magnetic diffusion occur. Once the
density reaches $\rho_{\rm g}\sim 10^{-13}~{}{\rm g~{}cm}^{-3}$ (or $n_{\rm
g}\sim 10^{10}~{}{\rm cm}^{-3}$ ), the gas becomes optically thick to the dust
thermal emission, and the compressional heating overtakes the radiative
cooling. The effective polytropic index increases to $\Gamma_{\rm eff}=5/3$,
which is larger than the critical value of $4/3$, and the pressure gradient
force begins to balance the gravitational force. As a result, the pressure
supported first hydrostatic core (or first core) forms (_Larson_ , 1969;
_Masunaga et al._ , 1998; _Commerçon et al._ , 2011, 2012; _Tomida et al._ ,
2013; _Vaytet and Haugbølle_ , 2017; _Bhandare et al._ , 2018), whose size and
mass are determined by the Jeans length $\sim 1$ to $10$ au and Jeans mass
$\sim 10^{-2}$ to $10^{-1}\thinspace M_{\odot}$ (_Larson_ , 1969; _Masunaga et
al._ , 1998).

In the first core, a dramatic increase in resistivity happens, which is
crucial for magnetic field evolution. As shown in Figure 6 (and Equation 10),
the ambipolar resistivity begins to increase around the density of $\rho_{\rm
g}\gtrsim 10^{-13}~{}{\rm g~{}cm}^{-3}$ (or $n_{\rm g}\gtrsim 10^{10}~{}{\rm
cm}^{-3}$), as density increases due to adsorption of ion/electron by dusts.
Hence, in the first core,

$\displaystyle\frac{t_{\rm ff}}{t_{\rm AD}}\sim 2.2\rho_{\rm g,10^{-12}~{}{\rm
g~{}cm}^{-3}}^{3/2}.$ (15)

and the diffusion timescale becomes shorter than the free-fall timescale at
$\rho_{\rm g}\gtrsim 10^{-12}~{}{\rm g~{}cm}^{-3}$ (or $n_{\rm g}\gtrsim
10^{11}~{}{\rm cm}^{-3}$). Furthermore, gas is supported by the pressure
gradient force, and the first core has a much longer lifetime than the free-
fall time. Thus, the magnetic diffusion timescale becomes significantly
shorter than the lifetime of the first core, its magnetic field weakens and
its mass-to-flux ratio increases mainly due to ambipolar diffusion ($\eta_{\rm
AD}$ is still larger than $\eta_{\rm Ohm}$ in $\rho_{\rm g}\lesssim
10^{-10}~{}{\rm g~{}cm}^{-3}$ or $n_{\rm g}\lesssim 10^{13}~{}{\rm cm}^{-3}$
_Tomida et al._ , 2015; _Tsukamoto et al._ , 2015b; _Masson et al._ , 2016;
_Tsukamoto et al._ , 2017b). Simultaneously, decoupling between the gas and
magnetic field in the first-core suppresses angular momentum removal by
magnetic tension and allows the first-core to retain its rotation, which is
essential for the formation of a circumstellar disk at the protostar formation
epoch (_Machida and Matsumoto_ , 2011; _Dapp et al._ , 2012; _Tsukamoto et
al._ , 2015b; _Wurster et al._ , 2018b; _Vaytet et al._ , 2018).

Figure 7 shows the density and plasma $\beta(=P_{\rm gas}/P_{\rm mag})$ (where
$P_{\rm gas}$ and $P_{\rm mag}$ gas and magnetic pressure) maps of the first
core formed in ideal MHD simulation and non-ideal MHD simulation (with Ohmic
dissipation and ambipolar diffusion). We can see the striking difference of
plasma $\beta$ in the first core between the two simulations. Without magnetic
diffusion, the plasma $\beta$ inside the first core is $\beta\sim 10$, and a
strong magnetic field retains in the first core. On the other hand, the plasma
$\beta$ at the center is $\beta\sim 10^{4}$ in non-ideal MHD simulations. This
significant difference induced by magnetic diffusion significantly alters the
fate of the first core, and the formation process of the protostars and
circumstellar disk.

Recent simulations also show that the magnetic field saturates in the first
core and has a universal value of $B\sim 0.1~{}\rm G$ (_Masson et al._ , 2016;
_Tsukamoto et al._ , 2017b; _Hennebelle et al._ , 2020; _Xu and Kunz_ , 2021).
Figure 8 shows the comparison of the magnetic field strength of the first-core
in ideal MHD and non-ideal MHD simulation (with ambipolar diffusion).
Ambipolar diffusion becomes stronger as the magnetic flux is brought into the
first core. This causes negative feedback on magnetic field amplification. As
a result, the magnetic field saturates at the specific value of $B\sim
0.1~{}\rm G$.

Figure 7:  Density (top panels) and plasma $\beta$ (bottom panels) map of the
first core in the ideal MHD (left panels) and non-ideal MHD (right panels)
simulations (_Tsukamoto et al._ , 2015b) from edge-on view. Red arrows show
the velocity field.  Figure 8:  Magnetic field strength as a function of the
density at the first-core formation epoch in ideal MHD (red) and non-ideal MHD
(blue) simulations at 2000 years after first core formation. The solid and
dashed black lines show $B\propto\rho^{2/3}$ and $B\propto\rho^{1/2}$,
respectively (from _Masson et al._ , 2016).  Figure 9:  Left panel shows the
first core (orange iso-density surface) and inner disc (red iso-density
surface) at $t=1.7$ yr after the formation of the protostar (from _Machida and
Matsumoto_ , 2011). The density distribution on the $x=0$, $y=0$ and $z=0$
plane is projected on to each wall surface. The velocity vectors on the $z=0$
plane are also projected on to the bottom wall surface. Right panel shows the
radial velocity just after the formation of the protostar (from _Tsukamoto et
al._ , 2015b). The red, orange, and blue lines show the radial velocity in
ideal MHD, non-ideal MHD with Ohmic dissipation, and non-ideal MHD with Ohmic
dissipation and ambipolar diffusion, respectively.  Figure 10:  Left and
middle panels show density isosurfaces around the newly born protostar in the
ideal MHD (left) and non-ideal MHD (middle) simulations. The magnetic field
morphology is shown as lines and are colored according to the magnitude of the
magnetic field (from _Vaytet et al._ , 2018). The right panel shows the
magnetic field profile and jet launching after the protostar formation (from
_Tomida et al._ , 2013). The edge of the right panel is $\sim 0.27$ au. The
high-density region ($\rho>10^{-5}~{}{\rm g~{}cm}^{-3}$ ) is visualized with
the orange surface. White arrows denote the direction of the fluid motion and
white lines show the magnetic field. Fast outflowing gas ($v_{z}>3~{}{\rm
km~{}s^{-1}}$ ) is volume rendered in pale yellow.

####  3.1.4 Second collapse and birth of protostar and circumstellar disk

The seeds of protostar and jet begin to sprout in the inner part of the first
core as it evolves. The first core adiabatically evolves and its central
temperature increases as $T\propto\rho^{\Gamma_{\rm eff}-1}$ (_Larson_ , 1969;
_Masunaga et al._ , 1998; _Vaytet et al._ , 2012; _Tomida et al._ , 2015).
When the central temperature reaches $T\sim 10^{3}$ K at $\rho_{\rm g}\sim
10^{-8}~{}{\rm g~{}cm}^{-3}$ (or $n_{\rm g}\sim 10^{15}~{}{\rm cm}^{-3}$), the
magnetic field and gas recouple due to thermal ionization of potassium and go
back to the ideal MHD regime but with a weak magnetic field ($\beta\gg 1$) due
to magnetic diffusion occurring at the first core scale. Subsequently, the
molecular hydrogen begins to dissociate, an endothermic reaction, at $\sim
2\times 10^{3}$ K and decreases the effective polytropic index $\Gamma_{\rm
eff}\sim 1.1$, which is smaller than the critical value. As a result, the
gravitational collapse in the first core resumes, and is called as the ”second
collapse”. The second collapse lasts until the density reaches $\rho_{\rm
g}\sim 10^{-2}~{}{\rm g~{}cm}^{-3}$ (or $n_{\rm g}\sim 10^{21}~{}{\rm
cm}^{-3}$), and the molecular hydrogen has wholly dissociated. Then the
effective polytropic index becomes $\Gamma_{\rm eff}\sim 5/3$, and the
pressure supported (second) core or protostar forms, whose mass is determined
by the Jeans mass at $\sim 10^{-3}$ to $10^{-2}\thinspace M_{\odot}$ (_Larson_
, 1969; _Masunaga and Inutsuka_ , 2000; _Vaytet and Haugbølle_ , 2017). This
is the birth of a protostar.

The magnetic field amplification during the second collapse determines the
magnetic field strength of the newly born protostar. Due to the conservation
of the magnetic flux, the magnetic field in the ideal MHD regime evolves as
$B\propto R^{-2}$. Thus, during the second collapse the central part of the
first core ($\sim 1$ au) collapses to the protostellar size of $0.01$ au, and
the magnetic field strength increases from the critical value of the magnetic
field in the first core, $|\mathbf{B}|\sim 10^{-1}~{}\rm G$ to $\sim
10^{3}~{}\rm G$, which is consistent with protostar observations (_Johns-Krull
et al._ , 2009). On the other hand, when we conduct the ideal MHD simulations
of the collapsing cloud core, the magnetic field strength of the central
protostar is more than ten times larger than that in the non-ideal MHD
simulations (_Tsukamoto et al._ , 2015b; _Vaytet et al._ , 2018; _Wurster et
al._ , 2018a; _Machida and Basu_ , 2019). Therefore, the Ohmic dissipation and
ambipolar diffusion are the key to solve the classical “magnetic flux problem”
(_Mouschovias et al._ , 1985) in star formation.

During the second collapse, most of the gas in the first core is distributed
around the protostar with angular momentum that is sufficient to allow the
centrifugal force to balance gravity. Therefore, most of gas in the first core
quickly transforms into a rotationally supported disk. This is the birth of
the circumstellar disk. Figure 9 shows this transformation. The left panel
shows the bird’s eye view of the forming protoplanetary disk (red region)
inside the first core (orange region) shortly after the formation of the
protostar. The right panel shows the infall velocity inside the first-core
shortly after the protostar formation. In the ideal MHD simulation (red line),
gas is directly accreted to the protostar ($\sim 0.01$ au), whereas in the
non-ideal MHD simulation, gas accretion stops at $1$ au– indicating that the
centrifugal force balances with gravity at $\sim 1$ au, and the protoplanetary
disk is forming. The size of the newly born circumstellar disk is
approximately 1 to 10 au, which corresponds to the size of the first core.

####  3.1.5 Outflow/jet launching from the newly born disk and protostar

The formation process of protostars and circumstellar disks described in the
previous section is also closely related to the formation of outflows and
jets. Protostellar outflows and jets have been ubiquitously observed in star-
forming regions. Wide-angle and slow-velocity flows are called ‘molecular
outflow’ or ‘low-velocity outflow’ in molecular line emissions. On the other
hand, well-collimated high-velocity flows are called ‘collimated jet’ or
‘optical jet’ and are observed by molecular and atomic line emissions from
radio to optical wavelengths. Hereafter, we refer to the former as ‘the
outflow’ and the latter as ‘the jet.’

Magnetically driven wind scenarios have been proposed as the driving mechanism
of these flows (e.g., _Pudritz and Norman_ , 1986). These winds have been
investigated in past theoretical and simulation studies, in which the outflow
and jet are magnetocentrifugally driven (_Blandford and Payne_ , 1982) or
driven by magnetic pressure gradient force (_Uchida and Shibata_ , 1985).
However, in these past studies, the settings for driving the outflow and jet
were highly idealized. Considerably idealized (or an unrealistic)
circumstellar disk and magnetic field configurations were setup for the
emergence of winds. For example, only poloidal fields (or dipole magnetic
fields) were assumed within a thin disk as the initial state in past studies.
The problem is that such an idealized condition (artificial distribution of
density or magnetic field) is not actually realized in the star formation
process.

_Tomisaka_ (1998, 2000, 2002) successfully reproduced the outflow in the star
formation simulation with a two dimensional ideal MHD code, for the first
time. These studies simulated the gravitational collapse process of star
forming cores, spatially resolving both prestellar core and protostar. In this
simulation, the outflow and jet naturally appear in gravitationally collapsing
clouds. Furthermore, driving the outflow and jet has been comprehensively
investigated in many star-formation simulations (_Machida et al._ , 2004,
2006; _Banerjee and Pudritz_ , 2006; _Machida et al._ , 2008; _Hennebelle and
Fromang_ , 2008; _Ciardi and Hennebelle_ , 2010; _Bate_ , 2011; _Price et al._
, 2012; _Joos et al._ , 2012; _Seifried et al._ , 2012b; _Tomida et al._ ,
2013; _Bate et al._ , 2014; _Lewis and Bate_ , 2017; _Kölligan and Kuiper_ ,
2018; _Wurster et al._ , 2018b; _Xu and Kunz_ , 2021). With these studies, the
driving mechanism of outflow and jet in the star formation process can be
established.

In a first core, the gravitational collapse slows and the collapse timescale
is longer than the rotation timescale. Thus, magnetic field lines are twisted
around the first core, and the low-velocity outflow emerges due to the
magnetocetrifugal force before the formation of the protostar. The outflow
driven by the first core continues to be produced by the rotationally
supported disk after the protostar has formed.

During the main accretion phase, the disk gas intermittently falls onto the
protostar due to the episodic accretion, inducing an episodic mass ejection or
a time-variable outflow (and jet) (_Machida and Basu_ , 2019). The outflow
physical parameters such as mass, momentum, momentum flux, kinetic energy and
kinetic luminosity, are in good agreement with observations, suggesting that
the molecular outflow is directly driven by the disk (e.g., _Machida and
Hosokawa_ , 2013). A wire-like (relatively strong), radially-distributed
magnetic field guides the outflow, which is mainly driven by the
magnetocentrifugal mechanism and creates a wide opening angle flow.

On the other hand, a weak magnetic field around the protostar, due to Ohmic
dissipation and ambipolar diffusion, is passively twisted by the rotation of
protostar and disk inner edge and generates a strong magnetic pressure
gradient force to drive the jet. Figure 10 illustrates this evolution. The
left panel shows the magnetic field configuration around the protostar just
after its formation in the ideal MHD simulation, and shows that an hourglass-
shaped magnetic field has been realized. This indicates that the strong
magnetic field suppresses the gas rotation and that the gas motion can not
twist the magnetic field. Conversely, the weak magnetic field due to magnetic
diffusion is easily twisted during the second collapse. The middle panel shows
the magnetic field configuration in the non-ideal MHD simulation, and shows
that the magnetic field is passively twisted by the rotation. This passive
twisting of the magnetic field drives the jets from the vicinity of the
protostar, as shown in the right panel. Therefore, magnetic diffusion also
plays an essential role in driving the protostellar jet. The mass-loss rate of
the jet obtained from the 3D simulation is $\sim 10^{-5}~{}\thinspace
M_{\odot}\thinspace{\rm yr^{-1}}$ (_Machida and Basu_ , 2019), and is in
agreement with the observed value in a protostar, $\sim 3\times
10^{-6}~{}\thinspace M_{\odot}\thinspace{\rm yr^{-1}}$ (_Lee_ , 2020).

Lastly, we comment on the classical scenario for driving the outflow. It had
been considered that the low-velocity outflow is entrained by the high-
velocity (primary) jet (so-called the entrainment scenario). In this scenario,
the jet appeared near the protostar and provided linear momentum to the
ambient gas through some mechanism (_Arce et al._ , 2007). The ambient gas
entrained by the jet is then observed as the low-velocity outflow. Recently,
observations of outflow rotation could resolve the long-standing debate on
what is driving the outflow. It had been considered that we would unveil the
outflow driving mechanism if this rotation could be measured. In the
entrainment scenario, the primary jet, which provides both linear and angular
momentum to the outflow (or the entrained gas), has only a small amount of
angular momentum, because the jet driving region is located near the
protostar. Thus, it is expected that the outflowing gas receives a very small
amount of angular momentum from the primary jet. On the other hand, in the
direct disk driven scenario proposed from the core collapse simulations, the
outflow should have a large amount of angular momentum because the outflow
driving region is much far from the protostar where the rotation velocity and
angular momentum are large (_Lee et al._ , 2021a).

###  3.2 Evolution of protoplanetary disks from Class 0 to the end of the
Class I phase in YSOs

At the time of PP VI, many researchers were seriously considering the
possibility that highly efficient magnetic braking completely suppresses the
disk formation in the early phase of protostar evolution (so-called magnetic
braking catastrophe (MBC) (see the review of PP VI _Li et al._ , 2014b). MBC
is, in the strict sense, resolved by the scenario described in §3.1.4 because
the circumstellar disk with a size of $\sim 1$ au simultaneously forms at the
formation epoch of protostar (Note that a disk with a size of $\sim 1$ au is
out of the scope of the studies of MBC by _Mellon and Li_ , 2008, 2009; _Li et
al._ , 2011, because the inner boundary is at $r\sim 6$ au in their
simulations).

However, theoretically, magnetic braking is estimated to be very efficient,
and findings of large disks with size of several $10$ to $100$ au in the
embedded phases by recent observational surveys (see section2.5.1) force
researchers to revisit the disk size evolution more precisely.

In this subsection, we review the physical mechanisms which weaken the
magnetic braking and may play key roles in determining the disk size evolution
during the early phases of star formation.

####  3.2.1 Misalignment between magnetic field and rotation vector

In many theoretical studies, it is assumed for simplicity that the angular
momentum of the parent cloud core is aligned with its magnetic field. However,
in real molecular cloud cores, the magnetic field and rotation vector are
often observed to be misaligned (_Hull et al._ , 2013, 2014; _Yen et al._ ,
2021b).

_Hennebelle and Ciardi_ (2009) first reported that the rotationally supported
disk more easily forms in cores with misalignment between the magnetic field
and the angular momentum vector. More quantitatively, _Joos et al._ (2012)
showed that the mean specific angular momentum of the central region in a
perpendicular core (in which the initial magnetic field and angular momentum
are perpendicular) is about two times larger than that in a parallel core (in
which the initial magnetic field and angular momentum are parallel). The
influence of misalignment is also investigated by _Li et al._ (2013). The
authors obtained results consistent with _Hennebelle and Ciardi_ (2009), and
concluded that disk formation becomes possible when $\mu\gtrsim 4$, where
$\mu$ is the mass-to-flux ratio of the core normalized by its critical value.

By examining the angular momentum evolution of fluid elements during the cloud
core collapse, _Tsukamoto et al._ (2018) elucidated three mechanisms by which
misalignment affects the angular momentum evolution (and ultimately the size)
of the disk,

  1. 1.

the selective accretion of fluid elements with large (small) angular momentum
to the central perpendicular (parallel) core;

  2. 2.

magnetic braking in the isothermal-collapse phase;

  3. 3.

magnetic braking in the disk;

The selective accretion is a process in which the magnetic tension suppresses
mass accretion from the direction perpendicular to the magnetic field and
selectively enhances mass accretion parallel to the field. As a result, the
fluid elements with a larger (small) angular momentum accrete onto the central
region in a perpendicular (parallel) core. This effect may play a role until
the envelope disappears. Thus, this effect can lead to the misinterpretation
of the impact of misalignment on magnetic braking efficiency on time scales
shorter than the time scale of envelope dissipation, and it makes the magnetic
braking appear stronger (weaker) in parallel (perpendicular) cores. (In
reality, it is just that the angular momentum of the accreted gas is smaller
(larger))

_Tsukamoto et al._ (2018) showed that the magnetic braking during the
isothermal collapse phase is stronger in perpendicular cores than in parallel
cores. This result is opposite to that expected from the previous studies such
as _Hennebelle and Ciardi_ (2009) and _Joos et al._ (2012), but is consistent
with the classical discussion of magnetic braking by _Mouschovias_ (1985) and
the simulation results of prestellar collapse phase (_Matsumoto and Tomisaka_
, 2004). The inconsistency between the trend of magnetic braking in the
isothermal collapse phase and the results of _Hennebelle and Ciardi_ (2009)
and _Joos et al._ (2012), which investigate the disk evolution after protostar
formation, may suggest that magnetic braking during the prestellar collapse
phase (or in the envelope) is not the dominant mechanism determining the
evolution of the disk size, otherwise the size of the disks should be opposite
in the parallel and perpendicular cores.

Of the three mechanisms, magnetic braking in the disk seems to play the
dominant role for disk size evolution in the main accretion phase because the
disk is supported by the centrifugal force and the magnetic field has a much
longer time than the local free-fall time to extract the gas angular momentum.
Furthermore, because the magnetic field fans out around the central region and
has an hour-glass-like geometry, the magnetic braking in the disk with the
parallel magnetic field can be enhanced (_Mouschovias_ , 1985; _Hennebelle and
Ciardi_ , 2009; _Tsukamoto et al._ , 2018). Note also that _Li et al._ (2013)
(see also _Hirano et al._ , 2020) point out that the angular momentum removal
by disk outflows also plays an important role in the parallel core which
further favors the growth of the disk in the misaligned core in which outflow
formation is suppressed (e.g., _Joos et al._ , 2012; _Li et al._ , 2013;
_Wurster et al._ , 2016; _Tsukamoto et al._ , 2017b). Note, however, that
_Marchand et al._ (2020) demonstrated that magnetic braking is primary and
outflows are secondary in angular momentum transport.

In summary, because mechanisms 1 and 3 work effectively during the (early)
main accretion phase, misalignment seems to promote disk growth. On the other
hand, in the prestellar collapse phase, the presence of misalignment seems to
enhance the angular momentum removal from the central region due to mechanism
2.

Note, however, that it has been pointed out that the impact of misalignment is
not so prominent once ambipolar diffusion is considered (_Masson et al._ ,
2016), probably because of the suppressed magnetic braking inside the disk. To
observationally confirm that misalignment plays a significant role in disk
evolution, it is important to investigate a correlation between outflow
activity (rather than direction), magnetic field orientation, and disk size.
Several studies suggest that misalignment suppresses outflow formation, and
that there is a correlation among the disk size, outflow activity, and
misalignment (_Li et al._ , 2013; _Hirano et al._ , 2020). If such correlation
is observed, it provides evidence that misalignment affects disk evolution.

####  3.2.2 Turbulence

_Santos-Lima et al._ (2012) suggested that turbulence in the cloud core
weakens magnetic braking. They compared the simulation results for a
coherently rotating core and a turbulent core and found that a rotationally
supported disk is formed only in the turbulent cloud core. Similar results
were obtained by _Seifried et al._ (2012a, 2013). _Santos-Lima et al._ (2012)
suggested that random motions due to turbulence cause small-scale magnetic
reconnection events and provide an effective magnetic diffusion that
efficiently removes magnetic flux, and disks with a size of $r\sim 100$ au are
formed even in the ideal MHD limit. However, their results are obtained with
relatively coarse numerical resolution of $r\gtrsim 1$ au. With such a coarse
resolutions in ideal MHD, the numerical magnetic diffusion can promote the
magnetic flux loss and can artificially suppress the magnetic braking.

_Joos et al._ (2013) investigated the impact of turbulence with a higher
numerical resolution of $\sim 0.4$ au. Their results suggests that the impact
of turbulence is limited. For example, they compared the mass-to-flux ratio of
the central region (e.g., $r\sim 100$ au) between cores with subsonic
turbulence and without turbulence, with a realistic magnetic field strength of
$\mu\leq 5$. Their results show that the difference is about the factor of two
and is not so significant. Furthermore, they showed that the mass of the disk
decreases as the numerical resolution increases. In other words, they point
out that the problem of numerical convergence cannot be resolved even with
resolutions below 1 au in the case of ideal MHD simulations with turbulence.

Furthermore, it is pointed out that the impact of turbulence becomes less
important once the non-ideal MHD effect is considered (_Lam et al._ , 2019;
_Wurster and Lewis_ , 2020). These results, as well as the analytic estimate
of turbulent (reconnection) diffusion rate in §3.1.2, suggest that the
turbulent diffusion may not play a significant role for the disk size
evolution.

####  3.2.3 Ohmic dissipation and ambipolar diffusion

So far, we have examined the mechanisms that weaken magnetic braking in the
ideal MHD limit. However, the ideal MHD approximation is not valid in real
YSOs because of their low ionization degree. Non-ideal MHD effects can
decouple the magnetic field and gas, and can decrease the magnetic braking
efficiency in the disk. Thus, non-ideal effects may affect the size evolution
of circumstellar disks. In this section, we review the influence of Ohmic
dissipation and ambipolar diffusion on early disk evolution.

In the context of disk size evolution, Ohmic dissipation becomes effective in
the dense inner region of the disk of $\rho_{\rm g}>10^{-10}~{}{\rm
g~{}cm}^{-3}$ (or $n_{\rm g}>10^{13}~{}{\rm cm}^{-3}$). _Machida et al._
(2011) used 3D simulations with Ohmic dissipation and found that the disk size
is $r\lesssim 10$ au in $t\lesssim 10^{4}$ yr after the formation of the
protostar (or Class 0 phase where the envelope is very massive). This result
suggests that it is still difficult to form disks with sizes of several tens
of au in the Class 0 phase by Ohmic dissipation alone.

Ambipolar diffusion is a more efficient magnetic diffusion process in the disk
and envelope. It has two important properties. One is that ambipolar
resistivity $\eta_{\rm AD}$ is much larger than the Ohmic resistivity
$\eta_{\rm Ohm}$ in almost the entire region of the disk. The ambipolar
resistivity becomes an increasing function of density at $\rho_{\rm g}\gtrsim
10^{-12}~{}{\rm g~{}cm}^{-3}$ (or $n_{\rm g}\gtrsim 10^{11}~{}{\rm cm}^{-3}$)
and dominates the Ohmic resistivity in $\rho_{\rm g}\lesssim 10^{-9}~{}{\rm
g~{}cm}^{-3}$ (or $n_{\rm g}\lesssim 10^{14}~{}{\rm cm}^{-3}$) (e.g., _Tomida
et al._ , 2015; _Tsukamoto et al._ , 2015b; _Marchand et al._ , 2016; _Zhao et
al._ , 2018b, see also Figure 6 and equations (9) and (10)). 3D simulations
have shown that ambipolar diffusion allows the formation of a relatively
extended disk with a size of $r\gtrsim 10$ au even in the early disk evolution
phase ($t\lesssim 10^{4}$ yr after protostar formation, _Masson et al._ ,
2016; _Zhao et al._ , 2018a; _Tsukamoto et al._ , 2020). They also showed that
the disk is massive and the spiral arms induced by gravitational instability
form despite a relatively strong magnetic field.

Since ambipolar diffusion is also effective in the envelope, the magnetic flux
brought into the central region by accretion is distributed around the disk.
It has been proposed that the accretion towards the magnetic flux tube results
in the formation of a shock (ambipolar diffusion shock _Li and McKee_ , 1996;
_Ciolek and Königl_ , 1998).

The resistivities ($\eta_{\rm Ohm},\eta_{\rm AD},$ and $\eta_{\rm Hall}$)
strongly depend on the dust properties as well as the cosmic-ray (CR)
ionization rate, and any change of these parameters is expected to affect the
disk size evolution (_Zhao et al._ , 2016; _Wurster et al._ , 2018c;
_Dzyurkevich et al._ , 2017; _Koga et al._ , 2019a; _Guillet et al._ , 2020a)

The CR ionization rate ($\zeta_{\rm CR}$) generally gauges the overall
abundances of charge species. The typical level of $\zeta_{\rm CR}$ in dense
cores (inferred from chemical analysis) ranges from a few times 10-18 s-1 to a
few times 10-16 s-1 (_Caselli et al._ , 1998; _Padovani et al._ , 2009; _Ivlev
et al._ , 2019), with $\zeta_{\rm CR}\approx 10^{-17}$ s-1 as the “standard”
rate. However, the current methods for constraining $\zeta_{\rm CR}$ rely on
the measured abundance of molecular ions such as HCO+, DCO+ and N2H+ (_Caselli
et al._ , 1998; _van der Tak and van Dishoeck_ , 2000; _Doty et al._ , 2002),
which can be largely uncertain due to the freeze-out of molecules in cloud
cores. Nevertheless, recent non-ideal MHD simulations show that disk formation
is possible for such a range of $\zeta_{\rm CR}$. If only ambipolar diffusion
and Ohmic dissipation are considered, a slightly higher CR ionization rate
($\zeta_{\rm CR}$ $\gtrsim$a few times $10^{-17}$ s-1) than the standard rate
at core scale can suppress the formation of disks with sizes of several tens
of au (_Zhao et al._ , 2016; _Kuffmeier et al._ , 2020).

Dusts plays an important role in both determining the abundance of ions and
electrons and in determining the conductivity. Dust adsorbs charged particles
and determines the ionization degree of the gas phase. On the other hand, a
large population of charged small grains ($\lesssim$100 Å) with Hall parameter
close to unity can determine the conductivities (_Li_ , 1999; _Padovani et
al._ , 2014; _Zhao et al._ , 2016; _Dzyurkevich et al._ , 2017), and decrease
the ambipolar and Hall resistivities at envelope and disk densities (_Zhao et
al._ , 2018b; _Koga et al._ , 2019b; _Marchand et al._ , 2020). The main
reason for the suppression of disk formation in early non-ideal MHD work
(_Mellon and Li_ , 2009; _Li et al._ , 2011) is largely attributed to the use
of ionization chemistry assuming the presence of a large number of small
charged grains. As shown in analytical work, the smallest grains are rapidly
depleted in cold dense environments (_Ossenkopf_ , 1993; _Hirashita_ , 2012;
_Köhler et al._ , 2012; _Guillet et al._ , 2020a; _Silsbee et al._ , 2020),
which is supported by the non-detection of spinning dust grain emission
(produced by small dust grains of $\lesssim$100 Å) in recent Galactic cold
core surveys (_Tibbs et al._ , 2016). Therefore, given the observational and
chemical constraints in cloud cores on the microphysical parameters
$\zeta_{\rm CR}$ and the minimum grain sizes, the formation of circumstellar
disks in low-mass protostellar cores should be relatively universal.

Although the microscopic physics which determines the magnetic diffusion rate,
and hence disk evolution, looks complicated, there is a simple essence that
determines disk size evolution under a strong magnetic field with magnetic
diffusion, which is the balance of coupling and decoupling between the
magnetic field and the gas in the disk. _Hennebelle et al._ (2016) showed that
ambipolar diffusion limits the disk size to a few tens of au during Class 0
phase by assuming that the generation timescale of toroidal magnetic field
equals to the ambipolar diffusion timescale and the magnetic braking timescale
equals to the rotation timescale. This analytic estimate reproduces well in
many simulations of disk evolution during the Class 0 phase (_Hennebelle et
al._ , 2020; _Mignon-Risse et al._ , 2021a).

The microscopic origin of ambipolar diffusion is ion-neutral drift. In other
words, if ambipolar diffusion works effectively, a velocity difference between
ions (e.g., H3+, HCO+, _Tassis et al._ , 2012; _Zhao et al._ , 2018b) and
neutral gases (e.g., CO) is expected to occur. Thus, observations of ion-
neutral drift are essential to test the importance of ambipolar diffusion.
Theoretical studies have suggested that the velocity difference is possibly
observable (_Zhao et al._ , 2018a; _Tsukamoto et al._ , 2020; _Marchand et
al._ , 2020) in the disk and inner envelope (i.e., $100$ to $1000$ au scale)
of late Class 0 to Class I YSOs. Although recent attempts to observe ion-
neutral drift have resulted in non-detection for B335 (_Yen et al._ , 2018),
possibly because it is too young (as shown in _Tsukamoto et al._ , 2020, ion-
neutral drift is more likely observable in more evolved objects), or because
it is highly ionized. Future observational attempts to detect ion-neutral
drift are extremely important.

####  3.2.4 Hall effect

Unlike other non-ideal effects, which are just magnetic diffusion, the Hall
effect has a unique feature. The Hall effect can actively induce a clockwise
rotation (when $\eta_{\rm Hall}<0$) around a magnetic field by generating a
toroidal magnetic field from the poloidal magnetic field (_Wardle and Ng_ ,
1999; _Braiding and Wardle_ , 2012a, b). During gravitational collapse, the
magnetic field is dragged toward the center and an hourglass-shaped magnetic
field structure is formed. At the “neck” of the hourglass-shaped magnetic
field, a toroidal current exists. Then, the Hall effect drags the magnetic
field toward the azimuthal direction as if the gas rotates with the velocity
$\mathbf{v}_{\rm H}=-\eta_{\rm Hall}(\nabla\times\mathbf{B})/B$ (see equation
(5)). The generated toroidal magnetic field exerts a toroidal magnetic tension
on the gas and induces the rotation. The combination of the rotation induced
by the Hall effect and the inherent rotation of the cloud core sometimes
assists the disk size growth, and causes various interesting velocity
structures in YSOs.

_Krasnopolsky et al._ (2011) used two-dimensional simulations to investigate
the impact of the Hall effect in the context of disk formation for the first
time. They focused on the dynamical behavior induced by the Hall effect and
showed that a circumstellar disk with a size of $r\gtrsim 10$ au can be formed
with sufficiently high $\eta_{\rm Hall}$. Another finding of _Krasnopolsky et
al._ (2011) is the formation of a counter-rotating envelope opposite to the
disk rotation. As a back reaction to the Hall effect induced rotation in the
midplane, angular momentum conservation causes the upper layer to rotate in
the opposite direction of the induced rotation. As a result, a counter-
rotating (upper) envelope emerges. _Li et al._ (2011) investigated the effect
of the Hall effect in two-dimensional simulations that included all the non-
ideal MHD effects using realistic magnetic diffusion coefficients. They
confirmed that Hall-induced rotation occurs even when other non-ideal effects
are considered. They also showed the formation of the counter-rotating region
opposite to the initial rotation in the upper region of the envelope at $\sim
100$ au.

_Tsukamoto et al._ (2015a) conducted three-dimensional simulations of disk
formation which included all the non-ideal effects as well as a radiative
transfer for the first time. They followed the evolution until shortly after
the formation of the protostar. They showed that a disk $\sim 20$ au in size
is formed at the protostellar formation epoch when the magnetic field and
initial rotation vector of the core are anti-parallel (in this case, the Hall
induced rotation and initial rotation has the same direction). On the other
hand, a disk $1$ au in size formed with the parallel configuration (in this
case, the Hall induced rotation and initial rotation has the opposite
direction). Because the Hall effect strengthens or weakens magnetic braking
depending on whether the angle between the magnetic field and the angular
momentum of the cloud core is acute or obtuse, they suggest that disk size
could be bimodal in the Class 0 phase with the Hall effect.

More recently, _Wurster et al._ (2016) investigated the impact of the Hall
effect several thousand years after the formation of the protostar with three-
dimensional simulations. They also confirmed that disk size could be bimodal.
_Marchand et al._ (2018, 2019) also reproduced the evolution induced by the
Hall effect described above. On the other hand, they pointed out that the Hall
effect is difficult to handle numerically, and that it is necessary to pay
attention to whether the conservation of angular momentum is satisfied in the
implementation. _Tsukamoto et al._ (2017b) investigated the impact of the Hall
effect in the misaligned core. They showed that, in a core with an acute angle
misalignment between the magnetic field and the angular momentum, the disk
size growth is suppressed, while in a core with an obtuse angle, the disk size
growth is promoted.

_Zhao et al._ (2020, 2021) investigated the impact of dust models on the Hall
effect and the resulting evolution of the disk. When Hall effect is included,
the upper limit of $\zeta_{\rm CR}$ for disk formation increases to a few
times $10^{-16}$ s-1, matching the upper range of the estimated $\zeta_{\rm
CR}$ in cloud cores. They also showed that the Hall effect could flip the
rotation direction at $\sim 100$ au from initial forward rotation to the
counter rotation, making a counter-rotating disk. This kind of counter-