# Recent progress with observations and models to characterize the magnetic
fields from star-forming cores to protostellar disks

Anaëlle Maury 1,2,∗, Patrick Hennebelle 1 and Josep Miquel Girart 3,4

###### Abstract

In this review paper, we aim at providing a global outlook on the progresses
made in the recent years to characterize the role of magnetic fields during
the embedded phases of the star formation process. Thanks to the development
of observational capabilities and the parallel progress in numerical models
capturing most of the important physics at work during star formation, it has
recently become possible to confront detailed predictions of magnetized models
to observational properties of the youngest protostars. We provide an overview
of the most important consequences when adding magnetic fields to state-of-
the-art models of protostellar formation, emphasizing their role to shape the
resulting star(s) and their disk(s). We discuss the importance of magnetic
field coupling to set the efficiency of magnetic processes, and provide a
review of observational works putting constraints on the two main agents
responsible for the coupling in star-forming cores: dust grains and ionized
gas. We recall the physical processes and observational methods allowing to
trace the magnetic field topology and its intensity in embedded protostars,
and review the main steps, success and limitations in comparing real
observations to synthetic observations from the non-ideal MHD models. Finally,
we discuss the main threads of observational evidence that suggest a key role
of magnetic fields for star and disk formation, and propose a scenario solving
the angular momentum for star formation, also highlighting the remaining
tensions that exist between models and observations.

##  1 Keywords:

Star formation, magnetic fields, protostars, MHD modeling, polarization

##  2 Introduction

The formation of stars takes place in filamentary molecular clouds, when the
high-density interstellar medium partly collapses and fragments into bound
starless dense cores. Depending on their properties, such cores can undergo
further collapse to form a protostar that will accrete its circumstellar
material (evolving through the Class 0, I, II and later stages) until reaching
the main sequence. Class 0 protostars are the first (proto)stellar objects,
observed only $t\lower 2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}5\times
10^{4}$ yr (André et al., 1993, 2000; Evans et al., 2009; Maury et al., 2011)
after their formation, while most of their mass is still in the form of a
dense core/envelope collapsing onto the central protostellar embryo. Embedded
Class 0 and Class I protostars are also accreting objects, as most of the
final stellar mass is assembled during those short phases. During this
accretion phase, the circumstellar gas must redistribute most of its initial
angular momentum outward or else centrifugal forces will soon balance gravity
and prevent inflow, accretion, and the growth of the star. This long-standing
“angular momentum problem” was estimated to be quite severe (Bodenheimer,
1995). Comparing observations of the specific angular momentum $J/M$ in star-
forming clouds prior to contraction ($J/M\sim
10^{21}\,\rm{cm}^{2}\,\rm{s}^{-1}$, Goodman et al. 1993) to the angular
momentum contained in a typical solar-type star ($J/M\sim
10^{15}\,\rm{cm}^{2}\,\rm{s}^{-1}$) indicates that, during the brief accretion
phase, the gas must transfer all but 1 millionth of its initial angular
momentum out of the accretion flow (Belloche, 2013). Exactly how the
circumstellar mass contained in a star-forming core is transferred to the
forming star, but not its associated angular momentum, has been an active
field of research in modern astronomy.

Wherever we have the means of observing them, magnetic fields are detected on
nearly all scales and across the full spectrum of astrophysical environments:
from our own Earth (Russell, 1991) to stars (Donati and Landstreet, 2009), the
Milky Way (Wielebinski, 2005) and cosmological structures (Kunze, 2013). As
regards star-formation processes, magnetic fields provide a mechanism for
launching and collimating outflows winds and jets (Blandford and Payne, 1982),
which are routinely observed around young stellar objects (YSOs), and magnetic
fields of typical strengths 10–100 $\mu$G are threading nearly all star-
forming clouds (Crutcher, 2012). Therefore, it is now widely accepted that
most star-forming cores are magnetized to some level. However, it is only
recently that the role of these magnetic fields could be investigated in
details.

Indeed, in the past decade, numerical models of star formation have been
gradually including most of the physical ingredients for a detailed
description of protostellar evolution in the presence of magnetic fields, such
as resistive MHD, radiative transfer and chemical networks, at all the
relevant scales. Moreover, observational capabilities resolving the internal
environment of star-forming cores, e.g., interferometry at (sub)millimeter
wavelengths, have started to produce detailed maps of the gas and dust
properties at the very small scales of protostellar cores, where material is
accreted into a stellar embryo, stored in a disk, and ejected under the form
of outflows and jets. The massive development of polarization capabilities on
telescopes probing the cold Universe, such as on the SubMillimeter Array
(SMA), then on the Atacama Large Millimeter Array (ALMA), and now also
developed at the IRAM NOrthern Extended Millimeter Array (NOEMA), have
produced sensitive observations of magnetic fields in protostellar cores with
a great level of detail. These are used as tools to put unprecedented
constraints on magnetic features the star-formation models should be able to
reproduce (e.g. Hull and Zhang, 2019).

Hence, the simultaneous major improvements of instrumental and computational
facilities have opened an era of detailed confrontation between observed
protostellar properties and magnetized models predictions. Testing the
detailed interplay of physics at work has allowed a major leap forward in our
understanding of star and disk formation processes, and provide a new detailed
scenario to describe the early stages of the formation of stars and their
planetary systems. In this review, we present a synthetic description of the
progresses made in the past decade regarding the properties and roles of
magnetic fields in shaping young protostars, their envelopes, disks, and
resulting stellar systems.

##  3 How magnetic field influences gravitational collapse: a theoretical
overview

This section is devoted to a review of the fundamental MHD processes relevant
in the context of dense core collapse and disk formation.

###  3.1 The fluid equations

The fluid equations are as follows. $\rho$, $P$, $\mathbf{v}$, $\mathbf{B}$,
and ${\rm\Phi}_{g}$ are, respectively, gas density, pressure, velocity,
magnetic field, and gravitational potential, while $\eta_{\rm O}$, $\eta_{\rm
H}$, and $\eta_{\rm AD}$ are the Ohmic, Hall and ambipolar diffusivities.

${\partial\rho\over\partial t}+\nabla\cdot(\rho\mathbf{v})=0~{},$ (1)

is the continuity equation which describes matter conservation.

$\displaystyle\rho\left({\partial\mathbf{v}\over\partial
t}+(\mathbf{v}\cdot\nabla)\mathbf{v}\right)$ (2) $\displaystyle=-\nabla
P+{1\over
4\pi}(\nabla\times\mathbf{B})\times\mathbf{B}-\rho\nabla{\rm\Phi}_{g}~{}$
$\displaystyle=-\nabla\left(P+{B^{2}\over 8\pi}\right)+{1\over
4\pi}\mathbf{B}.\nabla\mathbf{B}-\rho\nabla{\rm\Phi}_{g}~{},$

represents the momentum conservation. We see in particular that the Lorentz
force can be written as a magnetic pressure and a magnetic tension.

$\displaystyle{\partial\mathbf{B}\over\partial t}=\nabla\times(\mathbf{v\times
B})-{c^{2}\over 4\pi}\nabla$ (3) $\displaystyle\times\left\\{\eta_{\rm
O}\nabla\times\mathbf{B}+\eta_{\rm
H}(\nabla\times\mathbf{B})\times{\mathbf{B}\over B}\right.$
$\displaystyle\left.+\eta_{\rm AD}{\mathbf{B}\over
B}\times\left[(\nabla\times\mathbf{B})\times{\mathbf{B}\over
B}\right]\right\\}~{},$

is the Maxwell-Faraday equation. The three non-ideal MHD terms, namely the
Ohm, Hall and ambipolar diffusion contributions are accounted for.

While this set of equations is complete (after the inclusion of an energy
conservation equation, and an equation of state), it is nevertheless
enlightening to also discuss the equation describing the conservation of
angular momentum. It is obtained by combining the azimuthal component of the
momentum equation with the continuity equation (see for instance Joos et al.,
2012). The equation of conservation of angular momentum is

$\displaystyle{\partial_{t}\left(\rho rv_{\phi}\right)+\nabla\cdot
r}\bigg{[}\rho v_{\phi}{\bf v}$ (4)
$\displaystyle\left.+\left(P+\frac{B^{2}}{8\pi}-\frac{g^{2}}{8\pi
G}\right){\bf e_{\phi}}-\frac{B_{\phi}}{4\pi}{\bf B}+\frac{g_{\phi}}{4\pi
G}{\bf g}\right]=0,$

This equation reveals the existence of two torques able to transport the
angular momentum, namely $-r{B_{\phi}}{\bf B}/4\pi$, the magnetic torque and
$r{g_{\phi}}{\bf g}/(4\pi G)$, the gravitational torque. These two torques are
playing a fundamental role during the collapse regarding the issue of angular
momentum evolution. They have a similar expression, and in particular both
require a toroidal and a poloidal fields. However, there is an important
difference between the two because a toroidal magnetic field, $B_{\phi}$, can
be produced in an axi-symmetric cloud, once the cloud is in rotation. On the
contrary, a toroidal gravitational field, requires a non-axisymmetric density
distribution such as a spiral wave.

###  3.2 Magnetic support and magnetic compression: the pseudo-disk

The simplest effect magnetic field has on a cloud is through magnetic
pressure, which provides a support against gravity. Assuming field freezing
within a cloud of radius $R$, threaded by magnetic field, $B$, we have that
$\phi=\pi R^{2}B$, the magnetic flux, is conserved during collapse. It is easy
to calculate the ratio of magnetic over gravitational energy

$\displaystyle{E_{\rm mag}\over E_{\rm grav}}\propto{R^{3}B^{2}\over
GM^{2}/R}\propto{1\over G}\left({\phi\over M}\right)^{2}.$ (5)

Equation 5 shows that the energy ratio stays constant, as long as spherical
symmetry is maintained and that it is proportional to $\phi/M$, i.e. the ratio
of the magnetic flux over the cloud mass. Obviously there is a critical value
for $(M/\phi)_{cri\textdagger}$ above which magnetic field is strong enough to
prevent gravitational collapse. The latter is expected to be such as
$(M/\phi)_{cri\textdagger}\propto 1/\sqrt{G}$. A calibration has been
performed by Mouschovias and Spitzer (1976) using exact equilibrium and, it
has been inferred that

$\displaystyle\left({M\over\phi}\right)_{crit}={1\over\sqrt{G}}{c_{1}\sqrt{5}\over
3\pi},$ (6)

where $c_{1}\simeq 0.51$. It is then common to define $\mu$, the mass-to-flux
over critical mass-to-flux ratio. A value of $\mu=1$ indicates that magnetic
field and gravity compensate while a value of $\mu$ larger than 1, implies
that magnetic field cannot prevent collapse to occur. Another commonly used
closely related parameter has traditionally been called $\lambda$ and uses the
column density and magnetic field,

$\displaystyle\lambda=2\pi G^{1/2}{\Sigma\over B}.$ (7)

Due to the anisotropic nature of the magnetic field, and in particular to the
fact that the Lorentz force vanishes along the field lines, a magnetized cloud
does not collapse spherically but typically get flatten along the field lines.
It should be stressed however that as the collapse proceeds, the field lines
get strongly pinched by the differential motions, forming an hour glass shape.
This has two consequences. First, it creates a magnetic pressure force,
$-\partial_{r}(B_{z}^{2})/8\pi$, parallel to the equatorial plane and pointing
outwards. This typically slows down the collapse when magnetic field is strong
enough. Second, it creates another component of the magnetic force, which is
oriented along the z-axis and is equal to $-\partial_{z}(B_{r}^{2})/8\pi$.
However, for symmetry reasons, $B_{r}$ is usually vanishing in the equatorial
plane. Therefore, this force is pointing toward the mid-plane and is
compressing the gas. Compared to the hydrodynamical collapse, for which the
singular isothermal, $\rho=c_{s}^{2}/(2\pi Gr^{2})$ is usually an acceptable
approximation, particularly for low mass cores, this creates close to the
equatorial plane, a density enhancement which can be as high as a factor of
several. Because of its shape, this thin layer is usually called a pseudo-
disk. Exact solutions of this equilibrium, called the magnetized isothermal
toroids, have been obtained by Li and Shu (1996), while an approximation of
the density enhancement and the dense layer thickness is presented in
Hennebelle and Fromang (2008). It should be stressed that like for the
singular isothermal sphere, the resulting equilibrium is unstable to collapse
and thus, unless the core is magnetically dominated, the pseudo-disk
collapses. Indeed, the pseudo-disk, unlike a centrifugally supported disk, is
not a structure which is at mechanical equilibrium. It is essentially a
collapsing envelope that has been flattened by the magnetic field effect.

It is worth stressing that the aligned configuration, i.e., magnetic field and
rotation being initially parallel, which has generally been assumed in most
early calculations, is somehow peculiar because in this configuration only the
z-component of the field is not zero in the equatorial plane and therefore the
magnetic pressure is squeezing the gas along the z-axis. If magnetic field and
rotation are initially misaligned, the magnetic field lines get twisted by
rotation and the radial field does not vanish anymore in the equatorial plane.
This has as effect to somehow thicken the pseudo-disk (Hennebelle and Ciardi,
2009; Hirano et al., 2020).

###  3.3 Magnetic Braking

The magnetic braking acts through the magnetic torque $-rB_{\phi}{\bf B}$.
Since it is a local quantity, it is not straightforward to anticipate its
role, and it is enlightening to investigate some of its key properties thanks
to simple cases, as proposed by Mouschovias and Paleologou (1980) and more
recently by Joos et al. (2012) and Hirano et al. (2020).

####  3.3.1 Aligned rotator

Let us first consider a cloud of mass $M$, density $\rho_{c}$, radius $R_{c}$,
and half-height $Z$, surrounded by an external medium of density $\rho_{\rm
ext}$. The cloud is initially in rotation at a speed, $\Omega$, and for
simplicity the magnetic field ${\bf B}$ is considered uniform and parallel to
the rotation axis. The magnetic braking timescale, $\tau_{\parallel}$, is
estimated as the time needed for a torsional Alfvén wave to transfer the
initial angular momentum of the cloud to the external medium (Mouschovias,
1979):

$\rho_{\rm ext}v_{\rm A,ext}\tau_{\parallel}\sim\rho_{c}Z.$ (8)

The Alfvén speed in the external medium is given by $v_{\rm
A,ext}=B/\sqrt{4\pi\rho_{\rm ext}}$ , the mass of the core is $M\sim
2\pi\rho_{c}R_{c}^{2}Z$, while the magnetic flux is $\Phi_{B}\sim\pi
R_{c}^{2}B$. This leads to

$\tau_{\parallel}\sim\left(\frac{\pi}{\rho_{\rm
ext}}\right)^{1/2}\frac{M}{\Phi_{B}}.$ (9)

Therefore the magnetic braking timescale depends on the cloud environment
through the density of the external medium, $\rho_{\rm ext}$. This emphasizes
that magnetic braking represents a transfer of angular momentum between a
faster rotating cloud and a slowly rotating surrounding medium.

However, as discussed above, in a collapsing cloud, the magnetic field lines
are strongly bent and therefore in Eq. (8), one should take into account that
as the Alfvén waves propagate away from the clouds, the field lines are
fanning out. Moreover, at equilibrium the field lines are likely in
corotation, otherwise, the field toroidal component will be further growing.
This leads to modify Eq. (8) as

$\Omega R_{0}^{2}\ \pi R_{0}^{2}\ \rho_{\rm ext}v_{\rm
A,ext}\tau_{\parallel,fo}\sim\Omega R_{c}^{2}\ \pi R_{c}^{2}\ \rho_{c}Z,$ (10)

where $R_{0}$ is the typical distance between the field lines and the rotation
axis, in the external medium. We thus obtain the magnetic braking time for the
case of fan-out, $\tau_{\parallel,fo}$, which is given by

$\tau_{\parallel,fo}=\frac{\rho_{c}}{\rho_{\rm ext}}\frac{Z}{v_{\rm
A,ext}}\left(\frac{R_{c}}{R_{0}}\right)^{4}.$ (11)

It is similar to Eq. (8), apart from the term $(R_{c}/R_{0})^{4}$, whose
origin is twofold. First, because the field lines fan-out, the volume of the
external medium swept by the Alfvén waves is larger by $(R_{0}/R_{c})^{2}$
than when the field lines are straight. Second, because it is assumed that the
field lines are in co-rotation, the fluid elements which are attached to the
field lines have a specific angular momentum that increases like $R_{0}^{2}$.
This altogether leads to another factor $(R_{0}/R_{c})^{2}$.

Combining Eq. (11) with the expressions of the mass, $M\sim
2\pi\rho_{c}R_{c}^{2}Z$, the magnetic flux of the core $\Phi_{B}\sim\pi
R_{0}^{2}B_{\rm ext}=\pi R_{c}^{2}B_{c}$, and the expression for $v_{\rm
A,ext}$, we obtain (see, e.g. Mouschovias and Morton, 1985),

$\tau_{\parallel,fo}=\left(\frac{\pi}{\rho_{\rm
ext}}\right)^{1/2}\frac{M}{\Phi_{B}}\left(\frac{R_{c}}{R_{0}}\right)^{2}.$
(12)

Compared to the magnetic braking time with straight lines, the magnetic
braking timescale is thus significantly reduced since in a collapsing core
$R_{c}\ll R_{0}$ and $(R_{c}/R_{0})^{2}$ is expected to be a small number.

####  3.3.2 Perpendicular rotator ($\alpha=90^{\circ}$)

In the case of a perpendicular rotator, i.e., if magnetic field lines are
initially perpendicular to the rotation axis, the braking timescale
corresponds to the time it takes for the Alfvén waves to reach $R_{\perp}$,
the radius at which the angular momentum of the external medium is equal to
the initial angular momentum of the cloud. The Alfvén waves propagate in the
equatorial plane and sweep a cylinder of half-height $Z$ and radius
$R_{\perp}$, thus

$\rho_{\rm ext}(R_{\perp}^{4}-R_{c}^{4})\sim\rho_{c}R_{c}^{4}.$ (13)

Considering that the magnetic field is such that $B(r)\propto r^{-1}$, so that
$v_{\rm A}(r)=v_{\rm A}(R_{c})\times R_{c}/r$, the magnetic braking time is
then

$\tau_{\perp}=\int_{R_{c}}^{R_{\perp}}\frac{{\rm d}r}{v_{\rm A}(r)}\\\
=\frac{1}{2}\frac{R_{c}}{v_{\rm
A}(R_{c})}\left[\left(1+\frac{\rho_{c}}{\rho_{\rm
ext}}\right)^{1/2}-1\right],$ (14)

which leads to

$\tau_{\perp}\sim 2\left(\frac{\pi}{\rho_{c}}\right)^{1/2}\frac{M}{\Phi_{B}}.$
(15)

Comparing the braking timescales between the aligned configuration with
straight field lines and the perpendicular rotator, Eq. (9) and (15) gives

$\frac{\tau_{\parallel}}{\tau_{\perp}}=\frac{1}{2}\left(\frac{\rho_{c}}{\rho_{\rm
ext}}\right)^{1/2}.$ (16)

Since $\rho_{c}\gg\rho_{\rm ext}$, magnetic braking is found to be more
efficient in the perpendicular case. However, when taking into account the
fact that field lines are fanning out, the term $(R_{c}/R_{0})^{2}$ must be
taken into account. As an illustrative example, let us assume that the density
follows the singular isothermal sphere, i.e. $\rho_{\rm ext}(R_{0})\propto
R_{0}^{-2}$, and $\rho_{c}(R_{c})\propto R_{c}^{-2}$. With Eqs. (12) and (15),
we obtain

$\frac{\tau_{\parallel,fo}}{\tau_{\perp}}=\frac{R_{c}}{R_{0}}.$ (17)

Since $R_{c}/R_{0}\ll 1$, the magnetic braking time is thus shorter in an
aligned rotator than in a perpendicular one provided the field lines are
fanning out.

Numerical simulations of misaligned collapsing cores have been performed by
Matsumoto and Tomisaka (2004), Joos et al. (2012), Tsukamoto et al. (2018) and
Hirano et al. (2020) where analysis of the angular momentum distribution
through the core were also performed. The reported results have been in
apparent contradiction. While Matsumoto and Tomisaka (2004) and Tsukamoto et
al. (2018) concluded that the angular momentum in the core was higher in the
aligned configuration than in the perpendicular one, Joos et al. (2012) and
Hirano et al. (2020) concluded the opposite. The origin of this apparent
contradiction lays on the different timescale that are being probed in these
studies. Matsumoto and Tomisaka (2004) and Tsukamoto et al. (2018) analyse
their simulations at a relatively early time, that is to say before any disk
forms. They also analyse the angular momentum within very dense material.
However, Joos et al. (2012) and Hirano et al. (2020) investigate later times,
after a disk has formed and looked at the angular momentum distribution
through the core. The most likely explanation is that at early times and for
the densest material, the field lines have not been strongly stretched and
thus the braking time is shorter in the perpendicular configuration than in
the parallel one. However, as time goes on, the material that falls in the
central region, say the star/disk material, was initially located further away
and thus the corresponding field lines are more and more bent leading to a
stronger braking.

##  4 The formation of centrifugally supported disks

In recent years, most of the efforts regarding the collapse of dense cores has
been devoted to the study of disk formation. This has appeared to be a complex
topic, which in particular relies on the assumptions made regarding the
microphysics. Here we follow a progressive approach starting from the basics.

###  4.1 What is the origin of angular momentum in star formation models?

The very first question to be asked when discussing rotationally supported
disks is obviously what is the value of angular momentum to be considered and
where it comes from. Traditionally the working assumption has been that
angular momentum was inherited from large scales and that it was conserved, at
least to some degree, during the collapse. While this picture is widely
accepted, it should be stressed that, in fact, it partly relies on the
underlying assumption that the collapsing cloud is axisymmetric, in which case
the distinction between radial and azimuthal velocity fields can be made
rigorously. Moreover, observationally infall and rotational motions can be
made easily. However, in the general case, clouds are not axisymmetric and
this has important consequences. First, while angular momentum is still
conserved with respect to the center of mass, the latter is usually not a
relevant point because it is not the collapse center. Moreover, the star that
forms is not attached to the center of mass, but its position moves as it
evolves, creating small offsets between the global center of mass of the
envelope+star system, and the star itself. Therefore, angular momentum is not
a conserved quantity with respect to the star. From the inertial forces, a
torque is actually operating. This implies that angular momentum may not
necessarily need to be inherited from the large scales. Indeed, Verliat et al.
(2020) investigated the collapse of a non-axisymmetric cloud which initially
had no motion and therefore no angular momentum. They show that in this
configuration, disks would form as well.

In the rest of the section, we will not consider this scenario further, but it
will be discussed later in this review.

###  4.2 Disk formation in hydrodynamical models

Let us consider a rotating and axisymmetric cloud without magnetic field. In
such circumstance, angular momentum is a conserved quantity. Let $M_{*}$ be
the stellar mass and $j$, the fluid particle specific angular momentum. The
centrifugal radius, $r_{d}$, is such that centrifugal and gravitational forces
compensate each other, leading to

$\displaystyle r_{d}={j^{2}\over GM_{*}},$ (18)

which clearly shows that disk formation is directly related to angular
momentum distribution.

Let us consider a cloud with a density profile initially proportional to
$1/r^{2}$ (e.g. Shu, 1977) leading to a mass, $M(R_{0})$, within radius
$R_{0}$, such that $M(R_{0})\propto R_{0}$. On the other-hand, the specific
angular momentum is given by $j(R_{0})=R_{0}^{2}\Omega$. Therefore, we get
$j(R_{0})\propto M(R_{0})^{2}$. The centrifugal radius for a fluid particle
initially located in $R_{0}$ can therefore be expressed as
$r_{d}=j(R_{0})^{2}/(GM(R_{0}))\propto M(R_{0})^{3}$ (Terebey et al., 1984).
On the other hand, considering a uniform density, $M(R_{0})\propto R_{0}^{3}$
and $j\propto M(R_{0})^{2/3}$ which gives $r_{d}\propto M(R_{0})^{1/3}$. If
for simplicity a constant accretion rate is assumed, the disk is found to grow
like $t^{3}$ when the density is initially in $r^{-2}$ and to grow like
$t^{1/3}$ when it is uniform.

From these two examples, it is seen that the angular momentum distribution
sensitively determines the disk radius. To get a quantitative estimate useful
for reference, let $\rho_{0}$ be the cloud density, and $\Omega_{0}$ the
angular rotation velocity. For a fluid particle initially located at radius
$R_{0}$, the centrifugal radius is given by

$\displaystyle{r_{\mathrm{d,hydro}}\,\simeq\,{\Omega_{0}^{2}R_{0}^{4}\over
4\pi/3\rho_{0}R_{0}^{3}G}\,=\,3\beta R_{0}}$ (19) $\displaystyle=106\,{\rm
AU}\,{\beta\over 0.02}\,\left({M\over
0.1M_{\odot}}\right)^{1/3}\left({\rho_{0}\over 10^{-18}{\rm
g\,cm}^{-3}}\right)^{-1/3},$

where $\beta=R_{0}^{3}\Omega_{0}^{2}/3GM$ is the ratio of rotational over
gravitational energy. Note that observationally, cores have been initially
inferred to have a typical $\beta\simeq 0.02$ (Goodman et al., 1993; Belloche,
2013), which is the value used for reference.

###  4.3 Disk formation in ideal MHD models

The first collapse calculations, which have been performed with a magnetic
field, assumed to be parallel to the rotation axis (e.g. Allen et al., 2003;
Galli et al., 2006; Price and Bate, 2007; Hennebelle and Fromang, 2008; Mellon
and Li, 2008). A surprising conclusion has been that even with relatively
modest magnetic fields, typically corresponding to $\mu$ as high as 5-10, the
disk formation was nearly suppressed, a process which has been called
catastrophic braking. The reason of this behavior can be understood by using
simple orders of magnitude.

The magnetic braking and the rotation time, which are most important, are
given by

$\displaystyle\tau_{\rm br}$ $\displaystyle\simeq$ $\displaystyle{\rho
v_{\phi}4\pi h\over B_{z}B_{\phi}},$ (20) $\displaystyle\tau_{\rm rot}$
$\displaystyle\simeq$ $\displaystyle{2\pi r\over v_{\phi}}~{},$ (21)

where $h$ is the scale height of the disk. If the rotation time is longer than
the braking time, a fluid particle may rotate a few times before it loses a
significant amount of angular momentum. Since the magnetic torque is
proportional to $B_{\phi}$, we need to estimate it, which can be accomplished
using the Maxwell-Faraday equation. Essentially, $B_{\phi}$ is produced by the
twisting of the poloidal magnetic field by the differential rotation. As long
as ideal-MHD holds, it grows continuously with time, and we have

$\displaystyle B_{\phi}\simeq\tau_{\rm br}{B_{z}v_{\phi}\over h}.$ (22)

This leads to

$\displaystyle{\tau_{\rm br}\over\tau_{\rm rot}}$ $\displaystyle\simeq$
$\displaystyle\left({M_{*}\over M_{\rm d}}\times{h\over
r_{d}}\right)^{1/2}{G^{1/2}\Sigma\over 2B_{z}}$
$\displaystyle\simeq\left({M_{*}\over M_{\rm d}}{h\over
r_{d}}\right)^{1/2}{\lambda_{eff}\over 4\pi},$

where we have assumed $v_{\phi}\simeq\sqrt{GM_{*}/r}$, $M_{*}$ being the mass
of the central star and where $M_{\rm d}=\pi r_{d}^{2}\rho h$ is the disk
mass, $\Sigma=2h\rho$, is the disk column density and $\lambda_{eff}=2\pi
G^{1/2}\Sigma/B$ is the mass to flux ratio. This shows that modest magnetic
intensities corresponding to values as high as few times $4\pi$, the magnetic
braking timescale is shorter than the rotation time and therefore should
prevent or severely limit the formation of centrifugally supported disk in
good agreement with what has been inferred from the simulations.

###  4.4 Disk formation with non-ideal MHD

Since observational estimates of the typical values of $\mu$ in dense cores
indicate values of a few (Crutcher, 2012), a very efficient magnetic braking
capable of suppressing disk formation for most observed cores, is clearly in
tension with observations. As mentioned in Sect. 3.3, misalignment between
magnetic field and rotation (Joos et al., 2012; Li et al., 2013; Gray et al.,
2018; Hirano et al., 2020) could help to alleviate the problem by reducing the
efficiency of magnetic braking. Indeed, disk formation is reported in all
studies which have considered misalignment for values of $\mu$ on the order of
several, though the exact value of $\mu$ at which disk would form, varies
between studies. We stress however that when a disk forms, if ideal MHD
applies, the toroidal magnetic field within the disk is continuously amplified
and therefore the disk quickly becomes thick and inflated (Hennebelle and
Teyssier, 2008). This emphasizes the need to consider non-ideal MHD.

A very important property of ideal MHD is that the magnetic flux through a
given fluid particle is conserved. When non-ideal MHD processes are important,
significant deviations from flux freezing can be produced, and this may
qualitatively change the impact of magnetic braking. Two main classes of
process may lead to strong departures from flux freezing. First, if the
coupling between the magnetic field and the neutrals is poor and second, if
the flow is strongly turbulent. Let us stress that the latter corresponds to a
high magnetic Reynolds number, while the former would correspond to a low
magnetic Reynolds number.

####  4.4.1 The possible role of turbulence

Turbulence violates the frozen-in condition of ideal MHD flows because it
drives reconnection and thus, the diffusion of the magnetic field lines. Since
the original proposition made by Lazarian and Vishniac (1999), this has been
extensively studied and demonstrated numerically (e.g. Kowal et al., 2009;
Eyink et al., 2013; Santos-Lima et al., 2021).

The role that turbulence may have on disk formation has been carefully
investigated by Santos-Lima et al. (2012), Seifried et al. (2012) and Joos et
al. (2013). In all these studies, it has been concluded that the efficiency of
magnetic braking is reduced by turbulence, though the proposed explanations
were not identical. Santos-Lima et al. (2012) emphasized the role of turbulent
reconnection which transports magnetic field outwards therefore leading to a
reduced magnetic torque, while Joos et al. (2013) stressed that the
misalignment induced by turbulence may add up to the reconnection diffusion
triggered by turbulence in reducing magnetic braking efficiency. Seifried et
al. (2012) proposed that it may be the coherence of the averaged magnetic
torque that is reduced by the fluctuating magnetic field, again induced by
turbulence. While it is clear that reconnection diffusion is occuring in
simulations which include turbulence (see Santos-Lima et al., 2013, for a
discussion), it remains difficult to estimate the respective role played by
these three effects, which all concur to favor disk formation.

####  4.4.2 Low ionisation and high resistivities

One of the most obvious solutions to the so-called magnetic braking
catastrophe is to be searched in high magnetic resistivities. This has been
early proposed by Galli et al. (2006) where Ohmic dissipation was envisioned
to limit magnetic braking. A long series of increasingly realistic
calculations has since been performed by several authors. A comprehensive
description of the various calculations performed can be found in Zhao et al.
(2020).

For instance, Dapp and Basu (2010) show that when Ohmic dissipation is
included in their calculation a small disk forms but not when they assume
ideal MHD. This is because at small scale, the magnetic field is efficiently
diffused out and therefore the braking time typically becomes longer than the
rotation time. Several works have performed simulations that include ambipolar
diffusion, (Dapp et al., 2012; Tomida et al., 2013, 2015; Masson et al., 2016;
Zhao et al., 2016; Hennebelle et al., 2020) finding that centrifugally
supported disks of few tens of AU always form. Note that on the contrary,
Wurster (2016) and Lam et al. (2019) conclude that ambipolar diffusion is not
sufficient to produce a disk.

Analytical arguments to predict the radius of the disks have been developed by
Hennebelle et al. (2016). The effect of ambipolar diffusion, $\tau_{\rm
diff}$, is determined by a characteristic timescale which describes how fast
$B_{\phi}$ is diffused out of the disk

$\displaystyle\tau_{\rm diff}$ $\displaystyle\simeq$ $\displaystyle{4\pi
h^{2}\over c^{2}\eta_{\rm AD}}{B_{z}^{2}+B_{\phi}^{2}\over
B_{z}^{2}}\simeq{4\pi h^{2}\over c^{2}\eta_{\rm AD}}~{}.$ (24)

To get a stationary $B_{\phi}$ within the disk, an equilibrium between
generation and diffusion must occur. Combining Eqs. (22) and (24), as well as
Eq. (20) with Eq. (21) together with an estimate of the density at the edge of
the disk,

$\displaystyle\rho(r)=\delta{C_{\rm s}^{2}\over 2\pi Gr^{2}}\left(1+{1\over
2}\left({v_{\phi}(r)\over C_{\rm s}}\right)^{2}\right).$ (25)

where $\delta$ is a coefficient on the order of a few, the following
expression is inferred

$\displaystyle r_{ad}\simeq 18\,{\rm
AU}\times\delta^{2/9}\left({\eta_{\mathrm{AD}}\over 7.16\times 10^{18}\,{\rm
cm^{2}~{}s^{-1}}}\right)^{2/9}$ (26) $\displaystyle\times\left({B_{z}\over
0.1\,{\rm G}}\right)^{-4/9}\left({M_{\mathrm{d}}+M_{*}\over
0.1M_{\odot}}\right)^{1/3}.$

This expression has been compared with a broad set of MHD simulations
(Hennebelle et al., 2016, 2020) and an agreement within a factor of about 2
has been inferred for the simulations in which the magnetic field is strong
enough. It shows in particular that the disk size grows with the magnetic
resistivity and also with the stellar mass.

The influence of the Hall effect (see Eq. 3) has been the subject of several
studies. Let us first note a peculiarity of the Hall term. Flipping the sign
of ${\bf B}$ in Eq. (3), we see that all terms, but the Hall one, change their
sign. Physically, this is because the Hall term describes the generation of
the magnetic field induced by the motion of the charged particles induced by
the Lorentz force. This implies that two configurations, i.e., the aligned and
anti-aligned cases (${\bf\Omega}.{\bf B}>0$ and $<$ 0 respectively) have to be
distinguished. As a consequence, several teams have found that in the aligned
configuration, the magnetic braking is enhanced and, on the contrary, it is
reduced. This includes the work of Braiding and Wardle (2012) in which
analytical solutions are being used, Tsukamoto et al. (2015), Wurster (2016)
which perform 3D smooth particle hydrodynamical simulations and Marchand et
al. (2019) which perform adaptive mesh refinement calculations. Due to these
different magnetic braking efficiencies, these authors find that the disks
that form in the aligned case are smaller, than the disks which form when the
Hall effect is not accounted for, which are themselves smaller than the disks
which form in the anti-aligned configurations. This has led to the idea of the
possible existence of a bimodal population of disks. Lee et al. (2021c) have
extended the study of Hennebelle et al. (2016) to include the Hall effect,
predicting disks of similar size to that inferred in the simulations.
Tsukamoto et al. (2017) present calculations of misaligned configuration and
conclude that the impact of anti-alignment persists, even so the differences
between the 45∘ and $135^{\circ}$ is significantly reduced compared to the
differences between 0∘ and $180^{\circ}$.

Importantly, Zhao et al. (2020) and Zhao et al. (2021) have run 2D simulations
for longer period of time and conclude that the external part of the disks
formed in the anti-aligned configuration, tend to disappear, leading to a few
tens of AU disks. This would suggest that the bimodal distribution could be a
transient feature.

Finally, we stress that both the high resistivities and the reconnection
diffusion induced by turbulence could contribute simultaneously or in
different situations. For instance, if in some dense cores, the ionisation
fraction is high, the resistivities could be lower and the reconnection
diffusion possibly dominant. However, so far the studies which have considered
in isolated dense cores, both high resistivities and turbulence, have
concluded that the latter does not substantially modify the formation of disk
(e.g. Hennebelle et al., 2020; Wurster and Lewis, 2020).

##  5 Tracing magnetic fields with photons around protostars or at core scales

The current perspective on magnetic fields properties in star-forming cores
has been established observationally by the analysis of polarized light
resulting from the interaction of the B fields with the surrounding gas
molecules and dust grains. Several observational techniques have been
developed: we detail them concisely here below, then describe how these
measurements are used to obtain quantitative constraints from observed
objects, and how the physical processes they rely on have been implemented in
radiative transfer models to produce synthetic observations.

###  5.1 Observational signatures of magnetic fields in protostars

Most of the observational signatures of the presence of the magnetic field in
the interstellar medium are directly related with polarization. However, there
are other mechanisms that can also produce polarization and are not related
with the presence of magnetic fields. Here on, we focus on the magnetic fields
signatures expected in molecular clouds at core’s scales, the possible caveats
and other sources of polarization for each case:

####  5.1.1 Zeeman effect

Molecular rotational lines split in submagnetic levels under the presence of a
magnetic field. The frequency separation of these levels is proportional to
the magnetic field strength and the magnetic dipole moment. For most
molecules, the magnetic dipole moment is very small, making the Zeeman
splitting undetectable, except for the maser lines (e.g., Alves et al., 2012;
Crutcher and Kemball, 2019). However, molecules with an unpair electron have
relatively large magnetic moments, yielding to significant Zeeman splitting.
The Zeeman molecules that are abundant in molecular clouds are OH, CN, SO, CCH
and CCS. The splitting can be measured through circular polarization, but in
almost all cases not in total intensity. This is because the Zeeman splitting
in frequency is typically only of the order of Hz $\mu$G-1 (Bel and Leroy,
1989, 1998; Shinnaga and Yamamoto, 2000; Uchida et al., 2001; Turner and
Heiles, 2006; Cazzoli et al., 2017), much smaller than typical linewidths in
molecular clouds for the expected magnetic field strength ($\lesssim$ few mG).
The circular polarization is proportional to the magnetic field strength
component along the line-of-sight, the intensity derivative and the specific
Zeeman splitting for the observed line. The molecules with hyperfine structure
(CCH, CN) are more adequate to observe the Zeeman splitting because different
hyperfine levels have different Zeeman splitting values. This allows to
separate the possible instrumental polarization, which does not distinguish
between hyperfine lines, from the Zeeman effect. There have been several
attempts to detect Zeeman effect at core scales, mostly through CN rotational
transitions, but there are only a handful of detections reported in the
literature (Crutcher et al., 1999; Crutcher, 1999; Levin et al., 2001;
Falgarone et al., 2008; Maury et al., 2012; Pillai et al., 2016; Nakamura et
al., 2019), and none at disk scales (Vlemmings et al., 2019; Harrison et al.,
2021).

Caveats: This is the only method that can provide a direct estimation of the
field strength along the line of sight. Nevertheless, in order to be able to
derive the role of the magnetic fields at the core and disk scale, special
care has to be made on where the emission of the Zeeman molecules arise. These
are radical molecules, so they are chemically active and their abundances may
change significantly depending on the environment conditions. Thus, in
prestellar cores, CN appears to behave as other N-bearing molecules and thus
survives at high densities (Hily-Blant et al., 2008). On the contrary, CCH,
CCS and SO appear to deplete toward the core’s center (Tafalla et al., 2006;
Padovani et al., 2009b; Juárez et al., 2017; Seo et al., 2019). For massive
cores, all these molecules but SO appear to be anticorrelated with the dust
emission at scales of $\lesssim 0.05$ pc (Dirienzo et al., 2015; Paron et al.,
2021). The distribution of these molecules in the planet-forming disk is also
complex and, combined with the expected magnetic field configuration, makes
the interpretation of Zeeman observations difficult (Mazzei et al., 2020).

####  5.1.2 Linear polarization of molecular lines

Rotational levels split into magnetic sublevels under the presence of magnetic
fields. Transitions where the submagnetic level does not change will be
linearly polarized, with the polarization angle parallel to the magnetic
field. In the other cases, the polarization will be perpendicular to the
magnetic field. In most cases, collisions do not differentiate among magnetic
sublevels, so they populate the sublevels equally. That means that the net
linear polarization is zero. However, an anisotropic radiation field will
generate unbalance of the sublevels, giving a partially linearly polarized
emission. This process is known as the Goldreich-Kylafis (G-K) effect because
it was initially developed by Goldreich and Kylafis (1981). The level of
linear polarization produced by the G-K effect depends on various parameters,
such as the ratio of the collisional rate to the radiative decay rate and the
optical depth (Goldreich and Kylafis, 1981, 1982; Deguchi and Watson, 1984).

However, the polarization direction can be parallel or perpendicular to the
magnetic field. This means that the properties of the emission (optical depth,
velocity, and density gradients) should be analyzed to solve this ambiguity.
Once this is done, then this technique allows obtaining the magnetic field
morphology projected in the plane of the sky. Multi-transition observations of
the linearly polarized lines can be used to derive the field strength (Cortes
et al., 2005). The G-K has been detected in a small sample of cores (Girart et
al., 2004; Lai et al., 2003a; Cortes et al., 2005, 2008; Cortés et al., 2021),
in few disks (Stephens et al., 2020; Teague et al., 2021) and in some
molecular outflows (Girart et al., 1999; Ching et al., 2016; Lee et al.,
2018a). Differential collisions due to ambipolar diffusion could also produce
polarization in the ion molecules Lankhaar and Vlemmings (2020a).

Caveats: The properties of linearly polarized emission from a molecular line
produced by the G-K can be altered if there is a foreground molecular
component at the same velocity (Hezareh et al., 2013). This is the Anisotropic
Resonant Scattering (ARS) effect, which not only alters the properties of the
linear polarization, but it leaks a fraction of this, generating circular
polarization (Houde et al., 2013, 2022). There is some evidence that this may
happen (Chamma et al., 2018). Sensitive observations of all Stokes parameters
are needed to correct for this effect and obtain the unaltered original
signature of the G-K effect. In addition, the detection of this non-Zeeman
circular polarization could be used to measure the magnetic field strength
(Houde et al., 2013). Molecular ions could also have collisionally-driven
linear polarization produced by ambipolar drifts (Lankhaar and Vlemmings,
2020a). The resulting polarization has an angle perpendicular to the magnetic
field direction. This effect could be distinguished from G-K by using
optically thin molecular ion lines.

####  5.1.3 Dust polarization

Since the early 1950s we know that interstellar grains are partially aligned
with the magnetic fields (Davis and Greenstein, 1951). The major axis of the
grains are aligned perpendicular to the magnetic field, yielding to linear
polarization in the dust emission with an angle perpendicular to the magnetic
field projected in the plane of the sky. There have been some proposed
mechanisms that allow the grain angular momentum to align with the magnetic
field (e.g. see review by Lazarian et al., 2015). Radiative torques (RATs) are
thought to be the most efficient way to help grains to align with the magnetic
field. Some predictions of this theory have been confirmed observationally
(Alves et al., 2014; Jones et al., 2015). However, recent results with ALMA
shows that at core scales RATs are not enough to explain the observed dust
polarization properties (Le Gouellec et al., 2020). The dust emission appears
to be polarized at significant levels ($\gtrsim 1$ %) in a significant high
fraction of observed cores, and therefore it is the most used technique in the
millimeter through far-IR wavelengths to study the magnetic fields at core and
disk scales (e.g., Girart et al. 2009, Pattle et al. 2017, Alves et al. 2018,
Sanhueza et al., 2021, and for a more complete references see Hull and Zhang,
2019 and Pattle et al., 2022).

Caveats: At disks scales, grain growth is so important that self-scattering
produced by large grains appears to be the dominant polarization mechanism
(e.g., Kataoka et al., 2015; Yang et al., 2017; Kirchschlager and Bertrang,
2020), although other mechanisms have also been proposed to generate
polarization at disk scales (Tazaki et al., 2017; Kataoka et al., 2019;
Brunngräber and Wolf, 2019). Of special interest for environments containing
large grains, if feasible, is the the alignment of these grains with magnetic
fields in the Mie regime, when grains start to have a size similar to or
larger than the wavelength of the incident light (Guillet et al., 2020a). At
the core scales, in the densest part where the dust emission may be optically
thick, especially at the shortest wavelengths, the polarization pattern may be
altered or even reversed (Liu et al., 2016; Ko et al., 2020).

####  5.1.4 Ion to neutral velocity drift

The ionization fraction in molecular clouds is tiny, $\sim 10^{-6}$, but the
ions and neutrals are well coupled. However, small kinematic differences are
expected due to diffuse processes such ambipolar diffusion (Houde et al.,
2000; Li and Houde, 2008; Falceta-Gonçalves et al., 2010). This effect can be
observed as differences between neutrals and ions, in their line width and in
the velocity maps. HCO+ and HCN lines with the same rotational level from
their different isotopologues are ideal to test this because they have similar
excitation conditions. The line width differences have been observed in
several cores (Houde et al., 2000; Lai et al., 2003b). In a magnetized
turbulent medium the linewidth difference should change with the length scale,
which has been observed at $\sim 0.1$ pc scales, allowing to measure the
turbulent dissipation scale and the magnetic field strength (Li and Houde,
2008; Hezareh et al., 2010, 2014). Recent observations of NH3 and N2H+, at
$\lesssim 0.1$ pc scales, show that, contrarily of what is expected in AD, the
ion linewidth is systematically broader than that of the neutral by 20%
(Pineda et al., 2021).

Caveats: The expected velocity drift is very difficult to measure at small
scales typical of inner envelopes, as the velocity offset is expected to be of
the order of the best spectral resolution currently available from typical
instruments and the velocity field is complex (Yen et al., 2018; Cabedo et
al., 2022). The main limitation of this method is to ensure that the selected
neutral and ion species trace the same gas, or are affected by optical depth
effects (e.g., Pratap et al., 1997; Jørgensen et al., 2004; Girart et al.,
2005; Zinchenko et al., 2009).

####  5.1.5 Observational techniques to infer magnetic field properties

Here, we describe the main techniques that allow to infer the magnetic field
properties from observations, independently of the use of the radiative
transfer tools that are described in the next subsection. The most popular
technique is the Davis-Chandrasekhar-Fermi (CDF) equation (Davis and
Greenstein, 1951; Chandrasekhar and Fermi, 1953) which allows estimating the
magnetic field strength in the scenario of small perturbation due to the
anisotropic motions (such as turbulence). The equation relates the magnetic
field in the plane-of-the sky, $B_{\rm pos}$ with the gas density, $\rho$, the
non-thermal velocity dispersion along the line-of-sight, $\delta{\rm v}$ and
the dispersion of the polarization angles of in the plane of the sky,
$\delta{\rm\theta}$: $B_{\rm pos}=\sqrt{4\pi\rho}\,\delta{\rm
v}/\delta{\rm\theta}$ (Chandrasekhar and Fermi, 1953). There have been several
works to account for the limitations of this technique, such as line-of-sight
smearing or cases with large dispersion, that in general leads to an
overestimation of the field strength (Ostriker et al., 2001; Heitsch et al.,
2001; Falceta-Gonçalves et al., 2008; Cho and Yoo, 2016; Skalidis and Tassis,
2021; Liu et al., 2021). A more sophisticated method is the use of the
structure function, which allows separating the turbulent component from the
smooth variations of the polarization angles to the large scale field
(Hildebrand et al., 2009; Houde et al., 2009, 2011, 2016). There are other
(semi) analytical expressions that allow to evaluate the relevance of magnetic
fields for the specific hour glass configuration (Ewertowski and Basu, 2013;
Myers et al., 2018).

###  5.2 Polarized radiative transfer tools for protostellar environments

In this section, we describe how physical processes responsible for producing
polarized light as a result of the interaction of magnetic fields with
protostellar material are implemented in radiative transfer codes and coupled
to the numerical models for star and disk formation. We focus on two codes
which are widely used by the community interested in star formation: POLARIS
for dust grain alignment (Reissl et al., 2016), and PORTAL for spectral lines
(Lankhaar and Vlemmings, 2020b). While a complete description of the
polarization-inducing processes go beyond the scope of this review, we briefly
describe them below.

####  5.2.1 Radiative transfer of emission from magnetically aligned grains

The three-dimensional continuum radiative transfer RT code POLARIS (Reissl et
al., 2016) uses 3D models of astrophysical structures to produce synthetic
maps of the polarized dust continuum emission, allowing several flavours of
alignment mechanisms for the dust grains. The photon propagation within the 3D
model is implemented following a Monte-Carlo photon transfer scheme. The
interaction of the incoming radiation with dust grains is determined in each
cell, and depends on mostly of the cross-section of extinction, the dust
density. Depending on the local physical conditions such as the dust grain
albedo, POLARIS either computes the scattering, or the absorption with
instantaneous re-emission, then a new wavelength is calculated and the dust
thermal energy of the cell is adjusted. In this computation of photon
propagation and dust heating mode, POLARIS assumes that dust grains are
spherical: this step thus allows to derive the dust temperature in each cell
with a limited computational effort. The alignment probability of dust grains
in each cell is calculated, accounting for the imperfect internal alignment
and the imperfect alignment between the dust grain’s angular momentum and
magnetic field. Then, the anisotropy of the radiation field at any given
wavelength $\lambda$ is calculated in each cell which allows determining the
alignment of the different dust grains due to radiative torques (RATs, see
previous section). Since a distribution of dust grain sizes is considered
(usually following a classical MRN, after Mathis et al. 1977, distribution
consisting of power-laws of separate populations of bare spherical silicate
and graphite grains), the RAT alignment in given irradiation and B-field
conditions depends also on the effective grain radius $a_{align}$. Dust grains
larger than $a_{align}$ are considered aligned (with a fraction depending on
the high-J attractor point $f_{high-J}$, which can be manually set to any
value between 0 and 1), and contribute to the polarization of the dust thermal
emission. Several studies have used the POLARIS code to predict and confront
observations of polarized dust emission, from molecular cloud conditions (Liu
et al., 2021; Reissl et al., 2021) down to disk conditions (Brunngräber and
Wolf, 2021). Note that a few studies have used radiative transfer tools
developed in a recent past, such as Dustpol (Padovani et al., 2012; Lee et
al., 2017) but here we focus on the most recent results, and hence on POLARIS
because it is the most widely used RT code for dust polarization nowadays. An
example of synthetic observations of the polarized dust emission, assuming
different alignment conditions for the dust from the same numerical model of a
collapsing core, is shown in Figure 1. Details and results of the works
specifically dedicated to protostars are described in the next Section 5.3
below.

####  5.2.2 Radiative transfer of emission from magnetically-sensitive
molecular lines

The POLARIS code can also be used to simulate the polarization of spectral
lines due to the Zeeman effect, thanks to the ZRAD extension (Brauer et al.,
2017). This is based on the line RT algorithm Mol3D (Ober et al., 2015), and
makes use of atomic and molecular parameters such as the energy levels and
transitions taken from the Leiden Atomic and Molecular Database (LAMDA), the
Landé factors of the involved energy levels, the line strengths of the allowed
transitions between Zeeman sublevels, and emitting radius of the molecular
species. It has been used to test the robustness of Zeeman measurements to
infer magnetic field strengths in molecular clouds, showing that while the gas
density affects little the uncertainty of the measurements, strong variations
in the LOS component of the gas velocity and of the magnetic field strength
significantly impact the precision of the method (Brauer et al., 2017).

PORTAL is an adaptive three-dimensional polarized line radiative transfer
model that considers the local anisotropy of the radiation field as the only
alignment mechanism of molecular or atomic species to calculate the aligned
molecular or atomic states. It allows simulating the polarization of photons
produced from line emission through a magnetic field of arbitrary morphology.
While it can be run in stand-alone mode, PORTAL can also process the outputs
of 3D radiative transfer codes on regular grids. It has been used, coupled to
the parallelized non-LTE 3D line radiative transfer code Line Emission
Modelling Engine (LIME, Brinch and Hogerheijde 2010), to predict spectral line
polarization of different molecular transitions arising from protoplanetary
disks, at the ALMA wavelengths (Lankhaar et al., 2022).

###  5.3 Synthetic observations of B-fields from protostellar models: methods

The question of the physical processes causing the polarization of photons
from star-forming structures, both from the dust and from molecular species,
has been a long-standing one. Some simple analytical models assuming a quasi-
perfect alignment of dust grains with magnetic field lines, for example, have
been proposed in the past, and sometimes even successfully reproduced some
observations (Padovani et al., 2012). The development of more detailed
physical models of grain alignment have allowed more predictive studies of the
polarized dust emission arising from magnetically-aligned grains, and
quantitative confrontation to observations performed towards protostellar
envelopes.

The combination of polarized radiative transfer, as described above, to state-
of-the-art numerical models of protostellar formation can be used to compare
model predictions to actual observations. The general process is quite
standard. First, one selects the outputs corresponding the best to the
properties of the observed object to be compared to. Usually, a numerical
simulation output can be extracted as a datacube containing the gas density,
the gas pressure (or equivalently the gas temperature), the three components
of the velocity field, as well as the three components of the magnetic field.
These gas properties are extracted and put in a suitable format to be post-
processed with a radiative transfer code, such as Polaris. The central
radiation source is modelled as a blackbody of chosen luminosity that allows
to reproduce the observed bolometric luminosity of the studied object. Note
that, to facilitate the propagation of photons and avoid missed photon
packages from the central source due to long computational times, many authors
choose to artificially empty a small central sphere of radius a few au around
the sink particle, as in Valdivia et al. (2019). The radiative transfer can
also include an external isotropic radiation field of strength that
illuminates the external layers of the core. The dust properties are normally
taken from tabulated values and in most studies assumed uniform throughout the
model, with a gas-to-dust ratio of $100$, and a typical composition and size
distribution reproducing observations of dust in the ISM (Mathis et al., 1977;
Hildebrand and Dragovan, 1995). The radiative transfer calculation is done via
Monte Carlo methods or ray tracing methods. For example, the POLARIS code
computes first the propagation of photons alongside the dust temperature via a
MCMC analysis, and then solves the grain alignment equations, using the
density, radiation field, dust grain properties and temperature, in each cell
of the grid. Grain alignment can be assumed to be perfect, or to depend on
local conditions, as, e.g., when assuming RAT alignment (Hoang and Lazarian,
2014). The outputs are maps of the Stokes I, Q and U from the dust thermal
emission, which can be further convoluted by instrumental effects (e.g., the
simobs task in CASA to simulate ALMA datasets) and compared to observations.

Figure 1: Synthetic maps of the $1.3\ \mathrm{mm}$ dust continuum emission.
Overlaid are the polarization vectors rotated by $90^{\circ}$, obtained with
POLARIS either considering perfect alignment of the dust grains (right) or RAT
grain alignment (left), from Kuffmeier et al. (2020).

##  6 Observed protostellar properties

Measuring the properties of the gas, the dust, and the magnetic field in
embedded protostars is cornerstone to inform the magnetized models of star
formation, test their predictions, but also understand the role of non-ideal
effects in disk formation and evolution. We recall here below some key
observational results which relate directly to these questions.

###  6.1 Dust Grains

Because they are the seeds from which planet formation processes are triggered
in circumstellar disks, and because they are also key agents in the efficiency
of coupling the magnetic field to the gas, characterizing dust grains is a
cornerstone in building a comprehensive scenario of star formation.

In the far-IR and longer wavelengths, the opacity of astrophysical dust scales
with wavelength as $\kappa_{\nu}\propto\nu^{\beta}$, where $\beta$ is the
emissivity spectral index of dust emission. $\beta$ reflects how emissive dust
grains are, and is therefore, commonly used to characterize dust grains in
astrophysical structures. It is measured by comparing the relative intensity
of dust thermal emission when observed at different wavelengths: indeed, if
the dust emission is optically thin, the ratio of flux densities at different
wavelengths only depends on $\beta$. In the diffuse interstellar medium,
Planck studies have shown the dust has $\beta$ $\sim$ 1.6 (Planck collab.
2014, Juvela et al. 2015). Observational works show that significant
variations in $\beta$ is found at protostellar core scales, with millimetre-
wavelength $\beta$ values much lower than this typical value, however (Martin
et al., 2012; Chen et al., 2016; Sadavoy et al., 2016; Bracco et al., 2017;
Van De Putte et al., 2020).

While studies dedicated to resolving the dust properties at small radii in
embedded protostars (Jørgensen et al., 2007; Kwon et al., 2009; Chiang et al.,
2012; Miotello et al., 2014) started before the advent of large millimetre
interferometers such as NOEMA and ALMA, the development of these two
observatories has allowed a significant leap forward in this area of research.
Li et al. (2017), Galametz et al. (2019), Tychoniec et al. (2020) and Bouvier
et al. (2021) have measured the dust emissivity spectral index in relatively
large samples of embedded protostars, finding $\beta$ values ranging between 0
and 2, with a majority of the protostellar dust exhibiting $\beta<1$ at
envelope radii $\sim 500$ au. Figure 2 shows in its left panel the
observations of millimeter dust emissivity indices measured by Galametz et al.
(2019) in the CALYPSO survey, finding that most protostars show significantly
shallow dust emissivities at envelope radii $\sim 100-500$ au, with $\beta<1$.
Galametz et al. (2019) did an analysis of their interferometric data in the
visibility space, allowing to also distinguish a radial evolution of the dust
emissivity, with many objects showing a decrease of $\beta$ with decreasing
envelope radius. These observations also confirm it is the dust pertaining to
the innermost envelope, and not to the unresolved disks at smaller scales,
which is different from the dust typically observed in the diffuse ISM. In
Class I protostars, less observations sensitive to the inner envelope regions
are available: despite the poor statistics they also tend to show low dust
emissivity in a majority of objects, with sometimes extremely low values
($\beta<0.5$) at disk scales (see e.g. Harsono et al., 2018; Nakatani et al.,
2020; Zhang et al., 2021).

Other effects affecting the millimetre flux could be responsible for the low
dust opacities found, such as anomalous emission from spinning dust grains
(Planck Collaboration et al., 2011), contamination by non-thermal emission or
the dust self-scattering (Liu et al., 2019). At the typical envelope scales,
however, both the effects of optical depth and non-thermal emission are
expected to be negligible at millimetre wavelengths (Tychoniec et al., 2020).

Another possible explanation for the observed difference in the dust
emissivity $\beta$ could stem from different dust grain composition compared
to that of the diffuse ISM. For example, it has been suggested that grains
with higher ratio of carbonaceous to silicate would exhibit lower $\beta$
(Jones et al., 2017; Ysard et al., 2019; Zelko and Finkbeiner, 2020).
Similarly, grains that are less compact and more fluffy could also be
associated to low emissivity (Brunngräber and Wolf, 2021; Köhler et al.,
2008). These explanations are not favoured as the unique cause for the
observed low emissivities, however, because of the low values found by
observations (around 0) can not be matched by current dust models, even with
significant changes of composition and compactness. Indeed, laboratory studies
probing different interstellar dust analogues suggest that dust emissivities
$<$1 at millimetre wavelengths can only be produced by grains larger than 100
$\mu$m (Demyk et al., 2017b, a; Ysard et al., 2019) in the typical physical
conditions reigning in protostellar envelopes.

Another independent thread of evidence for the presence of relatively large
grains in Class 0 envelopes and disks stems from polarimetric observations (Le
Gouellec et al., 2019; Lee et al., 2021a) and their comparison to synthetic
observations (Valdivia et al., 2019; Le Gouellec et al., 2020), performed as
described in 5.3. They show that relatively large grains are required to
reproduce the observed level of polarization fractions (see an example in the
right panel of Figure 2). Indeed, if dust grains align following the radiative
torques (RATs) theory, the radiation field in deeply embedded protostellar
environments is not prone to align the small grains where polarization of the
dust mm emission is routinely detected. Only synthetic observations performed
with dust grain size distributions including grains up to
$a_{\mathrm{max}}\sim 20~{}\mathrm{\mu m}$ produce polarized dust emission at
levels similar to those currently observed in solar-type protostars, at
(sub-)millimetre wavelengths.

Figure 2: Left: Observed millimeter dust emissivity index in a sample of young
protostars from Galametz et al. (2019). All protostellar envelopes probed at
radii $\sim 500$ au show lower values than the progenitor diffuse ISM. Right:
Radial profile of the polarization fraction from the $1.3~{}\mathrm{mm}$ dust
emission predicted from RAT alignment of dust grains in a typical low-mass
protostellar envelope (Valdivia et al., 2019). Large ($>10\,\mu$m) dust grains
are required to match the observations of the polarized emission (grey area).

###  6.2 Fraction of ionized gas

Characterizing the ionization of the gas in the dense envelope material is
critical to set constraints on the coupling of the magnetic field with the
infalling-rotating envelope gas, and the role of diffusive processes, such as
ambipolar diffusion or reconnection diffusion, to counteract the outward
transport of angular momentum from the infalling-rotating envelope due to
B-fields.

Cosmic rays (CRs), mostly relativistic protons, are the dominant source of
ionization in relatively dense molecular gas where ultraviolet radiation
cannot penetrate (Grenier et al., 2015). Protostars are deeply embedded
sources, where the gas ionization fraction can only be inferred using indirect
chemical signatures. A handful of measurements were obtained in Class 0
objects at core scales (typical densities $n_{\rm H_{2}}\sim 10^{4}$ cm-3):
they suggest typical cosmic ray ionization rates $\chi_{e}\sim
10^{-17}-10^{-15}\,\rm{s}^{-1}$ with large uncertainties (Ceccarelli et al.,
2014; Podio et al., 2014; Favre et al., 2017, 2018). In Class I protostars,
only a few studies have been carried out, using the abundance of HCO+ to
estimate gas ionization from cosmic rays in two young disks around $\xi\sim
10^{-17}s^{-1}$, and suggest it is lower in the disk compared to the inner
envelope (Harsono et al., 2021; van ’t Hoff et al., 2022). Such low values may
suggest it is unlikely that the accretion through the disk to the central star
could be driven by magneto-rotational instabilities. In the Class 0 protostar
B335, Cabedo et al. (2022) used deuteration detected in molecular line
emission maps to characterize the ionization of the gas at envelope radii
$\lesssim$ 500 au (typical densities $n_{\rm H_{2}}\sim 10^{6}$ cm-3), and
found large cosmic ray ionization rate $\zeta$ between $10^{-16}$ and
$10^{-14}$ s-1. Such high values seem inconsistent with the fiducial value if
the interstellar cosmic rays flux is responsible for the gas ionization, as it
should be efficiently attenuated while penetrating into the dense cores
(Padovani et al., 2018). The observations of Cabedo et al. (2022) also show
the CRs ionization rate is increasing at small envelope radii, toward the
central protostellar embryo. Several theoretical works have investigated the
role of shocks at the protostellar surface and magnetic mirroring within the
jets as efficient forges to accelerate locally low-energy cosmic rays, and
their role in increasing the ionization rate of the shielded protostellar
material (Padovani et al., 2015; Silsbee et al., 2018; Padovani et al., 2021;
Fitz Axen et al., 2021). In B335, it seems that local acceleration of CRs, and
not the penetration of interstellar CRs, may be responsible for the gas
ionization at small envelope radii. This would imply that, in the inner
envelope, the collapse transitions from non-ideal to a quasi-ideal MHD once
the central protostar starts ionizing its surrounding gas, and very efficient
magnetic braking of the rotating-infalling protostellar gas might then take
place.

###  6.3 Gas kinematics: angular momentum and mass infall rates

The dynamical role of the magnetic field can also be assessed by careful
examination of the distribution of angular momentum associated to protostellar
gas, and measurement of the mass transported inwards, from envelope to disk
scales (Pineda et al., 2019; Galametz et al., 2020; Yen et al., 2021b). This
is not simple from an observational point of view, as the velocity field of
the molecular gas is much less organized than expected from the simple
monotoneous collapse of axisymmetric rotating cores in many protostars. Recent
studies have revealed envelopes with complete reversal of the velocity fields,
or containing multiple velocity components (Gaudel et al., 2020; Maureira et
al., 2017) at radii $r<5000$ au. Figure 3 shows a few recent examples of the
detection of well-developed asymmetric features from envelopes to disk scales,
which may trace preferred pathways funneling the accretion, such as streamers
(Pineda et al., 2020; Chen et al., 2021) or supersonic infall along outflow
cavity walls (Cabedo et al., 2021).

Figure 3: Streamer structures seen at different scales and towards different
embedded protostars. Top: IRAS 03292+3039 in Perseus (Pineda et al., 2020).
Middle: Serp-SMM1 in Serpens (Le Gouellec et al., 2019). Bottom: [BHB2007] 11
in the Pipe (Alves et al., 2019).

Mass infall rates from large to small scales are difficult to estimate, as the
complex kinematics of envelopes surrounding embedded protostars produce
convoluted signatures whose degeneracies can only be lifted by a joint
analysis of a variety of tracers probing widely varying density conditions and
spatial scales. Double-peaked profiles in spectral lines from molecular
species (due to self-absorption) with a brighter blueshifted peak are
sometimes used as a signpost of probable infall motions (e.g., Choi et al.,
1995; Mardones et al., 1997; Evans et al., 2015; He et al., 2015; Yang et al.,
2021). These profiles are used to derive infall rates, $\dot{M}_{\rm inf}$,
following the expression: $\dot{M}_{\rm inf}=(\Omega/2)V_{\rm inf}\,(R_{\rm
c})\,\mu\,m_{\rm H}\,\langle n_{\rm H_{2}}\rangle\,R^{2}_{\rm c}$, where
$\Omega$ is the solid angle subtended by the core center over which infall
occurs, $R_{\rm c}$ is the core’s radius, $V_{\rm inf}\,(R_{\rm c})$ is the
infall velocity at this radius and $\langle n_{\rm H_{2}}\rangle$ is the
core’s average volume density (e.g. Beltrán et al., 2022). Surprisingly, few
of these line profiles are observed towards embedded protostars at core scales
(Mottram et al., 2013, 2017). Interferometric observations resolving the inner
envelopes also revealed only a handful of them, for which modeling suggests
typical mass infall rates $\sim 10^{-5}$ $\rm{M}_{\odot}$ per year at $\sim
500$ au scales in Class 0 protostars (Pineda et al., 2012; Evans et al., 2015;
Su et al., 2019). Class I protostars in Ophiuchus have been surveyed with
ALMA, exhibiting typical mass accretion rates a few $10^{-7}$ $\rm{M}_{\odot}$
per year (Artur de la Villarmois et al., 2019). In the well-characterized
Class I protostar TMC1-A, the observed mass infall rate is measured to be
larger than the value towards Ophiuchus Class I, estimated around $\sim
10^{-6}$ $\rm{M}_{\odot}$ per year. This protostar also shows a decelerating
infall velocity towards small radii, and an infall velocity $\sim 0.3$ times
smaller than the free-fall velocity yielded by the dynamical mass of the
protostar (Aso et al., 2015). The accretion rates derived in high mass star
forming cores are significantly larger, typically between $\sim 10^{-4}$ to
$10^{-2}$ $\rm{M}_{\odot}$ per year at scales of few hundred to few thousands
of au (e.g., Beltrán et al., 2006; Qiu et al., 2011; Wyrowski et al., 2012;
Liu et al., 2013, 2020). In the case of the G31.41+.31, high angular
resolution multi-line inverse P-Cygni measurements indicate infall
acceleration at decreasing radius (Beltrán et al., 2022).

A key figure to assess the magnitude of rotational energy contained in
protostellar cores, and compare these rotational motions to model predictions,
is the specific angular momentum of the gas $j_{\rm{spe}}=r\times v_{\phi}$.
High angular resolution observations of molecular line emission in Class 0
protostars have revealed that the gas contained in envelopes has a specific
angular momentum $j_{\rm{spe}}\sim 10^{20}$
$\mathrm{cm}^{2}\,{{\rm{s}}}^{-1}\,$ at 1000 au (Yen et al., 2015a; Tatematsu
et al., 2016; Pineda et al., 2019; Gaudel et al., 2020; Heimsoth et al., 2021;
Hsieh et al., 2021). In the literature, it is common to find that the
conservation of angular momentum in an infalling rotating envelope should
produce a radial profile of rotational velocity $v_{\phi}\propto r^{-1}$,
because $j_{\rm{spe}}=r\times v_{\phi}$ should be constant. While if the
initial rotation profile at the beginning of collapse follows a
$v_{\phi}\propto r^{-1}$ relationship, this is true, it is not in most cases.
Indeed, an infalling envelope initially in solid body rotation ($v_{r}\propto
r^{-1/2}$ and $v_{\phi}\propto r$) should exhibit $j_{\rm{spe}}\propto r^{2}$.
If gas particles conserve their specific angular momentum during the collapse,
the slope of the radial profile $j_{\rm{spe}}(r)$ should be conserved with
time as they move inwards, and protostellar envelopes should thus be observed
with $j_{\rm{spe}}\propto r^{2}$ down to the disk sizes. However, this is not
what is observed. At envelope radii between 1000 and 10000 au, $j_{\rm{spe}}$
scales as a power-law with radius, $j_{\rm{spe}}\propto r^{n}$ with $n\sim
1.6-1.8$. Interestingly, as pointed out by Gaudel et al. (2020), this scaling
is closer to the trend expected from the dissipation cascade of Kolmogorov-
like turbulence than from solid-body rotation. This may suggest that large-
scale ISM turbulence or turbulence locally driven around/by protostars is
actually responsible for the observed angular momentum in outer envelopes.
Moreover, resolved observations have identified a break in the specific
angular momentum radial profiles, from the steep outer profile
$j_{\rm{spe}}(r)\propto r^{1.6}$ at envelope radii $r>1600$ au, to a quasi
flat $j_{\rm{spe}}(r)\sim\rm{cte}$ inner profile at radii $<1600$ au (Gaudel
et al., 2020). If angular momentum was conserved during the collapse, it
suggests that the initial angular momentum profile in these cores was, also,
mostly flat and would rule out that cores are build with solid body rotation
as initial conditions.

Taken altogether, these observed features suggest that a complex interplay of
magnetic fields, turbulence, and gravity are responsible for organizing the
collapse along preferential directions, and that the angular momentum
contained at small scales does not seem closely related to the gas flows
observed at larger envelope radii.

###  6.4 Magnetic fields at core’s scales

High angular resolution observations at (sub-)mm wavelengths show that the
thermal dust emission probing the envelope-to-disk scales ($\sim$ 50-10,000
au) is polarized at a few percent level. This has been observed both in low
and high mass star forming regions (see e.g. Matthews et al., 2009; Zhang et
al., 2014; Galametz et al., 2018; Hull and Zhang, 2019; Beltrán et al., 2019;
Sanhueza et al., 2021; Eswaraiah et al., 2021). Considering that the polarized
flux is a quantity that is prone to cancellation if the polarization angle is
highly disorganized along the line of sight, observations of rather large
polarization fractions suggests the magnetic field lines underlying the
alignment of the protostellar dust remain at least partly organized inside
star-forming cores (Le Gouellec et al., 2020). Indeed, these observations have
already yielded to a large statistical sample to look for morphological trends
at core scales. This is even taking into account that in many (mostly pre-
ALMA) observations the detected signal was not extended enough to infer a
clear, well resolved, morphology. In many dense cores, the magnetic lines are
often quite organized. Indeed, the expected hourglass configuration expected
in the collapse of cores threaded by a uniform magnetic field has been
detected (see e.g. Schleuning, 1998; Girart et al., 2006, 2009; Qiu et al.,
2014; Kandori et al., 2017; Maury et al., 2018; Beltrán et al., 2019; Redaelli
et al., 2019; Kwon et al., 2019). This configuration implies that either
magnetic lines efficiently resist the angular momentum due to rotation or that
the latter is weak, otherwise the field lines would show a clear toroidal
component. However, there are few cores where a spiral pattern suggests that
the kinetic energy associated to angular momentum may dominate (Lee et al.,
2019; Beuther et al., 2020; Sanhueza et al., 2021). In some cases (Hull et
al., 2017b), the observations reveal quite disorganized B-field topologies,
which suggest weak-field conditions, and the possible role of turbulence or
anisotropic gas flows in organizing the magnetic field lines.

Moreover, there are two interesting features that ALMA observations have
revealed. One is the detection of significant polarization along the outflow
cavity walls, as in some cases this is the only place where polarization is
indeed detected (Le Gouellec et al., 2019; Hull et al., 2020). The other is
the presence of well organized polarized filaments with the magnetic field
along the filament (Le Gouellec et al., 2019; Takahashi et al., 2019; Hull et
al., 2020). This may be related with accretion streamers that have been
detected recently (Yen et al., 2019; Alves et al., 2020; Pineda et al., 2020),
although kinematic information is needed to confirm this scenario.

(Koch et al., 2014) compiled the magnetic fields properties of 50 low and high
mass protostellar cores. They propose four types of morphology (see also Koch
et al., 2013): (1) magnetic field mostly aligned along the minor axis of the
core, (2) magnetic field mostly aligned along the major axis of the core, (3)
hourglass and quasi-radial fields, and (4) irregular shapes. This correlation
was also confirmed by (Zhang et al., 2014), which also found that the magnetic
fields in massive dense core scales are either parallel or perpendicular to
the parsec–scale magnetic fields. However, there are cases where this
correlation at different scales does not hold (Girart et al., 2013; Hull et
al., 2017a). Despite the overall trends where magnetic fields appear to be
coherent with respect to larger scales and core orientation, there is no
correlation with molecular outflow direction (Hull et al., 2013; Zhang et al.,
2014). However, there are other works that found some correlation between the
outflow direction and the magnetic field, although the alignment is far from
being perfect (Galametz et al., 2020; Yen et al., 2021a). These two works show
that, on one hand, the misalignment between the outflow and magnetic field
direction appears to be correlated with the amount of angular momentum
(Galametz et al., 2020), and, on the other hand, the observed misalignment is
not sufficient to reduce the efficiency of magnetic braking (Yen et al.,
2021a).

Most of the observations have relied on indirect methods to derive the
magnetic field strength from the (mostly dust) polarization observations (see
Section 5.1). In spite of not being very accurate, these methods give a good
approximation of the magnetic field strength. Thus, comparison between semi-
analytical models or MHD simulations with polarization observations have found
good agreement with the values obtained using different Davis-Chandrasekhar-
Fermi approximations (Gonçalves et al., 2008; Frau et al., 2011; Juárez et
al., 2017; Maury et al., 2018; Beltrán et al., 2019). Overall, the DCF shows
that star forming cores appear to be supercritical by a factor of about 2, and
that these cores are gravitationally bound (Myers and Basu, 2021). Measuring
directly the (line-of-sight) strength of the magnetic field is very difficult,
and there is a small amount of positive detection at core scales (Crutcher et
al., 1999; Falgarone et al., 2008; Crutcher et al., 2009; Pillai et al., 2016;
Nakamura et al., 2019). DR21(OH) is probably the only core where three
different techniques have been used to derive the field strength (Crutcher et
al., 1999; Hezareh et al., 2010; Girart et al., 2013). For the few sources
with magnetic field strength derived from both (OH) Zeeman and dust
polarization, the ratio of the plane-of-sky magnetic field component to its
line-of-sight component, $4.7\pm 2.8$, is larger than the expected average
value for random orientations (Myers and Basu, 2021). However, we should note
that the OH Zeeman observations probably trace lower densities than (sub)mm
dust observations. In any case, using both data sets, the magnetic fields
appear to increase with volume density as $B\propto n^{2/3}$, indicating that
magnetic field strength is significant but not enough to avoid the
gravitational collapse (Myers and Basu, 2021). Finally, it is important to be
aware that the interpretation of the observational data is sometimes a source
of vivid debate (Mouschovias and Tassis, 2010; Crutcher et al., 2010; Tassis
et al., 2014), and the application at core scales should be taken with caution
(Reissl et al., 2021; Liu et al., 2021).

###  6.5 Protostellar disks

One of the major predictions of magnetized models regards the properties of
rotationally-supported disks: embedded young disks are expected to be compact
and dust-rich. In this section, we briefly summarize recent constraints
brought by observations of the sizes and masses of protostellar disks.

Maury et al. (2010, 2019) used the NOEMA CALYPSO survey, at 1.3 mm and 2.7 mm
to characterize the disk properties in twenty-six Class 0 and I protostars.
Modeling the millimeter dust continuum emission with a combination of envelope
and disk contributions directly in the visibility space, they find 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. The VANDAM survey (Segura-Cox et al., 2018; Tobin et
al., 2020) used VLA then ALMA observations of the mm dust continuum emission
to characterize the radii of Class 0 and Class I disk sizes in Perseus and
Orion. Their most recent results in Orion performing multi-wavelength analysis
(25 Class 0 sources, 44 Class I sources) report smaller median dust disk radii
than their preliminary analysis, with Class 0 disks having radii $\sim 35$ au
in Orion (Sheehan et al., 2022), and no statistically significant difference
between the properties of Class 0 and Class I disks. This later point may be
specific to Orion, however, as our census of all embedded disks radii from the
literature presented in Figure 4 does show a difference, with Class I disks
being statistically more extended. Cox et al. (2017) and Encalada et al.
(2021) used 2D Gaussian fits applied to the ALMA dust continuum images of
Class I protostars in Ophiuchus and find respectively a median dust disk
radius from 12.6 au to $\sim 23.5$ au. These measurements are also included in
Figure 4.

Figure 4: Class 0 (in red) and Class I (in green) protostellar disk radii
observed, from the dust continuum emission at millimeter wavelengths
($\lambda<2.7$mm). Most measurements stem from the CALYPSO survey (Maury et
al., 2019) and the VANDAM survey (Sheehan et al., 2022). Other sources from
Oya et al. (2014); Lee et al. (2018b); Yen et al. (2015b, 2017); Cox et al.
(2017); Aso et al. (2015); Ohashi et al. (2014); Brinch and Jørgensen (2013);
Harsono et al. (2014); Miotello et al. (2014); Sheehan and Eisner (2017);
Encalada et al. (2021). The mean dust disk radii for each Class are indicated
as horizontal lines (48 au for Class 0 and 86 au for Class I).

.

These aforementioned studies rely mainly on the analysis of the thermal dust
emission, but protostellar disks can also be identified thanks to kinematic
signatures. Inside the centrifugal radius, the gas rotates with nearly-
Keplerian motions as the pressure and centrifugal acceleration balance the
radial gravitational acceleration. Measuring the size of rotationally-
supported disks in embedded protostars needs to distinguish the disk emission
from that of the infalling envelope, kinematically. Identifying the transition
from envelope kinematics dominated by infalling gas ($v_{r}\propto r^{-1/2}$
and $v_{r}>v_{\phi}$) to gas contained in a rotationally supported disk, with
$v_{\phi}(\rm{r})\propto\sqrt{(G\times M_{\star+disk})/r}$ and
$v_{r}<<v_{\phi}$, is thus, in theory, a simple way of measuring disk gaseous
sizes. In practice, the simultaneous contribution of gas kinematics from
different origins, chemistry effects on molecular gas tracers, and projection
effects of asymmetric gas motions make it a hard task to clearly measure the
rotation velocity $v_{\phi}$ in protostellar envelopes down to the disk outer
radius. Only few protostars show clear signatures of Keplerian motions that
can be used to characterize their disk: in the few objects that have an
estimate of the dusty disk sizes, the gaseous disk size and dusty disk sizes
are similar, within the relatively large uncertainties associated (Ohashi et
al., 2014; Yen et al., 2017; Bjerkeli et al., 2019; Maret et al., 2020;
Harsono et al., 2014; Chou et al., 2014; Yen et al., 2014; Aso et al., 2015).

Protostellar disk masses are usually measured from the dust emission,
estimating dust masses from a set of hypothesis on the dust properties then
correcting by a gas-to-dust ratio to obtain gas masses. The choice of
wavelength(s) for the dust observations is crucial: for example, in Perseus,
Tychoniec et al. (2020) measure median dust masses of the embedded disks $\sim
5\times 10^{-4}{\rm\,\thinspace M_{\odot}}$ for Class 0 and $2\times
10^{-4}{\rm\,\thinspace M_{\odot}}$ for Class I from the VLA dust continuum
emission at centimetre wavelengths. In comparison, dust masses from sub-
millimetre dust emission in ALMA bands towards the same population of disks in
Perseus are significantly lower, $1.4\times 10^{-4}{\rm\,\thinspace
M_{\odot}}$ and $4\times 10^{-5}{\rm\,\thinspace M_{\odot}}$ for 38 Class 0
and 39 Class I respectively. These lower dust disk masses are in better
agreement with the dust masses found in Orion embedded disks: it suggests that
dust masses extrapolated from long wavelengths may be significantly over-
estimated, due to the unknown emissivity and size distribution of dust grains
in the dense environments that are disks. In Ophiuchus, the mean Class I disk
dust mass is found to be significantly lower, $9\times 10^{-6}{\rm\,\thinspace
M_{\odot}}$ (Williams et al., 2019; Encalada et al., 2021): it is still about
5 times greater than the mean Class II disk mass in the same region, but the
dispersion in each class is so high that there is a large overlap between the
two distributions. Sheehan et al. (2022) used multi-wavelength analysis of the
SED towards Orion protostars. These estimates are also associated to large
error bars, but the multi-wavelength analysis they performed should allow to
better discriminate the disk from the envelope contribution and hence provide
more robust results. They find median dust mass $2\times
10^{-5}{\rm\,\thinspace M_{\odot}}$ for the Class 0 protostars, and $1.5\times
10^{-5}{\rm\,\thinspace M_{\odot}}$ for the Class I protostars, pointing
towards much smaller embedded disk masses and less clear mass differences
between the evolutionary stages than suggested by previous simplistic
estimates made from single wavelength analysis. Finally, when the temperature
of embedded disks can be investigated, observational studies suggest these
disks may be warmer than their older counterparts, with typical temperatures
around 20-30 K (van’t Hoff et al., 2020; Zamponi et al., 2021).

For the high mass stars, the rapid dynamical evolution and the larger
distances make the detection of the rotationally supported disk more
difficult, since they are in most cases surrounded by very massive molecular
envelopes. This makes the identification of the disks more controversial.
Resolving their velocity structures, masses, and sizes has been made possible
only recently thanks to ALMA. Relatively isolated disks are very rare, such as
Orion I, which has unique features possibly related to a violent multiple star
interaction event (Plambeck and Wright, 2016; Hirota et al., 2017; Ginsburg et
al., 2019; Wright et al., 2022). Other clear cases of disks around massive
stars, that appear to be a scaled up version of the ones around low-mass
stars, are GGD MM1 (Girart et al., 2018; Añez-López et al., 2020b),
G11.92-0.61 MM1 (Ilee et al., 2018), and G17.64+0.16 (Maud et al., 2019), with
very massive (2–5 ${\rm\,\thinspace M_{\odot}}$), dense and hot ($\gtrsim$400
K) disks. However, there are many cases reported in the literature where
Keplerian velocity patterns can be fitted towards disk-like structures, which
can extend from few hundreds to a thousand au (e.g. Sánchez-Monge et al.,
2013; Johnston et al., 2015; Cesaroni et al., 2017; Beuther et al., 2017;
Girart et al., 2017; Tanaka et al., 2020; Williams et al., 2022). In most
cases, a detailed measurement of the gas temperature and surface density is
needed to check the stability of such structures. The presence of disks around
massive stars has also been found through near/mid-IR interferometry (Kraus et
al., 2010; Frost et al., 2019).

These two works suggest that the observed massive disks are a scaled up
version of low-mass disks, but need to be confirmed with larger samples.

###  6.6 Fragmentation into multiple systems

Measuring the multiplicity of Class 0 and I solar-type protostellar systems
has been challenging and the focus of many observational works.
Submillimetre/millimetre observations are the only reliable tool for
characterizing Class 0 and I multiplicity, as near-infrared emission towards
embedded protostars suffer from significant extinction due to the envelopes,
and contamination by scattered light, while near-infrared observations are
usually robust at characterizing multiplicity in more evolved YSOs (Duchêne
and Kraus, 2013). The observed faction of binary and multiple systems in low
mass protostars are high, and similar to the multiplicity of main sequence
stars in the field. On average, a multiplicity percentage of 64% is found
among Class 0 protostars with linear separations in the range 50 - 5000 au
(e.g., Looney et al. 2000; Maury et al. 2010; Enoch et al. 2011; Tobin et al.
2013; Bouvier et al. 2021), whilst it is between 18% and 47% for Class I
sources with linear separations in the range 45 - 5000 au (Haisch et al.,
2004; Duchêne et al., 2004, 2007; Connelley et al., 2008a, b). For example, in
the Orion molecular cloud, observations of 300 Class 0, I, or Flat Spectrum
YSOs found a total of 85 multiple systems at separations $\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}10000$ au, 58 multiples at
separations $\;\buildrel<\over{\sim}\;$1000 au, and only 47 multiples with
maximum separations less than 500 au (Tobin et al., 2022). No statistically
significant difference in the separations with evolutionary stages during the
protostellar stages were found in the most complete studies so far, while
tentative evidence is found that field stars and non-embedded YSOs may feature
a lower number of wide (e.g., $a>1000$ au) multiple systems than their
embedded counterparts, in Orion (Tobin et al., 2022). Note that this result
may be significantly driven by incompleteness at sampling wide systems around
the most evolved YSOs and field stars: this limitation has now been partially
lifted thanks to increasing astrometric precision and led, for example, to the
discovery of a new large population of wide systems in Taurus within the range
1–60 kAU (Joncour et al., 2017). Finally, no statistically significant
difference was found regarding the multiplicity fractions observed between
regions of high and low YSO density, also in Orion.

Figure 5: Composite images of the G14.225-0.506 massive star forming region
(Busquet et al., 2016; Santos et al., 2016). Top left panel: R band optical
polarization vectors (red segments) overlaid on Herschel $250\mu$m image
overlapped (from Santos et al. 2016). Central panels: SOFIA/HAWC$+$ $200\mu$m
images (beam $14\mbox{${}^{\prime\prime}$}$) of the Northern (top) and
Southern (bottom) hubs, with black segments showing the magnetic field
direction (F. Santos, private communication). Right panels: Submillimeter
Array (SMA) images of the 1.2 mm emission toward the centre of the Northern
(top) and Southern (bottom) hubs (Busquet et al., 2016), with orange segments
showing the magnetic field direction (Añez et al. in prep.).

Massive stars have a predilection for forming in clustered environments with
other protostars, and their young embedded counterparts are thus often studied
in cluster star-formation modes (Cyganowski et al., 2017). High mass star
forming cores observed at high angular resolution appear to show different
level of fragmentation (Beuther and Schilke, 2004; Rodón et al., 2012; Palau
et al., 2013; Busquet et al., 2016; Sánchez-Monge et al., 2017; Liu et al.,
2019; Sadaghiani et al., 2020). The process of fragmentation is hierarchical
(Beuther and Schilke, 2004; Beuther et al., 2015) and appears to proceed down
to disk scales (Beuther et al., 2017; Busquet et al., 2019). The first
systematic study of the level of fragmentation down to scales of $\sim 1000$
au suggested that this could be set by the initial magnetic field/turbulence
balance (Palau et al., 2013). Subsequent studies found that the fragmentation
level was only mildly correlated with the density profile, and that it is
consistent with thermal or turbulent Jeans fragmentation (Palau et al., 2014,
2015, 2018; Beuther et al., 2018; Liu et al., 2019). However, in a more recent
work when the fragmentation level is compared with the magnetic field
properties derived from the dust polarization, there is a tentative
correlation of the fragmentation level with the mass-to-flux-ratio (see Fig.
5, and Busquet et al., 2016; Añez-López et al., 2020a; Palau et al., 2021).

##  7 Discussion: constraints on magnetized models from the observations

###  7.1 Magnetic fields

The observations of (sub-)millimeter polarized dust emission show that the
magnetic field is detected in all dense environments producing stellar
embryos. Assessing whether the observations can be trusted to infer
statistically robust constraints on the magnetic fields threading protostellar
cores, and compared to B-fields in models, requires the analysis of synthetic
observations from magnetized models. Le Gouellec et al. (2020) and Valdivia et
al. (2022) have post-processed outputs from non-ideal MHD models of
protostellar evolution with the Ramses code (Fromang et al., 2006) (see also
the work of Kuffmeier et al. 2020 in Figure LABEL:fig:synthobs-kuffi). Their
work analyses the resulting synthetic polarized dust emission maps, which are
compared to the true B-field properties in the models and to observations.
Valdivia et al. (2022) show that measurements of the line-of-sight averaged
magnetic field line orientation using the polarized dust emission are precise
enough to recover the mean field lines distribution with accuracy $<15\deg$
(typical of the error on polarization angles obtained with observations from
large mm polarimetric facilities such as ALMA) in about $95\%$ of the
independent lines of sight peering through protostellar envelopes at all
radii. When focusing on the smaller envelope radii $<500~{}\mathrm{au}$, where
the magnetic field lines are more likely perturbed from the initially smooth
configuration by infall and outflow and opacity effects kick in, $75\%$ of the
lines of sight still give robust results. Finally, Valdivia et al. (2022) and
Le Gouellec et al. (2020) find that the polarization fraction is correlated to
the degree of organization of the magnetic field along the line of sight,
confirming that the behavior observed at larger scales, in less dense gas,
holds true down to the scales of the envelopes of protostars. Le Gouellec et
al. (2020) find that the dust alignment efficiency does not significantly vary
with local gas density in their observations, and that the synthetic maps only
can reproduce the observed polarimetric data when significant irradiation from
the central protostar is included. Lam et al. (2021) have used numerical
models compared to ALMA observations of the polarized dust emission by Cox et
al. (2018): comparing polarization fractions they also conclude that dust
polarization at ALMA wavelengths is most likely due to magnetically aligned
grains in inner envelopes and dust scattering in disks.

Figure 6: Left: Density (color and contours) and velocity (arrows)
distributions in models from Hirano et al. (2020), implementing different
angles between the magnetic field initial direction and the initial rotation
axis of the core, at $t_{\rm ps}=0$, $1000$, and $5000$ yr after protostar
formation. The box size is $780$ au. Right: Three-dimensional structures for
the same models, highlighting two isodensity contours in purple and yellow,
and one isovelocity contour of radial velocity $v_{\rm rad}=2\,~{}{\rm
km~{}s^{-1}}$ (in red). Note the absence of an outflow in the case of
orthogonal configuration (third column): this case thus seems unrealistic
considering the prevalence of outlows observed around low-mass protostars.

Observationally, the correlation between outflow axis and the mean magnetic
field direction has received a lot of attention because it is believed that
the efficiency of magnetic braking to redistribute angular momentum and
prevent the growth of disks to large radii depends on the configuration of the
core’s magnetic field with respect to the core’s rotation axis when the
collapse starts (Joos et al., 2012; Hirano et al., 2020), as portrayed in Fig.
6. Using an SMA survey of twenty low-mass protostars, Galametz et al. (2018,
2020) has found that the protostellar envelopes tend to have a higher angular
momentum associated to rotation on 5000 au scales if the mean envelope
magnetic field measured at similar scales is misaligned with the rotational
axis of the core, assumed to coincide with the outflow axis. On the other
hand, observations analyzed in Yen et al. (2021b) have shown no correlation of
the dust continuum protostellar disk radii with the misalignment between the
magnetic fields and outflow axes in Orion A cores. Protostellar gas kinematics
from the SMA MASSES (sample of 32 protostellar envelopes in Perseus) survey
brings further support to the scenario proposed by Galametz et al. (2020),
with a significant correlation between the rotational velocity gradients (at
1000 au, normalized by the infalling velocity gradients) and the misalignment
angles between the magnetic fields and outflows (Gupta et al., 2022). At the
very small envelope radii, the mean B-field direction are 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, which may suggest the existence of a non-negligible toroidal
field morphology (caused by the core/disk rotation) in addition to the
poloidal one caused by the envelope-disk accretion.

These results may be confronted to models where the magnetic field is
efficient at reducing the amount of angular momentum transmitted to the inner
envelope scales ($<1000$ au), inhibiting the formation of large hydro-like
disks. In some cases, numerical simulations have shown tentative evidence of
better alignment of core-scale B fields and outflows in the more magnetized
models (Lee et al., 2017). However, these observational results 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, at a few
hundreds of au scales, as discussed in recent theoretical studies following
the disk evolution and angular momentum evolution in star-forming cores
(Verliat et al., 2020; Xu and Kunz, 2021; Lee et al., 2021b). Future
observations will undoubtedly bring more constraints on the dynamical role of
B-fields to regulate the protostellar collapse and set the pristine disk
properties, with the upcoming large surveys thanks to the development of
polarimetric capabilities and tools to model, e.g., polarized dust emission.
Observations, with e.g. the large radio interferometers and next generation of
mid-infrared facilities, of the small scale structures of these young disks,
still largely unresolved spatially, will also shed light on this question.

###  7.2 Coupling the magnetic field to protostellar material

Observations of the low dust emissivities and high polarization fractions at
mm wavelengths in embedded protostars suggest that partial grain growth, maybe
up to grain sizes $>100$ $\mu$m could have already occurred during the first
$0.1$ Myr of the star formation process. If these low emissivities and high
polarization fractions do indeed trace a population of large dust grains in
the inner layers of protostellar envelopes less than 0.1 Myrs old, the
timescales to grow grains up to sizes $>100$ $\mu$m in dense star-forming
material may need to be revised. For example, considering porous dust grains
may help in growing grains to sub-millimeter sizes in a few dynamical
timescales during the protostellar collapse (Ormel et al., 2009). Also, since
small grains are not only coupled to the gas but also to the magnetic field, a
magnetized collapse may also help segregating the grains with different
charges, and ultimately shortening the grain coagulation timescale (Hoang,
2021). Also, the presence of rather large grains at early stages in the
protostellar envelopes may increase the ability of protostellar disks to be
efficient forges to build up even bigger dust grains, that are re-injected in
envelopes thanks to protostellar winds and outflows (Wong et al., 2016; Bate
and Lorén-Aguilar, 2017; Lebreuilly et al., 2020; Ohashi et al., 2021;
Tsukamoto et al., 2021). Figure 7 shows an artist view of such circulation
processes potentially affecting dust grains during the protostellar phase.

Figure 7: Artist impression of the dust growth and recycling from disk to
envelope thanks to outflowing gas during the protostellar stage. Modified from
an original illustration in Tsukamoto et al. (2021)

Moreover, observational evidences of grain growth during the embedded stages
also bear strong consequences on the efficiency of magnetic fields to regulate
the transport of angular momentum. Indeed, whereas in the diffuse interstellar
medium, electric charges are carried by electrons and protons, in the dense
cores and particularly at high densities, the small dust grains are the main
charge carriers (Nishi et al., 1991; Nakano et al., 2002; Zhao et al., 2016).
The disappearance of small dust grains while forming bigger grains increases
magnetic resistivities (Nishi et al., 1991; Guillet et al., 2020a) and
consequently let more B flux leak outwards during the collapse of the gas from
the envelope onto the star-disk system (Zhao et al., 2021; Guillet et al.,
2020b). Hence the disk properties may also depend strongly on the dust
properties in the inner envelope.

The influence of the grain distribution on disk formation has been
investigated through numerical simulations by Zhao et al. (2016). By removing
the population of very small grains, the authors conclude that the ambipolar
diffusion is enhanced by 1-2 orders of magnitude, see Figure 8. As expected
the numerical simulations reveal that indeed the centrifugally supported
disks, which form, sensitively depend on the presence of the very small
grains. For instance, for a particular set of parameters (with a relatively
strong field aligned with the rotation axis), no disk would form when a MRN
grain distribution is assumed while disks of several tens of AU radius form
when a truncated MRN distribution is employed. Let us stress that the presence
of very small grains in dense cores is presently poorly constraint. From a
theoretical point of view, Guillet et al. (2020b) have shown that the latter
may be efficiently removed by coagulation due to the drift between different
dust species induced by ambipolar diffusion (see also Silsbee et al., 2020).
To what extent this population could not be replaced, for instance by
fragmentation of bigger grains, remains to be investigated. At disk scales,
where the large densities are favorable for rapid dust growth, up to
$a_{d}\sim 1~{}{\rm mm}$, the magnetic resistivity can become many orders of
magnitude bigger if small grains are rare, weakening ambipolar diffusion and
recoupling the magnetic field and the gas.

Resistivities are also strongly dependent on the ionisation rate. Whereas most
of the calculations assumed a value of a few $\zeta=10^{-17}$ s-1 inferred
from typical cosmic ray galactic abundances (Padovani et al., 2009a), more
extreme ionisation rates and their consequences on disk formation, have been
recently explored. Wurster et al. (2018) performed simulation with $\zeta$
varying from $10^{-13}$ to $10^{-24}$ s-1. They concluded that for
$\zeta>10^{-14}$ s-1, the results are essentially identical to ideal MHD while
for $\zeta<10^{-24}$ s-1, they are close to hydrodynamics. Kuffmeier et al.
(2020) specifically explored the disk radius and mass dependence on the
ionisation rate, varying it by a factor of 10 from $\zeta=10^{-17}$ to
$\zeta=10^{-16}$ s-1. Performing bi-dimensional simulations with a mass-to-
flux of 2.5, they found that with the highest $\zeta$, no disk forms while a
few tens of AU one forms with their smallest $\zeta$. They further propose
that cosmic ray abundances may control disk mass and size and explain the
differences between disks observed in various regions (e.g. Cazzoletti et al.,
2019).

Figure 8: Change of MHD resistivities when removing the very small grains,
from Zhao et al. (2016).

###  7.3 Gas kinematics: rotation, mass infall and outflow rates

Observations do not suggest that cloud rotation is transmitted to star-forming
cores (Tatematsu et al., 2016; Hsieh et al., 2021). At core scales, both
hydrodynamical and magnetized models of core formation and evolution have
shown that they do not produce rotating cores with the high angular momentum
values measured by e.g. Goodman et al. (1998) at 0.1pc scales. Recent
observations and models suggest that the angular momentum measured at core
scales (e.g. $\sim 5000$ au) could be due to non-axisymmetric motions
associated to, e.g., turbulent and gravitational processes (Burkert and
Bodenheimer, 2000; Offner et al., 2008; Dib et al., 2010; Kuznetsova et al.,
2019; Verliat et al., 2020). Hence, large scales velocity gradients in
envelopes should not be interpreted automatically as organized rotation: for
characterizing the motions responsible for the angular momentum observed, and
as illustrated in Figure 9, resolved observations and their comparisons to
model predictions are key. Finally, the angular momentum of the gas measured
at envelope radii $<1000$ au would predict disk whose sizes are broadly
consistent with the observed ones, although observations reveal a tentative
decrease of $j_{\rm{spe}}$ in inner envelopes, which may trace a decrease of
angular momentum at small radii (Gaudel et al., 2020). Observations of complex
gas kinematics in protostellar envelopes, from disturbed core-scale velocity
gradients to localized streamers (see Figure 3) connecting the envelope to the
disk scales confirm the wealth of observational evidence that the monolithic
vision of collapse which prevailed in the past decades needs to be revised.

Figure 9:  Observations and models of the angular momentum at core scales and
disk scales, from Belloche (2013); Gaudel et al. (2020); Dib et al. (2010);
Verliat et al. (2020).

The rather high values of mass infall rates found in Class 0 protostars seem
at first sight inconsistent with observed protostellar luminosities. While the
bolometric luminosities of Class 0 protostars are typically a few
$\rm{L}_{\odot}$ (with large variations from object to object), such accretion
rates should produce a accretion luminosities a few tens of
$\rm{L}_{\odot}$Ṫhis is the well known luminosity problem (Dunham and
Vorobyov, 2012), which was proposed to be solved with a scenario of quiet
accretion including sporadic large accretion bursts. It remains unclear how
such rare events could be captured so efficiently by observations measuring
the gas accretion rates, as a recent ALMA survey of protostars in Perseus
suggest the frequency of the bursts decrease with protostellar evolution, from
a burst every 2400 yr in the Class 0 stage to 8000 yrs in the Class I stage
(Hsieh et al., 2019). However, this scenario finds support in observations of
molecular species: 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). It may also
be that our lack of understanding of the physical conditions at the very small
scales where most of the radiation is reprocessed to produce the observed
luminosity, prevents us from properly model the expected luminosities, or that
the gas mass infall rates measured at few hundreds of au in the envelopes are
not representative of the true accretion rates at much smaller radii, onto the
protostellar objects. Supporting this last hypothesis, it is interesting to
note that MHD models following the evolution of low-mass cores find typical
mass accretion rates $\sim 10^{-6}$ $\rm{M}_{\odot}$ per year, with large
variations (Hennebelle et al., 2020; Kuznetsova et al., 2020). The global mass
accretion rate from the disk to the central star is lower when the disk
structure is properly modeled, and the short-term variation (bursts) of the
mass accretion is also highly attenuated with increased resolution (Kuffmeier
et al., 2018; Lee et al., 2021b). It seems therefore that either the
protostellar accretion rates obtained from observations are over-estimated,
which could be due to sampling the inner envelope and not the disk-star
connection, or the models predict slower mass accretion than protostars really
do accrete. We stress that the existence of preferential directions along
field lines, and the development of highly non-axisymmetric collapse may be
responsible, in magnetized models, to regulate the mass accretion rate
directly from the inner envelope to the central star, without the need to
transit through the disk. In their MHD models of the evolution of massive
cores into protostars Mignon-Risse et al. (2021) finds average mass accretion
rates on the central object around $\sim 10^{-3}$ $\rm{M}_{\odot}$ per year,
which is also about an order of magnitude lower than estimates from the
observations towards similarly massive protostars.

###  7.4 The formation of disks

Recent observations benefiting from increased spatial resolution and
sensitivities have firmly established that most disks around embedded
protostars are compact and not as massive as early studies had suggested. The
current disk radii estimates from the literature are shown in Figure 4. Such
compact disks are difficult to produce in large fractions with purely
hydrodynamical analytical models of disk formation conserving angular
momentum, but could be a natural outcome of magnetized models of disk
formation and models of anisotropic collapse (Hennebelle et al., 2020;
Kuznetsova et al., 2022). Obtaining self-consistent disk populations from
numerical simulations is quite challenging because it requires to be able to
treat simultaneously spatial scales sufficiently small to resolve the
protostellar disks while in the same time advance sufficiently the clump
scales to allow the formation of a statistically significant number of disks.
For this reason, only two studies have been reporting disk population self-
consistently formed from numerical simulations. Bate (2018) presents Smooth
particle hydrodynamical simulations of a 500 $M_{\odot}$ clump including
radiative transfer, producing a disk population whose sizes ranges from about
10 to few 100 AU. The author compares the mass distribution inferred from
simulations with several observational surveys and concludes that the disk
formed in the simulations are up to 10 times more massive. Lebreuilly et al.
(2021) present three adaptive mesh refinement simulations of 1000 $M_{\odot}$
clump which treat radiative stellar feedback and both ideal and non-ideal MHD.
The spatial resolution is up to 1 AU. Figure 10 shows for the three runs, the
mass distribution and the radius distribution, respectively. For comparison,
the data of VLA Class 0 disk mass distribution (purple, Tychoniec et al.,
2020) have been reported, as well as the disk radius inferred by the Calypso
(Maury et al., 2019) and VANDAM projects (Segura-Cox et al., 2018; Tobin et
al., 2020). Altogether and as already suggested from individual studies that
compared HD, MHD, and non-ideal MHD numerical models for single objects, the
hydrodynamical disks appear to be bigger than the MHD ones and overall in less
good agreement with observations than when magnetic field is included,
particularly regarding the disk radius distribution. Whereas there is some
scatter in the distributions, these first population studies, properly taking
into account the initial conditions, appear to be promising tools for future
comparisons to observations.

Figure 10: Left panel: Distributions of disk sizes obtained for the three disk
population synthesis models presented in Lebreuilly et al. (2021) (note that
the sizes are normalized so that their peak in the y-axis is 1). The mass of
the miminum mass for the Solar nebula (MSMN) is indicated as a dashed line.
Right: Cumulative distribution function of disk radius for the three models
presented in Lebreuilly et al. (2021).

A local origin of the angular momentum building rotationally-supported disks
has been proposed by some models: for example, Verliat et al. (2020) have
shown that the observations of angular momentum in protostellar envelopes can
be satisfactorily reproduced if the disk formation results from anisotropies
in the local velocity field, and not a consequence of the transfer of angular
momentum from organized large scale core rotation. As shown in Figure 9, some
observational evidence may support that scenario, with a good agreement
between those models and the observations of the specific angular momentum
found in envelopes, or more recently for example the observations of the L1489
protostar suggesting only the inner part of the envelope, at radii
$<4000-6000$, may be directly involved in forming the central star (Sai et
al., 2022). This would imply that the material participating directly to
building the disk and the star contains much less angular momentum than the
values measured at core scales, lessening the angular momentum problem for
star formation.

Although it was not shown observationally in the most embedded disks, models
suggest it may be that the big dust drifts even during the early disk phases,
leading to different apparent disk sizes recovered at different wavelengths.
The uncertainties regarding dust properties in these specific conditions also
affect the dust masses and hence the disk masses that are estimated in these
objects. If observations suggesting young embedded disks are warm (20-30 K,
van’t Hoff et al. 2020; Zamponi et al. 2021) are confirmed, it could suggest
gas kinematics within the disc play a significant role in heating the disk, as
the protostellar radiation is expected to be highly extinct at such very high
densities. For example, Zamponi et al. (2021) compare observations with
synthetic observations of MHD protostellar disk models formed after the
collapse of a dense core, and suggest that heating due to gravitational
instabilities in the disk is able to generate dust temperatures in agreement
with observational constraints in the IRAS 16293 Class 0 disk.

Figure 11: Observation and best model to match the properties in the B335
protostar, from Maury et al. (2018) and new ALMA observations (in prep.). 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 in B335 is being pulled in the
dominant direction of the collapse, but yet is strong enough ($\mu\sim 6$) to
counteract partially 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.

The exact role of magnetic fields in setting the disks sizes and masses during
the embedded phases can only be quantitatively addressed thanks to the
comparison of observed properties to the outcomes of models. Maury et al.
(2018) have shown that the B field geometry, 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 in B335 is being pulled in the
dominant direction of the collapse, but yet is strong enough ($\mu\sim 6$) to
counteract partially the transfer of angular momentum inwards. In B335, they
show the magnetic field is very likely to regulate the formation processes of
the protostellar disk, constraining the size of the protostellar disk to $<20$
au (see Figure 11). The role of the magnetic field in setting the disk size in
B335 has also been discussed in subsequent studies (Yen et al., 2019; Bjerkeli
et al., 2019; Imai et al., 2019). However, new observations by Cabedo et al.
(2022) suggest that the gas at small envelope radii is strongly ionized in
B335, and points towards conditions typical of ideal MHD. It may very well be
that the star-disk building phase consists in a succession of non-ideal and
ideal MHD conditions in the “life of the protostar”. It would be possible to
grow a quite large disk if it is formed before the onset of high ionization
around the protostar (due to the protostar itself and the production of CR in
the B-field lines around the protostar), while if the disks are still compact
when the local ionization starts to increase at a fast pace, it may be more
difficult to form large disks in the second half of the protostellar phase.

Figure 12: Lebreuilly et al. (2021): A sample of disks formed in three massive
clump simulations, hydrodynamics, ideal MHD and non-ideal MHD (ambipolar
diffusion). Hydrodynamical disks tend to be bigger and ideal MHD ones smaller
than when ambipolar diffusion is included.

###  7.5 Influence of magnetic field on the formation of multiple systems

The fragmentation of low mass cores has been the subject of several studies
(see e.g. Matsumoto and Hanawa, 2003; Goodwin et al., 2004). Several modes of
fragmentation have been identified. Generally speaking, it is induced by the
density fluctuations that are generated during the collapse on one hand by
gravo-turbulence processes within the envelope and on the other hand through
the formation of massive centrifugally supported unstable disks. The outcome
of fragmentation therefore entirely depends on the initial conditions,
particularly rotation and turbulence initial values, but also on the amplitude
of the initial density perturbations. It has been generally found however that
under typical conditions and in the absence of magnetic field, a solar mass
dense core tends to produce few fragments (say 2-10).

Several studies have been dedicated to the influence of magnetic field in this
process (e.g. Machida et al., 2008; Hennebelle and Teyssier, 2008; Commerçon
et al., 2010; Wurster et al., 2017; Wurster and Bate, 2019; Hennebelle et al.,
2020), covering a broad range of parameters in terms of magnetic field
intensity, rotation and initial turbulence. From these studies, clear trends
can be inferred. First, the higher the rotational and/or turbulent energy, the
more prone to fragmentation is the core. Numerical models of turbulent dense
cores suggest that the directions of the specific angular momentum axis tends
to vary as a function of envelope scales (Joos et al., 2013; Matsumoto et al.,
2017), and that misaligned systems can form in turbulent conditions. Some
numerical models also suggest that turbulent velocity fluctuations around
protostellar cores may be responsible for setting the rotational motions of
these cores (Misugi et al., 2019). However, magnetic field has a drastic
influence and in the case of low mass cores, mass-to-flux ratios as high as
$\mu>5-10$ may entirely suppress the formation of fragments. This is because
in low mass cores, turbulence is at best trans-sonic and the density
fluctuations induced by turbulence remain limited. Therefore, the dominant
fragmentation mode is through the formation of massive disks. However, in the
presence of magnetic field the disk sizes and masses are much smaller, and
therefore they tend to be stable. Another interesting trend is that the cores
in which the magnetic field is initially perpendicular to the rotation axis
are less prone to fragmentation than when it is aligned. One important fact to
be taken into account however is that the amplitude of the density
perturbations is also an important parameter, and the mentioned trends have
been inferred for modest density fluctuations (typically below 10$\%$). When
the density fluctuations have a large amplitude, say of about 50$\%$, they are
sufficiently unstable to collapse individually even in the absence of
rotation. In this case, magnetic field is unable to impede fragmentation
(Hennebelle and Teyssier, 2008; Wurster et al., 2017). This would imply that
the fragmentation is driven by the larger scales and the assembling process of
the dense cores.

Figure 13: Models of the gravitational collapse of a $300{\rm\,\thinspace
M_{\odot}}$ clump, with gas temperature T = 20 K, and Mach number 6.4, from
Fontani et al. (2018). Column density maps of two models are shown in the
upper row, in the presence of a strong (left) magnetic field and a weak one
(right). The lower row shows ALMA synthetic observations of the models using
the 280 GHz dust continuum emission. Fewer fragments are seen in the more
magnetized case.

The influence of the magnetic field on the fragmentation of massive clumps has
also been investigated. Because of the large amplitude of the density
fluctuations, the impact of magnetic field on the fragmentation is relatively
less important than that found in the case of low mass cores. For instance,
Hennebelle et al. (2011) and Myers et al. (2013) found that in the presence of
a significant magnetic field (e.g a few mG for mass-to-flux ratios around 2),
the number of self-gravitating fragments is reduced by a factor of about 2.
Fontani et al. (2016) and Fontani et al. (2018) have carried out observations
and simulations of massive clumps. They have studied and compared the
fragmentation properties of the clumps from observations and simulations using
synthetic images that mimic the observations as shown in Fig.13. They
concluded that the fragmentation properties within the observations are better
reproduced by the significantly magnetized models (typically $\mu=2$) than by
the weakly magnetized ones.

##  8 Summary

Understanding how protostars and planet-forming disks form are cornerstone
questions for us to understand both the stellar populations setting the
evolution of our Universe and the conditions responsible for producing the
planetary systems observed around most stars. The recent progresses made
through observations and simulations suggest that magnetic field is modifying
in depth the outcome of the collapse and the formation of stars, by adding an
extra support to the gas, by reducing the angular momentum available to build
the disks, by launching outflows and even possibly favoring the dust grain
coagulation into pebbles. Observations suggest that the so-called magnetic
braking catastrophe is an issue that is solved, alleviated by implementing
realistic physical conditions (turbulence, anisotropies of the density field,
non-homogeneous initial B field due to conditions of core assembly and local
environment, gas ionization and dust properties) in MHD models.

If the magnetized scenario we propose is common, recent work suggest that the
angular momentum problem for star formation may be actually ”solved” not by
the formation of large disks but by the combination of (i) lack of organized
rotation motions at large envelope radii, (ii) the inefficient angular
momentum transport due to magnetic braking in the inner envelope (and angular
momentum removed through rotating outflows generated by the presence of the
magnetic field) and (iii) a local origin of the angular momentum incorporated
in the star-disk system.

Major questions however remain to be solved for the next generations of
astrophysicists. First, our understanding of the origin of the angular
momentum that eventually build the disks is very incomplete. Whereas some of
the gas momentum could obviously be inherited from larger scales, inertial
processes naturally produce angular momentum in a non-axisymmetric system,
opening the possibility to disks weakly related to large scale angular
momentum. This possible scenario questions the effective role of magnetic
braking at envelope scales to set the disk sizes, as local processes may be
more important in shaping the outcome of star and disk formation. That means
to better understand the transport of the magnetic field by processes like
reconnection diffusion and ambipolar diffusion and their importance on
regulating angular momentum transport and disk formation in the different
stages. Characterizing the properties of the compact protostellar disks, with
very high angular resolution observations, will also be a key to set
constraints and refine our models. Second, this scenario deeply relies on the
efficiency of magnetic fields to couple to the gas reservoirs. This efficiency
is driven by a complex set of physical conditions, which remain up to this day
largely unconstrained. Our current knowledge of magnetic resistivities remains
thus hampered by very large uncertainties. It is therefore key, in the future,
that observations characterize both the gas and dust properties in embedded
protostars, and are compared to models so we know whether magnetic braking is
much more efficient in locations, and epochs, that are critical for setting
the disk and stellar properties. Answering these questions will be a
fascinating challenge that will require us to produce robust measurements of
the magnetic intensity and topology in large samples of objects, but also find
clever ways to access the gas ionisation fraction and dust grain properties in
young embedded protostars.

## Acknowledgments

AM acknowledges support from the European Research Council (ERC) under
Starting grant number 679937 – ‘MagneticYSOs’. This work was also partially
supported by the program Unidad de Excelencia María de Maeztu
CEX2020-001058-M. JMG also acknowledges support by the grant
PID2020-117710GB-I00 (MCI-AEI-FEDER,UE). PH acknowledges support from the
European Research Council synergy grant ECOGAL (Grant : 855130)

## References