The Optically Thick Rotating Magnetic Wind from a Massive White Dwarf Merger Product

We basically combine the classical Weber–Davis stellar wind solution (Weber & Davis 1967) and the optically thick nova wind model (Kato & Hachisu 1994), both of which assume a stationary state. As the Weber–Davis solution, we only consider the wind in the equatorial plain emanating from the surface of a rigidly rotating star with a mass M_* and an angular frequency Ω. The flux freezing condition together with Faraday's law gives the flux conservation of the radial magnetic field B_r and a relation with the toroidal magnetic field B_ϕ as

$$\begin{eqnarray}&&{{ \mathcal F }}_{B}={r}^{2}{B}_{r}=\mathrm{const}.,\end{eqnarray} \tag{ 1 }$$

and

$$\begin{eqnarray}&&\displaystyle \frac{{B}_{\phi }}{{B}_{r}}=\displaystyle \frac{{v}_{\phi }-r{\rm{\Omega }}}{{v}_{r}}.\end{eqnarray} \tag{ 2 }$$

Here v_r and v_ϕ are the radial and azimuthal velocity of the wind, respectively, and r denotes the radial coordinate. The mass and angular momentum conservations are described as

$$\begin{eqnarray}&&\displaystyle \frac{\dot{M}}{4\pi }=\rho {v}_{r}{r}^{2}=\mathrm{const}.,\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{ \mathcal L }={{rv}}_{\phi }-\left(\displaystyle \frac{{{rB}}_{r}{B}_{\phi }}{4\pi \rho {v}_{r}}\right)=\mathrm{const}.,\end{eqnarray} \tag{ 4 }$$

by introducing the mass-loss rate $\dot{M}$ and the specific angular momentum ${ \mathcal L }$, where ρ is the mass density of the wind. The coupling between the gas and radiation is treated by means of the flux-limited diffusion approximation, where the temperature gradient can be described as

$$\begin{eqnarray}&&\displaystyle \frac{{dT}}{{dr}}=-\displaystyle \frac{\kappa \rho {L}_{\mathrm{rad}}}{16\pi {ac}\lambda {T}^{3}{r}^{2}},\end{eqnarray} \tag{ 5 }$$

where κ is the opacity, L_rad the radiation luminosity, a the radiation constant, c the speed of light, T the temperature, and λ the flux limiter. We calculate κ = κ(ρ, T) using the OPAL code (Iglesias & Rogers 1996), assuming the metal abundance consistent with the J005311 wind, and $\lambda =\lambda (\rho ,\kappa ,d\mathrm{ln}T/{dr})$ following Levermore & Pomraning (1981). The momentum equation of the gas in the radial direction is given by

$$\begin{eqnarray}&&\begin{array}{l}{v}_{r}\displaystyle \frac{{{dv}}_{r}}{{dr}}+\displaystyle \frac{1}{\rho }\displaystyle \frac{{{dP}}_{\mathrm{gas}}}{{dr}}-\displaystyle \frac{\kappa {L}_{\mathrm{rad}}}{4\pi {r}^{2}c}+\displaystyle \frac{{{GM}}_{* }}{{r}^{2}}\\ \qquad -\ \displaystyle \frac{{v}_{\phi }{}^{2}}{r}+\displaystyle \frac{{B}_{\phi }}{4\pi \rho r}\displaystyle \frac{d}{{dr}}({{rB}}_{\phi })=0.\end{array}\end{eqnarray} \tag{ 6 }$$

The first four terms are identical to the spherical radiation driven wind; the first, second, third, and fourth terms correspond to gas advection, gas pressure gradient, radiation pressure gradient, and gravitational acceleration, respectively. The gas pressure is given by ${P}_{\mathrm{gas}}=\rho {k}_{{\rm{B}}}T/(\mu {m}_{{\rm{u}}})$, where μ is the mean molecular weight, k_B the Boltzmann constant, and m_u the atomic mass unit. The fifth and sixth terms in Equation (6) are the radial acceleration produced by the centrifugal force and the magnetic sling effect, respectively. The energy conservation equation is given by

$$\begin{eqnarray}&&{v}_{r}\displaystyle \frac{d{\varepsilon }_{\mathrm{gas}}}{{dr}}+{P}_{\mathrm{gas}}{v}_{r}\displaystyle \frac{d}{{dr}}\left(\displaystyle \frac{1}{\rho }\right)=-\displaystyle \frac{1}{4\pi {r}^{2}\rho }\displaystyle \frac{{{dL}}_{\mathrm{rad}}}{{dr}},\end{eqnarray} \tag{ 7 }$$

where ${\varepsilon }_{\mathrm{gas}}=3{k}_{{\rm{B}}}T/(2\mu {m}_{{\rm{u}}})$ is the gas energy density. Note that we neglect the convective luminosity in Equation (7). In general, there exists a convective interface between the carbon-burning region and the wind region. However, the convective luminosity is relevant only in the vicinity of the surface region as in the case of nova winds with a similar mass-loss rate (Kato & Hachisu 1994). Also in our case, the convective motion might be suppressed by strong magnetic fields (e.g., Valyavin et al. 2014).

Using Equations (1)–(4), Equations (6) and (7) can be transformed into

$$\begin{eqnarray}&&\begin{array}{l}\left({v}_{r}^{2}-\displaystyle \frac{{k}_{{\rm{B}}}T}{\mu {m}_{{\rm{u}}}}-\displaystyle \frac{{A}_{\phi }^{2}{v}_{r}^{2}}{{v}_{r}^{2}-{A}_{r}^{2}}\right)\displaystyle \frac{r}{{v}_{r}}\displaystyle \frac{{{dv}}_{r}}{{dr}}=\displaystyle \frac{\kappa {L}_{\mathrm{rad}}}{4\pi {rc}}\\ \qquad +\ \displaystyle \frac{{k}_{{\rm{B}}}}{\mu {m}_{{\rm{u}}}}\left(\displaystyle \frac{{dT}}{d\mathrm{log}r}+2T\right)\\ \qquad -\,\displaystyle \frac{{{GM}}_{* }}{r}+{v}_{\phi }^{2}+2{v}_{r}{v}_{\phi }\displaystyle \frac{{A}_{r}{A}_{\phi }}{{v}_{r}^{2}-{A}_{r}^{2}}\end{array}\end{eqnarray} \tag{ 8 }$$

and

$$\begin{eqnarray}&&\displaystyle \frac{d\varepsilon }{{dr}}=\displaystyle \frac{\kappa {L}_{\mathrm{rad}}}{4\pi {r}^{2}c},\end{eqnarray} \tag{ 9 }$$

with

$$\begin{eqnarray}&&{A}_{r}=\displaystyle \frac{{B}_{r}}{\sqrt{4\pi \rho }}\end{eqnarray} \tag{ 10 }$$

being the radial Alfvén velocity,

$$\begin{eqnarray}&&{A}_{\phi }=\displaystyle \frac{{B}_{\phi }}{\sqrt{4\pi \rho }}\end{eqnarray} \tag{ 11 }$$

being the azimuthal Alfvén velocity, and

$$\begin{eqnarray}\begin{array}{rcl}\varepsilon & = & \displaystyle \frac{{L}_{\mathrm{rad}}}{\dot{M}}+\displaystyle \frac{1}{2}({v}_{r}{}^{2}+{v}_{\phi }{}^{2})+\displaystyle \frac{5}{2}\displaystyle \frac{{kT}}{\mu {m}_{u}}\\ & & -\ \displaystyle \frac{{{GM}}_{* }}{r}-r{\rm{\Omega }}{v}_{\phi }+{ \mathcal L }{\rm{\Omega }},\end{array}\end{eqnarray} \tag{ 12 }$$

respectively. In total, we have four constraint equations (Equations (1)–(4)) and three ordinary differential equations (Equations (5), (8), and (9)) for seven variables $(\rho ,{v}_{r},{v}_{\phi },{B}_{r},{B}_{\phi },T,{L}_{\mathrm{rad}})$.

A rotating magnetic wind solution has to satisfy the regularity condition at the slow, Alfvén, and fast points (e.g., Lamers & Cassinelli 1999). It is convenient to normalize variables with the quantities at the Alfvén radius r_A where the Alfvén Mach number becomes unity, i.e., v_r = A_r. From Equations (1), (3), and (10), the velocity can be written as

$$\begin{eqnarray}&&{v}_{{\rm{A}}}={A}_{r}({r}_{{\rm{A}}})=\displaystyle \frac{{{ \mathcal F }}_{B}^{2}}{\dot{M}{r}_{{\rm{A}}}^{2}}.\end{eqnarray} \tag{ 13 }$$

Normalizing r and v_r with r_A and v_A as x ≡ r/r_A and u ≡ v_r/v_A, ρ, B_r, v_ϕ, and B_ϕ can be described as

$$\begin{eqnarray}&&\rho =\displaystyle \frac{\dot{M}}{4\pi {v}_{{\rm{A}}}{r}_{{\rm{A}}}^{2}}\displaystyle \frac{1}{{{ux}}^{2}},\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&{B}_{r}=\displaystyle \frac{{{ \mathcal F }}_{B}}{{r}_{{\rm{A}}}^{2}}\displaystyle \frac{1}{{x}^{2}},\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&{v}_{\phi }={r}_{{\rm{A}}}{\rm{\Omega }}\displaystyle \frac{x(1-u)}{1-{x}^{2}u},\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&{B}_{\phi }=-\displaystyle \frac{{{ \mathcal F }}_{B}{\rm{\Omega }}}{{r}_{{\rm{A}}}{v}_{{\rm{A}}}}\displaystyle \frac{1-{x}^{2}}{x(1-{x}^{2}u)}.\end{eqnarray} \tag{ 17 }$$

We note that the angular momentum can be also simplified as ${ \mathcal L }={r}_{{\rm{A}}}^{2}{\rm{\Omega }}$. The regularity of v_ϕ and B_ϕ and also Equation (8) at r = r_A can be guaranteed by introducing the slope ${du}/{dx}{| }_{r={r}_{{\rm{A}}}}$ as an additional parameter. In the limit of $x\to 1$ and $u\to 1$, v_ϕ and B_ϕ become

$$\begin{eqnarray}&&{v}_{\phi }({r}_{{\rm{A}}})={r}_{{\rm{A}}}{\rm{\Omega }}\displaystyle \frac{{du}/{dx}{| }_{r={r}_{{\rm{A}}}}}{2+{du}/{dx}{| }_{r={r}_{{\rm{A}}}}},\end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray}&&{B}_{\phi }({r}_{{\rm{A}}})=-\displaystyle \frac{{{ \mathcal F }}_{B}{\rm{\Omega }}}{{r}_{{\rm{A}}}{v}_{{\rm{A}}}}\displaystyle \frac{2}{2+{du}/{dx}{| }_{r={r}_{{\rm{A}}}}}.\end{eqnarray} \tag{ 19 }$$

By solving Equations (5), (8), (9), and (14)–(17) with fixing (${r}_{{\rm{A}}},{du}/{dx}{| }_{r={r}_{{\rm{A}}}},\dot{M},T({r}_{{\rm{A}}}),{L}_{\mathrm{rad}}({r}_{{\rm{A}}}),{{ \mathcal F }}_{B},{\rm{\Omega }},{M}_{* }$), we can obtain one numerical solution. Most of the numerical solutions passing r = r_A are not a physical wind solution; a wind solution also has to pass both the slow and fast points where the coefficient of dv_r/dr in Equation (8) becomes zero, i.e., $\{{v}_{r}^{2}-{k}_{{\rm{B}}}T/(\mu {m}_{{\rm{u}}})\}({v}_{r}^{2}-{A}_{r}^{2})-{A}_{\phi }^{2}{v}_{r}^{2}\,=\,0$. For a solution to be regular at these points, the right-hand side of Equation (8) has to be zero there. This gives two additional constraints on the wind solution. Here we change r_A and ${du}/{dx}{| }_{r={r}_{{\rm{A}}}}$ until finding a wind solution for a given parameter set of ($\dot{M},T({r}_{{\rm{A}}}),{L}_{\mathrm{rad}}({r}_{{\rm{A}}}),{{ \mathcal F }}_{B},{\rm{\Omega }},{M}_{* }$).

We have five additional constraints to be satisfied by a wind model for WD J005311. First, the mass-loss rate has to be consistent with the observed value:

$$\begin{eqnarray}&&\dot{M}\gtrsim {\dot{M}}_{\mathrm{obs}}.\end{eqnarray} \tag{ 20 }$$

Second, the terminal velocity of the wind has to be consistent with the observed value:

$$\begin{eqnarray}&&{v}_{r}(\infty )\gtrsim {v}_{\infty ,\mathrm{obs}}.\end{eqnarray} \tag{ 21 }$$

Note that we only consider the equatorial direction, for which the mass-loss rate and radial velocity are the largest. As in the case of the Weber–Davis solution, the terminal velocity of our wind solution is approximately given by the so-called Michel velocity (Michel 1969):

$$\begin{eqnarray}&&\begin{array}{l}{v}_{r}(\infty )\approx {\left(\displaystyle \frac{{{ \mathcal F }}_{B}^{2}{{\rm{\Omega }}}^{2}}{\dot{M}}\right)}^{1/3}\sim {\rm{30,000}}\,\mathrm{km}\,{{\rm{s}}}^{-1}{\left(\displaystyle \frac{{B}_{* }}{{10}^{8}{\rm{G}}}\right)}^{2/3}\\ \times \,{\left(\displaystyle \frac{{R}_{* }}{1500\mathrm{km}}\right)}^{4/3}{\left(\displaystyle \frac{{\rm{\Omega }}}{1{{\rm{s}}}^{-1}}\right)}^{2/3}{\left(\displaystyle \frac{\dot{M}}{3\times {10}^{-6}{M}_{\odot }{\mathrm{yr}}^{-1}}\right)}^{-1/3},\end{array}\end{eqnarray} \tag{ 22 }$$

where B_* denotes the magnetic field at the surface, i.e., B_* = B_r(R_*). Thus, Equation (21) gives an additional constraint on ${{ \mathcal F }}_{B}{\rm{\Omega }}$. Third and fourth, the radiation temperature and luminosity have to be consistent with the observed values:

$$\begin{eqnarray}&&T({r}_{\mathrm{ph},\mathrm{obs}})\approx {T}_{\mathrm{eff}},\end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}&&{L}_{\mathrm{rad}}({r}_{\mathrm{ph},\mathrm{obs}})\approx {L}_{\mathrm{bol}}.\end{eqnarray} \tag{ 24 }$$

With using the flux-limited diffusion approximation, there is no perfect definition of the photosphere; the approximation can be only justified for the optically thick and thin limits. In our wind model for WD J005311, the decoupling between gas and radiation occurs at around the Alfvén radius, where the bulk of the wind acceleration completes. Thus, instead of Equations (23) and (24), we set the conditions on the temperature and luminosity at the Alfvén radius, ${T}_{{\rm{A}}}\equiv T({r}_{{\rm{A}}})$ and ${L}_{\mathrm{rad},{\rm{A}}}\equiv {L}_{\mathrm{rad}}({r}_{{\rm{A}}})$, as

$$\begin{eqnarray}&&{T}_{{\rm{A}}}\gtrsim {T}_{\mathrm{eff}},\end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray}&&{L}_{\mathrm{rad},{\rm{A}}}\approx {L}_{\mathrm{bol}},\end{eqnarray} \tag{ 26 }$$

with

$$\begin{eqnarray}&&{\left[\displaystyle \frac{{L}_{\mathrm{rad},{\rm{A}}}}{\pi {{acT}}_{{\rm{A}}}^{4}}\right]}^{1/2}\lt {r}_{{\rm{A}}}\lt {r}_{\mathrm{ph},\mathrm{obs}}.\end{eqnarray} \tag{ 27 }$$

The lower limit of r_A comes from the upper limit of the radiation luminosity, i.e., ${L}_{\mathrm{rad}}\lt \pi {r}_{\mathrm{ph},\mathrm{obs}}^{2}{{acT}}_{\mathrm{eff}}^{4}$. Fifth, we have an inner boundary condition; at the surface of the degenerate core (r = R_*), the wind is powered by the nuclear burning and the energy generation rate should be equal to the radiation luminosity:

$$\begin{eqnarray}&&{L}_{\mathrm{nuc}}(\rho ({R}_{* }),T({R}_{* }),{R}_{* })\approx {L}_{\mathrm{rad}}({R}_{* }).\end{eqnarray} \tag{ 28 }$$

For evaluating the left-hand side, we use the analytic fitting formula for the energy generation rate by the carbon burning (Kippenhahn et al. 2012):

$$\begin{eqnarray}&&\begin{array}{l}{\varepsilon }_{\mathrm{CC}}\approx 5.49\times {10}^{43}\,\mathrm{erg}\ {{\rm{s}}}^{-1}\ {{\rm{g}}}^{-1}\,{f}_{\mathrm{CC}}\rho {X}_{{\rm{C}}}^{2}{T}_{9}^{-3/2}{T}_{9\alpha }^{5/6}\\ \qquad \times \ \exp [-84.165/{T}_{9\alpha }^{1/3}],\end{array}\end{eqnarray} \tag{ 29 }$$

$$\begin{eqnarray}&&\times {\left[\exp (-0.01{T}_{9\alpha }^{4})+5.56\times {10}^{-3}\exp (1.685{T}_{9\alpha }^{2/3})\right]}^{-1},\end{eqnarray} \tag{ 30 }$$

where T₉ = (T/10⁹ K) and ${T}_{9\alpha }={T}_{9}/(1+0.0067{T}_{9})$. For simplicity, we fix the screening factor as f_CC = 0.5 and the impacts of neutrino cooling are neglected.

After putting the conditions given by Equations (20), (21), (25), (26), and (28), the residual model parameters are the mass M_* and (equatorial) radius R_* of the degenerate core. When we specify the M_*–R_* relation, a sequence of wind solutions for WD J005311 can be obtained. To this end, we calculate the two-dimensional equilibrium structure of rotating ONe WD using the self-consistent field scheme (Fujisawa 2015). We assume the uniform rotation, and use the zero-temperature equation of state with Z = 9.5 (Salpeter 1961) since we are interested in the ONe core after the Kelvin–Helmholtz contraction. Note that the magnetic field can be safely neglected for B_* < 10⁹ G (Fujisawa et al. 2012). In this case, an equilibrium structure can be obtained for a given angular frequency and central density. The results are shown in Figure 2. Each solid line indicates the M_*–R_* relation with each of the different angular frequencies ranging from Ω = 0 s⁻¹ to Ω = 6 s⁻¹. The dotted line corresponds to the cases with the critical rotation velocity. When the rotation is negligible, i.e., Ω ≲ 0.1 s⁻¹, R_* ranges from ∼1 × 10⁸ cm to ∼5 × 10⁸ cm, and the maximum mass is M_ch ∼ 1.38 M_⊙. For a given M_*, the near critical case has a larger R_* than the non-rotating case by a factor of ∼2. For Ω ≳ 2 s⁻¹, the maximum mass of the WD becomes larger than M_ch at most by ≲10%. As a reference, we show the cases with the central density of ρ_c = 10⁹ g cm⁻³ with the dotted line.

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The calculation method is as follows.

• 1.  
  First, we fix $\dot{M}$, ${{ \mathcal F }}_{B}{\rm{\Omega }}$, T_A, ${L}_{\mathrm{rad},{\rm{A}}}$ so that Equations (20)–(21) and (25)–(26) are satisfied. The observed terminal velocity and mass-loss rate should be regarded as the lower limits of ${v}_{r}(\infty )$ and $\dot{M}$. Here, we study three cases with $({v}_{r}(\infty ),\dot{M})=(2.4\times {10}^{9}\,\mathrm{cm}\,{{\rm{s}}}^{-1},6\,\times {10}^{-6}\,{M}_{\odot }\,{\mathrm{yr}}^{-1})$, $(1.9\times {10}^{9}\,\mathrm{cm}\,{{\rm{s}}}^{-1},6\times {10}^{-6}\,{M}_{\odot }\,{\mathrm{yr}}^{-1})$, and $(2.4\times {10}^{9}\,\mathrm{cm}\,{{\rm{s}}}^{-1},3\times {10}^{-6}\,{M}_{\odot }\,{\mathrm{yr}}^{-1})$.
• 2.  
  We fix M_* and Ω. We numerically integrate differential Equations (5), (8), and (9) with the fourth-order explicit Runge–Kutta method both inward and outward from the Alfvén point. The constraint equations described with dimensionless variables u and x as Equations (14)–(17) are consistently solved. We tune (r_A, ${du}/{dx}{| }_{r={r}_{{\rm{A}}}}$) until we obtain a wind solution that passes the slow and fast points.
• 3.  
  If the wind solution satisfies the M_*–R_* relation, we adapt it as a wind solution for WD J005311. If not, we change M_* or Ω (but still fixing ${{ \mathcal F }}_{B}{\rm{\Omega }}$) and repeat the above calculations. We explore a range of 1 M_⊙ < M_* < 1.45 M_⊙ and Ω < 6 s⁻¹.

When fixing the observed parameters, ${v}_{\infty ,\mathrm{obs}},{\dot{M}}_{\mathrm{obs}},{T}_{\mathrm{eff}}$, and L_bol, and the M_*–R_* relation, the wind solution for WD J005311 can be uniquely determined, namely, M_*, B_*, and Ω of this object can be determined. We search solutions by taking uncertainties in the observed parameters and limitations in our model into account as described in Section 2.3.

Let us first describe the basic properties of the wind solutions. In Figure 3, we show the case with M_* = 1.25 M_⊙, R_* = 3.3 × 10⁸ cm, B_* = 4.2 × 10⁷ G, Ω = 0.5 s⁻¹, ${v}_{r}(\infty )\,=2.4\times {10}^{9}\,\mathrm{cm}\,{{\rm{s}}}^{-1}$, and $\dot{M}=6\times {10}^{-6}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$ that corresponds to the massive end of the allowed parameter region (see Figure 5). Other wind solutions in Figure 5 basically have the same overall features as Figure 3. The slow, Alfvén, and fast points are at r_s = 6.0 × 10⁸ cm, r_A = 5.9 × 10⁹ cm, and r_f = 3.8 × 10¹² cm, respectively, indicated by the asterisks. The photospheric radius of the wind solution is at r_ph ≈ r_A ∼ 10¹⁰ cm. The top left panel shows the radial and azimuthal velocity profile in linear scale. In the left panel of Figure 4, we also plot other velocities including the radial and azimuthal Alfvén velocities, the corotation velocity with the degenerate core, the adiabatic sound velocity of the gas, and the escape velocity. In the radial direction, the gas is initially subsonic and mainly accelerated by the thermal pressure up to the slow point. On the other hand, in the azimuthal direction, the velocity is already trans-sonic at the surface since the gas corotates with the rapidly rotating star. In this region, v_ϕ ∝ r. For r_s < r < r_A, the wind is further accelerated in the radial direction mainly by the magnetic sling effect until the inertia of the wind becomes comparable to the magnetic field. After passing the Alfvén point, the radial velocity approaches to the azimuthal Alfvén velocity with the additional acceleration by the pressure gradient of the azimuthal magnetic field and finally passes the fast point. The top right panel of Figure 3 shows the magnetic field strength of the wind solution. The radial field is simply proportional to r⁻² due to the constraint equation (Equation (1)). Although the constraint equation of the azimuthal field (Equation (17)) is a bit complicated, the solution turns out to be approximately proportional to r⁻¹. Note that the azimuthal field becomes dominant at slightly beyond the slow point.

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In Figure 3, the density (middle left), temperature (middle right), and radiation luminosity (bottom left) profiles are also shown. At the surface where carbon burns, the density and temperature are ρ(R_*) ∼ 7 × 10⁴ g cm⁻³ and T(R_*) ∼ 7 × 10⁸ K, respectively. The radiation luminosity stays almost constant; it slightly decreases with radius for R_* < r < r_s where the wind is accelerated by the thermal pressure gradient. For ${r}_{{\rm{s}}}\lt r\lesssim {r}_{\mathrm{ph}}$, the gas expands almost adiabatically, i.e., ρT³ ∝ r⁰. Beyond the photosphere, the gas and radiation are decoupled and the temperature becomes constant. The right panel of Figure 4 shows the profile of the energy density of the wind solution. The black solid line is the effective total energy density (Equation (12)). We also plot each component in Equation (12): the magnetic field energy, the kinetic energy, the radiation energy, the gas thermal energy, and the gravitational energy. The energy density is dominated by the magnetic field energy and stays almost constant even taking into account the radiation loss. The kinetic energy grows, i.e., the wind is accelerated, by consuming the thermal energy for R_* < r < r_s and the magnetic field energy for r > r_s.

Let us now discuss the allowed parameter region of WD J005311. Figure 5 shows sequences of wind solutions. The left panel shows the surface magnetic field and spin angular frequency, and the right panel shows the mass and radius. The circle, triangle, and square symbols represent the cases with $({v}_{r}(\infty ),\dot{M})=(2.4\times {10}^{9}\,\mathrm{cm}\,{{\rm{s}}}^{-1},6\times {10}^{-6}\,{M}_{\odot }\,{\mathrm{yr}}^{-1})$, (1.9 × 10⁹ cm s⁻¹, 6 × 10⁻⁶ M_⊙ yr⁻¹), and (2.4 × 10⁹ cm s⁻¹, 3 × 10⁻⁶ M_⊙ yr⁻¹), respectively. The colors correspond to the y-axis of the right panel; redder (bluer) points correspond to larger (smaller) mass cases. In each sequence, a more massive WD has a faster spin. This can be understood as follows. In the case of magnetic rotating wind, the radiation luminosity is almost constant with radius, ${L}_{\mathrm{rad}}({R}_{* })\approx {L}_{\mathrm{rad}}({r}_{\mathrm{ph}})$ (see the bottom panel of Figure 3). Thus, the observed bolometric luminosity sets the nuclear burning rate on the surface through Equation (28). Since the nuclear burning rate is very sensitive to the temperature, it is almost fixed as T(R_*) ∼ 7 × 10⁸ K, and consequently the gas energy density on the surface is almost fixed. Since the quasi-hydrostatic equilibrium is achieved in the radial direction around the surface (see Figure 3), the energy density is directly connected to the effective surface gravity $\approx {{GM}}_{* }/{R}_{* }^{2}-{R}_{* }{{\rm{\Omega }}}^{2}$. In general, a more massive WD has a smaller radius, thus a larger gravity. In order to compete with it, a faster spin is required. By combining the M_*–R_* relation with this condition on the effective surface gravity, a relation between M_*, R_*, and Ω is almost fixed. This is shown in the right panel; different sequences align in a line even though they have different $\dot{M}$ and ${v}_{r}(\infty )$.

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There exist high- and low-mass ends in each sequence of solutions in Figure 5. The high-mass ends are set by Equation (27); from Equations (13) and (21), ${r}_{{\rm{A}}}\,={({{ \mathcal F }}_{B}^{2}/\dot{M}{v}_{{\rm{A}}})}^{1/2}={[{v}_{r}{(\infty )}^{3}/{v}_{A}{{\rm{\Omega }}}^{2}]}^{1/2}$. Thus, for a fixed ${v}_{r}(\infty )$, the Alfvén radius becomes smaller for a larger Ω or larger M_*, and eventually the condition ${r}_{{\rm{A}}}\gt {({L}_{\mathrm{rad},{\rm{A}}}/\pi {{acT}}_{{\rm{A}}}^{4})}^{1/2}$ breaks down. This physically means that a more massive and compact WD satisfying Equations (20) and (21) requires having a radiation luminosity larger than the observed one. The upper limit of the mass is larger for a sequence with a larger ${v}_{r}(\infty )$. We find that M_* ≲ 1.3 M_⊙ as long as assuming that ${v}_{r}(\infty )$ is non-relativistic. The low-mass end is also set by Equation (27); the smaller the WD mass becomes, the larger r_A becomes, and the condition ${r}_{{\rm{A}}}\lt {r}_{\mathrm{ph},\mathrm{obs}}$ breaks down. Even if accepting ${r}_{{\rm{A}}}\gt {r}_{\mathrm{ph},\mathrm{obs}}$ and taking smaller M_*, we find that the carbon burning no longer ignites on the surface for M_* ≲ 1.1 M_⊙. This may be also conflict with the observed chemical abundance of the WD J005311 wind. We conclude that the observed properties of WD J005311 can be explained by the rotating magnetic wind from an ONe WD with ${M}_{* }=1.1\mbox{--}1.3\,{M}_{\odot }$, B_* = (2–5) × 10⁷ G, and Ω = 0.2–0.5 s⁻¹.