Modeling of Thermal Emission from ULX Pulsar Swift J0243.6+6124 with General Relativistic Radiation MHD Simulations

The basic equations are as follows (see, e.g., Takahashi et al. 2018): the mass conservation law,

$$\begin{eqnarray}&&{\left(\rho {u}^{\mu }\right)}_{;\mu }=0,\end{eqnarray} \tag{ 1 }$$

the energy-momentum conservation laws for magnetohydrodynamics (MHD),

$$\begin{eqnarray}&&{\left({T}^{\mu \nu }\right)}_{;\nu }={G}^{\mu },\end{eqnarray} \tag{ 2 }$$

the energy-momentum conservation laws for the radiation field,

$$\begin{eqnarray}&&{\left({R}^{\mu \nu }\right)}_{;\nu }=-{G}^{\mu },\end{eqnarray} \tag{ 3 }$$

the induction equations,

$$\begin{eqnarray}&&{\partial }_{t}\left(\sqrt{-g}{B}^{i}\right)+{\partial }_{j}\left\{\sqrt{-g}\left({b}^{i}{u}^{j}-{b}^{j}{u}^{i}\right)\right\}=0,\end{eqnarray} \tag{ 4 }$$

where ρ is the proper mass density, u^μ is the four-velocity of the gas, Bⁱ is the magnetic field vector in the laboratory frame, b^μ is the magnetic four-vector in the fluid frame, and g is the determinant of the metric, $g=\det ({g}_{\mu \nu })$. The energy-momentum tensor of ideal MHD T^{μ ν} consists of that of fluid ${{T}_{\mathrm{gas}}}^{\mu \nu }$ and electromagnetic field M^{μ ν} :

$$\begin{eqnarray}&&{{T}_{\mathrm{gas}}}^{\mu \nu }=(\rho +e+{p}_{\mathrm{gas}}){u}^{\mu }{u}^{\nu }+{p}_{\mathrm{gas}}{g}^{\mu \nu },\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{M}^{\mu \nu }={b}^{2}{u}^{\mu }{u}^{\nu }+{p}_{\mathrm{mag}}{g}^{\mu \nu }-{b}^{\mu }{b}^{\nu },\end{eqnarray} \tag{ 6 }$$

where e is the internal energy density, p_gas = (Γ − 1)e is the gas pressure (Γ = 5/3), and p_mag = b²/2 is the magnetic pressure in the fluid frame. The energy-momentum tensor of the radiation field in the M1 formalism is given by (Sadowski et al. 2013)

$$\begin{eqnarray}&&{R}^{\mu \nu }=\left(\bar{E}+{p}_{\mathrm{rad}}\right){{u}_{{\rm{R}}}}^{\mu }{{u}_{{\rm{R}}}}^{\nu }+\displaystyle \frac{1}{3}\bar{E}{g}^{\mu \nu }.\end{eqnarray} \tag{ 7 }$$

Here, $\bar{E}$, ${p}_{\mathrm{rad}}=\bar{E}/3$, and ${{u}_{{\rm{R}}}}^{\mu }$ are the radiation energy density, the radiation pressure in the radiation rest frame, and the four-velocity of the radiation rest frame, respectively. The radiation four-force G^μ , which describes the interaction between the ideal MHD and radiation field, is defined as

$$\begin{eqnarray}\begin{array}{rcl}{G}^{\mu } & = & -\rho {\kappa }_{\mathrm{abs}}\left({R}^{\mu \alpha }{u}_{\alpha }+4\pi \hat{B}{u}^{\mu }\right)\\ & & -\rho {\kappa }_{\mathrm{sca}}\left({R}^{\mu \alpha }{u}_{\alpha }+{R}^{\alpha \beta }{u}_{\alpha }{u}_{\beta }{u}^{\mu }\right)+{{G}_{\mathrm{comp}}}^{\mu },\end{array}\end{eqnarray} \tag{ 8 }$$

where κ_abs and κ_sca are the opacity for free–free absorption and isotropic electron scattering, respectively:

$$\begin{eqnarray}&&{\kappa }_{\mathrm{abs}}=6.4\times {10}^{22}\rho {T}_{{\rm{e}}}^{-3.5}\,\,\,[{\mathrm{cm}}^{2}\,{{\rm{g}}}^{-1}],\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&{\kappa }_{\mathrm{sca}}=0.4\,\,\,[{\mathrm{cm}}^{2}\,{{\rm{g}}}^{-1}].\end{eqnarray} \tag{ 10 }$$

Here, T_e is the electron temperature. The blackbody intensity is given by $\hat{B}=a{{T}_{{\rm{e}}}}^{4}/4\pi$, where a is the radiation constant. In this study, we consider the thermal Comptonization defined as follows (Fragile et al. 2018; Utsumi et al. 2022):

$$\begin{eqnarray}&&{{G}_{\mathrm{comp}}}^{\mu }=-{\kappa }_{\mathrm{sca}}\rho \hat{E}\displaystyle \frac{4k({T}_{{\rm{e}}}-{T}_{{\rm{r}}})}{{m}_{{\rm{e}}}}{u}^{\mu }.\end{eqnarray} \tag{ 11 }$$

Here, $\hat{E}$ is the radiation energy density in the fluid frame, ${T}_{r}={(\hat{E}/a)}^{1/4}$ is the radiation temperature, and m_e is the electron rest mass. We take T_e = T_g for simplicity, where T_g is the gas temperature. The gas temperature can be derived from

$$\begin{eqnarray}&&{T}_{{\rm{g}}}=\displaystyle \frac{{\mu }_{{\rm{w}}}{m}_{{\rm{p}}}{p}_{\mathrm{gas}}}{\rho k},\end{eqnarray} \tag{ 12 }$$

where m_p is the proton mass, k is the Boltzmann constant, and μ_w = 0.5 is the mean molecular weight. We consider the mean-field dynamo in our simulations (Sadowski et al. 2015).

In this study, we take M_NS = 1.4M_⊙ and r_NS = 10 km, where M_NS and r_NS are the mass and radius of the NS, and consider the NSs with dipole magnetic fields (Wasserman & Shapiro 1983). The dipole magnetic field strength at the NS surface B_NS applied in this study, which is relatively weaker than the typical value observed in the X-ray pulsar, 10¹¹⁻¹³ G (see Section 4.5), is described in Table 1. Table 1 also reports the maximum gas density of the initial torus ρ₀, the end time of the simulation t_end, the time after which the mass outflow rate is almost constant t_eq, and numerical grid points (N_r , N_θ ). The light-crossing time for the gravitational radius of the NS, r_g = M_NS = 2.1 km, is denoted by t_g. The computational domain consists of r = [r_NS, r_out], where r_out = 2100 km, and θ = [0, π]. The size of the radial grid exponentially increases with r, and the size of the polar grid is uniform (Takahashi & Ohsuga 2017). Multipole components of the NS magnetic field are not considered. GR-RMHD simulations considering the multipole components are left as important future work, although the GR-MHD simulations employing quadrupole fields were recently performed by Das et al. (2022). The rotation of the NSs is ignored since the rotation period of the NSs observed in ULXPs, 1–10 s, is much longer than the rotational timescale of the accretion disk, ∼10⁻² s, even within r ∼ 100 km. There remains the possibility that ULX is a millisecond pulsar (Kluzniak & Lasota 2015), which we will discuss later. We assume that the magnetic axis coincides with the rotation axis of the accretion disk. The convergence of the simulation results is confirmed using models B1e10D001, B1e10D001_a, B1e10D001_b, and B1e10D001_c. These models have the same initial parameters except for the resolution.

We initially set the equilibrium torus given by Fishbone & Moncrief (1976), as can be seen in Figure 1, where $(R,z)=(r\sin \theta ,r\cos \theta )$. The gas pressure p_gas inside the torus is replaced with p_gas + p_rad by assuming a local thermodynamic equilibrium, T_g = T_r. The inner edge of the torus and the maximum pressure radius are set to 210 km and 304.5 km, respectively. The poloidal-loop magnetic fields inside the torus are considered in addition to the dipole magnetic field of the NS. The vector potential of this loop magnetic field is proportional to ρ. The embedded loop magnetic field is antiparallel to the dipole magnetic field at the inner edge of the torus (Romanova et al. 2011; Parfrey & Tchekhovskoy 2017; Takahashi & Ohsuga 2017). We set the maximum (p_gas + p_rad)/p_mag to 100 inside the torus. We give a perturbation on p_gas + p_rad by 10% to break an equilibrium state of the torus.

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The NS and torus are initially surrounded by a relatively hot and low-density corona,

$$\begin{eqnarray*}\begin{array}{rcl}\rho & = & \max \left[{\rho }_{\mathrm{flr}},{\rho }_{{\rm{c}}}{\left(\displaystyle \frac{{r}_{\mathrm{NS}}}{r}\right)}^{-5}\right],\\ {p}_{\mathrm{gas}} & = & \max \left[{p}_{\mathrm{flr}},\min \left[\displaystyle \frac{k\rho {T}_{{\rm{g}},\max }}{{\mu }_{{\rm{w}}}{m}_{{\rm{p}}}},\displaystyle \frac{k{\rho }_{c}{T}_{\mathrm{init}}}{{\mu }_{{\rm{w}}}{m}_{{\rm{p}}}}{\left(\displaystyle \frac{{r}_{\mathrm{NS}}}{r}\right)}^{-5}\right]\right],\end{array}\end{eqnarray*}$$

where ρ_c is determined by ${\sigma }_{{\rm{c}}}={B}_{\mathrm{NS}}^{2}/(4\pi {\rho }_{{\rm{c}}})={10}^{3}$, and T_init is set to 5 × 10¹¹ K. The floor value of the gas density ρ_flr and gas pressure p_flr are described in Section 2.3. The radiation energy density of the corona is $\bar{E}=a{\bar{T}}_{\mathrm{rad}}^{4}$, where ${\bar{T}}_{\mathrm{rad}}={10}^{5}\,{\rm{K}}$. The velocity of the gas uⁱ and radiation rest frame ${u}_{{\rm{R}}}^{i}$ are set to zero in the corona. The present result is not affected by the corona gas, which is set initially. This is because these hot gases are blown away by the radiation force from the accretion disk after some time has elapsed since the simulations began.

We adopt the outgoing boundary at r = r_out, and the reflective boundary at θ = 0, π. We assume that the inner boundary, r = r_NS, is the boundary at which the gas can be swallowed by the NS but the energy cannot be swallowed by the NS (Ohsuga 2007). At inner ghost cells, ρ, p_gas, and $\bar{E}$ are the free boundary, and uⁱ and ${u}_{{\rm{R}}}^{i}$ are set to zero. For the magnetic field, the radial component is fixed to be the dipole field (Wasserman & Shapiro 1983), while the tangential component is set according to the free boundary condition (Parfrey & Tchekhovskoy 2017; Abarca et al. 2021). The numerical flux in mass conservation law, energy conservation law for MHD, and energy conservation law for the radiation field are zero at the NS surface. The numerical flux of the induction equation is also set to zero at the NS surface to satisfy the ideal MHD condition. We also impose the following condition on the gas density and pressure at the NS surface:

$$\begin{eqnarray}&&\rho (t)=\min \left[{\rho }_{* }(t),\,\rho (t=0)\right],\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&{p}_{\mathrm{gas}}(t)=\displaystyle \frac{k\rho (t){T}_{{\rm{g}},* }(t)}{\mu {m}_{{\rm{p}}}},\end{eqnarray} \tag{ 14 }$$

where ρ_* and T_g,* are the gas density and gas temperature calculated from the conservative variables, respectively. In this case, the gas density is set to the initial value by keeping the gas temperature constant at the NS surface. The amount of reduced kinetic and thermal energy densities by applying Equations (13) and (14) is added to the radiation energy density R^t _t . Here, the kinetic and thermal energy densities are defined as follows:

$$\begin{eqnarray}&&{U}_{\mathrm{kinetic}}=\rho {u}^{t}({u}_{t}-\sqrt{-{g}_{\mathrm{tt}}}),\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&{U}_{\mathrm{thermal}}=(e+{p}_{\mathrm{gas}}){u}^{t}{u}_{t}+{p}_{\mathrm{gas}}{g}_{t}^{t}.\end{eqnarray} \tag{ 16 }$$

The mass reduced by the treatment of Equation (13) is considered to be swallowed by the NSs.

We impose a floor condition on ρ and p_gas to solve the GR-RMHD equations stably:

$$\begin{eqnarray}&&{\rho }_{\mathrm{flr}}=\max \left[\displaystyle \frac{{b}^{2}}{{\sigma }_{\max }},\,{10}^{-4}{\rho }_{0}{\left(\displaystyle \frac{r}{{r}_{{\rm{g}}}}\right)}^{-1.5}\right],\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&{p}_{\mathrm{flr}}={10}^{-6}\left({\rm{\Gamma }}-1\right){\rho }_{0}{\left(\displaystyle \frac{r}{{r}_{{\rm{g}}}}\right)}^{-2.5},\end{eqnarray} \tag{ 18 }$$

where ${\sigma }_{\max }$ is set to 10³. The upper and lower limits of the gas temperature are set to 5 × 10¹¹ K and 5 × 10⁵ K, respectively.

The gas density should be low near θ = 0, π because the strong magnetic pressure and tension due to the open poloidal magnetic field lines inhibit the mass flow toward the rotation axis in the very vicinity of the rotation axis. In the numerical simulation, however, the gas reaches the region of θ = 0, π due to the numerical diffusion. Then, the unphysical mass outflow driven by the radiation force is formed close to the polar axis. We decided to ignore the gas–radiation interaction to avoid this problem in this region. Practically, we set the absorption and scattering opacity to zero at σ > σ_rad, where σ = b²/ρ is the magnetization. In this study, we set σ_rad = 10 in all models. This value is set so that the opacity near the rotation axis is zero while the radiation from the dense accretion flows at the NS surface can be solved.

We use the numerical data from period t_eq to t_end shown in Table 1 for investigating a quasi-steady-state structure in the present work, since the flows become quasi-steady and the mass outflow rate is almost constant after t_eq. First, we introduce the quasi-steady-state structure of the accretion flows and outflows around the magnetized NS. Figure 2 shows the time-averaged overview of models B1e10D001 (left) and B1e10D01 (right) in the quasi-steady state. Colors represent ρ in panels (a) and (c), and $\bar{E}$ in panels (b) and (d). The white region with r < 10 km is the NS, and its center is located at the origin, (R, z) = (0, 0). The magnetic field lines are shown by cyan solid lines. We find that the accretion disk is formed near the equatorial plane at 20 km ≲ R ≲ 200 km (the green region in panels (a) and (c)). In panels (b) and (d), it is clear that $\bar{E}$ is large in the accretion disk. The accretion column is formed near the NS magnetic poles, r ≲ 20 km, the details of which are described in Figure 3. Streamlines follow the poloidal velocity vectors in panels (a) and (c). It can be seen that the outflows emanate from the disk toward the low-density region. In addition, turbulent motions exist inside the accretion disk. The radiative flux is represented by streamlines in panels (b) and (d). We can see that the outward radiative flux from the accretion disk appears. The accretion disk, the outflows, and the outward radiative flux from the accretion disk are common to all models. Both ρ and $\bar{E}$ in the accretion disk tend to increase as ρ₀ becomes large. Actually, both of ρ and $\bar{E}$ in model B1e10D01 (ρ₀ = 0.1 g cm⁻³) are an order of magnitude higher than those in model B1e10D001(ρ₀ = 0.01 g cm⁻³).

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We can see that the gas accretes onto the NS around the pole outside the accretion column (r ≲ 100 km and θ ≲ 40°, 140° ≲ θ in panel (a)). This is because the interaction between the gas and radiation is artificially switched off at the high-σ region (σ > σ_rad) in our simulation to maintain numerical stability. The radiation force does not work around the polar axis, and the gas falls down to the NS. We note that the density in this region is too small to contribute to the mass accretion rate, and the effects on the overall structure are negligibly small.

Next, we explain the dependence of the magnetospheric structure on ρ₀. Colors in Figure 3 show ρ, and magnetic field lines are represented by cyan lines. We take models B1e10D001, B1e10D01, and B1e10D1_np. We plot the time-averaged magnetization, $\left\langle \sigma \right\rangle =\left\langle {b}^{2}\right\rangle /\left\langle \rho \right\rangle =10$, by yellow contours. Here, the angle brackets denote the time-averaged value. The magnetization $\left\langle \sigma \right\rangle$ on the high-density side of the yellow line is < 10. In the left panel (model B1e10D001), the high-density accretion column, around (R, z) = (5 km, 12 km), is formed (one-side accretion). Although not shown in Figure 3, the same is true of models B3e9D001, B1e10D002, B3e10D001, B3e10D01, and B3e10D02. In the middle panel (model B1e10D01), the accretion columns are formed at (R, z) = (6 km, ± 10 km), which is true of models B1e10D004, B3e10D04, and B3e10D1. At the truncation radius, which is the radius of the inner edge of the accretion disk, the gas begins to move away from the equatorial plane toward the NS poles. The truncation radius is R ∼ 25 km in model B1e10D001 and R ∼ 13 km in model B1e10D001, and approximately corresponds to the position where $\left\langle \sigma \right\rangle =10$ at the equatorial plane. The density inside this radius is <1% of the density of the disk, except inside the accretion column. Also, the truncation radius is roughly comparable to the magnetospheric radius, r_M, ∼27 km for model B1e10D001 and ∼13 km for model B1e10D01. In the present work, r_M is defined as the maximum radius of the region, where ${\left\langle \beta \right\rangle }_{{u}_{\phi }}\lt 1$, inside the maximum pressure radius of the initial torus, 304.5 km. Here, ${\left\langle \beta \right\rangle }_{{u}_{\phi }}$ is u_ϕ -weighted plasma beta,

$$\begin{eqnarray}\begin{array}{rcl}{\left\langle \beta \right\rangle }_{{u}_{\phi }} & = & \displaystyle \frac{1}{{\displaystyle \int }_{0}^{\pi }\left\langle {u}_{\phi }\right\rangle \sqrt{{g}_{\theta \theta }}d\theta }\\ & & \times \,{\displaystyle \int }_{0}^{\pi }\displaystyle \frac{\left\langle {p}_{\mathrm{gas}}\right\rangle +\left\langle {p}_{\mathrm{rad}}\right\rangle }{\left\langle {p}_{\mathrm{mag}}\right\rangle }\left\langle {u}_{\phi }\right\rangle \sqrt{{g}_{\theta \theta }}d\theta .\end{array}\end{eqnarray} \tag{ 19 }$$

Although p_rad is the radiation pressure in the radiation rest frame, it is approximately equal to the radiation pressure in the fluid frame since the MHD and radiation field inside the accretion disk are in local thermal equilibrium, and therefore the radiation rest frame coincides with the fluid frame. We note that the gas pressure is negligible in the super-Eddington accretion disk (Takahashi & Ohsuga 2017), so that p_mag ∼ p_rad at ${\left\langle \beta \right\rangle }_{{u}_{\phi }}=1$. The coincidence between the truncation radius and r_M indicates that the truncation radius is determined by the balance between p_mag and p_rad. As can be seen from the left and middle panels in Figure 3, the truncation radius decreases as ρ₀ increases.

In model B1e10D1_np, since the radiation pressure of the accretion disk is larger than the magnetic pressure of the dipole field even at the NS surface, the accretion disk reaches the NS surface without being truncated and no accretion column is formed. In this case, the region where ${\left\langle \beta \right\rangle }_{{u}_{\phi }}\gt 1$ reaches the NS surface, and r_M cannot be determined. Although not shown in this figure, accretion columns are not formed in model B3e9D01_np for the same reason. Based on the hypothesis that the radiation from the accretion column is the origin of the X-ray pulse, models B3e9D01_np and B1e10D1_np are inappropriate for explaining ULXPs. Incidentally, the reason why the one-side accretion occurs in model B1e10D001 is described in Section 4.4.

Table 2 lists the time-averaged value of the mass accretion rate ${\dot{M}}_{\mathrm{in}}$ at r = 11 km, mass outflow rate ${\dot{M}}_{\mathrm{out}}$ at r = r_out, radiative luminosity L_rad at r = r_out, and kinetic luminosity L_kin at r = r_out of each model. The Eddington mass accretion rate is denoted by ${\dot{M}}_{\mathrm{Edd}}={L}_{\mathrm{Edd}}/{c}^{2}$. The mass accretion rate and outflow rate are defined as

$$\begin{eqnarray}&&{\dot{M}}_{\mathrm{in}}(r)=-\int \min [\rho {u}^{r},0]\sqrt{-g}d\theta d\phi ,\end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}&&{\dot{M}}_{\mathrm{out}}(r)=\int \max [\rho {u}^{r},0]\sqrt{-g}d\theta d\phi .\end{eqnarray} \tag{ 21 }$$

The radiative and kinetic luminosity can be calculated using following formula (Sadowski et al. 2016):

$$\begin{eqnarray}&&{L}_{\mathrm{rad}}(r)=-\int \min [{R}_{t}^{r},0]\sqrt{-g}d\theta d\phi ,\end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}&&{L}_{\mathrm{kin}}(r)=-\int \min [\rho {u}^{r}\left({u}_{t}+\sqrt{-{g}_{{tt}}}\right),0]\sqrt{-g}d\theta d\phi .\end{eqnarray} \tag{ 23 }$$

As can be seen from this table, ${\dot{M}}_{\mathrm{in}}$, ${\dot{M}}_{\mathrm{out}}$, L_rad, and L_kin tend to be large as ρ₀ increases. We also find that L_kin/L_rad tends to be large as ${\dot{M}}_{\mathrm{in}}$ increases. Indeed, the mass accretion rate is $54{\dot{M}}_{\mathrm{Edd}}$ and L_kin/L_rad ∼ 0.083 in the case of model B1e10D001, mass accretion rate is $500{\dot{M}}_{\mathrm{Edd}}$ and L_kin/L_rad ∼ 0.86 in the case of model B1e10D01, and mass accretion rate is $4700{\dot{M}}_{\mathrm{in}}$ and L_kin/L_rad ∼ 1.2 in the case of model B1e10D1_np. That is, the kinetic luminosity is comparable to the radiation luminosity for the model with high mass accretion rate. These trends are also reported in nonrelativistic radiation hydrodynamics simulation (Ohsuga 2007).

Table 2. Time-averaged Value for the Mass Accretion Rate, Mass Outflow Rate, Radiative Luminosity, and Kinetic Luminosity

	$\left\langle \right.{\dot{M}}_{\mathrm{in}}(r=11\,\mathrm{km})\left.\right\rangle$	$\left\langle \right.{\dot{M}}_{\mathrm{out}}({r}_{\mathrm{out}})\left.\right\rangle$	$\left\langle {L}_{\mathrm{rad}}({r}_{\mathrm{out}})\right\rangle$	$\left\langle {L}_{\mathrm{kin}}({r}_{\mathrm{out}})\right\rangle$	Tbb	rbb
Model	[${\dot{M}}_{\mathrm{Edd}}$]	[${\dot{M}}_{\mathrm{Edd}}$]	[LEdd]	[LEdd]	[107 K]	[km]
B3e9D001	32	36	3.0	0.15	0.72	57
B3e9D01_np	220	2700	130	47	0.63	230
B1e10D001	54	74	4.1	0.34	0.71	58
B1e10D002	110	110	9.5	1.0	0.73	65
B1e10D004	160	730	21	7.1	0.66	130
B1e10D01	500	2600	42	36	0.66	240
B1e10D1_np	4700	23000	500	410	0.50	1100
B3e10D001	98	170	5.5	0.89	0.75	50
B3e10D01	720	2900	140	56	0.69	270
B3e10D02	650	4600	150	37	0.63	250
B3e10D04	1500	13000	320	210	0.59	660
B3e10D1	2300	28000	490	460	0.55	1300

Note. The mass accretion rate is measured at r = 11 km, and the others are calculated at r = r_out.

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Figure 4 shows r_M and the time derivative of the NS rotation period (spin-up rate) $\dot{P}$ estimated using our numerical models as a function of the mass accretion rate. Points in Figure 4(a) indicate the resulting r_M for B_NS = 3.3 × 10⁹ G (red), 10¹⁰ G (blue), and 3.3 × 10¹⁰ G (black), in which all models have accretion columns. The magnetospheric radius r_M decreases with increasing ${\dot{M}}_{\mathrm{in}}$ and increases with increasing B_NS. Dashed lines indicate the analytical solution for r_M, which can be calculated assuming that the magnetic pressure of the dipole field balances with the radiation pressure of the accretion disk (Takahashi & Ohsuga 2017, see also Appendix B for details):

$$\begin{eqnarray}&&\begin{array}{c}{r}_{{\rm{M}}}\sim 2.0\times {10}^{6}\,[\mathrm{cm}]\\ \,\times {\left(\frac{\alpha }{0.1}\right)}^{2/7}{\left(\frac{{\dot{M}}_{\mathrm{in}}}{{10}^{2}{\dot{\ M}}_{\mathrm{Edd}}}\right)}^{-2/7}{\left(\frac{{B}_{\mathrm{NS}}}{{10}^{10}\,{\rm{G}}}\right)}^{4/7}\\ \,\times {\left(\frac{{M}_{\mathrm{NS}}}{1.4{M}_{\odot }}\right)}^{-3/7}{\left(\frac{{r}_{\mathrm{NS}}}{10\,\mathrm{km}}\right)}^{12/7},\end{array}\end{eqnarray} \tag{ 24 }$$

where α is the viscous parameter (Shakura & Sunyaev 1973), and we take α = 0.1. We can find that the resulting r_M is consistent with the analytical one. We note that the results of models B1e10D1_np and B3e9D01_np are not plotted since r_M is not determined in these models as mentioned above.

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We estimate $\dot{P}$ as follows:

$$\begin{eqnarray}&&\dot{P}=\displaystyle \frac{\left\langle \dot{L}\right\rangle }{{M}_{\mathrm{NS}}{l}_{\mathrm{NS}}}P,\end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray}&&\dot{L}=\int {M}_{\phi }^{r}\sqrt{-g}d\theta d\phi .\end{eqnarray} \tag{ 26 }$$

(Shapiro & Teukolsky 1986; Takahashi & Ohsuga 2017). Here, ${l}_{\mathrm{NS}}=2\pi {r}_{\mathrm{NS}}^{2}/P$ and P are the specific angular momentum and rotation period of the NS, respectively. For numerical estimation, we assume P = 9.8 s, which corresponds to that observed in Swift J0243.6+6124. Figure 4(b) shows that $\dot{P}$ is in the range from −10⁻⁹ to −10⁻⁷ s s⁻¹, which is not inconsistent with the observed value of Swift J0243.6+6124, $\dot{P}\sim -{10}^{-8}\,{\rm{s}}\,{{\rm{s}}}^{-1}$ (see Section 3.3 for details). It can also be clearly seen that the larger ${\dot{M}}_{\mathrm{in}}$ and higher B_NS lead to large $\dot{P}$. Dashed lines indicate the analytical solution for $\dot{P}$, which can be calculated assuming that the Keplerian angular momentum at r_M is transported to the NS without dissipation (Takahashi & Ohsuga 2017; see also Appendix B):

$$\begin{eqnarray}\begin{array}{rcl}\dot{P} & \sim & -2.2\times {10}^{-9}\,[{\rm{s}}\cdot {{\rm{s}}}^{-1}]\\ & \,\times & {\left(\displaystyle \frac{\alpha }{0.1}\right)}^{1/7}{\left(\displaystyle \frac{{\dot{M}}_{\mathrm{in}}}{{10}^{2}{\dot{M}}_{\mathrm{Edd}}}\right)}^{6/7}{\left(\displaystyle \frac{{B}_{\mathrm{NS}}}{{10}^{10}\,{\rm{G}}}\right)}^{2/7}\\ & \,\times & {\left(\displaystyle \frac{{M}_{\mathrm{NS}}}{1.4{M}_{\odot }}\right)}^{2/7}{\left(\displaystyle \frac{{r}_{\mathrm{NS}}}{10\,\mathrm{km}}\right)}^{-8/7}{\left(\displaystyle \frac{P}{10\,{\rm{s}}}\right)}^{2}.\end{array}\end{eqnarray} \tag{ 27 }$$

The resulting $\dot{P}$ is roughly comparable to the analytical one.

In Figure 5, we show that powerful outflows are launched from the super-Eddington accretion disk and are driven by radiation and centrifugal forces. The top panel in Figure 5 shows time-averaged ${\dot{M}}_{\mathrm{in}}$ (thick solid line), ${\dot{M}}_{\mathrm{out}}$ (dashed line), and net flow rate ${\dot{M}}_{\mathrm{net}}={\dot{M}}_{\mathrm{in}}-{\dot{M}}_{\mathrm{out}}$ (thin solid line) as a function of r in the case of model B1e10D01. The position of r_M in this model is represented by a vertical line. This panel indicates that outflows mainly originate from the accretion disk. This is clearly understood from the fact that ${\dot{M}}_{\mathrm{out}}$ increases significantly with increasing r, and ${\dot{M}}_{\mathrm{out}}$ for r < r_M is much smaller than that for r > r_M. The mass accretion rate ${\dot{M}}_{\mathrm{in}}$ decreases with decreasing r due to efficient mass ejection from the accretion disk. The mass outflow rate ${\dot{M}}_{\mathrm{out}}$ is much smaller than the mass accretion rate at r < r_M, and we find ${\dot{M}}_{\mathrm{net}}\sim {\dot{M}}_{\mathrm{in}}\gg {\dot{M}}_{\mathrm{out}}$ inside the magnetosphere.

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Next, we investigate the radial profile of forces acting on fluid elements. The steady-state axisymmetric equations of motion in the r- and θ-directions in the statistic observer frame are

$$\begin{eqnarray}&&{f}_{\mathrm{grav}}^{(r)}+{f}_{\mathrm{thermal}}^{(r)}+{f}_{\theta ,\mathrm{cent}}^{(r)}+{f}_{\phi ,\mathrm{cent}}^{(r)}+{f}_{\mathrm{rad}}^{(r)}+{f}_{\mathrm{mag}}^{(r)}+{f}_{\mathrm{adv}}^{(r)}=0,\end{eqnarray} \tag{ 28 }$$

$$\begin{eqnarray}&&{f}_{\mathrm{inertial}}^{(\theta )}+{f}_{\mathrm{thermal}}^{(\theta )}+{f}_{\phi ,\mathrm{cent}}^{(\theta )}+{f}_{\mathrm{rad}}^{(\theta )}+{f}_{\mathrm{mag}}^{(\theta )}+{f}_{\mathrm{adv}}^{(\theta )}=0,\end{eqnarray} \tag{ 29 }$$

where superscripts (r) and (θ) denote the r- and θ-components of the vectors in the static observer frame. The forces acting on fluid elements consist of the gravitational force f_grav, thermal force f_thermal, radiation force f_rad, Lorentz force f_mag, centrifugal force f_cent, inertial force f_inertial, and advection force f_adv. The centrifugal force is divided into two components, f_θ,cent and f_ϕ,cent, which are caused by the polar (θ-) and azimuthal (ϕ-) motions, respectively (see Appendix C for details). We define the total force f_tot as the sum of all forces without f_grav, and therefore $-{f}_{\mathrm{tot}}^{(r)}/{f}_{\mathrm{grav}}^{(r)}=1$ holds.

The middle panel in Figure 5 shows time- and shell-averaged mass flux weighted radial forces normalized by the gravitational force as a function of r, which can be calculated as follows:

$$\begin{eqnarray}&&{\left\langle {f}^{(r)}\right\rangle }_{\rho {u}^{r},\theta \phi }=\displaystyle \frac{1}{\left\langle \right.{\dot{M}}_{\mathrm{out}}(r)\left.\right\rangle }\left\langle \int {f}^{(r)}\max [\rho {u}^{r},\,0]\sqrt{-g}d\theta d\phi \right\rangle .\end{eqnarray} \tag{ 30 }$$

The bottom panel in Figure 5 shows the time-averaged mass flux weighted z-component of the forces normalized by the vertical gravitational force,

$$\begin{eqnarray}&&{\left\langle {f}^{(z)}\right\rangle }_{\rho {u}^{r}}=\displaystyle \frac{\left\langle \max [\rho {u}^{r},0]{f}^{(z)}\right\rangle }{\left\langle \max [\rho {u}^{r},0]\right\rangle },\end{eqnarray} \tag{ 31 }$$

$$\begin{eqnarray}&&{f}^{(z)}={f}^{(r)}\cos \theta -{f}^{(\theta )}\sin \theta ,\end{eqnarray} \tag{ 32 }$$

at the disk surface. The polar angle of the disk surface is defined as θ = π/2 − Θ_H, where

$$\begin{eqnarray}&&{{\rm{\Theta }}}_{{\rm{H}}}=\left\langle \sqrt{\displaystyle \frac{\int \rho {(\pi /2-\theta )}^{2}\sqrt{{g}_{\theta \theta }}d\theta }{\int \rho \sqrt{{g}_{\theta \theta }}d\theta }}\right\rangle .\end{eqnarray} \tag{ 33 }$$

It is clear that the outflows are mainly driven by the radiation force for 20 km < r < 50 km, where the radiation force is stronger than the other forces in both the r- and z-directions. For 50 km < r < 200 km, the gas is launched from the accretion disk by the radiation force and moves outward by centrifugal force. The Lorentz force is dominant at r < r_M, but as already mentioned, most of the outflows are of disk origin (r > r_M). The thermal force, centrifugal force due to the motion in the θ-direction, advection force, and inertial force are significantly small over the whole region. The radial distribution of forces for r > r_M in models B3e9D001, B3e9D01, B1e10D001, B1e10D002, B1e10D004, B3e10D001, B3e10D01, B3e10D02, B3e10D04, and B3e10D1 is similar to that in model B1e10D01. The radial profile of forces outside r = r_NS in models B3e9D01_np and B1e10D1_np, where the magnetosphere is not formed, is the same as that in the disk region outside r = r_M in model B1e10D01.

In Figure 6, we show the gas temperature distribution and the location of the photosphere. In all models, the gas temperature tends to be high near the z-axis and in the equatorial plane at r < 750 km. Red lines indicate the photospheres, $\left\langle {\tau }_{\mathrm{eff}}\right\rangle =1$, where $\left\langle {\tau }_{\mathrm{eff}}\right\rangle$ is the time-averaged effective optical depth measured as,

$$\begin{eqnarray}&&\left\langle {\tau }_{\mathrm{eff}}(r)\right\rangle ={\int }_{r}^{{r}_{\mathrm{out}}}\left\langle \rho \right\rangle \sqrt{\left\langle {\kappa }_{\mathrm{abs}}\right\rangle (\left\langle {\kappa }_{\mathrm{abs}}\right\rangle +{\kappa }_{\mathrm{sca}})}\sqrt{{g}_{{rr}}}{dr},\end{eqnarray} \tag{ 34 }$$

where $\left\langle {\kappa }_{\mathrm{abs}}\right\rangle =6.4\times {10}^{22}\left\langle \rho \right\rangle {\left\langle {T}_{{\rm{g}}}\right\rangle }^{-3.5}\,[{\mathrm{cm}}^{2}\,{{\rm{g}}}^{-1}]$. As can be seen in this figure, the photosphere extends to surround the disk region. The larger the mass accretion rate, the wider the region surrounded by the photosphere. This can be understood from a comparison of models B1e10D001, B1e10D01, and B1e10D1_np. Also, within the range of parameters employed in the present study, the position of the photosphere does not depend much on B_NS. It is clear from the fact that the photosphere of models B1e10D01 and B3e10D01 hardly changes (see the red lines in Figure 6). In all models, the gas temperature on the photosphere is ∼10⁷ K over a wide range of polar angles. From this, it is expected that blackbody radiation with a temperature of ∼10⁷ K can be observed.

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Hereafter, we define the thermalization radius r_th as the radius where $\left\langle {\tau }_{\mathrm{eff}}\right\rangle =1$. That is, the radius of the photosphere is represented as r_th. In Figure 7, we plot r_th (upper panel) and the time-averaged gas temperature at the photospheres T_th (middle panel) as a function of polar angle θ. In model B1e10D1_np, which has the largest ${\dot{M}}_{\mathrm{in}}$ in our models, r_th ≳ 1000 km at 30° < θ < 150°. Models with smaller ${\dot{M}}_{\mathrm{in}}$ tend to have smaller r_th. The thermalization radius r_th is slightly less than 1000 km at 40° < θ < 140° in models B1e10D01 and B3e10D01 and is ∼500 km at 60° < θ < 120° in model B1e10D001. The thermalization radius cannot be determined close to the rotation axis (θ ≲ 30°, 150° ≲ θ) since the gas is optically thin and $\left\langle {\tau }_{\mathrm{eff}}\right\rangle \lt 1$ even at the NS surface. The range of θ, within which r_th can be obtained, narrows as ${\dot{M}}_{\mathrm{in}}$ decreases. In all models, T_th is almost constant at the angular range where r_th can be determined. The bottom panel in Figure 7 shows the polar angle dependence of the radial gas velocity at the photosphere, v^(r)(r_th), where v^(r) = u^(r)/u^(t). We can see v^(r)(r_th) > 0 on the photosphere, which indicates that most of the photosphere is located inside the optically thick outflows. Hereafter, we refer to such an outflow as an effectively optically thick outflow (Urquhart & Soria 2016).

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Here, we estimate the blackbody temperature T_bb by taking θ-averaged T_th on the photosphere as

$$\begin{eqnarray}&&{T}_{\mathrm{bb}}=\displaystyle \frac{1}{\int \sqrt{-g({r}_{\mathrm{th}}(\theta ),\theta )}d\theta }\int {T}_{\mathrm{th}}(\theta )\sqrt{-g({r}_{\mathrm{th}}(\theta ),\theta )}d\theta .\end{eqnarray} \tag{ 35 }$$

This integration is performed on the surface where the thermalization radius can be estimated. The corresponding blackbody radius r_bb can be calculated as

$$\begin{eqnarray}&&{r}_{\mathrm{bb}}={\left[\displaystyle \frac{{\left\langle {L}_{\mathrm{ISO}}\right\rangle }_{\theta }}{4\pi \sigma {T}_{\mathrm{bb}}^{4}}\right]}^{1/2},\end{eqnarray} \tag{ 36 }$$

where the θ-averaged isotropic luminosity ${\left\langle {L}_{\mathrm{ISO}}\right\rangle }_{\theta }$ is

$$\begin{eqnarray}&&{\left\langle {L}_{\mathrm{ISO}}\right\rangle }_{\theta }=-\displaystyle \frac{4\pi {r}_{\mathrm{out}}^{2}}{\int \sqrt{-g({r}_{\mathrm{out}},\theta )}d\theta }\int \left\langle {R}_{t}^{r}({r}_{\mathrm{out}})\right\rangle \sqrt{-g({r}_{\mathrm{out}},\theta )}d\theta .\end{eqnarray} \tag{ 37 }$$

Here, θ integration is also conducted only in the range where the thermalization radius can be obtained. The resulting T_bb and r_bb are presented in Table 2. The blackbody temperature is not so sensitive to the accretion rate and is ∼10⁷ K, which is consistent with the blackbody temperature observed in Swift J0243.6+6124 at the super-Eddington phase (Tao et al. 2019). The resulting r_bb is roughly fitted as

$$\begin{eqnarray}&&{r}_{\mathrm{bb}}=3.2{\left(\displaystyle \frac{\dot{M}}{{\dot{M}}_{\mathrm{Edd}}}\right)}^{0.71}\,[\mathrm{km}].\end{eqnarray} \tag{ 38 }$$

Tao et al. (2019) showed that the thermal photons would come from r = 100–500 km for Swift J0243.6+6124. Using Equation (38), our model can explain this radius when the accretion rate is $130{\dot{M}}_{\mathrm{Edd}}\lt {\dot{M}}_{\mathrm{in}}\lt 1200{\dot{M}}_{\mathrm{Edd}}$. In the models treated in the present study, there is no clear result that r_bb depends on B_NS. We will discuss this point later.

We run three models, B1e10D001_a, B1e10D001_b, and B1e10D001_c, which have the same initial parameters as model B1e10D001 except for the resolution, to assess the effect of the resolution. We confirm that r_M and ${\dot{M}}_{\mathrm{out}}$ of models B1e10D001_a and model B1e10D001_b are almost the same as those of model B1e10D001, while those of model B1e10D001_c are different from those of model B1e10D001. The resolution of the simulation models adopted in this study is the same as or higher than that of model B1e10D001_b. Therefore, our simulation results are independent of the resolution.

Here, we suggest how to constrain B_NS using analytical solutions of r_M and $\dot{P}$ and the conditions under which effectively optically thick outflows, which can reproduce the blackbody radiation observed in Swift J0243.6+6124, occur.

It is necessary to simultaneously satisfy r_M < r_co, where r_co is the corotation radius, and r_M can be defined (that is, r_M > r_NS) in ULXPs. The condition of r_M < r_co is for gas to accrete onto the NS (Illarionov & Sunyaev 1975). Otherwise, the gas would all be blown away due to the propeller effect. The accretion columns are formed in the case that the second condition is satisfied (see Figure 3 for details). Substituting Equations (24) and (27) into these conditions, we get

$$\begin{eqnarray}\begin{array}{rcl}{B}_{\mathrm{NS}} & \lt & 1.6\times {10}^{14}\left[{\rm{G}}\right]\\ & \,\times & {\left(\displaystyle \frac{\alpha }{0.1}\right)}^{-1/2}{\left(\displaystyle \frac{\dot{P}}{-{10}^{-8}\,{\rm{s}}\,{{\rm{s}}}^{-1}}\right)}^{1/2}\\ & \,\times & \left(\displaystyle \frac{{M}_{\mathrm{NS}}}{1.4{M}_{\odot }}\right){\left(\displaystyle \frac{{r}_{\mathrm{NS}}}{{10}^{6}\,\mathrm{cm}}\right)}^{-2},\end{array}\end{eqnarray} \tag{ 39 }$$

and

$$\begin{eqnarray}\begin{array}{rcl}{B}_{\mathrm{NS}} & \gt & 7.5\times {10}^{9}\,\left[{\rm{G}}\right]\\ & \,\times & {\left(\displaystyle \frac{\alpha }{0.1}\right)}^{-1/2}{\left(\displaystyle \frac{\dot{P}}{-{10}^{-8}\,{\rm{s}}\,{{\rm{s}}}^{-1}}\right)}^{1/2}{\left(\displaystyle \frac{{M}_{\mathrm{NS}}}{1.4{M}_{\odot }}\right)}^{1/2}\\ & \,\times & {\left(\displaystyle \frac{{r}_{\mathrm{NS}}}{{10}^{6}\,\mathrm{cm}}\right)}^{-1/2}{\left(\displaystyle \frac{P}{10\,{\rm{s}}}\right)}^{-1}.\end{array}\end{eqnarray} \tag{ 40 }$$

If the magnetic field strength at the NS surface satisfies the inequalities of Equations (39) and (40), then the column accretion onto the magnetic poles of the NS appears, and such objects can be the candidates for the ULXPs.

Next, we consider the emergence of the outflows from the super-Eddington disks. The radiatively driven outflows are thought to be launched in the slim disk region, which is inside the spherization radius (or trapping radius), r_sph. This is discussed in Shakura & Sunyaev (1973) and revealed by the numerical simulations (Kitaki et al. 2021; Yoshioka et al. 2022). The spherization radius is roughly estimated as ${r}_{\mathrm{sph}}=(3/2)({\dot{M}}_{\mathrm{in}}/{\dot{M}}_{\mathrm{Edd}}){r}_{{\rm{g}}}$. Since the disk is truncated at r = r_M, the ULXPs accompanying the outflows appear if the condition of r_M < r_sph is satisfied in addition to r_NS < r_M < r_co (Mushtukov et al. 2019). Using Equations (24) and (27), r_M < r_sph becomes

$$\begin{eqnarray}\begin{array}{rcl}{B}_{\mathrm{NS}} & \lt & 1.5\times {10}^{12}\,\left[{\rm{G}}\right]\\ & \,\times & {\left(\displaystyle \frac{\alpha }{0.1}\right)}^{-1/2}{\left(\displaystyle \frac{\dot{P}}{-{10}^{-8}\,{\rm{s}}\,{{\rm{s}}}^{-1}}\right)}^{3/2}\\ & \,\times & \left(\displaystyle \frac{{M}_{\mathrm{NS}}}{1.4{M}_{\odot }}\right){\left(\displaystyle \frac{P}{10\,{\rm{s}}}\right)}^{-3}.\end{array}\end{eqnarray} \tag{ 41 }$$

Equations (39)–(41) are the general conditions to explain the ULXPs, while we concentrate on the discussion about Swift J0243.6+6124 in the following. As described in Section 3.2, the range of the mass accretion rate that can reproduce the blackbody radiation observed in this object is $130{\dot{M}}_{\mathrm{Edd}}\lt {\dot{M}}_{\mathrm{in}}\lt 1200{\dot{M}}_{\mathrm{Edd}}$ from the relation in Equation (38). Using Equation (27), this condition is transformed as

$$\begin{eqnarray}\begin{array}{rcl}{B}_{\mathrm{NS}} & \lt & 9.8\times {10}^{11}\,[{\rm{G}}]\\ & \,\times & {\left(\displaystyle \frac{\alpha }{0.1}\right)}^{-1/2}{\left(\displaystyle \frac{\dot{P}}{-{10}^{-8}\,{\rm{s}}\,{{\rm{s}}}^{-1}}\right)}^{7/2}{\left(\displaystyle \frac{{M}_{\mathrm{NS}}}{1.4{M}_{\odot }}\right)}^{-1}\\ & \,\times & {\left(\displaystyle \frac{{r}_{\mathrm{NS}}}{{10}^{6}\,\mathrm{cm}}\right)}^{4}{\left(\displaystyle \frac{P}{10\,{\rm{s}}}\right)}^{-7},\end{array}\end{eqnarray} \tag{ 42 }$$

$$\begin{eqnarray}\begin{array}{rcl}{B}_{\mathrm{NS}} & \gt & 1.3\times {10}^{9}\,[{\rm{G}}]\\ & \,\times & {\left(\displaystyle \frac{\alpha }{0.1}\right)}^{-1/2}{\left(\displaystyle \frac{\dot{P}}{-{10}^{-8}\,{\rm{s}}\,{{\rm{s}}}^{-1}}\right)}^{7/2}{\left(\displaystyle \frac{{M}_{\mathrm{NS}}}{1.4{M}_{\odot }}\right)}^{-1}\\ & \,\times & {\left(\displaystyle \frac{{r}_{\mathrm{NS}}}{{10}^{6}\,\mathrm{cm}}\right)}^{4}{\left(\displaystyle \frac{P}{10\,{\rm{s}}}\right)}^{-7}.\end{array}\end{eqnarray} \tag{ 43 }$$

Among ULXPs with outflows, objects that meet the conditions of Equations (41) and (42) would have effectively optically thick outflows, and the blackbody radiation with r_bb > 100 km would appear in the radiation spectrum.

Figure 8 summarizes the conditions shown above in the $\dot{P}-{B}_{\mathrm{NS}}$ plane. Here, we assume P = 9.8 s, which is the rotation period observed in Swift J0243.6+6124 (Doroshenko et al. 2018; Chen et al. 2021). We take α = 0.1, M_NS = 1.4M_⊙, and r_NS = 10 km. The red shaded region indicates the ULXP with outflow, which satisfies both r_NS < r_M < r_co and r_M < r_sph. The ULXP without outflow is in the white region, where r_NS < r_M < r_co is satisfied but r_M < r_sph is not satisfied. The condition of ULXPs such as r_M > r_co (propeller regime) or r_M < r_NS (nonpulsating regime) is not satisfied in gray shaded regions. In the blue hatched region where the accretion rate is $130{\dot{M}}_{\mathrm{Edd}}\lt {\dot{M}}_{\mathrm{in}}\lt 1200{\dot{M}}_{\mathrm{Edd}}$, effectively optically thick outflows with r_bb = 100–500 km and T_bb ∼ 10⁷K appear so that the blackbody radiation, as observed in Swift J0243.6+6124, is thought to be reproduced. Here, it is confirmed that the models that can successfully reproduce such effectively optically thick outflows and accretion columns (B1e10D004, B1e10D01, B3e10D01, and B3e10D02) are located in the blue hatched region (filled circles). In contrast, other models are located outside the blue hatched region (filled squares). In observations of Swift J0243.6+6124, P is almost constant, while $\dot{P}$ is reported to be −2.22 × 10⁻⁸ s s⁻¹ (Obs. 1), −1.75 × 10⁻⁸ s s⁻¹ (Obs. 2), and −6.8 × 10⁻⁹ s s⁻¹ (Obs. 3; Doroshenko et al. 2018; Chen et al. 2021). These observed spin-up rates are denoted by black dashed lines in Figure 8. In Obs. 1 and 2, the observed isotropic luminosity exceeds L_Edd, and the thermal emission of which the observed blackbody radius is 100–500 km is observed (obsID: 90302319004, 90302319006 and 90302319008 in Tao et al. 2019). Then, the magnetic field strength is restricted to be 2 × 10¹⁰ G < B_NS < 5 × 10¹² G for Obs. 1 and 10¹⁰ G < B_NS < 4 × 10¹² G for Obs. 2, which are represented by black solid lines. In Obs. 3, on the other hand, the luminosity is lower than L_Edd, and the thermal emission is not observed. Such features are realized in the condition that the magnetic field is in the range of 3 × 10¹¹ G < B_NS < 10¹⁴ G (see black solid line for Obs. 3). Therefore, the magnetic field strength that satisfies the three observations is between 3 × 10¹¹ G and 4 × 10¹² G. Using Equation (27), the mass accretion rate corresponding to this range is estimated to be $200{\dot{M}}_{\mathrm{Edd}}\lesssim {\dot{M}}_{\mathrm{in}}\lesssim 500{\dot{M}}_{\mathrm{Edd}}$ for Obs. 1 and 2, and $60{\dot{M}}_{\mathrm{Edd}}\lesssim {\dot{M}}_{\mathrm{in}}\lesssim 100{\dot{M}}_{\mathrm{Edd}}$ for Obs. 3. Since the luminosity has been reported to be ∼10³⁹ erg s⁻¹ at the super-Eddington phase in Swift J0243.6+6124 (Obs.1 and 2), the radiative efficiency of this object is 1%–5%, which is smaller than the standard accretion efficiency. This indicates that Swift J0243.6+6124 is observed at a relatively large viewing angle within the range where the pulse emission is detected. The magnetic field strength evaluated in the present study is not inconsistent with previous studies (Tsygankov et al. 2018; Doroshenko et al. 2020). Tsygankov et al. (2018) estimated the upper limit for B_NS in Swift J0243.6+6124, <6 × 10¹² G, from the fact that the transition to the propeller regime was not detected even at a luminosity of 6 × 10³⁵ erg s⁻¹. Doroshenko et al. (2020) concluded that B_NS in Swift J0243.6+6124 is in (3–9) × 10¹²G and likely at the lower limit of this range. We note that B_NS here is assumed not to change during the observation. This assumption would be reasonable since it has been pointed out that the NS magnetic field strength does not change for a sufficiently long period, >1 Myr, based on the observations of some X-ray pulsars and the ULXP, NGC 1313 X-2 (Makishima et al. 1999; Sathyaprakash et al. 2019). Although the NSs might have multipole magnetic fields, we focus on the case of the dipole magnetic fields in this study. We will discuss multipole fields later.

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In all of our models, the radiation is highly beamed by the outflows from the accretion disk. The amplification factor $\left\langle {L}_{\mathrm{iso}}\right\rangle /\left\langle {L}_{\mathrm{rad}}\right\rangle \equiv 1/b$ exceeds 100 at maximum, which is higher than that indicated by King & Lasota (2019; typically 1/b ∼ 1–10; see also Takahashi & Ohsuga 2017; Abarca et al. 2021). The cause of this difference is probably the magnetic field strength. The difference will be smaller if we employ a strong B_NS by which r_M nearly equals r_sph, although r_M is much smaller than r_sph in the present study. As r_M approaches r_sph, the outflow rate would decrease since the outflows mainly occur in r_M < r < r_sph. In this case, the outflow can no longer effectively collimate the radiation, and therefore 1/b would decrease (Abarca et al. 2021). Although simulating r_M ∼ r_sph is difficult and beyond the scope of the present study (see also Section 4.5), we plan to conduct such simulations in the future.

Here, we show, based on the present simulations, that the ULXPs NGC 5907 ULX1 and NGC 1313 X-2 are thought to be powered by the highly super-Eddington accretion onto NSs with relatively weak magnetic fields. In NGC 5907 ULX1, the mechanical power (${L}_{\mathrm{mec}}^{\mathrm{obs}}$), which is evaluated from the observed nebula emission of the ULX bubbles, is reported to 1.3 × 10⁴¹ erg s⁻¹ (Belfiore et al. 2020), and is comparable to $\left\langle {L}_{\mathrm{kin}}\right\rangle$ for the higher-mass accretion rate model (B3e10D1), ∼8 × 10⁴⁰ erg s⁻¹. In addition, the isotropic luminosity $\left\langle {L}_{\mathrm{ISO}}\right\rangle$ in model B3e10D1 at θ ∼ 30° or 150° is almost the same as the observed X-ray luminosity, ${L}_{\mathrm{ISO}}^{\mathrm{obs}}\sim 2.2\times {10}^{41}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ (Israel et al. 2017a), where the isotropic luminosity is calculated as $\left\langle {L}_{\mathrm{ISO}}\right\rangle =-4\pi {r}_{\mathrm{out}}^{2}\left\langle {R}_{t}^{r}\right\rangle$. Thus, NGC 5907 ULX1 probably has a mass accretion rate of $\gt {10}^{3}{\dot{M}}_{\mathrm{Edd}}$ and is viewed from the polar angle 20°–30° away from the z-axis. The magnetic field strength of the NS is stronger than 10¹⁰ G. This is because $\left\langle {L}_{\mathrm{kin}}\right\rangle$ and $\left\langle {L}_{\mathrm{ISO}}\right\rangle$ of model B1e10D1_np are almost the same as those of model B3e10D1, but no accretion column appears in this model.

Also, $\left\langle {L}_{\mathrm{kin}}\right\rangle$ for our slightly lower-mass accretion models (B1e10D01, B3e10D01, and B3e10D02) are comparable to ${L}_{\mathrm{mec}}^{\mathrm{obs}}$ in NGC 1313 X-2, ∼10⁴⁰ erg s⁻¹ (Pakull & Grisé 2008). In these models, the viewing angle, at which L_ISO is consistent with ${L}_{\mathrm{ISO}}^{\mathrm{obs}}\sim (1.4\mbox{--}2.0)\times {10}^{40}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$, is about 30°–40° measured from the z-axis. Therefore, NGC 1313 X-2 would be almost a face-on object with a mass accretion rate of several ${10}^{2}{\dot{M}}_{\mathrm{Edd}}$. Since no accretion column is formed in model B3e9D01_np, of which the kinetic and isotropic luminosity are similar to those of the above three models, B_NS would be stronger than 3.3 × 10⁹ G.

According to our numerical models, both NGC 5907 ULX1 and NGC 1313 X-2 would also have effectively optically thick outflows with r_bb > 100 km. Indeed, the thermal emission, probably originated from outflows, was reported in NGC 1313 X-2 (Qiu & Feng 2021). However, such emission is not detected in NGC 5907 ULX1. This inconsistency can be resolved by conducting simulations of a large r_M. If B_NS is so strong that r_M is comparable to r_sph, then the outflow rate is thought to decrease, and thermal emission is not detected. In this case, 1/b decreases as discussed in Section 4.1, and the observer's viewing angle should be smaller than that estimated above for $\left\langle {L}_{\mathrm{ISO}}\right\rangle$ to be consistent with ${L}_{\mathrm{ISO}}^{\mathrm{obs}}\sim (1.4-2.0)\times {10}^{41}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$. However, increasing B_NS may change the intrinsic luminosity, and therefore simulations with a high B_NS are necessary to perform. Postprocessing radiative transfer simulations are needed to obtain detailed radiation spectra and compare the observation (Kawashima et al. 2012; Kitaki et al. 2017; Narayan et al. 2017).

On the other hand, no nebula has been detected around Swift J0243.6+6124. The reason for this is that the object is a transient source. In order to form a nebula extending over 100 pc, an energy injection, ∼10⁴⁰ erg s⁻¹, from the central object is needed to last for at least 10⁵ yr (Pakull & Grisé 2008; Belfiore et al. 2020).

Our simulations are consistent with the hypothesis proposed by Urquhart & Soria (2016) in which ULXs are observed as ultraluminous supersoft sources (ULSs) by an edge-on observer, as far as the angular range of τ_eff > 1 and its accretion rate dependence are concerned. According to their model, the optical depth of the outflows τ_eff is larger than unity at 35° < θ < 145° for ${\dot{M}}_{\mathrm{in}}\sim 100{\dot{M}}_{\mathrm{Edd}}$, and the polar angle at which τ_eff > 1 widens as the mass accretion rate increases. These features are also obtained in our simulations. Indeed, simulated outflows are effectively optically thick at polar angles > 40° away from the z-axis for models B1e10D01 and B3e10D01 (see Figures 6 and 7 for details). The mass accretion rates in these models are $500{\dot{M}}_{\mathrm{Edd}}$ and $720{\dot{M}}_{\mathrm{Edd}}$, respectively (Table 2). This angular range and the mass accretion rates do not so contradict the suggestion by Urquhart & Soria (2016) mentioned above. Other models, in which the accretion rate is $\sim {10}^{2-3}{\dot{M}}_{\mathrm{Edd}}$ and the accretion columns form, show approximately the same results. In addition, the angular range where τ_eff > 1 widens with an increase in the mass accretion rate (see Figure 7). However, the blackbody temperature estimated from the simulation, ∼10⁷ K, is higher than that observed in the ULSs, ∼10⁶ K. Such a discrepancy might be resolved by the simulations with the initial torus located far away since the winds launched from the outer part of the accretion disks work to expand the photosphere and lower the temperature there (we will discuss this point later). If this is the case, our simulations are consistent with the hypothesis in which ULSs are edge-on sources of ULXs.

The accretion to only one of the poles (one-side accretion) occurs in models with a relatively large truncation radius (B3e9D001, B1e10D001, B1e10D002, B3e10D001, B3e10D01, and B3e10D02) since the distorted magnetic field lines inhibit the gas accretion onto the opposite side. Initially, the shape of the NS magnetic field is symmetric with respect to the equatorial plane. Then, the magnetic field on the equatorial plane is perpendicular to the equatorial plane. However, when the gas accumulates at the truncation radius and flows toward either pole, the magnetic field lines are distorted (see magnetic field lines in the left panel of Figure 3). The magnetic field on the equatorial plane at the truncation radius will be tilted with respect to the equatorial plane. Since the gas tends to move to the side with the lower gravitational potential, the gas flows in the direction in which the preceding gas accretes. The accretion column on the opposite side is less likely to be formed. This one-side accretion is more pronounced for the model with a large truncation radius (B3e9D001, B1e10D001, B1e10D002, B3e10D001, B3e10D01, and B3e10D02). In contrast, for the case where the truncation radius is very small, the gas on the surface of the thick disk accretes to the opposite pole, forming two accretion columns (models B1e10D01, B1e10D004, B3e10D04, and B3e10D1). The one-side accretion described above was also shown by some MHD simulations (see, e.g., Lii et al. 2014; Takasao et al. 2022). We discuss the importance of three-dimensional simulations to study the detailed structure of accretion flow in the subsequent section.

Three-dimensional simulations of nonaxisymmetric accretion flows onto a rotating NS have yet to be performed. In this study, we assume an axisymmetric structure, in which the magnetic axis of the NS coincides with the rotation axis of the accretion disk, and perform two-dimensional simulations. However, the gas is thought to accrete onto the NS surface through multiple streams due to the magnetic Rayleigh–Taylor or interchange instability (Kulkarni & Romanova 2008). Furthermore, the pulse emission is produced by the misalignment between the magnetic axis and rotation axis of the NS. Thus, we need to treat a nonaxisymmetric structure with three-dimensional simulations. The misalignment of the two axes might hinder the one-side accretion since the gas in the accretion disk would preferentially accrete to the closest pole (see, e.g., Romanova et al. 2003). In addition, if the NS rotates as fast as millisecond pulsars, the outflow rate might increase due to the effective acceleration by a rapidly rotating magnetosphere of the NS (e.g., Lovelace et al. 1999). Since the size of the photosphere increases as the mass outflow rate increases, the gas temperature decreases. As a result, r_bb increases, and B_NS is larger than that estimated in Figure 8. Simulations taking account of the rotation of the NS have been performed by Parfrey & Tchekhovskoy (2017) and Das et al. (2022).

In our simulations, the gas–radiation interaction is switched off in the very vicinity of the rotation axis near the NS where σ > σ_rad (see Section 2.3). This treatment would not affect the resulting size and temperature of the photosphere in Section 3.2. In the present simulations, the radiatively driven outflows are launched from the accretion column base as well as the accretion disk (see also Abolmasov & Lipunova 2023). However, the opacity of free–free absorption of such outflows is quite small due to the low density and high temperature. This leads to the smaller size of the photosphere for θ ≲ π/6, 5π/6 ≲ θ compared with that formed by outflows from the accretion disk (r_th ∼ several 100 km; see the red lines in Figure 6). If we do not ignore the gas–radiation interaction near the rotation axis, the outflowing gas launched from the accretion column is more effectively accelerated by the radiation force near the rotation axis. This further reduces the gas density and size of the photosphere. Therefore, neglecting the gas–radiation interaction in the vicinity of the rotation axis near the NS does not affect the size and temperature of the photosphere presented in Section 3.2.

We need to investigate the influence of the boundary condition. In this study, we assumed that the kinetic and thermal energy reaching the inner boundary (NS surface) is immediately converted into radiative energy. Under this boundary condition, the energy carried to the NS core is ignored. In reality, a fraction of the energy reaching the NS surface is converted into the thermal energy of the NS atmosphere. The heat flux from the atmosphere to the core occurs due to thermal conduction (see Wijnands et al. 2017, and references therein). Then, the radiative energy generated on the NS surface might decrease, leading to a decrease in radiative luminosity. Also, the accretion flow geometry might change if we set different boundary conditions (see Figure 9 in Appendix A for details). Investigation of how the accretion flow depends on the boundary conditions is left as future work.

GR-RMHD simulations of large-scale accretion flows are also needed to evaluate r_bb and T_bb, more accurately. It has been pointed out that the simulations in which the initial torus is located inside the spherization radius overestimate the mass outflow rate (see Kitaki et al. 2021; Yoshioka et al. 2022). Therefore, the accretion rate to reproduce effectively optically thick outflows would be higher than that estimated in the present study. This point should be clarified in long-term simulations with the initial torus placed far enough away from the spherization radius. In such simulations, the disk wind is expected to blow from the outer region of the accretion disks (r ≲ r_sph). The blackbody radius would be larger, and the blackbody temperature would be smaller in the region near the equatorial plane. If this is the case, edge-on observers might identify the objects as the ULSs.

In the present study, we adopt a relatively weak magnetic field of the NS 3.3 × 10⁹⁻¹⁰ G compared to the typical X-ray pulsar 10^11–13 G. As B_NS increases while the accretion rate remains fixed, the amount of the outflowing gas probably reduces since the area where radiatively driven outflows mainly occur, $\pi ({r}_{\mathrm{sph}}^{2}-{r}_{{\rm{M}}}^{2})$, decreases. This leads the photosphere to be small and the gas temperature at the photosphere to be high. It is expected that the blackbody radius tends to decrease with an increase in B_NS. In this case, the magnetic field strength of Swift J0243.6+6124 estimated in Section 3.3 would decrease. Although r_bb may depend on B_NS as described above, r_bb is a function of mass accretion rate only (see Table 1 and Equation (38)) in our results since simulations of r_M ≪ r_sph are performed. In addition, it has been reported that when B_NS is strong enough that r_M > r_sph, r_M is no longer dependent on B_NS (Chashkina et al. 2017). We need to perform simulations for the strong B_NS case.

The M1 closure, which is employed in the present study, leads to unphysical solutions when the system is optically thin and the radiation fields are anisotropic (see, e.g., Ohsuga & Takahashi 2016). In our simulations, especially in the models where the one-side accretion occurs, the radiation fields are quite anisotropic. In order to accurately calculate the radiation fields, we have to solve the radiative transfer equation. Such radiation MHD simulations, in which the radiative transfer equation is solved, are performed by Jiang et al. (2014), Ohsuga & Takahashi (2016), Asahina et al. (2020), Zhang et al. (2022), and Asahina & Ohsuga (2022).

The modeling of Swift J0243.6+6124 considering multipolar magnetic field components is to be further investigated. Although we constrain the dipole magnetic field strength at the NS surface, 3 × 10¹¹ G < B_NS < 4 × 10¹² G, in the present work (see also Tsygankov et al. 2018; Doroshenko et al. 2020), a cyclotron resonance scattering feature (CRSF) corresponding to 1.6 × 10¹³ G was reported by Kong et al. (2022). This discrepancy is resolved if CRSF originates from multipole magnetic fields, as has already been suggested by some authors (see, e.g., Israel et al. 2017a). In the case that the multipole magnetic field component is dominant over the dipole component, the accretion flow geometry is expected to be more complex (Long et al. 2007; Das et al. 2022), and the position of the magnetospheric radius changes. We plan to perform the GR-RMHD simulations of super-Eddington accretion flows onto the NS with multipole magnetic fields.

Our models might exhibit a high polarization degree, as recently observed in the X-ray binary (Ratheesh et al. 2023; Veledina et al. 2023). The hard X-ray photons produced in the accretion column base are scattered by the inner wall of the funnel, which consists of the effectively optically thick outflows, and they pass through the low-density region near the rotation axis. If such photons are observed, a high polarization degree might be detected. Indeed, Ratheesh et al. (2023) reported that the polarization degree increases from approximately 6% at 2 keV to 10% at 8 keV. Postprocess polarized radiative transfer simulations are needed to compare our models with such observations.