# Super-Eddington Magnetized Neutron Star Accretion Flows: a Self-similar
Analysis

Ken Chen School of Astronomy and Space Science, Nanjing University, Nanjing
210023, China  Zi-Gao Dai Department of Astronomy, University of Science and
Technology of China, Hefei 230026, China<EMAIL_ADDRESS>
###### Abstract

The properties of super-Eddington accretion disks exhibit substantial
distinctions from the sub-Eddington ones. In this paper, we investigate the
accretion process of a magnetized neutron star (NS) surrounded by a super-
Eddington disk. By constructing self-similar solutions for the disk structure,
we study in detail an interaction between the NS magnetosphere and the inner
region of the disk, revealing that this interaction takes place within a thin
boundary layer. The magnetosphere truncation radius is found to be
approximately proportional to the Alfvén radius, with a coefficient ranging
between $0.34-0.71$, influenced by the advection and twisting of a magnetic
field, NS rotation, and radiation emitted from an NS accretion column. Under
super-Eddington accretion, the NS can readily spin up to become a rapid
rotator. The proposed model can be employed to explore the accretion and
evolution of NSs in diverse astrophysical contexts, such as ultraluminous
X-ray binaries or active galactic nucleus disks.

Compact object (288); Neutron stars (1108); Accretion (14)

##  1 Introduction

Accretion is a widespread process occurring in various neutron star (NS)
systems, such as newborn NSs within supernova envelopes (e.g. Chevalier,
1989), NSs in high-mass X-ray binaries (e.g. Reig, 2011), and NSs in low-mass
X-ray binaries (e.g. Bhattacharya & van den Heuvel, 1991), among others,
contributing significantly to the evolution of the mass, spin, and magnetic
field properties of the NSs (e.g. Konar, 2017; Igoshev et al., 2021; Abolmasov
et al., 2024), while concurrently producing observable radiation (e.g.
Mushtukov & Tsygankov, 2022; Ascenzi et al., 2024). Therefore, accretion plays
a crucial role in the NS populations (Harding, 2013). The accretion of the NS
is distinctive. Due to the inherent magnetization of the NS, accreting matter
is blocked by the NS magnetosphere from falling freely onto its solid surface,
but can penetrate into the magnetosphere via instability and subsequently
flows along the field lines toward the NS magnetic poles (e.g. Bozzo et al.,
2008; Shakura & Postnov, 2017; Fornasini et al., 2023). Additionally, a
circum-NS disk can form when the accretion material possesses angular
momentum, leading to a slightly more complicated accretion process (e.g.
Romanova & Owocki, 2015).

So far, the majority of studies have primarily focused on sub-Eddington
magnetized NS accretion systems, characterized by a thin and approximately
Keplerian rotating circum-NS disk. However, there exist astrophysical
scenarios that involve super-Eddington disk-accreting NSs. Recently, the
discovery of ultraluminous X-ray pulsars (ULXPs) with apparent luminosities
ranging between $10^{39}-10^{41}\,\rm{erg}\,\rm{s}^{-1}$, although with
uncertain geometrical beaming (Lasota & King, 2023), implies the existence of
super-Eddington NS accretion systems (Fabrika et al., 2021; King et al.,
2023). Besides, NSs are predicted to widely reside in active galactic nucleus
(AGN) disks (e.g. McKernan et al., 2020; Tagawa et al., 2021; Perna et al.,
2021), where the environment is extremely dense and differentially rotating
(e.g. Sirko & Goodman, 2003), thereby leading to a super-Eddington disk
accretion of the NS (e.g. Pan & Yang, 2021; Chen et al., 2023). In contrast to
the standard thin accretion disk, the super-Eddington disk exhibits
distinctive properties, in that it is geometrically thick, sub-Keplerian in
rotation, and advection-dominated in cooling (e.g. Abramowicz et al., 1988);
additionally, the accretion process is always accompanied by a powerful
outflow to reduce the inward inflow mass rate (Blandford & Begelman, 1999; Gu,
2015). Therefore, it is imperative to conduct a comprehensive study of the
super-Eddington magnetized NS accretion system, enabling us to explore
specific properties of NS accretion and the evolution of the NS driven by a
super-Eddington flow. Several works have both analytically and numerically
investigated the accretion structure of this system (e.g. Ohsuga, 2007;
Takahashi & Ohsuga, 2017; Chashkina et al., 2017, 2019; Abarca et al., 2021;
Inoue et al., 2023).

In this paper, we investigate the super-Eddington accretion process of a
magnetized NS by constructing a generalized super-Eddington circum-NS disk
with varying strengths of the disk wind, particularly focusing on an
interaction between the NS magnetosphere and the inner disk. A schematic
diagram of the accretion system is shown in Figure 1. This paper is organized
as follows. In Section 2, we present a self-similar solution to describe the
structure of a super-Eddington disk. The specific magnetosphere-disk
interaction process, as well as the resulting NS accretion properties, are
shown in Section 3. Several discussions are provided in Section 4. Our
conclusions are summarized in Section 5. The terms $c$ and $G$ in this paper
denote the speed of light and the gravitational constant, respectively. The
mass of an NS is taken to be $1.4M_{\odot}$ in the paper.

Figure 1:  Schematic diagram illustrating the super-Eddington disk accretion
process of a magnetized rotating NS, where the certain object is an NS,
$\Omega_{\rm{NS}}$ is the NS spin rate, $R_{\rm{M}}$ denotes the magnetosphere
truncation radius, $\Delta R$ represents the thickness of the interaction
boundary layer, and $R_{\rm{o}}$ is the disk outer boundary, which is
simplified set to be smaller than $R_{\rm{tr}}$, indicating that the launching
of disk winds occurs throughout the entire disk region. Some NS closed
magnetic field lines, depicted by the blue dashed line, would undergo inward
advection within the super-Eddington disk. Consequently, these field lines
become concentrated at the magnetosphere-disk interaction layer, as indicated
by the blue line, leading to an increase in magnetic pressure. It should be
noted that the schematic serves solely for illustrative purposes and does not
accurately depict the relative scale of the system components, while also the
interaction between the NS open field lines and the disk winds is disregarded.

##  2 Super-Eddington accretion disk

###  2.1 Disk Equations

In a super-Eddington accretion disk, internal photons are trapped because of a
dense gas obstructing their vertical diffusion. A large amount of photons and
energy can be advected inward with the gas inflow, rather than escape as
radiation. When the timescale of photon vertical diffusion exceeds that of
disk accretion, the radiation-effective flow converts to be radiation-
ineffective roughly at the trapping radius (e.g. Kato et al., 2008; Kitaki et
al., 2021):

$R_{\rm{tr}}=3\dot{m}hR_{\rm{g}},$ (1)

where $\dot{m}=\dot{M}/\dot{M}_{\rm{Edd}}$ is the dimensionless mass inflow
rate of the disk, $\dot{M}_{\rm{Edd}}=L_{\rm{Edd}}/c^{2}$ is the Eddington
limit accretion rate, $L_{\rm{Edd}}=4\pi GMm_{p}c/\sigma_{T}=1.26\times
10^{38}m\,\rm{erg\,s}^{-1}$, $m=M/M_{\odot}$ is the dimensionless mass of the
NS, $m_{p}$ is the proton mass, $\sigma_{T}$ is the Thompson cross section,
$h=H/R$ is the height ratio of the disk, and $R_{\rm{g}}=GM/c^{2}$ is the
gravitational radius. Inside $R_{\rm{tr}}$, the trapped photons exert strong
radiation pressure, which can induce disk winds to remove mass, angular
momentum, and energy from the accretion disk (Ohsuga & Mineshige, 2007).

Containing the contribution of an outflow, the equation of continuity is
written as

$\frac{1}{R}\frac{d}{dR}\left(R\Sigma V_{\rm{R}}\right)+\frac{1}{2\pi
R}\frac{d\dot{M}_{\rm w}}{dR}=0\ ,$ (2)

where $\Sigma=2\rho H$, $\rho$, and $V_{\rm{R}}$ are the surface density,
density, and radial inflow velocity of the disk, respectively, and the outflow
mass-loss rate $\dot{M}_{\rm{w}}$ is

$\dot{M}_{\rm w}(R)=\int_{R_{\rm{in}}}^{R}4\pi R^{\prime}\dot{m}_{\rm
w}(R^{\prime})dR^{\prime}\ ,$ (3)

where $R_{\rm{in}}$ denotes the radius of the disk inner edge and
$\dot{m}_{\rm w}$ is the mass-loss rate per unit area from each disk face.

The integrated radial momentum equation and the azimuthal equation of motions,
respectively, can be written by

$V_{\rm R}\frac{dV_{\rm R}}{dR}+\left(\Omega_{\rm
K}^{2}-\Omega^{2}\right)R+\frac{1}{\rho}\frac{dP}{dR}=0\ ,$ (4)
$-\frac{1}{R}\frac{d}{dR}\left(R^{3}\Sigma V_{\rm
R}\Omega\right)+\frac{1}{R}\frac{d}{dR}\left(R^{3}\nu\Sigma\frac{d\Omega}{dR}\right)-\frac{\left(lR\right)^{2}\Omega}{2\pi
R}\frac{d\dot{M_{\rm w}}}{dR}=0\ ,$ (5)

where $P$ and $\nu=-\alpha P/(\rho Rd\Omega/dR)$ are the pressure and
viscosity of the disk, $\alpha$ is the viscosity parameter (Shakura & Sunyaev,
1973), and $\Omega_{\rm K}$ is the Keplerian angular velocity. In the last
term of Equation (5), $l\geqslant 1$ quantifies the specific angular momentum
carried away by the disk wind, where $l=1$ represents the outflow removing the
local disk angular momentum at its originating radius (e.g. Knigge, 1999;
Proga, 2000).

The energy equation is

$Q_{\rm{vis}}=Q_{\rm adv}+Q_{\rm rad}+Q_{\rm w}\ ,$ (6)

where $Q_{\rm{vis}}$, $Q_{\rm adv}$, and $Q_{\rm rad}$ are the viscous heating
rate, the advection cooling rate, and the radiation cooling rate per unit
area, respectively, for which the expressions are as follows (e.g. Abramowicz
et al., 1995; Sirko & Goodman, 2003):

$Q_{\rm{vis}}=\nu\Sigma\left(R\frac{d\Omega}{dR}\right)^{2}\ ,$ (7) $Q_{\rm
adv}=\Sigma V_{\rm R}T\frac{dS}{dR}=\Sigma V_{\rm
R}\left(\frac{1}{\gamma-1}\frac{dc_{\rm s}^{2}}{dR}-\frac{c_{\rm
s}^{2}}{\rho}\frac{d\rho}{dR}\right)\ ,$ (8) $Q_{\rm rad}=\frac{16}{3}\sigma
T^{4}\left(\tau+\frac{4}{3}+\frac{2}{3\tau}\right)^{-1}\ ,$ (9)

where $S$ is the specific entropy, $\gamma$ is the ratio of specific heat,
$c_{\rm s}=(P/\rho)^{1/2}$ is the sound speed, and $\tau=\kappa\rho H$ is the
optical depth, for which the electron scattering is included as the main
opacity source, $\kappa=\kappa_{\rm{es}}=0.4$. The quantity $Q_{\rm w}$ in
Equation (6) represents the energy carried away by the outflow, which can be
expressed by

$Q_{\rm w}=2\eta\left(\frac{1}{2}\dot{m}_{\rm w}V_{\rm K}^{2}\right)\ ,$ (10)

where a factor of 2 signifies the outflow that is emitted bilaterally from the
accretion disk, $\eta$ is a specific energy parameter to determine the outflow
strength, and $V_{\rm K}$ is the Keplerian velocity.

Due to the presence of the disk wind, the mass inflow rate would reduce
inward. Numerical simulations show that the reduction roughly follows a power
law within $R_{\rm{tr}}$ (e.g. Yang et al., 2014; Kitaki et al., 2021;
Yoshioka et al., 2022; Yang et al., 2023), so we take the mass inflow rate
varying with radius as (Blandford & Begelman, 1999):

$\dot{M}=-2\pi R\Sigma
V_{\rm{R}}=\dot{M}_{\rm{o}}\left(\frac{R}{R_{\rm{o}}}\right)^{s}\ ,$ (11)

for a disk with $R_{\rm{o}}<R_{\rm{tr}}$, where $\dot{M}_{\rm{o}}$ is the mass
accretion rate at the disk outer boundary $R_{\rm{o}}$, and the power-law
index $s$ is a free parameter, constrained within $0\leqslant s\leqslant 1$,
which approaches zero in the absence of wind, leading to the disk satisfying
the slim solution (e.g. Abramowicz et al., 1988; Narayan & Yi, 1995; Watarai &
Fukue, 1999). Assuming that the complete reduction of mass inflow rate is
solely attributed to the disk wind, then, from Equation (2) and (11), we can
get

$\dot{m}_{\rm w}=\frac{s\dot{M}}{4\pi R^{2}}.$ (12)

###  2.2 Self-similar solutions

Although numerical simulations are the more reliable approach for
investigating the complicated super-Eddington accretion process around compact
objects (e.g. Ohsuga et al., 2005; Jiang et al., 2014; Sądowski et al., 2015;
Kitaki et al., 2018; Takahashi et al., 2018; Abarca et al., 2018; Hu et al.,
2022), analytic solutions are valuable, as they can roughly match the
structure of the accretion systems observed in simulations (e.g. Jiao et al.,
2015). Moreover, these solutions offer convenience in directly constructing
circum-CO disks under varying system parameters (e.g. Blandford & Begelman,
1999; Begelman, 2012; Gu, 2015; Ghoreyshi & Shadmehri, 2020; Zahra Zeraatgari
et al., 2020). Because a super-Eddington accretion disk is radiation-
ineffective, the radiation cooling term is of minor significance in Equation
(6) (e.g. Wu et al., 2022). If we ignore $Q_{\rm rad}$, a set of self-similar
variables–$\rho_{\rm{s}}\varpropto R^{s-3/2}$, $P_{\rm s}\varpropto
R^{s-5/2}$, $V_{\rm{R,s}}\varpropto R^{-1/2}$, $H_{\rm{s}}\varpropto R$, and
$\Omega_{\rm{s}}\varpropto r^{-3/2}$–can simultaneously satisfy the disk
Equations (2), (4), (5), and (6). Considering the vertical force balance, the
disk height can be expressed as $H=c_{\rm s}/\Omega_{\rm K}$ (e.g. Kato et
al., 2008). By combining these five equations, a solution to the disk
structure can be obtained.

Figure 2: Properties of a super-Eddington accretion disk under self-similar
solutions. The left panel represents the disk thickness $h=H/R$ as a function
of the inflow power-law index $s$ considering various system parameters (as
shown in the inset), where the fiducial ones are $\alpha=0.1$, $\gamma=4/3$,
$l=1$, and $\eta=1$. The right panel shows the variation of $h$ and $s$ as a
function of $\lambda$, where $\lambda=s/(H/R)$ qualitatively reflects the
impact of advection on the disk.

The self-similar solution is:

$\rho_{\rm{s}}=\frac{\left(1+2s-2l^{2}s\right)\dot{M}_{\rm
o}\omega}{4\pi\left(1+2s\right)\alpha(GM)^{1/2}R_{\rm
o}^{3/2}h^{3}}\left(\frac{R}{R_{\rm o}}\right)^{s-3/2},$ (13)
$P_{\rm{s}}=\frac{\left(1+2s-2l^{2}s\right)\dot{M}_{\rm
o}(GM)^{1/2}\omega}{4\pi\left(1+2s\right)\alpha R_{\rm
o}^{5/2}h}\left(\frac{R}{R_{\rm o}}\right)^{s-5/2},$ (14)
$V_{\rm{R,s}}=-\frac{\left(1+2s\right)\alpha(GM)^{1/2}h^{2}}{\left(1+2s-2l^{2}s\right)\omega
R_{\rm o}^{1/2}}\left(\frac{R}{R_{\rm o}}\right)^{-1/2},$ (15)
$H_{\rm{s}}=hR,$ (16) $\Omega_{\rm{s}}=\omega\frac{(GM)^{1/2}}{R_{\rm
o}^{3/2}}\left(\frac{R}{R_{\rm o}}\right)^{-3/2},$ (17)

where

$\omega=\frac{1}{2}\left\\{\left(2s-5\right)h^{2}+\left[\left(\left(5-2s\right)^{2}-\frac{8\alpha\left(1+2s^{2}\right)}{1+2s-2l^{2}s}\right)h^{4}+4\left(2s-5\right)h^{2}+4\right]^{1/2}+2\right\\}^{1/2},$
(18)

and $h$ is solved numerically via Equation (6), i.e.,

$\frac{3\left(1+2s-2l^{2}s\right)}{4\left(1+2s\right)}\left\\{\left(2s-5\right)h^{2}+\left[\left(2+\left(2s-5\right)h^{2}\right)^{2}-\frac{8\left(2s+1\right)^{2}\alpha^{2}h^{4}}{\left(1+2s-2l^{2}s\right)^{2}}\right]^{1/2}+2\right\\}+\left(3-2s-\frac{2}{\gamma-1}\right)h^{2}-\eta
s=0.$ (19)

As indicated by Equations (13)-(17), the disk properties are influenced by
$s$, $\alpha$, $\gamma$, $l$, and $\eta$. To investigate these dependencies,
we take $h$ as an indicator and explore its variation, which is shown in the
left panel of Figure 2. First, the super-Eddington accretion disk is
geometrically thick, with $h>0.1$. The value of $h$ decreases with the
increase of $s$, which can be attributed to the enhanced mass loss resulting
from a larger $s$. Consequently, more energy is carried away by the disk wind,
leading to a reduction in the thickness of the resultant disk. Second, for a
large value of $l$ or $\eta$, the maximum value of $s$ in a valid solution is
less than $1$, since a disk would not exist if the loss of angular momentum or
energy via the disk wind were to exceed its total reserve. Third, the disk
structures are basically not influenced by $\alpha$, but exhibit sensitivity
to $\gamma$. In the subsequent calculations, we set $\alpha=0.1$,
$\gamma=4/3$, $l=1$, and $\eta=1$.

Moreover, the selection of $s$ should be based on the disk properties rather
than being chosen randomly. Wu et al. (2022) adopt a reasonably appropriate
assumption that $s$ is proportional to $H/R$, i.e., $s=\lambda(H/R)$, where
$\lambda$ is a constant, considering that the disk thickness can reflect the
strength of advection. Still, $\lambda$ is uncertain, but it can be adjusted
to encompass all values of $s$, as shown in the right panel of Figure 2.
Therefore, we apply all physical values for $s$ from $0$ to $1$ to investigate
the comprehensive properties of the super-Eddington accretion disk.

##  3 Super-Eddington disk accretion of magnetized NS

The fundamental framework of disk accretion around a magnetized NS is widely
accepted in theory works (e.g. Pringle & Rees 1972; Lamb et al. 1973, and see
Lai 2014 for a review) and supported by numerical simulations (e.g. Romanova
et al., 2004, 2012; Zhu et al., 2024). According to this framework, the
circum-NS disk is disrupted by the NS magnetic field due to the high pressure
and stress it exerts at the magnetosphere truncation radius. Subsequently,
within an interaction boundary layer, the angular velocity of the inflow
transitions from the disk value to the NS spin rate through magnetic torque.
The gas, coupled with the rotating NS, is then loaded onto closed field lines
and funneled to the magnetic poles. Using the disk structures constructed in
the previous section, we investigate the specific properties of super-
Eddington accretion around the magnetized NS.

###  3.1 Magnetosphere Truncation Radius

The external magnetic field of an NS exhibits a complex structure superposed
by various multipoles (e.g. Bilous et al., 2019). Since the $n-$th multipole
field varies as $B_{n}\sim\mu_{n}/R^{n+2}$, where $\mu_{n}$ is the $n-$th
magnetic moment, in regions far from the NS surface, higher multipoles fall
off rapidly and the dipole dominates the overall field structure111The dipole
magnetosphere truncation radius $R_{\rm{M}}=3.9\times
10^{7}\,\rm{cm}\,k_{\rm{in}}\left(\mu/10^{30}\,\rm{G}\,\rm{cm}^{3}\right)^{4/7}\left(M/1.4M_{\odot}\right)^{-1/7}\left(\dot{M}_{\rm{in}}/10^{3}\dot{M}_{\rm{Edd}}\right)^{-2/7}$
is much larger than the NS radius, so the multipoles are dominant only if
$\mu_{n}\gg\mu_{\rm{dipole}}$. However, near the NS surface, the multipole
fields can be comparable or stronger than the dipole component. So, the
magnetic field at the NS surface exhibits a complex structure due to the
superposition of multipoles with varying tilt angles, guiding the funneled
inflow to a area wider than the pure dipole magnetic poles (e.g. Long et al.,
2007, 2008; Gregory & Donati, 2011). Consequently, the accretion streams would
overlap and heat various surface regions of the NS, thereby generating
anomalous multiwavelength radiation facilitated by the NS rotation (e.g.
Bilous et al., 2019). Additionally, in the case of super-Eddington accretion,
the funneled streams can form an optically thick envelope that can then
reprocess the photons emitted by the multipole accretion flows encircled in
the closed dipole fields, resulting in a more intricate radiation profile
(Mushtukov et al., 2019).. Therefore, at the magnetosphere truncation radius,
we adopt a dipole structure for the NS magnetic field with a field strength of
$B_{0}=\mu/R^{3}$, where $\mu=10^{30}\,\rm{G\,cm^{3}}$ is set. Simulations
indicate that the formation of a funnel stream occurs approximately at the
radius where magnetic and disk pressure reach an equilibrium (Kulkarni &
Romanova, 2013; Takahashi & Ohsuga, 2017). As the inflow mass rate of a super-
Eddington disk exhibits radial variation, we employ pressure balance as the
criterion for determining the magnetosphere truncation radius $R_{\rm{M}}$
222The ram pressure of the accreting gas is negligible compared to the thermal
pressure of the disk, as
$\rho_{\rm{s}}V_{\rm{R,s}}^{2}/P_{\rm{s}}=\left[\left(1+2s\right)\alpha
h/\left(1+2s-2l^{2}s\right)\omega\right]^{2}$ is much less than $1$, except in
the case where $\alpha\sim 1$ with $s\sim 0$. Therefore, we disregard the
contribution of this term to the pressure balance at $R_{\rm{M}}$., at which
the disk gas becomes trapped by field lines and the mass rate accreted onto
the NS is then $\dot{M}_{\rm{o}}(R_{\rm{M}}/R_{\rm{o}})^{s}$, i.e.:

$P_{\rm{B0}}=P_{\rm{s}},\ \ \ \frac{\mu^{2}}{8\pi
R_{\rm{M}}^{6}}=\frac{\left(1+2s-2l^{2}s\right)\dot{M}_{\rm
o}(GM)^{1/2}\omega}{4\pi\left(1+2s\right)\alpha R_{\rm
o}^{5/2}h}\left(\frac{R}{R_{\rm o}}\right)^{s-5/2},$ (20)

which leads to

$R_{\rm{M}}=k\left(\frac{\mu^{4}R_{\rm{o}}^{2s}}{GM\dot{M}_{\rm{o}}^{2}}\right)^{1/(7+2s)},$
(21)

where

$k=\left[\frac{(1+2s)\alpha h}{2(1+2s-2l^{2}s)\omega}\right]^{2/(7+2s)}$ (22)

describes the dependence of the magnetosphere truncation radius on the
properties of a super-Eddington accretion disk. More precisely, the truncation
is directly determined by the interaction between the NS magnetosphere and the
innermost region of the disk, therefore we also present $R_{\rm{M}}$ in a
widely accepted form (e.g. Ghosh & Lamb, 1979a, b):

$R_{\rm{M}}=k_{\rm{in}}\left(\frac{\mu^{4}}{GM\dot{M}_{\rm{in}}^{2}}\right)^{1/7},$
(23)

where the second factor on the right-hand side is the conventional expression
for the magnetosphere radius in the NS spherical accretion system (e.g. Elsner
& Lamb, 1977), i.e., the Alfvén radius, with
$\dot{M}_{\rm{in}}=\dot{M}(R_{\rm{M}})$ and

$k_{\rm{in}}=\left[\frac{(1+2s)\alpha h}{2(1+2s-2l^{2}s)\omega}\right]^{2/7}.$
(24)

To investigate the impact of the super-Eddington flow on the magnetized NS
accretion, we examine the correlation between $k$ or $k_{\rm{in}}$ and $s$ to
assess if it deviates significantly from the typical value $k_{\rm{in}}\sim
0.4-0.8$ (e.g. Long et al., 2005; Bessolaz et al., 2008; Kulkarni & Romanova,
2013; Zanni & Ferreira, 2013). As shown in Figure 3, the value of $k$ ranges
from $0.34$ to $0.46$, which is similar but relatively low compared to the
typical value, owing to the high pressure of the super-Eddington disk
resulting from significant energy storage and advection processes.
Furthermore, the variation trend of $k$ on $s$ shows an initial increase
followed by a subsequent decrease, which mirrors the behavior of the disk
pressure (not depicted in the paper, for conciseness). Notably, $k$ exhibits
minimal variation with the change in $s$, except for $s>0.9$, where a
substantial compression of the NS magnetosphere caused by high $P_{\rm{s}}$
leads to a reduction of $k$. Correspondingly, $k_{\rm{in}}$ varies with the
range of $0.25-0.43$, which is slightly lower than $k$. Moreover, taking
$\dot{M}_{\rm{in}}$ into account, $k_{\rm{in}}$ exhibits a monotonic decrease
as the value of $s$ increases.

Figure 3: Variation of magnetosphere truncation radius as a function of the
super-Eddington accretion disk’s index $s$, where $k$ and $k_{\rm{in}}$ are
given by Equation (22) and (24), respectively. In each panel, the blue line
shows the closed NS magnetic field lines initially threading the disk are
inefficiently advected inward with the inflow, while the disk truncation
results from the local pressure balance, i.e., $P_{\rm{B0}}=P_{\rm{s}}$; the
golden line shows these closed lines are concentrated at the magnetospheric
boundary to enhance the magnetic field strength, with the NS rotating at an
angular velocity with $R_{\rm{LC}}=2R_{\rm{M}}$, while the green line
correspondingly shows the case of a rotating NS with
$R_{\rm{LC}}=10R_{\rm{M}}$; the orange line shows the compressed dipole lines
are strongly twisted to produce a toroidal component with
$B_{\phi}=B_{\rm{z}}$, which provides extra magnetic pressure; and the purple
line considers the extra feedback of the radiation released near the NS on the
interaction boundary layer.

When the NS closed magnetic field lines thread the accretion disk, they
simultaneously undergo inward advection, along with inflow and outward
diffusion controlled by turbulence (e.g. Parfrey et al., 2017). The advection
of the magnetic field relies on the disk radial velocity $V_{\rm{R,s}}$, i.e.,
Equation (15), while its diffusion is determined by the magnetic diffusivity
$\eta$. In magnetorotational-instability-generated turbulence, $\eta$ can be
expressed as $\eta=\nu/Pr_{\rm{m}}$, where $\nu=2\alpha c_{\rm{s}}H/3\omega$
and the magnetic Prandtl number $Pr_{\rm{m}}\sim 1$ (e.g. Lesur & Longaretti,
2009; Guan & Gammie, 2009; Fromang & Stone, 2009), while the resulting
diffusion velocity can be approximated as $V_{\rm{diff}}\sim\eta/H$ (e.g.
Lubow et al., 1994; Cao, 2011). For a super-Eddington accretion disk, using a
self-similar solution, we find that $V_{\rm{R,s}}\gtrsim V_{\rm{diff}}$, owing
to an effective disk advection. Additionally, as the wind extracts extra
angular momentum from the disk, it would lead to $Pr_{\rm{m}}$ exceeding unity
(Parfrey et al., 2017). Consequently, the threading magnetic field is expected
to overcome diffusion and flow inside. To estimate the impact of the field
advection, we consider an ideal scenario where all closed magnetic field lines
originating from the NS and initially penetrating the accretion disk are
advected inward and concentrated at the magnetosphere-disk interaction layer.
Adopting the magnetic flux conservation

$\delta\left(B\times Area\right)=0,\ \ \ \delta B\pi
R^{2}+\frac{\mu}{R^{3}}\left(-2\pi R\delta R\right)=0,$ (25)

the compressed magnetic fields can be derived as

$\Delta B=\int^{R_{\rm{LC}}}_{R}\delta
B=\int^{R_{\rm{LC}}}_{R}\frac{2\mu}{R^{4}}dR=\frac{2}{3}\mu\left(\frac{1}{R^{3}}-\frac{1}{R_{\rm{LC}}^{3}}\right),$
(26)

where $R_{\rm{LC}}$ is the light cylinder radius, inside which the initial NS
magnetic field lines are closed. Therefore, the magnetic field at the
magnetosphere truncation radius can be estimated by superimposing the
compressed component onto the initial value, i.e.:

$B=B_{0}+\Delta
B=\frac{\mu}{R^{3}}\left(\frac{5}{3}-\frac{2}{3}\frac{R^{3}}{R_{\rm{LC}}^{3}}\right).$
(27)

Due to the concentration of the magnetic field, a higher magnetic pressure,
$P_{\rm{B}}=B^{2}/8\pi$, is generated, which pushes the disk outward,
consequently resulting in an enlarged magnetosphere truncation radius. Two
instances of a rotating NS with $R_{\rm{LC}}=2R_{\rm{M}}$ and
$R_{\rm{LC}}=10R_{\rm{M}}$ are illustrated in Figure 3, the corresponding
values of $k$ are raised to $0.41-0.58$ and $0.42-0.59$, respectively. The
similarity between the two values suggests that $k$ or $k_{\rm{in}}$ is not
significantly affected by the NS rotation, which is consistent with Equation
(27), where the second term, $\propto R_{\rm{LC}}^{-3}$, remains negligible
unless the NS rotates very fast, so that $R_{\rm{LC}}\sim R_{\rm{M}}$. The
compressed magnetic field asymptotically approaches $B\simeq 5/3B_{0}$.

Besides advection and diffusion, the field lines linking the NS and the disk
are twisted due to the differential rotation at each end of the footpoints,
resulting in the generation of a toroidal field component. The resultant
magnetic pressure drives field inflation and reconnection, opening the closed
lines to sever the NS-disk linkage, ultimately arresting the growth of the
toroidal field (Aly & Kuijpers, 1990; Lovelace et al., 1995; Parfrey et al.,
2016). Lai (2014) argues that the maximum toroidal twist is $\lvert
B_{\phi}/B_{z}\lvert_{\rm{max}}\sim 1$. By assuming $B_{\phi}=B_{z}$, the
total magnetic pressure at the magnetosphere-disk interaction layer increases
to $B^{2}/4\pi$, leading to a larger truncation radius. As shown in Figure 3,
the resulting values of $k$ and $k_{\rm{in}}$ are $0.49-0.71$ and $0.40-0.69$,
respectively, which are consistent with the typical values and agree with the
simulation results (e.g. Takahashi & Ohsuga, 2017), wherein $k_{\rm{in}}\sim
0.7$. However, our findings slightly deviate from those of Chashkina et al.
(2017, 2019), where the relative magnetosphere radius, i.e., $k$ and
$k_{\rm{in}}$, exhibits a significant dependence on the inner disk mass inflow
rate. We attribute this discrepancy primarily to the differences in the
boundary conditions employed. It is worth noting that despite the high
efficiency of advection, not all closed magnetic field lines initially
threading the accretion disk are swept inward because of magnetic inflation,
flux opening, and reconnection (Parfrey et al., 2017). The disk winds also
exert a notable impact on the magnetic field structure. Moreover, reconnection
prevents the twisted toroidal magnetic field from always remaining at its
maximum strength. Therefore, the actual values of $k$ and $k_{\rm{in}}$ should
be lower than those derived from the ideal conditions.

Additionally, when the super-Eddington accretion flow is funneled close to the
NS magnetic pole, it gives rise to the formation of an accretion column
(Mushtukov et al., 2015a), within which the released gas energy is converted
to radiation and subsequently escapes from its lateral surface (e.g. Mushtukov
et al., 2015b; Kawashima et al., 2016; Mushtukov et al., 2018). When these
photons from the NS illuminate the interaction boundary, an extra radiation
pressure is exerted upon it (e.g. Chashkina et al., 2019). Considering this
pressure component, the equilibrium criterion is modified to

$P_{\rm{B}}+P_{\rm{rad,NS}}=P_{\rm{s}},\ \ \
P_{\rm{rad,NS}}=\frac{L_{\rm{rad,NS}}}{4\pi R_{\rm{M}}^{2}c},$ (28)

where $L_{\rm{rad,NS}}\simeq GM\dot{M}_{\rm{in}}/R_{\rm{NS}}$ is the
luminosity of the radiation emitted from the accretion column,
$R_{\rm{NS}}=10^{6}\,\rm{cm}$ represents the NS radius, and the radiation is
simplified to be isotropic. The value of $k$ and $k_{\rm{in}}$ for a case of
contrast, as shown in Figure 3, ranges between $0.43-0.63$ and $0.34-0.60$,
respectively. These values are slightly higher than those obtained when
disregarding the contribution of the accretion column radiation, yet the
deviation remains within $5\%$, suggesting that the impact of this radiation
effect on disk truncation is insignificant.

In brief, the magnetosphere truncation radius, which is influenced by the
properties of the super-Eddington disk and determines the mass accretion rate
onto the NS, is found to be proportional to the Alfvén radius, as described by
Equations (22) and (24), where $k$ and $k_{\rm{in}}$ vary within the ranges of
$0.34-0.71$ and $0.25-0.69$, respectively, depending on various effects,
including the advection and twisting of the magnetic field, the rotation of
the NS, as well as the radiation emitted from the NS accretion column. The
resulting mass rate accreted onto the NS is approximated as
$\dot{M}_{\rm{in}}=\dot{M}_{\rm{o}}(R_{\rm{M}}/R_{\rm{o}})^{s}$.

###  3.2 Thickness of Interaction Boundary Layer

Figure 4: Relative thickness of the magnetosphere-disk interaction boundary
layer, i.e., $\Delta R/R_{\rm{M}}$, as a function of $s$ for a sample system
with $\dot{M}_{\rm{o}}=10^{4}\dot{M}_{\rm{Edd}}$ and
$R_{\rm{o}}=10^{4}R_{\rm{g}}$. The blue line represents the case of no
magnetic field advection within the disk and the NS without spinning, i.e.,
$R_{\rm{LC}}\rightarrow\infty$, while the golden line correspondingly shows
the case of $R_{\rm{LC}}=100R_{\rm{M}}$; the green line shows that the
magnetic fields are concentrated and $R_{\rm{LC}}=100R_{\rm{M}}$, the orange
line reduces the degree of field twisting to $B_{\phi}=0.5B_{z}$; and the
purple line speeds up the NS rotation to $R_{\rm{LC}}=10R_{\rm{M}}$. $\Delta
R/R_{\rm{M}}\ll 1$ always holds, indicating that the boundary layer is very
thin.

At the truncation radius, though the NS magnetic pressure hinders the disk
inflow, various instabilities can arise to facilitate further plasma
penetration into the magnetosphere (Romanova & Owocki 2015 and references
therein). The differential rotation between the NS and the magnetosphere
plasma causes a twisting of the field lines, in turn resulting in the
generation of magnetic torque to eliminate the rotational difference within a
boundary layer (e.g. Spruit & Taam, 1993; Rappaport et al., 2004; Lai, 2014).
In the case of a typical Keplerian disk, the interaction boundary layer is
thin (e.g. Kulkarni & Romanova, 2013; Čemeljić et al., 2023); for comparative
analysis, we explore the thickness of this layer under super-Eddington disk
conditions. Taking into account the contribution of both the twisting magnetic
field lines above and below the disk midplane, the magnetic torque per unit
area can be expressed as $R_{\rm{M}}B_{\phi}B_{z}/2\pi$, facilitating the
transition of the inflow angular velocity from the disk’s $\Omega_{\rm{s}}$ to
the NS spin rate $\Omega_{\rm{NS}}$ within the boundary layer, i.e.:

$2\pi R_{\rm{M}}^{2}\Delta R\frac{B_{\phi}B_{z}}{2\pi}=R_{\rm{M}}^{2}\Delta
RB_{\phi}B_{z}=\dot{M}_{\rm{in}}\left|R_{\rm{M}}^{2}\Omega_{\rm{s}}-(R_{\rm{M}}-\Delta
R)^{2}\Omega_{\rm{NS}}\right|,$ (29)

where $\Delta R$ is the layer radial thickness, and the absolute value
quantifies the overall change in the inflow’s specific angular momentum.

The relative thickness of the interaction boundary layer, $\Delta
R/R_{\rm{M}}$, under various conditions is shown in Figure 4; in all cases,
unless otherwise specified, we assume that the field twisting results in
$B_{\phi}=B_{z}$. When the NS spin rate is low, or even when the NS is
approximately nonrotating, i.e., $\Omega_{\rm{NS}}\sim 0$, the second term in
the absolute sign of Equation (29) can be neglected, and the layer thickness
is reduced to

$\Delta R=\frac{\dot{M}_{\rm{in}}\Omega_{\rm{s}}R_{\rm{M}}^{6}}{\mu^{2}},$
(30)

which is thin, with a relative thickness smaller than $0.1$. The variation
trend of $\Delta R/R_{\rm{M}}$ on $s$ exhibits a similar pattern to that of
$R_{\rm{M}}$ or $k$, initially increasing and then decreasing, attributed to
the layer thickness being determined by the torque exerted by the magnetic
fields, which is directly influenced by the magnetosphere truncation radius
through $B\propto R_{\rm{M}}^{-3}$. Spinning up the NS, for example to
$R_{\rm{LC}}=100R_{\rm{M}}$, diminishes the gap in angular velocity between it
and the disk, causing a thinner layer; as a comparison, for an NS with
$R_{\rm{LC}}=10R_{\rm{M}}$, its angular velocity closely matches that of the
disk, leading to an extremely thin boundary. The inclusion of field advection
and accumulation leads to a decrease in the relative layer thickness, as a
stronger field generates a larger torque; however, this effect is not highly
significant. On the contrary, when the degree of field twisting is low, for
instance, $B_{\phi}=0.5B_{z}$, the resulting magnetic torque is attenuated; to
fully transform the rotation of disk gas, a thicker layer is required, for
which the relative thickness still remains below $0.1$. Overall, the
interaction boundary layer between the NS magnetosphere and the super-
Eddington disk is thin, in turn ensuring the availability of Equation (29), as
both the NS magnetic field and the mass inflow rate remain relatively constant
within a narrow $\Delta R$.

##  4 Discussion

###  4.1 Propeller regime

In an NS disk accretion system, when the NS rotates slower than the disk at
its inner boundary, after being coupled with the magnetosphere, the braked
inflow can no longer counteract gravity and becomes channeled toward the NS
magnetic poles along the field lines. Conversely, when the NS is the faster
object, the disk gas is accelerated by the magnetic torque to gain angular
momentum, and the resulting enhanced centrifugal force hinders the subsequent
accretion and may even lead to the ejection of most of the mass at the
magnetosphere boundary, forming an outflow; this process is called the
propeller regime (e.g. Illarionov & Sunyaev, 1975; Lovelace et al., 1999;
Ustyugova et al., 2006). By defining a fastness parameter,
$\omega_{s}=\Omega_{\rm{NS}}/\Omega_{\rm{s}}(R_{\rm{M}})$, the NS is spun up
through gas accretion when $\omega_{s}<1$ and is spun down by interaction
torque when $\omega_{s}>1$. Consequently, the NS reaches an equilibrium state
approximately at $\omega_{s}\simeq 1$ (e.g. Long et al., 2005; Dai & Li, 2006;
Lai, 2014) and the corresponding NS spin period is

$P_{\rm{eq}}=\frac{2\pi}{\Omega_{\rm{s}}(R_{\rm{M}})}=2\pi\omega^{-1}k_{\rm{in}}^{3/2}\left(GM\right)^{-5/7}\left(\frac{\mu^{2}}{\dot{M}_{\rm{in}}}\right)^{3/7}=0.03\,\rm{s}\left(\frac{\mu}{10^{30}\,\rm{G}\,\rm{cm}^{3}}\right)^{6/7}\left(\frac{M}{1.4M_{\odot}}\right)^{-5/7}\left(\frac{\dot{M}_{\rm{in}}}{10^{3}\dot{M}_{\rm{Edd}}}\right)^{-3/7},$
(31)

where the constant $2\pi\omega^{-1}k_{\rm{in}}^{3/2}$ varies in the range
$1-3$ as a function of $s$. The timescale for the accretion-induced spinup of
an initially nonrotating NS to reach its equilibrium state can be roughly
estimated as

$I\Delta\Omega_{\rm{NS}}\simeq\dot{M}_{\rm{in}}R_{\rm{M}}^{2}\Omega_{\rm{s}}\left(R_{\rm{M}}\right)\Delta
t,$ (32)

where $I\sim MR_{\rm{NS}}^{2}$ is the moment of inertia of the NS, leading to

$\Delta t\simeq\frac{I}{\dot{M}_{\rm{in}}R_{\rm{M}}^{2}}=1.7\times 10^{3}\
\rm{yr}\,k_{\rm{in}}^{-2}\left(\frac{\mu}{10^{30}\,\rm{G}\,\rm{cm}^{3}}\right)^{-8/7}\left(\frac{M}{1.4M_{\odot}}\right)^{9/7}\left(\frac{R_{\rm{NS}}}{10\
\rm{km}}\right)^{2}\left(\frac{\dot{M}_{\rm{in}}}{10^{3}\dot{M}_{\rm{Edd}}}\right)^{-3/7},$
(33)

within which the accretion process has negligible impact on the mass and
radius of the NS. Correspondingly, the NS spinup rate is

$\frac{2\pi I\dot{P}}{P^{2}}\simeq\dot{M}_{\rm{in}}\sqrt{G\rm{M}R_{\rm{M}}},\
\ \ \dot{P}=9.8\times
10^{-10}\,\rm{s}\,\rm{s^{-1}}k_{\rm{in}}^{1/2}\left(\frac{\mu}{10^{30}\,\rm{G}\,\rm{cm}^{3}}\right)^{2/7}\left(\frac{M}{1.4M_{\odot}}\right)^{-4/7}\left(\frac{R_{\rm{NS}}}{10\
\rm{km}}\right)^{2}\left(\frac{\dot{M}_{\rm{in}}}{10^{3}\dot{M}_{\rm{Edd}}}\right)^{6/7}\left(\frac{P}{1\,\rm{s}}\right)^{2}.$
(34)

Therefore, in the case of super-Eddington accretion, an NS can readily undergo
rapid spinup given a high mass accretion rate, as observed in ULXP sources,
such as the NGC 5907 ULXP with an X-ray luminosity $\geqslant 10^{41}\
\rm{erg}\ \rm{s}^{-1}$ (Israel et al., 2017), provided that the luminosity
accurately reflects the actual accretion rate333 Among the $O(10^{3})$
observed ULX candidates (Walton et al., 2022), only a few sources exhibit
pulsations, providing confirmation of their central object being an NS, as
shown in Table $2$ of King et al. (2023). Due to the limited number of ULXPs,
conducting statistical studies to investigate the general properties of these
sources is currently unfeasible. The observed X-ray luminosity of the ULXPs
ranging between $10^{39}-10^{41}\,\rm{erg}\,\rm{s^{-1}}$ indicates a range of
the maximum NS accretion rate $10^{2}-10^{4}\dot{M}_{\rm{Edd}}$. For a strong
magnetic field of $\mu\gtrsim 10^{31}\,\rm{G}\,\rm{cm}^{3}$ (e.g. Chashkina et
al., 2019), the equilibrium period is approximately $P_{\rm{eq}}\sim
0.08-0.58\,\rm{s}$, considering the geometrical beaming would decrease
$\dot{M}_{\rm{in}}$ to further raise the period to $O(1)\,\rm{s}$. Despite
$P_{\rm{eq}}=O(10^{-2})\,\rm{s}$, the lifetime of the NS accreting system in
ULX mode remains uncertain (Fabrika et al., 2021), thereby precluding its
eventual evolution toward an equilibrium state. The properties of the ULXP
systems deserve further investigation., or within AGN disk environments, where
the NS accretion rate can reach $10^{5}-10^{6}\dot{M}_{\rm{Edd}}$ (e.g. Chen
et al., 2023). Moreover, the magnetic field experiences decay throughout the
NS accretion process (e.g. Mukherjee, 2017; Konar, 2017; Igoshev et al.,
2021), resulting in a reduction of the equilibrium spin period $P_{\rm{eq}}$
due to an increased inflow-specific angular velocity at smaller $R_{\rm{M}}$.
However, the decrease in inflow angular momentum leads to slower spinup, as
indicated by Equation (33) and (34). Besides, through the analysis of ULXPs,
one can derive $P$ and $\dot{P}$ while estimating $\dot{M}_{\rm{in}}$ for the
system, thereby establishing a lower limit on the strength of the NS magnetic
field. The properties of the super-Eddington disk wind slightly influence the
estimation of the NS magnetic field strength, as indicated by Equation (34),
where $\mu\propto k_{\rm{in}}^{-7/4}$ while $k_{\rm{in}}$ is not significantly
relying on $s$. But the geometrical beaming originating from the wind leads to
an overestimation of $\dot{M}_{\rm{in}}$, thus $\mu$ is significantly
underestimated, since $\mu\propto\dot{M}_{\rm{in}}^{-3}$.

###  4.2 Error from radiation cooling

Figure 5: Radiation cooling as a percentage of total disk energy,
$Q_{\rm{rad}}/(Q_{\rm{vis}}+Q_{\rm{rad}})$, in the
$R_{\rm{o}}-\dot{M}_{\rm{o}}$ parameter space in the case of $s=0.5$, where
the energy rates are divided from the self-similar solutions, roughly
describing the derivation of the solutions ignoring radiation cooling from the
real disk. The magenta line and the region to the tight represent the case of
$R_{\rm{o}}\geqslant R_{\rm{tr}}$, where the disk photons can efficiently
diffuse out and the disk wind would fail to launch, therefore the super-
Eddington disk solution is invalid and the disk transitions to be
geometrically thin.  Figure 6: Radiation cooling ratio
$Q_{\rm{rad}}/(Q_{\rm{vis}}+Q_{\rm{rad}})$ as a function of $s$, where three
cases are taken as examples: $\dot{M}_{\rm{o}}=10^{3}\dot{M}_{\rm{Edd}}$ with
$R_{\rm{o}}=10^{3}R_{\rm{g}}$, $\dot{M}_{\rm{o}}=10^{4}\dot{M}_{\rm{Edd}}$
with $R_{\rm{o}}=10^{4}R_{\rm{g}}$, and
$\dot{M}_{\rm{o}}=10^{5}\dot{M}_{\rm{Edd}}$ with
$R_{\rm{o}}=10^{4}R_{\rm{g}}$.

To construct a self-similar super-Eddington accretion disk, we have neglected
the radiation cooling term in Equation (6). In this subsection, we investigate
whether $Q_{\rm{rad}}$ is indeed a small factor that does not significantly
impact the disk structure. Roughly speaking, the self-similar disk should
produce some additional energy, except its initial $Q_{\rm{vis}}$ for the
runaway radiation. So, using Equation (9), we calculate the radiation cooling
that the self-similar disk is supposed to generate and compare it with the
total energy of the disk encompassing this energy, i.e.,
$Q_{\rm{vis}}+Q_{\rm{rad}}$. Figure 5 and 6 illustrate
$Q_{\rm{rad}}/(Q_{\rm{vis}}+Q_{\rm{rad}})$ as a function of
$R_{\rm{o}}-\dot{M}_{\rm{o}}$ and $s$, respectively. From the figures, the
ratio is consistently much less than unity, indicating that radiation cooling
would not significantly impact the structure of the super-Eddington disk,
except for the case of $R_{\rm{o}}\sim R_{\rm{tr}}$ yet
$\dot{M}_{\rm{o}}/\dot{M}_{\rm{Edd}}\lesssim 1000$, where the disk size
becomes relatively small, hindering efficient photon advection within it. Both
increasing $\dot{M}_{o}$ and decreasing $R_{\rm{o}}$ to achieve $R_{\rm{o}}\ll
R_{\rm{tr}}$ can mitigate the impact of radiation cooling, thereby promoting
self-similarity in super-Eddington disks; two concrete disk structures are
shown in the Appendix A to illustrate this property.

The disk radiation cooling, in addition, does not exert a significant effect
on the accretion process of the NS. The magnetosphere truncation radius is
directly determined by the disk pressure, which only exhibits a slight
deviation from the self-similar value when considering radiation effects, as
shown in Figure 7, as $R_{\rm{M}}$. Or, more intuitively, the inclusion of
radiation cooling requires an increase of disk inflow mass rate to generate
more energy. Given that $Q_{\rm{rad}}$ is much lower than $Q_{\rm{vis}}$, the
alteration in the mass rate would thus be substantially smaller than
$\dot{M}_{\rm{o}}$; since $R_{\rm{M}}\propto\dot{M}_{\rm{o}}^{2/(7+2s)}$, its
relative change can be neglected.

###  4.3 Caveat

In this paper, we construct the radial structure of the super-Eddington disk
and adopt the vertical hydrostatic equilibrium assumption. Although the
resulting self-similar solution can conveniently provide the disk properties
under given initial conditions, they should be regarded as approximations. In
reality, the analytical solution for the disks should incorporate a three-
dimensional or at least two-dimensional structure and discard the vertical
force balance assumption, as the formation of disk winds indicates the dynamic
nature of the vertical gas; only by constructing a stratified disk structure
can we better predict the properties of the disk wind and inflow (e.g. Cao &
Gu, 2022; Jiao, 2023).

Besides, when investigating the magnetosphere-disk interaction, the NS
magnetic field is simplified as a dipole. However, the presence of an
accretion disk would result in the compression of the magnetosphere into a
nondipole shape, thereby impacting the disk truncation (e.g. Aly, 1980;
Kulkarni & Romanova, 2013). In addition, accretion onto the NS solid surface
introduces radiation feedback, which not only induces an outward displacement
of the disk inner boundary, as shown in Figure 3, but also brings about
significant alterations to the disk vertical structure (e.g. Takahashi &
Ohsuga, 2017; Takahashi et al., 2018). Furthermore, the disk wind would
interact with the open field lines (e.g. Mushtukov & Portegies Zwart, 2023),
leading to reciprocal influences on their respective properties, which would
thereby alter the magnetic field structure and affect the disk cooling and
angular momentum transfer.

In a word, to provide a more precise depiction of the super-Eddington
magnetized NS accretion system, it is imperative to construct a two-
dimensional disk and explore the intricate interplay among the disk, the NS
magnetic field, the disk wind, and the NS radiation, which will be studied
elsewhere.

##  5 Summary

In this work, we have explored the properties of a super-Eddington magnetized
NS accretion system. To comprehensively analyze the system, we have developed
general self-similar solutions for the disk structure by considering variable
properties of the disk wind, including its mass rate, specific energy, and
angular momentum. Extending the disk toward the vicinity of the NS, we have
investigated the specific interaction between the disk and the NS
magnetosphere, finding that the entire accretion process exhibits similar
properties to a typical system involving a sub-Eddington Keplerian disk, where
the magnetosphere truncation radius is approximately proportional to the
Alfvén radius, and the interaction boundary layer is thin. The magnetosphere
truncation is slightly influenced by variations in the super-Eddington disk
wind, unless the wind exhibits significant strength; conversely, it is
predominantly affected by the advection and twisting of the NS magnetic
fields. Additionally, the rotation of the NS and the radiation feedback from
its accretion column are secondary factors. The resulting proportional
coefficient varies within the range of $0.34-0.71$. Although containing
simplification, we suggest that our model can provide a preliminary
description of the accretion system, thus serving as a foundation for
constructing a more accurate system structure or for studying the evolution of
NSs under super-Eddington disk accretion.

This work was supported by the National SKA Program of China (grant No.
2020SKA0120302), and the National Natural Science Foundation of China (grant
No. 12393812).

##  Appendix A Numerical Solutions of Super-Eddington Accretion Disk

Figure 7: The deviation between numerical and analytic solutions of super-
Eddington accretion disks, taking a self-similar solution as the boundary
conditions. In each panel, the blue lines show the case of
$\dot{M}_{\rm{o}}=10^{3}\dot{M}_{\rm{Edd}},R_{\rm{o}}=10^{3}R_{\rm{g}}$, and
the golden lines show the case of
$\dot{M}_{\rm{o}}=10^{4}\dot{M}_{\rm{Edd}},R_{\rm{o}}=10^{3}R_{\rm{g}}$; the
dashed lines are the self-similar solutions ignoring radiation cooling, and
the solid lines are the structures derived from numerically solving the disk
equations directly. All the disk parameters are dimensionless, with
$\dot{m}_{\rm{o}},\rho,P,h,v_{\rm{r}}$, and $\Omega$ as units of
$\dot{M}_{\rm{Edd}},\dot{M}_{\rm{Edd}}/\sqrt{GMR_{\rm{g}}^{3}},\dot{M}_{\rm{Edd}}\sqrt{GM/R_{\rm{g}}^{5}},R_{\rm{g}},\sqrt{GM/R_{\rm{g}}}$,
and $\sqrt{GM/R_{\rm{g}}^{3}}$, respectively.

To directly demonstrate the impact of neglecting the radiation cooling term in
Equation (6) on disk structures, we employ the disk self-similar solutions at
the outer radius as boundary conditions and numerically solve differential
Equations (2)-(6) inward to determine the resultant disk structure. To specify
the property of the disk wind, we adopt $s=\lambda(H/R)$ with $\lambda=1$, and
the obtained value of $s$ is $0.412$. Two examples,
$\dot{M}_{\rm{o}}=10^{3}\dot{M}_{\rm{Edd}},R_{\rm{o}}=10^{3}R_{\rm{g}}$
representing scenarios where $R_{\rm{o}}\sim R_{\rm{tr}}$, and
$\dot{M}_{\rm{o}}=10^{4}\dot{M}_{\rm{Edd}},R_{\rm{o}}=10^{3}R_{\rm{g}}$
representing scenarios where $R_{\rm{o}}\ll R_{\rm{tr}}$, are shown in Figure
7. From the figure, the numerical solution exhibits minimal deviation from the
self-similar one when $R_{\rm{o}}\ll R_{\rm{tr}}$, as the contribution of the
radiation cooling is negligible, which is also illustrated in Figure 6. But
when $R_{\rm{o}}\sim R_{\rm{tr}}$, the proportion of the radiation cooling is
increased, resulting in a slight deviation of $\dot{m}_{\rm{o}},\rho$, and
$P$. Additionally, due to the crucial role of boundary conditions in solving
differential equations, selecting the self-similar values for inwardly
numerical calculations would lead to an accumulative deviation of
$h,v_{\rm{r}}$, and $\Omega$. On the whole, the analytic self-similar solution
exhibits a reasonable level of validity, while the numerical solution of a
super-Eddington accretion disk deserves further investigation.

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0 Research  Architecture of the WordStream Maker, with the integration of data
wrangling, natural language processing (NLP) extraction and computing text
characteristics for visualization.

# WordStream Maker: A Lightweight End-to-end Visualization Platform for
Qualitative Time-series Data

Huyen N. Nguyen
University of New Hampshire
Texas Tech University e-mail<EMAIL_ADDRESS>Tommy Dang
Texas Tech University  e-mail<EMAIL_ADDRESS>Kathleen A. Bowe
University of New Hampshire e-mail<EMAIL_ADDRESS>
###### Abstract

Whether it is in the form of transcribed conversations, blog posts, or tweets,
qualitative data provides a reader with rich insight into both the overarching
trends as well as the diversity of human ideas expressed through text.
Handling and analyzing large amounts of qualitative data, however, is
difficult, often requiring multiple time-intensive perusals in order to
identify patterns. This difficulty is multiplied with each additional question
or time point present in a data set. A primary challenge then is creating
visualizations that support the interpretation of qualitative data by making
it easier to identify and explore trends of interest. By combining the
affordances of both text and visualizations, WordStream has previously enabled
ease of information retrieval and processing of time-series text data, but the
data-wrangling necessary to produce a WordStream remains a significant barrier
for non-technical users. In response, this paper presents WordStream Maker: an
end-to-end platform with a pipeline that utilizes natural language processing
(NLP) to help non-technical users process raw text data and generate a
customizable visualization without programming practice. Lessons learned from
integrating NLP into visualization and scaling to large data sets are
discussed, along with use cases to demonstrate the usefulness of the platform.

Human-centered computingVisualizationVisualization systems and tools; Human-
centered computingVisualizationVisualization application domainsInformation
visualization;

##  1 Introduction

From everyday communication to analytical conversations, qualitative text data
is around us in all shapes and forms. Documents, news, and transcriptions of
audio and video recordings are examples of qualitative data; an area that has
gained huge growth in recent years–social media data–is also considered
qualitative data on a quantitative scale [10]. With the rapidly increasing
amount of data over time, the question is: how can we best leverage the data
that we already have to deal with on a daily basis into a more valuable asset?

Our previous effort to convey the evolution of topics over time in multiple
categories resulted in the WordStream visualization [6]. WordStream combines
the advantages of Wordle [24] and stream graph [11] into a hybrid
visualization to represent the topic streams, especially when they highly
fluctuate. By preserving the terms in the visualization, WordStream enables
both ease of information retrieval and information processing. However, the
production of WordStream is still difficult for non-technical users.
Furthermore, a recent work by Nguyen et al. [18], indicates a gap between data
visualization and qualitative research in data visualization literacy. Hence,
our primary aim is to build a visualization platform that helps bridge this
gap and ease the process of extracting insights from temporal patterns in text
data.

Transforming typical raw data into usable data involves the process of
cleansing, merging, formatting, extracting, and converting– generally known as
’data wrangling’ [13] that amounts up to 80% of the development time and cost
in data warehousing projects [8]. In qualitative data, this tedious task is
coupled with the context-laden, conceptual characteristics of the data itself,
making data processing increasingly challenging. With the advancement of
natural language processing (NLP), the complex textual structure can be broken
down and interpreted to assist humans in comprehension and communication.
However, applying NLP to process text data remains challenging to non-
technical users, especially when a large data set is involved. In WordStream
Maker, the processing of raw input data is embedded in the pipeline, and
parameters are available for customization on demand.

WordStream Maker is inspired by several interesting existing works. First is
the concept of the reusable chart, proposed by Mike Bostock [3], which
provides reconfiguration of a visualization model for customization
flexibility depending on the user’s needs. Following this concept, RAWGraphs
by Mauri et al. [15] introduces a visualization platform to create open
outputs, opening many possibilities for encoding data dimensions on uncommon
visual models. For static word clouds, the Word Cloud Generator developed by
Jason Davies [9] offers a straightforward way to generate such clouds
efficiently. However, there is still a lack of platforms for text data and
qualitative data in general, especially text data coupled with the time
element.

Therefore, we created an end-to-end platform to help process raw time-series
text data on the fly, extracting text characteristics with NLP, generating the
time-series visualization, and aiding users in customizing the representation.
The system is lightweight and runs completely on the end user’s browser; no
server side and database needed also help with data privacy. The lessons
learned from integrating NLP directly into visualization and scaling to large
data sets are also discussed.

The rest of the paper is structured as follows: Section 2 presents related
work to WordStream Maker. Section 3 describes WordStream Maker in more detail
in terms of architecture and user interface. Section 4 discusses the findings
with WordStream Maker. Finally, Section 5 concludes this paper and gives an
outlook for future work.

Figure 1: Visual interface of WordStream Maker.

##  2 Previous Work

In the state-of-the-art report on text data streams by Wanner et al. [25],
word clouds are considered to be the only visualization technique based on
textual data, while timeline and circular representations are more prevalent
to visualize the evolution of the data over time. A more recent work by Ottley
et al. [19] indicates that effective representations should take advantage of
the affordances of both text and visualization. Taking this into account, we
developed WordStream [6] as an effort to represent the topic streams in a
temporal manner.

The technique of WordStream has been applied in various domains to explore the
evolution of text data over time. The visualization was first introduced with
topics from IEEE VIS publications, Huffington Post collection, along with
other online blogs [6]. During the COVID-19 pandemic, the visualization is
integrated into a tool called CovidStream to monitor the evolution of emotions
associated with COVID-19 in Peru by Baca et al. [1]. The technique is also
employed to analyze emerging topics on social media: assisting earthquake
situational analysis [17, 5], cybersecurity [23] and natural resources [7].
Recent work shows the extensions of WordStream on educational assessments [18]
and in the search for potential crowdfunding investors [26].

Several directions are proposed to incorporate NLP into visual analytics. A
natural language interface (NLI) in interactive visualization systems can
offer graphical answers to address ambiguous questions from users [12, 21].
Narechania et al. [16] introduced NL4DV, a robust Python package assisting
visualization developers in creating new visualization NLIs and integrating
natural language input into their systems without prior knowledge in NLP.
While training and task framing remain challenges due to the characteristics
of natural language input [22], by applying automated text mining methods in
extracting and discovering knowledge in unstructured data sources [2], there
are many open opportunities in the intersection of natural language and
visualization.

##  3 WordStream Maker Platform

WordStream Maker is an open source web application111https://github.com/huyen-
nguyen/maker based on mainly two libraries: D3.js [4] for the creation of
WordStream visualization and Compromise [14] library for NLP engine.

###  3.1 Platform architecture

The overall architecture of WordStream Maker platform is presented in Figure
WordStream Maker: A Lightweight End-to-end Visualization Platform for
Qualitative Time-series Data. First, the primary input to the platform is a
raw time-series text file. Input data then goes through the main component of
WordStream Maker, consisting of three sub-components: data wrangling, NLP
extraction, and text characteristics computing. As the data source is
heterogeneous, data cleansing is crucial to ensure the data is credible and
usable. There are many cases where more than one row shares the same
timestamp; hence merging data is the next important step.

The next sub-component is the NLP engine. This module provides Part-of-speech
(POS) tagging, e.g., labeling whether a word is a noun, a verb, or other parts
of speech such as preposition or adverb. POS tagging not only helps to extract
the desired category but also makes it easier to remove stop words, which are
commonly used and often bear little meaning. These stop words are often
determiners, conjunctions, or prepositions; hence, we can skip these
categories by using POS tagging. Diving in the Named-entity Recognition (NER)
within the Noun category, three popular categories are chosen: Places, Person,
and Organization. The application of these NLP modules in visualization will
be discussed in Section 4. NLP engine also provides lemmatization, helping to
trace the root form of any given word. This function is very useful in
WordStream visualization; for example, the verb “study” in the present tense
(study/studying) or past tense (studied) conveys the same topic; hence should
be represented in the same root word “study”.

The last sub-component is text characteristics computing, which serves
directly to the visualization. The importance of a term can be represented by
frequency, as in Wordle and other word cloud formations. In WordStream Maker,
we use two other characteristics, namely sudden change and term frequency-
inverse document frequency (TF-IDF). While TF-IDF is a statistical measure
that evaluates the relevance of a word to a document in a collection of
documents [20], sudden change is calculated based on sharp changes in
frequency. Let $F_{1},F_{2},\ldots,F_{n}$ be the frequency of a word at $n$
different time points. The sudden change series ($S_{1},S_{2},\ldots,S_{n}$):
$S_{t}=\frac{(F_{t}+1)}{(F_{t-1}+1)}$.

###  3.2 User interface

The user interface of WordStream Maker is divided into three sections, as
shown in Figure 1: (1) Data loading and preview, (2) WordStream visualization,
and (3) Options for customization.

In the first section, users have the option of uploading a file containing
tabular data in the tabular format of comma-separated values (CSV) or tab-
separated values (TSV). For qualitative time-series data, the file must have
one column representing the time element and one representing the body of
text. There is also an option to load a sample data set for demonstration
purposes. After the data is loaded, the window on the right will render a
preview of input data as a table.

The second section is the WordStream visualization. At this point, the text
data has been completely processed and parsed, and keywords have been
extracted. The WordStream library will generate the visualization based on
extracted data, using a default configuration. The width of the stream
presents the total frequency, and the font size of each word corresponds to
its individual frequency (or other selected text characteristics).

The last section is for customizing the view: any adjustments will directly
affect the WordStream visualization above. This section is divided into three
groups: visualization-related, NLP-related, and text representation-related.
In the first group, users can adjust the minimum and maximum font size in the
WordStream, the number of words displayed in each stream for each time step,
and the sizes (width and height) of the view. In the second group, the user
can select to use POS tagging (default) or NER. POS tagging offers three
categories: Noun, Verb, and Adjective, while NER corresponds to three main
sub-categories: Person, Place, and Organization. In the last group,
representation of text characteristics, the three options are frequency
(default), sudden change, and TF-IDF.

###  3.3 Availability

WordStream Maker is available at https://huyen-nguyen.github.io/maker/.

##  4 Discussion

###  4.1 Use cases: Educational Assessment Data

The educational assessment data set [18] contains 63 entries with attributes
such as Course ID, Course Name, Prompt Text, Response Text. This data set
reflects the journal entries where students give answers to weekly prompts
made by instructors for class assessments. In this case, WordStream Maker is
used for exploration without specific visual analytics tasks.

With POS tagging and frequency option (Figure 2(a)), words are classified by
Noun, Verb, and Adjective, with their font size indicating their occurrences
in the text source. When we keep the frequency option and change POS tagging
to NER (Figure 2(b)), we can observe that: 1) the number of words falling into
the new categorization greatly declines, and 2) there are only two prominent
terms in the second view and they are of the same word “google”. This is
because POS tagging provides more generic categorization, and the only
organization that gains the most interest (based on frequency) is Google. When
we keep the NER categorization and change to sudden change representation
(Figure 2(c)), now the minor words in the previous view have the chance to
shine, as there is a sharp increase in the mention of other organizations such
as “microsoft”, “github”, “myspace”. This type of observation is useful in
data exploration, especially in the early stage of the analysis process when
the users may not be sure about what questions should be asked.

Figure 2: Journal data for education assessments throughout nine weeks.
Different combinations of Part-of-speech (POS) Tagging, Named-entity
Recognition (NER) in the NLP module, along with Frequency and Sudden Change in
the text characteristics module.

###  4.2 Data exploration

For different domains and different tasks, different ways of categorization
should be applied to the qualitative data. Another use case is presented in
Figure 3: Keywords from publications on protein pathways from 1996 to 2014.
Summarizing from previous study [17, 7, 18] and WordStream Maker, we observe
that: 1) NER categorization works best for data from social media or written
form of response. 2) POS tagging can be employed for various domains; however,
its generalization can be too broad to form a coherent story or specific
insight.

Figure 3: Pathway protein. Different combinations of Part-of-speech (POS)
Tagging and Named-entity Recognition (NER) with frequency.

###  4.3 Trade-offs

With the purpose of building a lightweight application, and the data
processing part is meant to be on the fly, the selection on the NLP engine
itself requires a modest-sized library. We go with Compromise due to the small
size of its minified version (250kb) and the diverse NLP functionalities it
provides. When it comes to tokenization– break a chunk of words into smaller
units, there are two options. One is to break the sentence into individual
words: a word is determined whenever we encounter the next whitespace. The
other way is to break sentences into noun chunks, each containing one or more
words; this way, we can preserve the meaning of the noun phrases. While the
former approach is more straightforward, the second approach helps to maintain
the semantics of text structure.

We run the experiment on a macOS Monterey 12.2.1 of 2.5GHz, and 16GB RAM. In
runtime, for a text file of size 1.5MB, containing more than 5000 rows, the
first approach takes less than 3 seconds to load data, clean data, and run NLP
extraction. However, the second approach takes 1 minute and 50 seconds, which
is very time-consuming. With the aim of a lightweight platform, we choose
simple tokenization at a small cost of semantics.

##  5 Conclusion and Future Work

In this paper, we present WordStream Maker, a lightweight web-based platform
to help non-technical users process raw text data and generate a customizable
visualization of WordStream without programming practice. The application is
open source and meant to be expandable. Outlook for future work will focus on
supporting more input types, user interactions, and applicability of
WordStream Maker in a variety of use cases.

###### Acknowledgements.

This project was supported by the National Aeronautics and Space
Administration (NASA) and the Gordon Research Conferences under NASA Award
#17-TWSC17-0055. This paper was completed during the first author’s internship
at the University of New Hampshire, whose warm hospitality and support were
greatly appreciated.

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