# The First Estimation of the Ambipolar Diffusivity Coefficient from Multi-
Scale Observations of the Class 0/I Protostar, HOPS-370

Travis J. Thieme Institute of Astronomy, National Tsing Hua University, No.
101, Section 2, Kuang-Fu Rd., Hsinchu 30013, Taiwan  Center for Informatics
and Computation in Astronomy, National Tsing Hua University, No. 101, Section
2, Kuang-Fu Rd., Hsinchu 30013, Taiwan  Department of Physics, National Tsing
Hua University, No. 101, Section 2, Kuang-Fu Rd., Hsinchu 30013, Taiwan
Academia Sinica Institute of Astronomy and Astrophysics, 11F of Astronomy-
Mathematics Building, AS/NTU, No.1, Sec. 4, Roosevelt Rd, Taipei 10617, Taiwan
Shih-Ping Lai Institute of Astronomy, National Tsing Hua University, No. 101,
Section 2, Kuang-Fu Rd., Hsinchu 30013, Taiwan  Center for Informatics and
Computation in Astronomy, National Tsing Hua University, No. 101, Section 2,
Kuang-Fu Rd., Hsinchu 30013, Taiwan  Department of Physics, National Tsing Hua
University, No. 101, Section 2, Kuang-Fu Rd., Hsinchu 30013, Taiwan  Academia
Sinica Institute of Astronomy and Astrophysics, 11F of Astronomy-Mathematics
Building, AS/NTU, No.1, Sec. 4, Roosevelt Rd, Taipei 10617, Taiwan  Yueh-Ning
Lee Department of Earth Sciences, National Taiwan Normal University, Taipei
116059, Taiwan  Center of Astronomy and Gravitation, National Taiwan Normal
University, Taipei 116059, Taiwan  Physics Division, National Center for
Theoretical Sciences, Taipei 106319, Taiwan  Sheng-Jun Lin Academia Sinica
Institute of Astronomy and Astrophysics, 11F of Astronomy-Mathematics
Building, AS/NTU, No.1, Sec. 4, Roosevelt Rd, Taipei 10617, Taiwan  Hsi-Wei
Yen Academia Sinica Institute of Astronomy and Astrophysics, 11F of Astronomy-
Mathematics Building, AS/NTU, No.1, Sec. 4, Roosevelt Rd, Taipei 10617, Taiwan

###### Abstract

Protostars are born in magnetized environments. As a consequence, the
formation of protostellar disks can be suppressed by the magnetic field
efficiently removing angular momentum of the infalling material. Non-ideal MHD
effects are proposed to as one way to allow protostellar disks to form. Thus,
it is important to understand their contributions in observations of
protostellar systems. We derive an analytical equation to estimate the
ambipolar diffusivity coefficient at the edge of the protostellar disk in the
Class 0/I protostar, HOPS-370, for the first time, under the assumption that
the disk radius is set by ambipolar diffusion. Using previous results of the
protostellar mass, disk mass, disk radius, density and temperature profiles
and magnetic field strength, we estimate the ambipolar diffusivity coefficient
to be $1.7^{+1.5}_{-1.4}\times 10^{19}\,\mathrm{cm^{2}\,s^{-1}}$. We quantify
the contribution of ambipolar diffusion by estimating its dimensionless
Elsässer number to be $\sim 1.7^{+1.0}_{-1.0}$, indicating its dynamical
importance in this region. We compare to chemical calculations of the
ambipolar diffusivity coefficient using the Non-Ideal magnetohydrodynamics
Coefficients and Ionisation Library (NICIL), which is consistent with our
results. In addition, we compare our derived ambipolar diffusivity coefficient
to the diffusivity coefficients for Ohmic dissipation and the Hall effect, and
find ambipolar diffusion is dominant in our density regime. These results
demonstrate a new methodology to understand non-ideal MHD effects in
observations of protostellar disks. More detailed modeling of the magnetic
field, envelope and microphysics, along with a larger sample of protostellar
systems is needed to further understand the contributions of non-ideal MHD.

Circumstellar disks (235) — Observational astronomy (1145) — Protostars (1302)
— Radio astronomy (1338) — Star formation (1569) — Young stellar objects
(1834)

††software: Astropy (Astropy Collaboration et al., 2013, 2018,
http://astropy.org), asymmetric uncertainty (Gobat, 2022), Matplotlib (Hunter,
2007, http://matplotlib.org/), proplot (Davis, 2021), Numpy (van der Walt et
al., 2011, http://numpy.org/)

##  1 Introduction

It has long been known that magnetic fields (B-fields) play a critical role in
regulating the formation of protostellar disks around low-mass protostars
(e.g., Mestel & Spitzer, 1956). Molecular cloud cores are observed to be
strongly magnetized, with normalized mass-to-flux ratios of $\mu\sim 2-10$
(e.g., Crutcher, 1999; Troland & Crutcher, 2008; Crutcher, 2012). Early ideal
magnetohydrodynamic (MHD) simulations show that rotationally supported disks
(RSDs) could not form due to magnetic braking efficiently transferring angular
momentum away from the collapsing central region in magnetized ($\mu\leq 10$)
dense cores (e.g., Allen et al., 2003; Matsumoto & Tomisaka, 2004; Banerjee &
Pudritz, 2006; Price & Bate, 2007; Hennebelle & Fromang, 2008; Mellon & Li,
2008; Joos et al., 2012). However, observational studies revealed the presence
of rotationally-supported Keplerian disks around several young, highly-
embedded protostars (e.g., Tobin et al., 2012; Murillo et al., 2013; Lee et
al., 2014; Yen et al., 2017; Ohashi et al., 2023). This contradiction between
observations and simulations was coined the so-called “Magnetic Braking
Catastrophe” and raised the fundamental question of how could these
protostellar disks form in such magnetized environments?

Non-ideal MHD effects, namely ambipolar diffusion (AD), Ohmic dissipation (OD)
and the Hall effect (HE), have been suggested as one possible route to
overcome magnetic braking and form a rotationally-supported protostellar disk
(e.g., Inutsuka et al., 2010; Li et al., 2011; Braiding & Wardle, 2012; Tomida
et al., 2015; Wurster et al., 2016, 2019; Wurster & Lewis, 2020; Wurster et
al., 2021). These non-ideal MHD terms describe the various regimes of coupling
of the ions, electrons and charged grains to the magnetic field, as well as
their interactions with the neutral particles (e.g., Wardle & Ng, 1999; Nakano
et al., 2002, see the recent reviews by Wurster & Li 2018; Zhao et al. 2020b;
Tsukamoto et al. 2023). In terms of relative importance, Ohmic dissipation is
efficient at high densities, such as the midplane of a protostellar disk,
while the Hall effect and ambipolar diffusion are more efficient at
intermediate and low densities, respectively, such as the upper disk layers
and in the protostellar envelope (e.g., Marchand et al., 2016; Wurster et al.,
2018a; Wurster, 2021). However, the Hall effect seems to be transient does not
last for long after the formation of a protostellar disk (Zhao et al., 2020b;
Lee et al., 2021b). While simulations clearly show the importance of non-ideal
MHD effects in the formation and evolution of protostellar disks, they have
yet to be quantified observationally. Yen et al. (2018) attempted to observe
the velocity drift between ions and neutral particles (ambipolar diffusion) in
the infalling envelope of a young Class 0 protostar, B335. However, no
velocity drift was detected and thus, it is important to look into other
possibilities on how non-ideal MHD effects can be quantified observationally.

In this paper, we aim to understand the role of ambipolar diffusion in
protostellar disk formation by using a methodology first developed by
Hennebelle et al. (2016), and later revisited by Lee et al. (2021b, 2024).
This methodology leads to an analytical equation describing the expected
protostellar properties, in particular the protostellar disk radius, due to
ambipolar diffusion (Hennebelle et al., 2016). The disk radius estimated with
this analytical equation ($R_{\mathrm{AD}}$) was found to be in good agreement
with the disk radius estimated from MHD simulations ($R_{\mathrm{sim}}$), with
$R_{\mathrm{sim}}/R_{\mathrm{AD}}\sim 1$ (Hennebelle et al., 2016, 2020;
Commerçon et al., 2022). Thus, by backwards engineering the equation, we can
estimate the ambipolar diffusivity coefficient, $\eta_{\mathrm{AD}}$, from
observable quantities under certain assumptions. Using multi-scale
observations of the young protostar, HOPS-370, we present a methodology to
estimate the ambipolar diffusivity coefficient for the first time, in order to
understand the role of ambipolar diffusion in the formation and evolution of
protostellar disks.

HOPS-370 is a Class 0/I protostar in the Orion A molecular cloud
(D$=392.8\,\mathrm{pc}$; Tobin et al. 2020a). Observations from the Herschel
Orion Protostar Survey (HOPS) constrain the bolometric luminosity
($L_{\mathrm{bol}}$) and temperature ($T_{\mathrm{bol}}$) to be
314$\,L_{\odot}$ and 71.5 K, respectively (Furlan et al., 2016). The
protostellar mass and disk properties were extensively studied by Tobin et al.
(2020b) as part of the VLA/ALMA Nascent Disk and Multiplicity (VANDAM) Survey
of Orion Protostars. By using MCMC radiative transfer modeling to fit the dust
continuum and several molecular lines, they found an average disk radius of
$94\,\mathrm{au}$, an average protostellar mass of $2.5\,M_{\odot}$, and a
disk mass of $0.035\,M_{\odot}$. More recently, Kao & Yen et al. (in prep.)
have derived the core-scale plane-of-sky magnetic field strength to be
$B_{\mathrm{pos}}=0.51\,\mathrm{mG}$. The combination of these derived
properties make HOPS-370 an ideal candidate for an initial study on the role
ambipolar diffusion plays in this source using this new methodology.

This paper is organized as follows. In Section 2, we describe our methodology
and assumptions to estimate $\eta_{\mathrm{AD}}$ at the edge of the HOPS-370
protostellar disk. Our resulting value of $\eta_{\mathrm{AD}}$ and a
comparison with a more theoretical non-ideal MHD estimate is given in Section
3. Several implications and uncertainties are discussed in Section 4. Section
5 summarizes our main results and discussions.

##  2 Methods

###  2.1 The Relation Between Protostellar Disk Properties and the Ambipolar
Diffusivity Coefficient

Here, we present an analytical equation that relates properties of the
protostellar disk at the disk-envelope interface to the ambipolar diffusion
coefficient. Hennebelle et al. (2016) were the first to derive such an
equation, however they make a number of simplifications to remove terms
related to the density and temperature, which differs from the modeling of
HOPS-370. A derivation is provided in Appendix A, while a summary and overview
is presented here. The main assumptions in this derivation are that

  1. 1.

ambipolar diffusion is the main diffusion process,

  2. 2.

the angular momentum is counteracted by magnetic braking resulting in the
advection and braking timescales to be of the same order,

  3. 3.

the toroidal field generated by differential rotation is offset by the
ambipolar diffusion in the vertical direction resulting in the Faraday
induction and vertical diffusion timescales to be of the same order,

  4. 4.

infalling and rotational velocities of the gas near the disk edge both scale
with the Keplerian velocity, and

  5. 5.

the gas is in vertical hydrostatic equilibrium.

These assumptions are likely valid in HOPS-370, as discussed in Section 4.1.
Under these assumptions, we derive a relationship between the ambipolar
diffusivity coefficient and observable quantities of

$\eta_{\mathrm{AD}}\simeq\frac{\delta_{r}\delta_{\phi}^{2}G^{1/2}C_{s}^{2}R_{d}^{1/2}(M_{\star}+M_{d})^{1/2}\rho}{B_{\phi}^{2}},$
(1)

where $\delta_{r}$ and $\delta_{\phi}$ are scaling factors for the infall and
rotational velocities, $G$ is the gravitational constant, $C_{s}$ is the
isothermal sound speed, $R_{d}$ is the disk radius, $M_{\star}+M_{d}$ is the
mass of the star+disk system, $\rho$ is the density at the disk-envelope
interface and $B_{\phi}$ is the toroidal (azimuthal) component of the magnetic
field strength at the edge of the disk. As shown in Appendix B, the global
magnetic field inclination with respect to the disk rotation axis has little
effect on the predicted ambipolar diffusivity coefficient. Thus, this
prescription should be considered generally valid regardless of the global
magnetic field orientation.

To simplify the use of our equation, we select several arbitrary normalization
constants to give

$\begin{split}\eta_{\mathrm{AD}}&\simeq 2.5\times
10^{17}\,\mathrm{cm^{2}\,s^{-1}}\left(\delta_{r}\delta_{\phi}^{2}\right)\\\
&\times\left(\frac{C_{s}}{200\,\mathrm{m\,s^{-1}}}\right)^{2}\left(\frac{R_{d}}{100\,\mathrm{au}}\right)^{1/2}\left(\frac{M_{\star}+M_{d}}{0.1\,M_{\odot}}\right)^{1/2}\\\
&\times\left(\frac{\rho_{d}}{1.8\times
10^{-15}\,\mathrm{g\,cm^{-3}}}\right)\left(\frac{B_{\phi}}{20\,\mathrm{mG}}\right)^{-2}.\end{split}$
(2)

In addition, a common normalization used in numerical simulations is to
multiply by $4\pi/c^{2}$, which was used by Hennebelle et al. (2016) in their
derivation to give $\eta_{\mathrm{AD}}$ in units of seconds. This
normalization produces a relation of

$\begin{split}\eta_{\mathrm{AD}}&\simeq
0.0035\,\mathrm{s}\left(\delta_{r}\delta_{\phi}^{2}\right)\\\
&\times\left(\frac{C_{s}}{200\,\mathrm{m\,s^{-1}}}\right)^{2}\left(\frac{R_{d}}{100\,\mathrm{au}}\right)^{1/2}\left(\frac{M_{\star}+M_{d}}{0.1\,M_{\odot}}\right)^{1/2}\\\
&\times\left(\frac{\rho_{d}}{1.8\times
10^{-15}\,\mathrm{g\,cm^{-3}}}\right)\left(\frac{B_{\phi}}{20\,\mathrm{mG}}\right)^{-2},\end{split}$
(3)

which will also be used in later comparisons.

As shown by Hennebelle et al. (2016), Hennebelle et al. (2020) and Commerçon
et al. (2022), the ratio of the disk radius measured in their numerical
simulations ($R_{\mathrm{sim}}$) to the theoretical disk radius predicted by
their ambipolar diffusivity equation ($R_{\mathrm{AD}}$) was
$R_{\mathrm{sim}}/R_{\mathrm{AD}}\sim 1$ (within a factor of $\simeq 2-3$) and
did not vary considerably over the evolution of the protostellar disks in
their simulations. Since our main assumptions are essentially the same, this
should still hold true even for our new relation. This will be explored in
more detail in a future paper. In the next sections, we describe each of the
variables used for our estimate of the ambipolar diffusivity coefficient at
the edge of the HOPS-370 protostellar disk for the first time. This estimation
is only possible due to the extensive modeling of HOPS-370 and its surrounding
environment from several different observational studies.

###  2.2 Previously Estimated Protostar+Disk Properties

Tobin et al. (2020b) derived several important properties of the protostar and
disk in HOPS-370. In this section, we describe their extensive molecular line
modeling in the context of the relevant values needed for our ambipolar
diffusivity coefficient estimation.

####  2.2.1 Protostellar Mass, Disk Mass and Disk Radius

The protostellar masses and disk radii are derived from 12 independent
molecular line fits (with a fixed temperature power-law index) using MCMC
radiative transfer fitting. They found the best fitting protostellar mass to
be between ranged between 1.8$\,M_{\odot}$ and 3.6$\,M_{\odot}$, with an
average protostellar mass of 2.5$\pm$0.2$\,M_{\odot}$. This protostellar mass
is the dynamical mass obtained from the Keplerian profile in the line fits.
For the disk radius, the best fits ranged between 70 au and 121 au, with an
average radius of 94.4$\pm$12.6 au. The uncertainties of these average values
were determined by using the median-absolute deviation (MAD) of their 12
molecular line fits and scaling them to correspond to one standard deviation
of the normal distribution. We adopt $R_{d}=94.4\pm 12.6\,$au and
$M_{\star}=2.5\pm 0.2\,M_{\odot}$ as the protostellar disk radius and
protostellar mass, respectively. It is important to note, the $R_{d}$ used for
comparison to $R_{\mathrm{AD}}$ in the numerical simulations by Hennebelle et
al. (2016) is defined by several conditions using an azimuthally-averaged
simulation snapshot: (1) the disk is Keplerian meaning the azimuthal velocity
is much greater than the radial velocity, (2) the disk is near hydrostatic
equilibrium meaning the azimuthal velocity is much greater than the vertical
velocity, (3) the disk is rotationally-supported meaning the rotational energy
is larger than the support from thermal pressure by some factor, (4) the disk
should be near the equitorial plane, and (5) a density threshold of
$n>10^{9}\,\mathrm{cm^{-3}}$ (Joos et al., 2012). We have assumed the best-fit
gas disk radius is equal to this radius. This is further explored in Section
3.3.3. The radius of the dust disk also modeled by Tobin et al. (2020b),
however the dust is potentially more prone to radial drift and/or optical
depth effects (e.g., Facchini et al., 2017), thus potentially underestimating
the actual extent of centrifugal support.

Additionally, several methods were used by Tobin et al. (2020b) to derive the
disk mass in HOPS-370. First, they used the continuum emission as 1.3 mm, 0.87
mm and 9 mm to derive a value for the disk mass under the assumptions of
isothermal and optically thin dust emission. The disk mass at each wavelength
was found to be 0.048$\,M_{\odot}$ at 0.87 mm, 0.084$\,M_{\odot}$ at 1.3 mm,
and 0.098$\,M_{\odot}$ at 9 mm. They also derive a disk mass from their MCMC
radiative transfer fitting of the 0.87 mm dust continuum emission. This method
resulted in a disk mass of $0.035^{+0.005}_{-0.003}\,M_{\odot}$, which is
slightly lower than the earlier estimations using the optically thin
assumption. The lower value is likely due to the maximum dust grain size fit
of the 0.87 mm emission being 432$\,\mu$m, meaning that the dust in the model
will radiate more efficiently than under the assumptions made for the
optically thin calculation. Thus, to be consistent with the dust grain
properties later used in our analysis (Section 3.3), we take the disk mass to
be $M_{d}=0.035^{+0.005}_{-0.003}\,M_{\odot}$ for our estimation. It is
important to mention that the uncertainty of the measured disk mass reported
by Tobin et al. (2020b) are the 1$\sigma$ statistical uncertainties from their
MCMC radiative transfer fitting. Thus, these uncertainties likely do not
reflect the entire uncertainty of the measured disk mass. Tobin et al. (2020b)
also fit for the disk mass in their 12 molecular line fits. However these
derived disk masses are highly sensitive to the chosen molecular abundances in
the fit, and may not be as reliable. This further motivates our choice to use
the best-fit disk mass estimated from the dust emission fitting.

####  2.2.2 The Temperature Distribution of the Disk

The gas temperature distribution of the HOPS-370 protostellar disk is modeled
using a parameterized equation given by

$T_{\mathrm{d}}(r)=T_{0}\left(\frac{r}{1\,\mathrm{au}}\right)^{-q},$ (4)

where $T_{0}$ is the gas temperature at 1$\,\mathrm{au}$ and $q$ is a power-
law index, which is fixed to be 0.35 in the 12 molecular line fits by Tobin et
al. (2020b). The best-fit average value of $T_{0}$ was found to be $980.0\pm
0.6\,\mathrm{K}$, where the errors are also found using the median absolute
deviation scaled to one standard deviation of the normal distribution. Using
the protostellar disk radius of $R_{d}=94.4\pm 12.6\,$au, we find the
temperature at the edge of the disk to be $T_{d}=199.0\pm
9.3\,\mathrm{K}$.111Uncertainties were propagated using the publicly hosted
python package: asymmetric_uncertainty (Gobat 2022;
https://github.com/cgobat/asymmetric_uncertainty). This package uses an
empirical/analytical function to model the error distributions.

With this gas temperature, the isothermal sound speed at the disk edge can be
estimated by

$C_{s}=\left(\frac{k_{\mathrm{B}}T_{d}}{\mu_{\mathrm{m}}m_{\mathrm{H}}}\right)^{0.5},$
(5)

where $k_{\mathrm{B}}$ is the Boltzmann constant, $\mu_{\mathrm{m}}=2.37$ is
the mean molecular weight for a molecular gas with solar metallicity, and
$m_{\mathrm{H}}$ is the mass of a hydrogen atom. The isothermal sound speed is
estimated to be $C_{s}=833.0\pm 19.5\,\mathrm{m\,s^{-1}}$, which is higher
than the typically assumed value of $200\,\mathrm{m\,s^{-1}}$ (e.g., Lee et
al., 2021b, 2024).

####  2.2.3 The Density at the Disk-Envelope Interface

The density at the disk-envelope interface can be estimated via two different
approaches. The first is by using the best fit values of the disk density
profile, while the second is by using the best fit values of the envelope
density profile, both modeled by Tobin et al. (2020b). We initially choose the
former approach, since the focus of the study by Tobin et al. (2020b) was on
the disk, and the observations taken likely resolve out most of the envelope
emission. However, as a comparison, we do explore the latter in Appendix C.
The disk density, which is related to the disk scale height and disk surface
density, was modeled using the molecular line emission by Tobin et al.
(2020b). The disk scale height ($h_{d}$) is given by

$h_{\mathrm{d}}(r)=\left(\frac{k_{\mathrm{B}}r^{3}T_{d}(r)}{GM_{\star}\mu_{\mathrm{m}}m_{\mathrm{H}}}\right)^{0.5},$
(6)

where $M_{\star}$ is the protostellar mass. The disk surface density
($\Sigma_{\mathrm{disk}}$) is given by

$\Sigma_{d}(r)=\Sigma_{0}\left(\frac{r}{r_{c}}\right)^{-\gamma}\exp\left[-\left(\frac{r}{r_{c}}\right)^{(2-\gamma)}\right],$
(7)

where $r_{c}$ is the critical radius of the disk ($r_{c}=R_{d}$ was assumed in
the molecular line fitting) and $\gamma$ is the surface density power-law
index. The normalization constant ($\Sigma_{0}$) is described by

$\Sigma_{0}=\frac{(2-\gamma)M_{d}}{2\pi r_{c}^{2}},$ (8)

where $M_{d}$ is the disk mass. The radiative transfer modeling of the 12
molecular line fits give an average value of the surface density power-law
index to be $\gamma=0.9\pm 0.2$. Finally, the disk volume density ($\rho_{d}$)
is expressed as

$\rho_{d}(r)=\frac{\Sigma_{d}(r)}{\sqrt{2\pi}\
h_{d}(r)}\exp\left(-\frac{1}{2}\left[\frac{z}{h_{d}(r)}\right]^{2}\right),$
(9)

where $z$ is the height above the disk midplane and the other parameters are
as described before. For simplicity, we approximate the density at the
midplane ($z=0$), which allows the exponential to go to 1 as the inner terms
go to 0. We are left with a simplified equation of

$\rho_{d}(r)=\frac{\Sigma_{d}(r)}{\sqrt{2\pi}\ h_{d}(r)},$ (10)

where we can then plug in our known values to calculate the approximate
density at the disk edge. By plugging in $r=R_{d}=94.4\pm 12.6\,\mathrm{au}$
and the other parameters previously mentioned, we find a disk scale height of
$h_{d}=16.2\pm 3.3\,\mathrm{au}$, a disk surface density of $\Sigma_{d}=2.2\pm
0.9\,\mathrm{g\,cm^{-2}}$, and a disk volume density of $\rho_{d}=3.7\pm
1.7\times 10^{-15}\,\mathrm{g\,cm^{-3}}$ at the edge of the disk.

###  2.3 Estimating the Magnetic Field Strength at the Edge of the Disk

Yen et al. (2021a) originally estimated the core-scale plane-of-sky magnetic
field strength using 850$\,\mu$m dust polarization legacy observations from
the Submillimetre Common-User Bolometer Array (SCUBA) Polarimiter (SCUPOL) on
the James Clerk Maxwell Telescope (JCMT). A magnetic field strength of
$B_{\mathrm{pos}}=0.54\pm 0.25\,\mathrm{mG}$ is derived for HOPS-370 using the
Davis-Chandrasekhar-Fermi (DCF) method (Davis, 1951; Chandrasekhar & Fermi,
1953). Updated observations have since been taken using the new SCUBA-2
detector and POL-2 polarimiter (Kao & Yen et al., in prep.), providing a new
and more precise magnetic field strength estimate of $B_{\mathrm{pos}}=0.50\pm
0.13\,\mathrm{mG}$ for HOPS-370. In addition to the magnetic field strength,
the average core mass and core density were estimated to be $M_{c}=37.0\pm
2.6\,M_{\odot}$ and $\rho_{c}=1.9\pm 0.2\times 10^{-18}\,\mathrm{g\,cm^{-3}}$,
respectively, within a core radius of $\sim 0.07\,\mathrm{pc}$. This is the
same radius in which the magnetic field strength was also estimated. In order
to scale this magnetic field strength from the core-scale to the edge of the
disk and obtain a value for $B_{\phi}$, several assumptions need to be made.

####  2.3.1 The Magnetic Field - Density Relation

The general form of the most commonly cited magnetic field-density ($B$-$n$)
relation is written as

$B=B_{0}\left(\frac{n}{n_{0}}\right)^{\kappa},$ (11)

where $B_{0}$ is the initial magnetic field strength to be scaled, $n$ and
$n_{0}$ are scaled and initial number densities, respectively, and $\kappa$ is
the power-law index (Crutcher et al., 2010; Crutcher & Kemball, 2019; Pattle
et al., 2023). For clouds undergoing spherical collapse with flux-freezing,
$\kappa$ is $\sim 2/3$ (Mestel, 1966), while collapse models with ambipolar
diffusion predict $\kappa$ evolves from 0 at the initial collapse to 0.5 in
the later stages (Mouschovias & Ciolek, 1999). Since our primary assumption is
that ambipolar diffusion is the main diffusion process, and HOPS-370 is an
evolved Class 0 protostar, we take $\kappa=0.5$. Thus, the total magnetic
field strength can be scaled by

$B_{\mathrm{tot,d}}=B_{\mathrm{tot,c}}\left(\frac{\rho_{\mathrm{d}}}{\rho_{\mathrm{c}}}\right)^{0.5},$
(12)

where $\rho_{c}$ and $\rho_{c}$ are the volume densities at the core and disk
scales, while $B_{\mathrm{tot,c}}$ and $B_{\mathrm{tot,d}}$ are the total
magnetic field strengths at the core and disk scales, respectively (hereafter,
referred to as the C04 method).

####  2.3.2 Magnetic Field Strength Scaling, Correction and Estimation

In order to estimate the magnetic field strength at the edge of the disk
($B_{\phi}$) as fairly as possible, we first convert the plane-of-sky magnetic
field strength ($B_{\mathrm{pos},c}$) to the total magnetic field strength
($B_{\mathrm{tot},c}$) using two different statistical relations for the sake
of completeness. We first use the relation derived from a sample of
observations (Crutcher et al., 2004), given as

$B_{\mathrm{tot}}=\left(\frac{4}{\pi}\right)B_{\mathrm{pos}},$ (13)

which gives a statistical average of the total magnetic field strength. Using
this relation, we derive a total magnetic field strength of
$B_{\mathrm{tot},c}=0.64\pm 0.16\,\mathrm{mG}$. Additionally, Liu et al.
(2021) derive the relation

$B_{\mathrm{tot}}=\sqrt{\frac{3}{2}}B_{\mathrm{pos}},$ (14)

using 3D MHD simulations and radiative transfer calculations to produce
synthetic polarization images to find a statistical average of the total
magnetic field strength. Using this relation, we derive a total magnetic field
strength of $B_{\mathrm{tot},c}=0.61\pm 0.16\,\mathrm{mG}$. These two values
are within error, and thus, indistinguishable for our purpose. We therefore
simply adopt the value using the statistical relation from Crutcher et al.
(2004) for the remainder of this paper.

We now scale our total core-scale magnetic field strength of
$B_{\mathrm{tot},c}=0.64\pm 0.16\,\mathrm{mG}$ down to the disk scales using
Equation 12. We find $B_{\mathrm{tot},d}=28.3\pm 9.9\,\mathrm{mG}$ using the
C04 method. Since $B_{\phi}$ should be the dominant magnetic field component
at the edge of the protostellar disk, we assume $B_{\phi}\sim
B_{\mathrm{tot,d}}$ in our estimations. This is discussed later in Section
4.1.2.

Table 1: Overview of parameters used for the estimation of
$\eta_{\mathrm{AD}}$

Parameter Description | Parameter | Value
---|---|---
Protostar + Disk Properties
Protostellar mass |  $M_{\star}$ ($M_{\odot}$) | $2.5^{+0.2}_{-0.2}$
Disk mass |  $M_{d}$ ($M_{\odot}$) | $0.035^{+0.005}_{-0.003}$
Disk radius |  $R_{d}$ (au) | $94.4^{+12.6}_{-12.6}$
Critical radius |  $r_{c}$ (au) | $=R_{d}$
Temperature at 1 au |  $T_{\mathrm{0}}$ (K) | $980.0^{+0.6}_{-0.6}$
Temperature power-law index | $q$ | $0.35$
Surface density power-law index | $\gamma$ | $0.9^{+0.2}_{-0.2}$
Disk temperature at $R_{d}$ |  $T_{d}$ (K) | $199.0^{+9.3}_{-9.3}$
Disk sound speed at $R_{d}$ |  $C_{s}$ (m s-1) | $833.0^{+19.5}_{-19.5}$
Disk scale height at $R_{d}$ |  $h_{d}$ (au) | $16.2^{+3.3}_{-3.3}$
Disk surface density at $R_{d}$ |  $\Sigma_{d}$ (g cm-2) | $2.2^{+0.9}_{-0.9}$
Disk volume density at $R_{d}$ |  $\rho_{d}$ ($10^{-15}$g cm-3) | $3.7^{+1.7}_{-1.7}$
Protostellar Core Properties
Core mass |  $M_{c}$ ($M_{\odot}$) | $37.0^{+2.6}_{-2.6}$
Core volume density |  $\rho_{c}$ ($10^{-18}$g cm-3) | $1.9^{+0.2}_{-0.2}$
Plane-of-sky B-field strength |  $B_{\mathrm{pos,c}}$ (mG) | $0.5^{+0.1}_{-0.1}$
Core to Disk Scale B-Field Properties
B-n relation power-law index | $\kappa$ | 0.5
Total Core B-field strength |  $B_{\mathrm{tot},c}$ (mG) | $0.6^{+0.2}_{-0.2}$
Total Disk B-field strength |  $B_{\mathrm{tot},d}$ (mG) | $28.3^{+9.9}_{-9.8}$

References. — Tobin et al. (2020b), Kao & Yen et al. (in prep.), this work.

###  2.4 Scaling Factors for the Infalling and Rotational Velocities

Here, we discuss the scaling factors of $\delta_{r}$ and $\delta_{\phi}$,
which describe the deviations of the infalling and rotational velocities,
respectively, from the Keplerian velocity (Equation A.9). As briefly mentioned
in Appendix A, recent MHD simulations of protostellar disk formation including
ambipolar diffusion find that $u_{\phi}$ is very close to Keplerian at the
disk edge ($\delta_{\phi}\gtrsim 0.9$), while $u_{r}$ is significantly less
($\delta_{r}\lesssim 0.5$) than the Keplerian velocity, possibly by even a
factor of a few, less than one order of magnitude (Lee et al., 2021a). For the
deviation of the rotational velocity from Keplerian, we will initially assume
$\delta_{\phi}=1$ as a conservative estimate, and since the modeling of the
HOPS-370 protostellar disk already assumes the rotational velocity structure
of the disk is Keplerian. For the deviation of the infall velocity from
Keplerian, it is less straight-forward but we can still make some estimates.
Recent observations of the young Class I protostar, L1489 IRS, revealed a so-
called “slow” infall, where the velocity structure of the infalling envelope
was modeled to be 2.5 times slower than freefall (Sai et al., 2022). If we use
the quantities for HOPS-370 and this assumption of
$v_{\mathrm{inf}}=0.4v_{\mathrm{ff}}$, we find $\delta_{r}\sim 0.6$. For a
conservative measure, we initially assume $\delta_{r}=0.8$. Modeling
$\delta_{r}$ in HOPS-370 would provide further constraints on our ambipolar
diffusivity coefficient, however, this is currently beyond the scope of this
paper and will be left for a future study. How these two values effect the
ambipolar diffusivity coefficient estimation is further explored in Section
3.3.3.

##  3 Results & Analysis

###  3.1 The First Estimation of the Ambipolar Diffusivity Coefficient from
Observations

In the previous sections, we obtained all of the necessary values needed to
estimate $\eta_{\mathrm{AD}}$ for the first time. An overview of all the
parameters obtained in the previous sections is shown in Table 2.3.2. We make
an estimation using the $B_{\phi}$ derived from the C04 method. We plug in the
values of

$\displaystyle\delta_{r}=0.8,$ $\displaystyle\delta_{\phi}=1.0,$
$\displaystyle C_{s}=833.0^{+19.5}_{-19.5}\,\mathrm{m\,s^{-1}},$
$\displaystyle R_{d}=94.4^{+12.6}_{-12.6}\,\mathrm{au},$ $\displaystyle
M_{\star}=2.5^{+0.2}_{-0.2}\,M_{\odot},$ $\displaystyle
M_{d}=0.035^{+0.005}_{-0.003}\,M_{\odot}$
$\displaystyle\rho_{d}=3.7^{+1.7}_{-1.7}\times 10^{-15}\,\mathrm{g\,cm^{-3}}$
$\displaystyle B_{\phi}=28.3^{+9.9}_{-9.8}\,\mathrm{mG},$

into the normalized ambipolar diffusivity coefficient equation (Equations 2
and 3) to obtain

$\displaystyle\eta_{\mathrm{AD}}$ $\displaystyle=1.7^{+1.5}_{-1.4}\times
10^{19}\,\mathrm{cm^{2}\,s^{-1}}$ $\displaystyle=2.4^{+2.1}_{-2.0}\times
10^{-1}\,\mathrm{s}.$

As this is the first ever estimation of the ambipolar diffusivity coefficient
from observations, there are no other observational values to compare with. In
the context of comparing to the value of this coefficient using a chemical
network, this will be explored in Section 3.3.

###  3.2 The Dimensionless Elsässer Number for Ambipolar Diffusion

The strength of non-ideal MHD effects are quantified through the dimensionless
Elsässer numbers, which for ambipolar diffusion is given by

$\mathrm{AM}=\frac{v^{2}_{\mathrm{A}}}{\eta_{\mathrm{AD}}\Omega_{\mathrm{K}}},$
(15)

where $v_{\mathrm{A}}$ is the Alfvén speed and $\Omega_{\mathrm{K}}$ is the
Keplerian rotation frequency (e.g., Wurster, 2021; Cui & Bai, 2021). The
Alfvén speed is defined as

$v_{\mathrm{A}}=\sqrt{\frac{B^{2}}{4\pi\rho}},$ (16)

which describes the speed of an MHD wave permeating through a dense medium.
Likewise, the Keplerian rotation frequency is defined as

$\Omega_{\mathrm{K}}=\sqrt{\frac{GM_{\star}}{r^{3}}}.$ (17)

Typically, $\mathrm{AM}\gg 1$ represents strong coupling between the magnetic
field and the neutral gas, while $\mathrm{AM}\lesssim 1$ indicates strong
magnetic diffusion (e.g., Wurster, 2021; Commerçon et al., 2022). We estimate
the dimensionless Elsässer number for ambipolar diffusion to be
$\mathrm{AM=1.7\pm 1.0}$. This shows we are likely in the regime of stronger
magnetic diffusion and indicates the importance of ambipolar diffusion in the
evolution of the HOPS-370 protostellar disk.

###  3.3 Comparing with the Non-Ideal MHD Coefficient and Ionisation Library
(NICIL)

The Non-Ideal MHD Coefficient and Ionisation Library
(NICIL)222https://bitbucket.org/jameswurster/nicil/src/master/ is a code to
calculate the diffusion coefficients for ambipolar diffusion
($\eta_{\mathrm{AD}}$), Ohmic dissipation ($\eta_{\mathrm{OD}}$) and Hall
effect ($\eta_{\mathrm{HE}}$) for MHD simulations using a chemical network
(Wurster, 2016, 2021). We aim to investigate whether our ambipolar diffusivity
coefficient is consistent with one calculated by NICIL. NICIL allows for
estimating these coefficients for different input parameters, such as density,
temperature and magnetic field strength. Additionally, parameters for the dust
grain size distribution and cosmic-ray ionization rate can be modified. First,
we describe the initial parameters used for several NICIL runs (Section
3.3.1). We attempt to emulate the conditions at the edge of the disk as
closely as possible by using the derived disk parameters and several different
assumptions for the magnetic field strength and the cosmic-ray ionization rate
(Section 3.3.2). We then explore several of the assumptions made during our
estimation of the ambipolar diffusivity coefficient to see how they affect the
value and its consistency with NICIL (Section 3.3.3). There are two files that
we modify for these different runs in NICIL: nicil.F90 and nicil_ex_eta.F90.
We assume a barotropic equation of state for all runs, since this is the same
assumption used in the numerical simulations by Hennebelle et al. (2016).

Figure 1: Comparison between our derived value of $\eta_{\mathrm{AD}}$ and the
NICIL calculated values of $\eta_{\mathrm{AD}}$ assuming a barotropic equation
of state and a constant magnetic field strength (C04 method). Our derived
$\eta_{\mathrm{AD}}$ is marked by the black circle and also printed in the top
right of each plot, along with the values used for the estimation just below.
(Left Column) Uses a constant (unattenuated) cosmic-ray ionization rate for
the NICIL calculation. (Right Column) Uses a varied (attenuated) cosmic-ray
ionization rate for the NICIL calculation. The dashed black lines indicate the
NICIL calculated $\eta_{\mathrm{AD}}$ based on the magnetic field strength
uncertainties. The shaded blue areas represent $\eta_{\mathrm{AD}}$ calculated
by NICIL for different ranges of $\zeta_{\mathrm{0}}$ between
$10^{-16}-10^{-18}\,\mathrm{s^{-1}}$ (darker shade) and
$10^{-15}-10^{-19}\,\mathrm{s^{-1}}$ (lighter shade). The mass volume density
and H2 number density are related by
$\rho=m_{\mathrm{H}}\mu_{\mathrm{H_{2}}}n_{\mathrm{H_{2}}}$, where
$m_{\mathrm{H}}$ is the mass of the hydrogen atom and $\mu_{H_{2}}$ is the
mean molecular weight per molecular hydrogen ($\mu_{\mathrm{H_{2}}}=2.8$).

####  3.3.1 Parameter Setup

We first describe several modifications made to the nicil_ex_eta.F90 test
script. This file contains the input parameters for the temperature, density
and magnetic field strength. We compute the barotropic equation of state over
the default temperature range of $10\,$K to $2\times 10^{5}\,$K and density
range of $10^{-22}\,\mathrm{g\,cm^{-3}}$ to $10^{0.5}\,\mathrm{g\,cm^{-3}}$.
For the magnetic field, we employ a constant (use_input_B = .true.) magnetic
field strength using a value of 28.3 mG (C04 method), which we estimated at
the edge of the disk. We also run using the upper and lower errors on the
magnetic field as the constant values to estimate an approximate error range
on the NICIL ambipolar diffusivity coefficient. Additionally, NICIL has the
option to vary the magnetic field using the function $B=1.34\times
10^{-7}\sqrt{n_{n}}\,\mathrm{G}$ (use_input_B = .false.). However, this
magnetic field strength comes from different underlying assumptions than what
we use and severely underestimates the magnetic field strengths compared to
what we find, so we do not compare with this case.

The dust grain and cosmic-ray ionization properties are then adjusted in the
nicil.F90 main script. We use the default gas-to-dust ratio of 100 and set the
number of grain size bins to 32. Tobin et al. (2020b) derive power-law slope
of the grain distribution to be $p=-2.63$ and the maximum grain size to be
$a_{\mathrm{max}}=432\,\mathrm{\mu m}$, while assuming the same minimum dust
grain size of $a_{\mathrm{min}}=0.005\,\mathrm{\mu m}$ used in their fitting.
Thus, we set these parameters in NICIL accordingly. The cosmic-ray ionization
rate in HOPS-370 is unknown, however the typical ISM value is usually quoted
to be $\zeta_{\mathrm{CR}}=10^{-17}\,\mathrm{s^{-1}}$ (e.g., Caselli et al.,
1998; McElroy et al., 2013). We initially set a constant cosmic-ray ionization
rate (zeta_of_rho = .false.) of
$\zeta_{\mathrm{CR}}=\zeta_{\mathrm{0}}=10^{-17}\,\mathrm{s^{-1}}$ in the
script. However, we also vary the cosmic-ray ionization rates between
$10^{-19}\,\mathrm{s^{-1}}<\zeta_{\mathrm{0}}<10^{-15}\,\mathrm{s^{-1}}$ as
another approximate “error” range. This should be a typical range in dense
molecular clouds inferred from chemical analyses (Caselli et al., 1998). In
addition, we also run using a varied cosmic-ray ionization rate (zeta_of_rho =
.true.), which mimics attenuated cosmic-rays via the relation of
$\zeta_{\mathrm{CR}}=\zeta_{\mathrm{0}}e^{-\Sigma/\Sigma_{\mathrm{CR}}}+\zeta_{\mathrm{min}}$.
In this case, we set $\zeta_{\mathrm{0}}=10^{-17}\,\mathrm{s^{-1}}$ and
$\zeta_{\mathrm{min}}=10^{-22}\,\mathrm{s^{-1}}$ (default value) in the
script. The the gas surface (column) density ($\Sigma$) is directly calculated
from several parameters when running the code. The cosmic-ray attenuation
depth ($\Sigma_{\mathrm{CR}}$) is a constant and kept at the default value of
96$\,\mathrm{g\,cm^{-2}}$. We also vary the cosmic-ray ionization rates
between
$10^{-19}\,\mathrm{s^{-1}}<\zeta_{\mathrm{0}}<10^{-15}\,\mathrm{s^{-1}}$ in
this case as well. We set the mean molecular weight to be $\mu=2.37$ for
consistency. All other parameters in both scripts are kept as the default
values.

####  3.3.2 Initial Comparison

We run NICIL using the aforementioned parameters and compare to derived values
of $\eta_{\mathrm{AD}}$ in Figure 1. The columns correspond to the two
different cosmic-ray ionization rate assumptions in NICIL. The derived value
of $\eta_{\mathrm{AD}}$ is shown in the top right corner of the left panel
plot along with the magnetic field strength from the C04 method. We describe
each case in more detail below. Our $\eta_{\mathrm{AD}}$ result surprisingly
consistent to the $\eta_{\mathrm{AD}}$ calculated by NICIL.

Constant $\mathbf{B}$ & $\mathbf{\zeta}_{\mathrm{\mathbf{CR}}}$: The left
panel of Figure 1 use a constant magnetic field strength (use_input_B =
.true.) of $B=28.3^{+9.9}_{-9.8}\,\mathrm{mG}$ from the C04 method (left
panel). In addition, we use a constant (unattenuated) cosmic-ray ionization
rate (zeta_of_rho = .false.). As mentioned in the previous section, we run for
the derived magnetic field strength of $28.3\,\mathrm{mG}$, and then perform
subsequent runs using the upper/lower errors on the magnetic field strength
(dashed black lines). We assumed the cosmic-ray ionization rate to be
$10^{-17}\,\mathrm{s^{-1}}$ for the previous three calculations, but also
varied it between $10^{-18}\,\mathrm{s^{-1}}\leq\zeta_{0}\leq
10^{-16}\,\mathrm{s^{-1}}$ (darker-blue shaded area) and
$10^{-19}\,\mathrm{s^{-1}}\leq\zeta_{0}\leq 10^{-15}\,\mathrm{s^{-1}}$
(lighter-blue shaded area) assuming the magnetic field strength of
$28.3\,\mathrm{mG}$. The derived $\eta_{\mathrm{AD}}$ using the C04 magnetic
field strength is surprisingly consistent with the results from NICIL. If the
infall velocity is much smaller than Keplerian rotation, then both values
would become more consistent (Section 3.3.3). The cosmic-ray ionization rates
have more of an affect on the predicted ambipolar diffusivity coefficient from
NICIL compared to the error on our magnetic field strength. Higher cosmic-ray
ionization rates correspond to smaller $\eta_{\mathrm{AD}}$ values and vice-
versa. The choice of the chemical network would also impact the estimation by
NICIL, and thus our overall comparison. This is discussed more in Section
4.1.3.

Figure 2: Comparison between the effect of different values of $\delta_{r}$
(top row), $\delta_{\phi}$ (middle row) and $R_{d}/R_{\mathrm{AD}}$ (bottom
row) on our derived $\eta_{\mathrm{AD}}$ and the NICIL calculated values of
$\eta_{\mathrm{AD}}$ assuming a barotropic equation of state, constant
magnetic field strength from the C04 method and constant (unattenuated)
cosmic-ray ionization rate. As in Figure 1, our derived $\eta_{\mathrm{AD}}$
is marked by the black circle and also printed in the top right of each plot,
while the black lines and shaded blue areas also have the same meanings as in
Figure 1.

Constant $\mathbf{B}$ & Varied $\mathbf{\zeta}_{\mathrm{\mathbf{CR}}}$: The
right panel of Figure 1 uses the same magnetic field parameters as the left,
however, the cosmic-ray ionization rate is attenuated (zeta_of_rho = .true.).
We do this in order to understand what effect this has on our derived
ambipolar diffusivity coefficient. As we can see, the attenuated cosmic-rays
only affect the very high densities $\gtrsim 10^{-12}\,\mathrm{g\,cm^{-3}}$,
where the ambipolar diffusivity coefficient begins to increase. It does not
affect the density regime in which our ambipolar diffusivity coefficient is
estimated. Thus, our derived $\eta_{\mathrm{AD}}$ is still consistent with
these results from NICIL.

####  3.3.3 Varied Parameters

There are several parameter assumptions made in our estimation using the
ambipolar diffusivity coefficient equation that could vary the resulting
$\eta_{\mathrm{AD}}$. First, our initial calculation assumes $\delta_{r}=0.8$
and $\delta_{\phi}=1.0$. Additionally, the results from Hennebelle et al.
(2016) and Hennebelle et al. (2020) indicate that the ratio of the disk radius
from their simulations to their predicted disk radius from their analytical
equation, $R_{d}/R_{\mathrm{AD}}$, could vary between 0.5 to 2, particularly
for lower mass cores. Therefore, we vary at each parameter individually, while
the others are kept at their initially assumed values, to see the magnitude in
difference for each. We compare to our NICIL run using constant $B$ from the
C04 method and constant $\zeta_{\mathrm{CR}}$ (Figure 2, left panel).

For $\delta_{r}$, we compare values of 0.8, 0.5 and 0.1 (Figure 2, top row).
As mentioned previously, $\delta_{r}$ could be a factor of a few lower than
the Keplerian velocity in simulations, and was found to be $\sim 0.6$ in
previous observations of the Class I protostar, L1489 IRS. As this value could
vary quite considerably depending on the environment, it will have more of an
impact on our estimated ambipolar diffusivity coefficient than
$\delta_{\phi}$, even though $\eta_{\mathrm{AD}}$ scales as
$\sim\delta_{\phi}^{2}$. In the case where $\delta_{r}=0.1$, it is much more
consistent with NICIL as the estimated ambipolar diffusivity coefficient is
about an order of magnitude lower, potentially hinting at the possibility of
slow infall in HOPS-370.

For $\delta_{\phi}$, we compare values of 1.0, 0.9 and 0.8 (Figure 2, middle
row). Since $\delta_{\phi}$ should be $\gtrsim 0.9$, it will not have too much
affect on our derived ambipolar diffusivity coefficient, which are all within
error in these cases. We only demonstrate $\delta_{\phi}=0.8$ as a more
extreme scenario, but still the effect is less than $\delta_{r}$ due to these
constraints. It would still be interesting to try to estimate if there is any
deviation from Keplerian rotation in the rotational velocity structure at the
edge of the disk, as it still lowers the value, if only even a little.

For $R_{d}/R_{\mathrm{AD}}$, we compare values of 0.5, 1.0 and 2.0 (Figure 2,
bottom row). As previously mentioned, the ratio of the actual disk radius to
the predicted disk radius due to ambipolar diffusion from Equation 1 could
vary between 0.5 to 2, particularly for lower mass cores (Hennebelle et al.,
2016, 2020). Although HOPS-370 is considered to be a more intermediate mass
Class 0/I, we would still like to investigate how the value varies between
these two extreme cases. The $R_{d}/R_{\mathrm{AD}}$ factor also doesn’t
really affect the calculation of $\eta_{\mathrm{AD}}$ too much, which is
similar to $\delta_{\phi}$. $R_{d}/R_{\mathrm{AD}}$ is slightly more
consistent with NICIL when $R_{d}/R_{\mathrm{AD}}>1$, while the right panel of
Figure 2 in Hennebelle et al. (2016) shows consistently lower
$R_{d}/R_{\mathrm{AD}}<1$ for protostars $M_{\star}+M_{d}<5\,M_{\odot}$. This
should be investigated in numerical simulations for a mass range around the
HOPS-370 protostar+disk mass as the spread can become quite noticeable when
zooming in on very low-mass simulations ($M_{\star}+M_{d}<0.5\,M_{\odot}$) in
the left panel of Figure 2 in Hennebelle et al. (2016).

##  4 Discussion

Figure 3: Relative comparison of the diffusivity coefficients for ambipolar
diffusion ($\eta_{\mathrm{AD}}$), Ohmic dissipation ($\eta_{\mathrm{OD}}$) and
the Hall effect ($\eta_{\mathrm{HE}}$) calculated by NICIL assuming a
barotropic equation of state, constant magnetic field strength and cosmic-ray
ionization rate. The symbols and labels have the same meaning as in Figure 1,
except we only show the lighter shaded blue area for $\zeta_{\mathrm{0}}$
between $10^{-15}-10^{-19}\,\mathrm{s^{-1}}$ for each coefficient. Our derived
value shows that HOPS-370 lies in an ambipolar diffusion dominated region.

###  4.1 Validity of Assumptions

Several assumptions are made in the derivation, estimation and comparison to
theoretical values of the ambipolar diffusivity coefficient. In this section,
we explore these assumptions in detail and discuss how they could effect our
results.

####  4.1.1 Derivation of the Ambipolar Diffusivity Coefficient Equation

In Section 2, we have listed several assumptions in the derivation of Equation
1. The first is that the main diffusion process is ambipolar diffusion. There
are many factors shown to alleviate the effects of magnetic braking to form
large, protostellar disks in MHD simulations. These mainly include non-ideal
MHD (e.g., Li et al., 2011; Dapp et al., 2012; Tsukamoto et al., 2015; Wurster
et al., 2016; Tsukamoto et al., 2017; Wurster et al., 2019; Zhao et al.,
2020a; Wurster et al., 2021), misalignment between the magnetic field and
rotation axis (e.g., Hennebelle & Ciardi, 2009; Li et al., 2013; Tsukamoto et
al., 2018; Hirano et al., 2020) and turbulence (e.g., Seifried et al., 2013;
Li et al., 2014; Seifried et al., 2015). From an observational standpoint,
there are several key results to consider. Magnetic field orientations in low-
mass protostars indicate that the field orientation is preferentially randomly
aligned with the rotation axis (e.g., Hull et al., 2013; Yen et al., 2021b)
However, recent results show no apparent correlation between the misalignment
angle of the magnetic field and apparent disk size measured from the dust
continuum (Yen et al., 2021a). Yen et al. (2021a) also conclude that the
turbulence measured from the non-thermal linewidth at core-scale does not
correlate with the apparent disk size either. Observations of angular momentum
profiles in protostellar envelopes do imply that there is some turbulence
present (Pineda et al., 2019; Gaudel et al., 2020; Sai et al., 2023), although
the level of turbulence is not directly quantified. Thus, non-ideal MHD likely
plays an important role in protostellar disk formation.

As far as which non-ideal MHD effect (ambipolar diffusion, Ohmic dissipation
or the Hall effect) is most important overall, simulations show ambipolar
diffusion is an efficient process in parts of the disk and envelope that can
regulate the properties of the disk (e.g., Tsukamoto et al., 2023). Ohmic
dissipation is only efficient at high densities and likely does not play much
of a role in the envelope itself (e.g., Marchand et al., 2016; Wurster et al.,
2018a; Wurster, 2021). The Hall effect has been shown to effectively disappear
shortly after the formation of the protostellar disk (Zhao et al., 2020b; Lee
et al., 2021b). Therefore, ambipolar diffusion may be the most important non-
ideal MHD effect, especially when the protostar + disk system becomes more
evolved. This is where comparing directly with non-ideal MHD simulations would
help us to understand the non-ideal MHD effects more deeply.

Since NICIL does also calculate the non-ideal MHD coefficients for Ohmic
dissipation ($\eta_{\mathrm{OD}}$) and the Hall effect ($\eta_{\mathrm{HE}}$),
it is interesting to compare them to the ambipolar diffusivity coefficient. We
show a comparison between each of the diffusivity coefficients from our NICIL
runs in Figure 3. We see that our derived $\eta_{\mathrm{AD}}$ is clearly in
an ambipolar diffusion dominated density regime, under our assumptions made
for our NICIL runs. This is in favor of the first assumption stated to derive
Equation 1, that ambipolar diffusion is the main diffusion process. The
coefficient for Ohmic dissipation starts to become prominent towards the
highest density regimes, which is consistent with previous findings (e.g.,
Marchand et al., 2016; Wurster et al., 2018a; Wurster, 2021). The Hall effect
seems to be dominant at intermediate density regimes, though it still shows
some contribution in the density regime where our ambipolar diffusion value is
calculated. We also note that in our NICIL run in the bottom left panel, the
Hall coefficient becomes negative at very low densities approaching
$10^{-18}\,\mathrm{g\,cm^{-3}}$ in the case of a low cosmic-ray ionization
rate of $\zeta_{0}=10^{-19}\,\mathrm{s^{-1}}$.

The cosmic-ray ionization rate will impact the efficiency of non-ideal MHD
effect (e.g., Wurster et al., 2018b; Kuffmeier et al., 2020), and thus should
be studied in the environment of HOPS-370 to further constrain the comparisons
with NICIL. The cosmic-ray ionization rate in the inner envelope of Class 0
protostar, B335, was previously found to be enhanced ($\zeta_{\mathrm{CR}}\sim
10^{-14}\,\mathrm{s^{-1}}$), which could explain the extremely small
($<10\,\mathrm{au}$) inferred protostellar disk (Cabedo et al., 2023).
Additionally, a new large scale study probing the NGC 1333 region of the
Perseus also finds an enhanced cosmic-ray ionization rate
($\zeta_{\mathrm{CR}}\gtrsim 10^{-16.5}\,\mathrm{s^{-1}}$) across the
molecular cloud, which is consistent with the small ($<50\,\mathrm{au}$) disks
in that region (Pineda et al., 2024).

Future studies also probing the ion-neutral drift in HOPS-370 could help to
further understand the role of ambipolar diffusion in this source. Yen et al.
(2018) tried to constrain the ion-neutral velocity drift in the young Class 0
protostar, B335, however, only an upper limit was obtained. This could be due
to B335 being too young, as some simulations have shown this velocity drift
could be more observable in more evolved Class 0/I protostars (e.g., Tsukamoto
et al., 2020). Since HOPS-370 is more evolved, it could be an ideal target for
this kind of study in the future.

Next, the angular momentum is counteracted by magnetic braking resulting in
the advection and braking timescales to be of the same order. The equations
for the advection and braking timescales are given by Equations A.1 and
Equations A.2, respectively. We make an estimation of the advection timescale
under the same assumption used for our ambipolar diffusivity estimate, where
$u_{r}=0.8v_{\mathrm{kep}}$, giving us a value of $\tau_{\mathrm{adv}}\sim
3.6\times 10^{9}\,\mathrm{s}$, which is also a lower limit. For the braking
timescale, since we do not directly know the poloidal component ($B_{z}$) of
the magnetic field strength, we make an approximation of $B_{z}B_{\phi}\approx
B_{\mathrm{tot}}^{2}$ and use our value derived from the C04 method. This
results in a lower limit of the braking timescale of $\tau_{\mathrm{br}}\sim
5.4\times 10^{8}\,\mathrm{s}$. These values are within one order of magnitude
difference, and show that this assumption can hold in HOPS-370. We note that
we use this exact assumption to estimate $B_{z}$ in Section 4.1.2, therefore,
using that value here presents a circular argument which is why we simply
estimate the lower limits for $\tau_{\mathrm{adv}}$ and $\tau_{\mathrm{br}}$.
Further modeling of the infall velocity structure and magnetic field
components ($B_{r}$, $B_{z}$, $B_{\phi}$) would be necessary to confirm.

Then, the toroidal field generated by differential rotation is offset by the
ambipolar diffusion in the vertical direction resulting in the Faraday
induction and vertical diffusion timescales to be of the same order. The
equations for the Faraday induction and vertical diffusion timescales are
given by Equations A.3 and Equations A.4, respectively. We can obtain a lower
limit approximation of the Faraday induction timescales by assuming $B_{z}\sim
B_{\phi}\sim B_{\mathrm{tot}}$. This gives a value of $\tau_{\mathrm{far}}\sim
4.9\times 10^{8}\,\mathrm{s}$. The vertical ambipolar diffusion timescales
would need to use our derived value, thus we check for self-consistency. We
find $\tau_{\mathrm{diff}}\sim 3.4\times 10^{9}\,\mathrm{s}$. Since varying
some of the parameters in Section 3.3.3 lower the value of
$\eta_{\mathrm{AD}}$, $\tau_{\mathrm{diff}}$ could also be considered as a
lower limit. Both values are within one order of magnitude difference, showing
that this assumption can hold in HOPS-370. Again, we do not use the $B_{z}$ in
Section 4.1.2 to avoid any circular arguments.

Additionally, the infalling and rotational velocities of the gas near the disk
edge both scale with the Keplerian velocity. The disk radius derived for
HOPS-370 from the radiative transfer modeling indicates that the rotational
velocity ($u_{\phi}$) should be Keplerian in nature. As for the infalling
velocity, further modeling needs to be done to see how much $u_{r}$ deviates
from Keplerian at the disk edge. For now, the assumption of the rotational
velocity holds, while the assumption for the infalling velocity should to be
further modeled.

Lastly, the gas near the disk edge has Keplerian velocity and is in vertical
hydrostatic equilibrium. Again, the gas disk in HOPS-370 is clearly resolved
by the observations by Tobin et al. (2020b) and the best-fit disk parameters
were found by fitting with radiative transfer models assuming Keplerian
rotation and hydrostatic equilibrium. Many previous studies have also clearly
resolved Keplerian rotating disks in young Class 0 and Class I protostars
(e.g., Tobin et al., 2012; Murillo et al., 2013; Yen et al., 2014, 2017;
Ohashi et al., 2023). Even if the assumptions in the fitting are wrong, they
are the same assumptions we use and we are still using a “best-fit” value,
which indicates this model does provide a good fit to the data. Therefore, the
values properties derived for the HOPS-370 protostellar disk clearly should
satisfy both assumptions.

####  4.1.2 Quantities and Relations used for the Ambipolar Diffusivity
Coefficient Estimation

Arguably the most important assumption in our estimation of
$\eta_{\mathrm{AD}}$ is how the envelope scales from core-scale down to the
edge of the protostellar disk. As previously stated, early theoretical works
predict $\kappa$ in Equation 4 to be $\sim 2/3$ for clouds undergoing
spherical collapse with flux-freezing (Mestel, 1966), while $\kappa\sim 0.5$
for a collapsing cloud with ambipolar diffusion (Mouschovias & Ciolek, 1999).
This was the basis of our initial assumptions, however, it is not so straight
forward. The recent review by Pattle et al. (2023) shows $\kappa$ derived from
observations of molecular clouds can vary quite a bit, possibly due to
different environmental factors. These observations probe the large scale
molecular clouds, filaments and cores whose magnetic field imprint could be
inherently different than the magnetic fields near a protostellar disk.
Additionally, a magnetic field - density relation recently derived by Lee et
al. (2024) for inside a collapsing, protostellar envelope is explored in
Appendix D. The magnetic field strength derived from this relation is
compatible with our estimates, however, needs to be further investigated due
to discrepancies in the model presumptions.

Observationally, Yen et al. (2023) recently derived a magnetic field - density
relation from the core to inner envelope scale in the young, Class 0 protostar
HH 211. They find $\kappa\sim 0.36$, which fits into the assumption that
ambipolar diffusion is playing a role to partially decoupled the magnetic
field from the neutral matter. Their inner envelope magnetic field strength
was derived using a force-balance equation (Koch et al., 2012), rather than
the DCF method. The core-scale magnetic field strength estimated by Kao & Yen
et al. (in prep.) was derived using the DCF method, which has several
uncertainties associated with it due to the assumptions of equipartition,
isotropic turbulence, projected polarization angle on the plane-of-sky, and
more (e.g., Liu et al., 2021, 2022a; Chen et al., 2022; Liu et al., 2022b;
Myers et al., 2023). These uncertainties may cause the DCF estimate to
overestimate the magnetic field strength, which would impact our ambipolar
diffusivity coefficient estimation. In the best case scenario, future
observations to derive the inner envelope strength near the disk in HOPS-370
could alleviate the need to even use a magnetic field - density relation.
Otherwise, if the magnetic field strength cannot be derived close enough to
the disk edge, it can still be estimated in the envelope to derive a magnetic
field - density relation, where the magnetic field strength could further be
scaled down to the edge of the disk.

We have also assumed that $B_{\mathrm{tot}}\approx B_{\phi}$, and that
$B_{\phi}$ is the dominant component of the magnetic field at the edge of the
protostellar disk. Our value for $B_{\mathrm{tot}}$ at the core-scale is a
statistical average based on a large sample of observations, which may or may
not necessarily be applied to only a single source. This is, however, the only
current way we obtain a total magnetic field strength from the plane-of-sky
magnetic field component and should be investigated further. To see whether
$B_{\phi}$ is really dominant in our case, we estimate $B_{z}=4.2\pm
2.7\,\mathrm{mG}$ using Equation A.5. This shows that $B_{\phi}$ is the
dominant component in our case, and thus is a reasonable assumption in our
ambipolar diffusivity coefficient estimation.

####  4.1.3 Comparison with NICIL and Input Values

While the cosmic-ray ionization rate and dust grain properties needed for
NICIL are not inherently part of our derived ambipolar diffusion equation,
they still play a role in the efficiency of non-ideal MHD diffusivities (e.g.,
Zhao et al., 2016; Dzyurkevich et al., 2017; Wurster et al., 2018b; Zhao et
al., 2018; Kuffmeier et al., 2020; Guillet et al., 2020; Zhao et al., 2021;
Kobayashi et al., 2023). Several studies have shown that disk formation is
suppressed in the presence cosmic-ray ionization rates higher than the
canonical value of $10^{-17}\,\mathrm{s^{-1}}$ in dense cores (e.g., Zhao et
al., 2016; Wurster et al., 2018b; Kuffmeier et al., 2020). Large numbers of
small dust grains can also influence the ionization degree, and thus the non-
ideal MHD diffusivities (e.g., Zhao et al., 2016; Dzyurkevich et al., 2017;
Zhao et al., 2018; Koga et al., 2019; Marchand et al., 2020). Tobin et al.
(2020b) do constrain the maximum grain size, while the minimum grain size is
set as a fixed parameter in their model fitting. We did explore how much the
minimum grain size affects the calculated $\eta_{\mathrm{AD}}$ from NICIL by
re-running our constant $B$ (C04 method) and constant $\zeta_{\mathrm{CR}}$
NICIL runs, for minimum grain sizes of $0.01\,\mathrm{\mu m}$,
$0.1\,\mathrm{\mu m}$ and $1.0\,\mathrm{\mu m}$. However, the difference was
indistinguishable, and thus, the resulting $\eta_{\mathrm{AD}}$ from NICIL may
rely more heavily on the choice of chemical network. We also checked if the
number of grain size bins used affected the results, but still the results did
not change. It is important to note that the derived dust grain properties are
that of the disk, and not the envelope. Also, there are currently no studies
exploring the cosmic-ray ionization rate in the disk or envelope of HOPS-370.
Determining the cosmic-ray ionization rate and dust grain properties in the
HOPS-370 protostellar envelope would allow for a better comparison to NICIL.
Our comparison with NICIL simply represents the closest theoretical scenario
we can achieve by using the values derived from observations. Therefore,
further constraints on the properties of the disk and envelope environment, as
well as, comparisons with actual non-ideal MHD simulations should be carried
out.

Figure 4: Schematic of the HOPS-370 protostellar system. (Left) 0.85 mm
continuum emission of the Orion A molecular cloud taken by the JCMT (Kao & Yen
et al., in prep.). The contour levels shown are 3, 5, 10, 15, 30, 50, 100, 300
and 500$\sigma$, where
$\sigma_{\mathrm{1.3mm}}=15.2\,\mathrm{mJy\,beam^{-1}}$. The location of
HOPS-370 is shown with a yellow star, with the protostellar class and distance
listed. (Right Top) 0.87 mm continuum emission of the protostellar disk around
HOPS-370 with self-contour levels of 3, 5, 10, 15, 30, 50 and 100$\sigma$,
where $\sigma_{\mathrm{0.87mm}}=0.39\,\mathrm{mJy\,beam^{-1}}$ (Tobin et al.,
2020b). CH3OH and SO integrated-intensity contours are shown in green and
orange, respectively, with contour levels of 3, 5, 10, 15, 30$\sigma$, where
$\sigma_{\mathrm{CH_{3}OH}}=26.2\,\mathrm{mJy\,beam^{-1}\,km\,s^{-1}}$ and
$\sigma_{\mathrm{SO}}=32.4\,\mathrm{mJy\,beam^{-1}\,km\,s^{-1}}$. These two
molecular lines were shown to trace the largest disk radius when modeled
together. The position of the continuum peak is marked with a yellow star.
(Right Bottom) The modeling and results of our ambipolar diffusivity
coefficient estimation. The best-fit disk density profile (for $z=0$; i.e. the
midplane) is shown in log scale, along with the best-fit quantities used in
Equation 2 and our estimated ambipolar diffusivity coefficient at the edge of
the disk.

##  5 Conclusion

We present the first estimation of the ambipolar diffusivity coefficient using
an analytical equation describing the protostar and disk properties due to
ambipolar diffusion. We show an illustrative schematic of the HOPS-370
protostellar system to bring together and summarize our results in the context
of the multi-scale observations needed for this study (Figure 4). The main
results of this paper are as follows:

  1. 1.

We derive a generalized analytical expression for the ambipolar diffusivity
coefficient in terms of observable quantities in protostellar environments. We
show that this relation should be valid, regardless of the global magnetic
field orientation with respect to the disk rotation axis.

  2. 2.

We make the first estimation of the ambipolar diffusivity coefficient to be
$\eta_{\mathrm{AD}}=1.7^{+1.5}_{-1.4}\times 10^{19}\,\mathrm{cm^{2}\,s^{-1}}$
at the edge of the HOPS-370 protostellar disk, under the assumption that the
magnetic field scales with density (Crutcher et al., 2004). We use the Alfvén
speed and Keplerian rotation frequency to estimate the dimensionless Elsässer
number for ambipolar diffusion to be $\mathrm{AM=1.7^{+1.0}_{-1.0}}$,
indicating that ambipolar diffusion is more dynamically important in the
region at the edge of the protostellar disk. Estimates of the ambipolar
diffusivity coefficient using the inner envelope density, rather than the
disk-edge density yields indistinguishable results.

  3. 3.

We use the Non-Ideal MHD Coefficient and Ionisation Library (NICIL) to
calculate the non-ideal MHD coefficients using the the physical conditions
observed in HOPS-370. We show that the ambipolar diffusivity coefficient from
NICIL using various magnetic field strength and cosmic-ray ionization
properties is consistent with our derived value. We vary the less certain
parameters of $\delta_{r}$, $\delta_{\phi}$ and $R_{d}/R_{\mathrm{AD}}$ in the
ambipolar diffusivity coefficient equation to find the derived value becomes
more consistent for decreasing $\delta_{r}$ and $\delta_{\phi}$ and increasing
$R_{d}/R_{\mathrm{AD}}$.

  4. 4.

We plot the Ohmic dissipation and Hall effect coefficients along side the
ambipolar diffusivity coefficient calculated by NICIL. We find that our
derived value shows HOPS-370 lies in an ambipolar diffusion dominated region.
This supports the main assumption in the derivation of Equation 1 that
ambipolar diffusion is the main diffusion process. When assessing the other
assumption made for our derivation of the ambipolar diffusivity equation, we
show that they should be valid for HOPS-370.

  5. 5.

We have demonstrated a new methodology for understanding the role of ambipolar
diffusion during protostellar disk evolution. Future studies including more
sources and more detailed modeling will help to fully understand the role of
non-ideal MHD effects in observations of the earliest stages of protostellar
disk formation and evolution.

## Acknowledgments

We thank the anonymous referee for their helpful comments and suggestions on
this manuscript. This work used high-performance computing facilities operated
by the Center for Informatics and Computation in Astronomy (CICA) at National
Tsing Hua University. This equipment was funded by the Ministry of Education
of Taiwan, the National Science and Technology Council of Taiwan, and National
Tsing Hua University. S.-P.L. and T.J.T. acknowledge grants from the National
Science and Technology Council (NSTC) of Taiwan 106-2119-M-007-021-MY3,
109-2112-M-007-010-MY3 and 112-2112-M-007-011. Y.-N.L. acknowledges support
from the National Science and Technology Council, Taiwan (NSTC
112-2636-M-003-001) and the grant for Yushan Young Scholar from the Ministry
of Education. S.-J.L. acknowledges the grants from the National Science and
Technology Council (NSTC) of Taiwan 111-2124-M-001-005 and 112-2124-M-001-014.
H.-W.Y. acknowledges support from the NSTC grant 110-2628-M-001-003-MY3 and
from the Academia Sinica Career Development Award (AS-CDA-111-M03).

##  Appendix A Derivation of the Ambipolar Diffusivity Coefficient Relation

The equation describing the disk radius due to ambipolar diffusion first
presented by Hennebelle et al. (2016), and later by Lee et al. (2021b, 2024),
make a number of simplifications. Here, we derive a new relationship between
the physical properties at the disk-envelope interface to the ambipolar
diffusivity coefficient, in order to better compare to more generalized models
that are used to fit observations, as in our case for HOPS-370. We follow the
prescriptions given by Lee et al. (2021b, 2024), where more detailed
explanations can be found. We assume ambipolar diffusion is the main diffusion
process, as discussed in Section 4.1.1.

First of all, the accretion of angular momentum onto the protostellar disk is
counteracted by magnetic braking to rapidly suppress the growth of the disk.
This results in an equilibrium condition at the disk-envelope interface
between the advection and magnetic braking timescales
($\tau_{\mathrm{adv}}\simeq\tau_{\mathrm{br}}$) given by

$\displaystyle\tau_{\mathrm{adv}}\simeq\frac{R}{u_{r}},$ (A.1)
$\displaystyle\tau_{\mathrm{br}}\simeq\frac{\rho u_{\phi}h}{B_{z}B_{\phi}},$
(A.2)

where $u_{r}$ and $u_{\phi}$ are the infalling and rotational velocities,
$B_{z}$ and $B_{\phi}$ are the poloidal (vertical) and toroidal (azimuthal)
magnetic field components, $R$ is the disk radius, $\rho$ is the density at
the disk-envelope interface and $h$ is the scale height of the disk at the
edge.

Next, $B_{\phi}$ is generated by the induction of $B_{z}$ through the
differential rotation of the protostellar disk is vertically diffused by
ambipolar diffusion. This results in another equilibrium condition between the
generation of $B_{\phi}$, which happens on the timescale of Faraday induction,
and the vertical ambipolar diffusion timescales
($\tau_{\mathrm{far}}\simeq\tau_{\mathrm{diff}}$) given by

$\displaystyle\tau_{\mathrm{far}}\simeq\frac{B_{\phi}h}{B_{z}u_{\phi}},$ (A.3)
$\displaystyle\tau_{\mathrm{diff}}\simeq\frac{h^{2}}{\eta_{\mathrm{AD}}},$
(A.4)

where $\eta_{\mathrm{AD}}$ is the ambipolar diffusivity coefficient.

Since $B_{\phi}$ should be the dominant component at the protostellar disk
edge, we solve our first equilibrium condition for $B_{z}$ in order to
substitute it into our second equilibrium equation, giving

$B_{z}\simeq\frac{\rho u_{\phi}u_{r}h}{RB_{\phi}}.$ (A.5)

Solving the second equilibrium equation in terms of our ambipolar diffusivity
coefficient and substituting in our new relation for $B_{z}$ gives

$\eta_{\mathrm{AD}}\simeq\frac{B_{z}u_{\phi}h^{2}}{B_{\phi}h}\simeq\frac{\rho
u_{\phi}^{2}u_{r}h^{2}}{RB_{\phi}^{2}}.$ (A.6)

We assume the rotational velocity ($u_{r}$) and infall velocity ($u_{\phi}$)
both scale with the Keplerian velocity ($v_{\mathrm{kep}}$) as

$\displaystyle u_{r}=\delta_{r}v_{\mathrm{kep}},$ (A.7) $\displaystyle
u_{\phi}=\delta_{\phi}v_{\mathrm{kep}},$ (A.8)

where $\delta_{r}$ and $\delta_{\phi}$ are the the scaling factors and
$v_{\mathrm{kep}}$ defined at the disk edge is

$v_{\mathrm{kep}}=\left(\frac{GM}{R}\right)^{1/2},$ (A.9)

where $G$ is the gravitational constant and $M=M_{\star}+M_{d}$ is the mass of
the star+disk system. From recent MHD simulations of protostellar disk
formation including ambipolar diffusion, $u_{\phi}$ is found to be very close
to Keplerian at the disk edge ($\delta_{\phi}\gtrsim 0.9$), while $u_{r}$ can
be significantly less ($\delta_{r}\lesssim 0.5$) than the Keplerian velocity,
possibly by even a factor of a few (Lee et al., 2021a, Section 2.4).
Substituting $u_{r}$ and $u_{\phi}$ into our ambipolar diffusivity coefficient
relation gives

$\eta_{\mathrm{AD}}\simeq\frac{\delta_{r}\delta_{\phi}^{2}G^{3/2}M^{3/2}\rho
h^{2}}{R^{5/2}B_{\phi}^{2}}.$ (A.10)

Assuming vertical hydrostatic equilibrium, the scale height is related to the
isothermal sound speed ($C_{s}$) as

$h=C_{s}\left(\frac{R^{3}}{GM}\right)^{1/2}.$ (A.11)

Now, we can replace $h$ in our ambipolar diffusivity coefficient equation to
get a final relation of

$\eta_{\mathrm{AD}}\simeq\frac{\delta_{r}\delta_{\phi}^{2}G^{1/2}C_{s}^{2}R^{1/2}M^{1/2}\rho}{B_{\phi}^{2}}.$
(A.12)

This expression should be valid regardless of the global magnetic field
orientation (see Appendix B for further discussion). We have left in the
density and sound speed terms, which deviates from the further simplifications
made by Hennebelle et al. (2016) and Lee et al. (2021b, 2024), since these
quantities can be modeled for protostellar disks from molecular line
observations.

##  Appendix B The Effects of Magnetic Field Inclination on the Ambipolar
Diffusivity Coefficient Estimation

For an inclined magnetic field, the equilibrium condition that needs to be
satisfied follows as

$\frac{B_{z}\sqrt{R}}{\sqrt{\eta_{\mathrm{AD}}\rho
u_{r}}}+\frac{B_{r}h}{\sqrt{\eta_{\mathrm{AD}}\rho u_{r}R}}=1,$ (B.1)

where $B_{r}$ is the radial component of the magnetic field strength and the
other symbols have the same meaning as in Appendix A (Lee et al., 2024). The
relationships between the $B_{r}$ and $B_{z}$ components are given by

$\displaystyle B_{r}=B_{0}\left(2/\pi\right)\sin i$ (B.2) $\displaystyle
B_{z}=B_{0}\cos i$ (B.3)

where $i$ is the magnetic field inclination with respect to the disk rotation
axis ($i=0^{\circ}$ means the magnetic field direction is aligned/parallel
withe disk rotation axis, i.e. the vertical case) and $B_{0}$ characterizes
the amount of magnetic flux that threads the disk region, while the local
field strength can be significantly enhanced by magnetic induction due to
vertical differential rotation (Lee et al., 2024). Using the relations of
$B_{z}$ and $B_{r}$, along with Equations A.7, A.8 and A.11, we can re-write
the previous equation as

$\frac{r^{3/4}B_{0}\cos
i}{\eta_{\mathrm{AD}}^{1/2}\rho^{1/2}\delta_{r}^{1/2}G^{1/4}M^{1/4}}+\frac{r^{5/4}B_{0}\left(2/\pi\right)\sin
i}{\eta_{\mathrm{AD}}^{1/2}\rho^{1/2}\delta_{r}^{1/2}G^{3/4}M^{3/4}C_{s}^{-1}}=1,$
(B.4)

where $M=M_{\star}+M_{d}$ is the total mass of the star+disk system. Re-
writing in terms of $B_{0}$, we find

$B_{0}=\eta_{\mathrm{AD}}^{1/2}\rho^{1/2}\delta_{r}^{1/2}\left[\frac{r^{3/4}\cos
i}{G^{1/4}M^{1/4}}+\frac{r^{5/4}\left(2/\pi\right)\sin
i}{G^{3/4}M^{3/4}C_{s}^{-1}}\right]^{-1},$ (B.5)

When the magnetic field is inclined, then the magnetic field strength derived
in Section 2.3.1 should not be assumed as one of the magnetic field
components, but rather regarded as the total magnetic field strength. We can
thus derive an ambipolar diffusivity equation in terms of the total magnetic
field strength. The total magnetic field strength ($B_{\mathrm{tot}}$) is the
sum of squares of all the components written as

$B_{\mathrm{tot}}^{2}=B_{r}^{2}+B_{z}^{2}+B_{\phi}^{2},$ (B.6)

where $B_{r}$ and $B_{z}$ can be substituted again using Equations B.2 and B.3
to give

$B_{\mathrm{tot}}^{2}=B_{0}^{2}\left[\left(2/\pi\sin i\right)^{2}+\left(\cos
i\right)^{2}\right]+B_{\phi}^{2}.$ (B.7)

Now, we substitute $B_{\phi}$ using our derived relationship in Equation A.12
to get

$\begin{split}B_{\mathrm{tot}}^{2}&=B_{0}^{2}\left[\left(2/\pi\sin
i\right)^{2}+\left(\cos i\right)^{2}\right]\\\
&+\delta_{r}\delta_{\phi}^{2}G^{1/2}C_{s}^{2}R^{1/2}M^{1/2}\rho\eta_{\mathrm{AD}}^{-1}.\end{split}$
(B.8)

We now have two equations (B.5 and B.8) with two unknowns ($B_{0}$ and
$\eta_{\mathrm{AD}}$). Thus, we substitute Equation B.5 into Equation B.8 to
remove $B_{0}$ and obtain a second-order polynomial ambipolar diffusivity
equation for an inclined magnetic field of

$\begin{split}&\eta_{\mathrm{AD}}^{2}\rho\delta_{r}\left[\frac{r^{3/4}\cos
i}{G^{1/4}M^{1/4}}+\frac{r^{5/4}\left(2/\pi\right)\sin
i}{G^{3/4}M^{3/4}C_{s}^{-1}}\right]^{-2}\\\ &\times\left[\left(2/\pi\sin
i\right)^{2}+\left(\cos i\right)^{2}\right]\\\
&-\eta_{\mathrm{AD}}B_{\mathrm{tot}}^{2}\\\
&+\delta_{r}\delta_{\phi}^{2}G^{1/2}C_{s}^{2}R^{1/2}M^{1/2}\rho=0,\end{split}$
(B.9)

which can be solved to find the ambipolar diffusivity coefficient. Using the
same values as in Section 3.1 and a magnetic field inclination with respect to
the disk rotation axis of $45\pm 22^{\circ}$ (Yen et al., 2021a), we find
$\eta_{\mathrm{AD}}=1.798^{+0.042}_{-0.007}\times
10^{19}\,\mathrm{cm^{2}\,s^{-1}}$, where the reported errors are only due to
the error on the magnetic field inclination angle. For one, this value is
extremely close to and within error of the value previously derived in Section
3.1. Additionally, the errors due to only the magnetic field inclination are a
few orders of magnitude smaller than in the previously derived value. This
show that the magnetic field inclination has essentially no effect on our
derived ambipolar diffusivity coefficient.

For completeness, we check the corresponding values of $B_{0}$, $B_{z}$,
$B_{r}$ and $B_{\phi}$ to see if which component of the magnetic field is
dominant. We use Equations B.2, B.3, B.5 and B.7 to estimate values of
$B_{0}\approx 5.5\,\mathrm{mG}$, $B_{z}\approx 3.9\,\mathrm{mG}$,
$B_{r}\approx 2.5\,\mathrm{mG}$ and $B_{\phi}\approx 28.0\,\mathrm{mG}$. This
shows that the $B_{\phi}$ component of the magnetic field still dominates even
when considering the orientation. Thus, our derived relation is considered to
be generalized and it is correct to assume $B_{\phi}\approx B_{\mathrm{tot}}$
in our initial assumptions (Section 2.3.1).

##  Appendix C Ambipolar Diffusivity Coefficient Estimation using Inner
Envelope Density

Figure C.1: Comparison between the best-fit disk and envelope volume density
profiles for HOPS-370 (Tobin et al., 2020b). The vertical dashed line
represents the radius of the Keplerian gas disk.

As described in Section 2.2.3, we could equally derive a density at the disk-
envelope interface from the best fit envelope volume density relation in Tobin
et al. (2020b). We stress that much of the envelope emission may be resolved-
out, which could effect the fitting results. However, it is interesting to
still investigate how the ambipolar diffusivity coefficient estimation is
effected using the current best-fit results. To model the envelope emission
using the molecular line data, Tobin et al. (2020b) uses the following
relation for the envelope

$\begin{split}\rho_{\mathrm{env}}(r)&=\frac{\dot{M}_{\mathrm{env}}}{4\pi}\left(GM_{\star}r^{3}\right)^{-1/2}\\\
&\times\left(1+\frac{\mu}{\mu_{0}}\right)^{-1/2}\left(\frac{\mu}{\mu_{0}}+2\mu_{0}^{2}\frac{R_{c}}{r}\right)^{-1},\end{split}$
(C.1)

where $\dot{M}_{\mathrm{env}}$ is the envelope-to-disk mass-accretion rate,
$\mu_{0}=\cos\theta_{0}$ is the initial polar angle of a streamline trajectory
out to $r\rightarrow\infty$, $\mu=\cos\theta$ is the polar angle along the
streamline trajectory, $R_{c}$ is the centrifugal radius where the infalling
material has sufficient angular momentum to maintain an orbit the central
protostar. We take the simplified case in the mid-plane of the inner envelope,
where $\theta_{0}=\theta=90^{\circ}$, which simplifies the equation to

$\rho_{\mathrm{env}}(r)=\frac{\dot{M}_{\mathrm{env}}}{4\pi}\left(2GM_{\star}r^{3}\right)^{-1/2}\left(1+2\frac{R_{c}}{r}\right)^{-1}.$
(C.2)

We show the best-fit disk and envelope density profiles from Tobin et al.
(2020b) in Figure C.1. We see a clear difference in densities between the disk
and envelope (density jump). We plug in $\dot{M}_{\mathrm{env}}=3.2\pm
0.6\times 10^{-5}\,M_{\odot}\,\mathrm{yr^{-1}}$ (no error bars are reported,
so we assume a 20% error), $M_{\star}=2.5\pm 0.2\,M_{\odot}$,
$r=R_{c}=R_{d}=94.4\pm 12.6\,\mathrm{au}$ (Tobin et al., 2020b), we find
$\rho_{\mathrm{env}}(r)=3.8\pm 1.2\times 10^{-16}\,\mathrm{g\,cm^{-3}}$. Using
our new inner envelope density, we re-apply the same steps as in Section 2.3.1
to scale the magnetic field from the core-scale density using the C04 method.
This gives us newly estimated magnetic field strength of
$B_{\mathrm{tot},e}=9.2\pm 2.8\,\mathrm{mG}$. We now plug in the values of

$\displaystyle\delta_{r}=0.8,$ $\displaystyle\delta_{\phi}=1.0,$
$\displaystyle C_{s}=833.0^{+19.5}_{-19.5}\,\mathrm{m\,s^{-1}},$
$\displaystyle R_{d}=94.4^{+12.6}_{-12.6}\,\mathrm{au},$ $\displaystyle
M_{\star}=2.5^{+0.2}_{-0.2}\,M_{\odot},$ $\displaystyle
M_{d}=0.035^{+0.005}_{-0.003}\,M_{\odot}$
$\displaystyle\rho_{d}=3.8^{+1.2}_{-1.2}\times 10^{-16}\,\mathrm{g\,cm^{-3}}$
$\displaystyle B_{\phi}=9.2^{+2.8}_{-2.8}\,\mathrm{mG},$

into the ambipolar diffusivity coefficient equation (Equation 3) to obtain

$\displaystyle\eta_{\mathrm{AD}}$ $\displaystyle=1.7^{+1.2}_{-1.2}\times
10^{19}\,\mathrm{cm^{2}\,s^{-1}}$ $\displaystyle=2.4^{+1.6}_{-1.6}\times
10^{-1}\,\mathrm{s}.$

The dimensionless Elsässer number is estimated to be
$\mathrm{AM}=1.7^{+0.8}_{-0.8}$ (C04 method). These results are
indistinguishable from the values calculated using the disk edge quantities.
Even though the envelope density is estimated to be an order of magnitude
lower than the disk edge, the magnetic field strength is also lower as a
result. Since $\eta_{\mathrm{AD}}$ has a dependence on $\sim\rho$ and $\sim
B_{\phi}^{-2}$, the values end up offsetting each other to give similar
estimates. This shows that either the disk or envelope density can be used
interchangeably to obtain a value for the ambipolar diffusivity coefficient.
We again check whether $B_{\phi}>B_{z}$ using Equation A.5, and find
$B_{z}\approx 1.38\,\mathrm{mG}$. Thus, $B_{\phi}$ is still the dominant
component, although it is more comparable to $B_{z}$ in this case.

##  Appendix D Magnetic Field Strength Estimation

Recently, Lee et al. (2024) derive a new analytical expression to describe how
the magnetic field should scale with density inside a collapsing protostellar
envelope for the first time. Considering the two density regimes ad the core
and disk scales, their magnetic field - density relation (Equation C7 in their
paper) can be simplified to

$B_{\mathrm{0,d}}=B_{\mathrm{0,c}}\left(\frac{M_{\star}+M_{\mathrm{d}}}{M_{\mathrm{c}}}\right)^{0.25}\left(\frac{\rho_{\mathrm{d}}}{\rho_{\mathrm{c}}}\right)^{0.525},$
(D.1)

which can also be used to scale the magnetic field strength down to inner
envelope/protostellar disk density regimes (hereafter, referred to as the L24
method). Here $B_{0}$ has the same meaning as in Appendix B and characterizes
the magnetic flux threading the disk. We assume the case of a vertical
magnetic field, since the effects of inclination are minimal on our estimates,
which gives $B_{0}\approx B_{z}$. Plugging in our known values of
$B_{\mathrm{tot,c}}$, $M_{\star}$, $M_{\mathrm{d}}$, $\rho_{\mathrm{d}}$ and
$\rho_{\mathrm{c}}$, we estimate $B_{\mathrm{0,d}}\approx
B_{z}=17.6^{+6.3}_{-6.2}\,\mathrm{mG}$. This is compatible with the
calculations in the main text within order of magnitude. The discrepancy
results from model presumptions that require further examination, while we do
not discuss the details.

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