RAPIDLY ACCRETING SUPERGIANT PROTOSTARS: EMBRYOS OF SUPERMASSIVE BLACK HOLES?

We calculate stellar evolution with mass accretion using the numerical codes developed in our previous work (see Omukai & Palla 2003; Hosokawa & Omukai 2009; Hosokawa et al. 2010, for details). The four stellar structure equations, i.e., equations of continuity, hydrostatic equilibrium, energy conservation, and energy transfer, including effects of mass accretion, are solved assuming spherical symmetry. We focus on the early evolution until slightly after the ignition of hydrogen fusion in this paper. To this end, the appropriate nuclear network for the thermo-nuclear burning of deuterium, hydrogen, and helium is considered.

The codes are designed to handle two different outer boundary conditions for stellar models: shock and photospheric boundaries. The shock boundary condition presupposes spherically symmetric accretion onto a protostar, whereby the inflow directly hits the stellar surface and forms a shock front (e.g., Stahler et al. 1980; Hosokawa & Omukai 2009). We solve for the structure of both the stellar interior and outer accretion flow. With this boundary condition, the photosphere is located outside the stellar surface where the accretion flow is optically thick to the stellar radiation. The photospheric boundary condition, on the other hand, presupposes a limiting case of mass accretion via a circumstellar disk, whereby accretion columns connecting the star and disk are geometrically compact and most of the stellar surface radiates freely (e.g., Palla & Stahler 1992; Hosokawa et al. 2010). In this case, we only solve the stellar interior structure without considering details of the accretion flow; the location of the photosphere always coincides with the stellar surface.

The different outer boundary conditions correspond to the two extremes of accretion flow geometries, or more specifically to different thermal efficiencies of mass accretion, which determine the specific entropy of accreting materials (e.g., Hosokawa et al. 2011a). The accretion thermal efficiency controls the entropy content of the star, which determines the stellar structure. With the shock boundary condition, the accreting gas obtains a fraction of the entropy generated behind the shock front at the stellar surface. The resulting thermal efficiency is relatively high ("hot" accretion).

For the photospheric boundary condition, on the other hand, the accreting gas is assumed to have the same entropy as in the stellar atmosphere. The underlying idea is that when the accreting gas slowly approaches the star via angular momentum transport in the disk, its entropy should be regulated to the atmospheric value. This is a limiting case of thermally inefficient accretion ("cold" accretion). In general, with even a small amount of angular momentum, mass accretion onto the protostar would be via a circumstellar disk, perhaps with geometrically narrow accretion columns connecting the disk with the star. For extremely rapid mass accretion, however, the innermost part of the disk becomes hot and entropy generated within the disk is advected into the stellar interior (e.g., Popham et al. 1993). Thus, cold accretion as envisioned for the photospheric boundary condition is not appropriate for the case of rapid mass accretion (also see discussions in Hartmann et al. 1997; Smith et al. 2011). We therefore expect that the shock boundary condition is a good approximation for the extremely high accretion rates considered here and mostly focus on stellar evolution with the shock boundary condition. We also consider a few cases with the photospheric boundary condition for comparison to test potential effects of reducing the accretion thermal efficiency (also see Section 2.2 below).

The cases considered are summarized in Table 1. In this paper, we only consider the evolution with constant accretion rates for simplicity. The adopted accretion rates range from 10⁻³ M_☉ yr⁻¹ to 1 M_☉ yr⁻¹. Stellar evolution with accretion rates less than 10⁻² M_☉ yr⁻¹ (cases MD1e3 and MD6e3) has been studied in detail in our previous work (e.g., Omukai & Palla 2003; Hosokawa & Omukai 2009). As described in Section 2.1 above, we calculate the protostellar evolution assuming shock outer boundary conditions for most of the cases. Cases MD3e1-HC10 and MD3e1-HC50 are the only exceptions, whereby we switch to the photospheric boundary condition after the stellar mass reaches 10 M_☉ and 50 M_☉, respectively, at a constant accretion rate of 0.3 M_☉ yr⁻¹. The underlying idea for switching the boundary at some moment is that the specific angular momentum of the inflow, and thus the circumstellar disk, grow with time, which reduces entropy of the accreted matter. The higher mass at switching point corresponds to the higher angular momentum of the parental core. As in our previous work, we start the calculations with initial stellar models constructed assuming that the stellar interior is in radiative equilibrium (e.g., Hosokawa & Omukai 2009). We adopt a slightly higher initial stellar mass of ∼1 M_☉ for stability reasons. The calculated initial stellar radii are also summarized in Table 1.

Table 1. Cases Considered

Case	$\dot{M}_*$	M*, 0	R*, 0	Notes	References
	(M☉ yr−1)	(M☉)	(R☉)
MD1e0	1.0	2.5	298.4		Section 3.2
MD3e1	0.3	2.0	238.3		Sections 3.2, 3.3
MD3e1-HC10	0.3	2.0 (10)	238.3 (437.1)	shock → photo. BC at M* = 10 M☉	Section 3.3
MD3e1-HC50	0.3	2.0 (50)	238.3 (826.8)	shock → photo. BC at M* = 50 M☉	Section 3.3
MD1e1	0.1	2.0	177.8	fiducial case	Sections 3.1, 3.2
MD6e2	0.06	1.0	118.6		Section 3.2
MD3e2	0.03	1.0	95.2		Section 3.2
MD6e3	0.006	1.0	52.3	also see Omukai & Palla (2003)	Section 3.2
MD1e3	0.001	0.05	12.5	also see Omukai & Palla (2003)	Sections 3.1, 3.2

Notes. Column 2: mass accretion rate, Column 3: initial stellar mass, Column 4: initial radius. For cases MD3e1-HC10 and MD3e1-HD50, the values when the boundary condition is switched are given in parentheses.

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We first consider the fiducial case (MD1e1), whereby the stellar mass increases with the constant accretion rate of $\dot{M}_*= 0.1 \,M_\odot \,{\rm yr}^{-1}$. The calculated evolution of the stellar interior structure is presented in Figure 1. We see that the stellar radius is very large and increases monotonically with the stellar mass. The stellar radius exceeds 10³ R_☉ when the stellar mass is M_* ≃ 45 M_☉ and reaches ≃ 6500 R_☉ at M_* ≃ 10³ M_☉. This evolution differs qualitatively from that calculated assuming a lower accretion rate 10⁻³ M_☉ yr⁻¹ as depicted in Figure 1 by the blue line (see also, e.g., Omukai & Palla 2003). At the lower accretion rate, the stellar radius initially increases with stellar mass but begins to decrease at M_* ≳ 6 M_☉.

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A key quantity for understanding the contrast between the two cases is the balance between the two characteristic timescales: the accretion timescale

$$\begin{equation} t_{\rm acc} \equiv \frac{M_*}{\dot{M}_*} \end{equation} \tag{ 1 }$$

and the Kelvin–Helmholtz (KH) timescale

$$\begin{equation} t_{\rm KH} \equiv \frac{G M_{\ast }^2}{R_{\ast } L_{\ast }}, \end{equation} \tag{ 2 }$$

where R_* and L_* are the stellar radius and luminosity, and G is the gravitational constant (e.g., Stahler et al. 1986; Omukai & Palla 2003; Hosokawa & Omukai 2009). In the early stage, during which the stellar radius increases with mass, the timescale balance is t_acc ≪ t_KH and radiative loss of the stellar energy is negligible (adiabatic accretion stage). However, the radiative energy loss becomes more efficient as the stellar mass increases. This is because opacity in the stellar interior, which is due to the free–free absorption following Kramers' law κ∝ρT^−3.5, decreases and the stellar luminosity L_* increases with the stellar interior temperature (and thus with its mass M_*). The upper panel of Figure 2 indeed shows that the maximum luminosity within the star L_max increases as a power-law function of M_*.

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This increase of L_* is consistent with the analytic scaling relation for radiative stars with Kramers' opacity, L∝M^11/2_*R_*^−1/2 (e.g., Hayashi et al. 1962). Our numerical results are well fitted by the analytic relation

$$\begin{equation} L_{\rm max} \simeq 0.6 \,L_\odot \left(\frac{M_*}{M_\odot } \right)^{11/2} \left(\frac{R_*}{R_\odot } \right)^{-1/2}. \end{equation} \tag{ 3 }$$

The increase of L_* causes an inversion of the timescale balance to t_acc > t_KH at low accretion rates. The star contracts by losing its energy (KH contraction stage), which is seen for M_* ≳ 6 M_☉. The opacity in the stellar interior has fallen down to the constant value of electron scattering. Figure 2 shows that, in this stage, luminosity takes its maximum value at the stellar surface and increases as L_* = L_max∝M³_*, which is valid for the constant opacity cases (e.g., Hayashi et al. 1962). The relation

$$\begin{equation} L_{\rm max} \simeq 10 \,L_\odot \left(\frac{M_*}{M_\odot } \right)^3 \end{equation} \tag{ 4 }$$

roughly agrees with our results. Temperature at the stellar center increases during the KH contraction stage. Hydrogen burning finally begins and the stellar radius begins to increase following the mass–radius relation of ZAMS stars for M_* ≳ 50 M_☉. Figure 2 shows that the stellar luminosity gradually approaches the Eddington limit

$$\begin{equation} L_{\rm Edd} (M_*) \simeq 3.8 \times 10^6 \,L_\odot \left(\frac{M_*}{100 \,M_\odot } \right). \end{equation} \tag{ 5 }$$

By contrast, there is no contraction stage for the case with a much higher accretion rate $\dot{M}_*= 0.1 \,M_\odot \,{\rm yr}^{-1}$ (MD1e1). Nevertheless, the evolution of the timescales still follows the picture described above (Figure 3(a)). We see that the timescale balance changes from t_acc < t_KH to t_acc > t_KH at M_* ≃ 40 M_☉. The protostar is in the adiabatic accretion stage for M_* ≲ 40 M_☉. The accretion luminosity $L_{\rm acc} \equiv G M_* \dot{M}_*/ R_*$ at this stage is much higher than the stellar luminosity L_*, since the luminosity ratio L_acc/L_* is equal to the timescale ratio t_KH/t_acc by definition. Stahler et al. (1986) derived the approximate analytic formulae describing radial positions of the stellar surface R_* and photosphere R_ph (located within the accretion flow) during the adiabatic stage:

$$\begin{equation} R_* \simeq 26 \,R_\odot \left(\frac{M_*}{M_\odot } \right)^{0.27} \left(\frac{\dot{M}_*}{10^{-3} \,M_\odot \,{\rm yr}^{-1}} \right)^{0.41}, \end{equation} \tag{ 6 }$$

$$\begin{equation} R_{\rm ph} \simeq 1.4 R_*. \end{equation} \tag{ 7 }$$

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The bottom panel of Figure 3 shows that these formulae still agree with our numerical results with 0.1 M_☉ yr⁻¹. Equation (6) shows that the stellar radius is larger for the higher accretion rate at the same mass. The larger radius is due to the higher specific entropy of accreting material and the resulting higher entropy content of the star (e.g., Hosokawa & Omukai 2009). Comparing two stars of the same mass, the one with a larger radius has a lower interior temperature, which then implies a higher opacity due to the strong T dependence of Kramers' law κ∝ρT^−3.5. For this reason, the adiabatic accretion stage is prolonged up to higher stellar masses for higher accretion rates. (See the Appendix for an analytic expression describing this dependence.) Figure 2(b) shows that the evolution of the stellar maximum luminosity L_max still obeys Equations (3)–(5). Unlike the case for 10⁻³ M_☉ yr⁻¹, however, it is only after L_max approaches the relation of L_max∝M³_* (Equation (4)) that t_KH becomes equal to t_acc. The rapid heat input via mass accretion prevents the star from losing internal energy until the stellar luminosity becomes sufficiently high.

The fact that t_KH is shorter than t_acc for M_* ≳ 40 M_☉ indicates that most of the stellar interior is contracting, as shown by the trajectories of the mass coordinates (dashed lines in Figure 1). The figure also shows that the bloated surface layer occupies only a small fraction of the total stellar mass. When the stellar mass is ≃ 300 M_☉, for example, the layer which has 30% of the total mass measured from the surface covers more than 98% of the radial extent. The star has a radiative core surrounded by an outer convective layer. Although the convective layer covers a large fraction of the stellar radius, even the central radiative core is much larger than a ZAMS star with the same mass (compare with the blue curve for M_* ≳ 40 M_☉ in Figure 1). Figure 4 shows the radial distributions of physical quantities, i.e., specific entropy, luminosity, temperature, and density, in the stellar interior. We see that the specific entropy is at its maximum value near the boundary between the radiative core and convective layer. The stellar entropy distribution is controlled by the energy equation,

$$\begin{equation} T \left(\frac{\partial s}{\partial t} \right)_M = \epsilon - \left(\frac{\partial L}{\partial M} \right)_t, \end{equation} \tag{ 8 }$$

where s is the specific entropy and is the energy production rate by nuclear fusion. For M_* ≳ 40 M_☉, most of the stellar luminosity comes from the release of gravitational energy. In fact, as seen in Figure 4(b), (∂L/∂M)_t > 0 in the radiative core, which means that the internal energy of the gas is decreasing. The local luminosity in the radiative core is close to the Eddington value given by Equation (5), using the mass coordinate M rather than the stellar mass M_* (Figure 4(b)). Note that the opacity in the radiative core is only slightly higher than that expected from electron scattering alone.

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In the outer parts of the star, where temperature and density are lower, however, opacity is higher than in the core because of bound–free absorption of H, He atoms, and the H⁻ ion. Energy transport via radiation is inefficient there, and most of the energy coming from the core is carried outward via convection. Figure 4 shows that the surface convective layer lies in the temperature and density range where T ≲ 10⁵ K and ρ ≲ 10⁻⁸ g cm⁻³, which is almost independent of the stellar mass. We also see that the specific entropy is not constant over the convective layer, decreasing toward the stellar surface (a so-called super-adiabatic layer). This is because convective heat transport is inefficient near the stellar surface. A part of the outflowing energy is absorbed there, as indicated by the fact that the surface layer has a negative luminosity gradient (∂L/∂M)_t < 0. This explains the high specific entropy in the outer convective layer.

The energy flux coming from the radiative core increases as the star grows in mass. As a result, the energy absorbed in the outer convective layer also increases with stellar mass, which raises the entropy peak at the bottom of the convective layer (Figure 4(a)). As the entropy increases, the outer layers expand, while the central part of the core contracts further. Therefore, the star assumes a more centrally condensed structure for the higher stellar mass.

The outermost part of the star has a density inversion, i.e., the density increases toward the stellar surface. Here, opacity assumes very high values because of H⁻ absorption. Radiation pressure is so strong that the hydrostatic balance is not achieved only with gravity; the additional inward force by the negative gas pressure gradient helps maintain the hydrostatic structure. Note, however, that this density inversion could be unstable in realistic multi-dimensions (see, e.g., Begelman et al. 2008).

Although deuterium burning is ignited when the stellar mass is ≃ 50 M_☉, its influence on the subsequent evolution is negligible (Figure 1). The total energy production rate by deuterium burning is approximately

$$\begin{eqnarray} &&L_{\rm D,st} \equiv \dot{M}_*\delta _{\rm D} = 1.5 \times 10^5 \,L_\odot \left(\frac{\dot{M}_*}{0.1 \,M_\odot \,{\rm yr}^{-1}} \right)\! \left(\! \frac{[{\rm D/H}]}{2.5 \times 10^{-5}} \!\right), \nonumber\\ \end{eqnarray} \tag{ 9 }$$

where δ_D is the energy available from deuterium burning per unit gas mass. Since this is much lower than the Eddington luminosity L_Edd in the mass range considered, energy production by deuterium burning contributes only slightly to the luminosity in the stellar interior. Hosokawa & Omukai (2009) showed that deuterium burning influences the stellar evolution only when the accretion rate is low $\dot{M}_*\lesssim 10^{-4} \,M_\odot \,{\rm yr}^{-1}$.

Figure 4(c) shows that temperature in the stellar interior increases with total mass. The central temperature reaches 10⁸ K when the stellar mass is ≃ 600 M_☉. Soon after that, hydrogen burning begins and a central convective core develops (Figure 1). This convective core can also be seen in the radial profiles for the 10³ M_☉ model in Figure 4 (indicated by magenta). The luminosity profile tells that most of the energy produced by hydrogen burning is absorbed within the convective core. The star still shines largely by releasing its gravitational energy even after hydrogen ignition.

When the stellar radius is sufficiently large, the accretion flow reaches the stellar surface before becoming opaque to the outgoing stellar light. In fact, soon after the end of the adiabatic accretion stage, the accreting envelope remains optically thin throughout (Figure 3(c)). We also see that the stellar effective temperature is almost constant at T_eff ≃ 5000 K during this period. In general, the stellar effective temperature never assumes a lower value due to the strong temperature dependence of H⁻ absorption opacity (e.g., Hayashi 1961). Stars that have a compact core and bloated envelope (e.g., red giants) commonly have an almost constant effective temperature, regardless of their stellar masses.

We now investigate how stellar evolution changes with the accretion rate. Figure 5 shows the evolution of the stellar radius for several cases, including $\dot{M}_*= 10^{-3} \,M_\odot \,{\rm yr}^{-1}$ (case MD1e3) and 0.1 M_☉ yr⁻¹ (case MD1e1) explained in Section 3.1. We see that, with accretion rates higher than 6 × 10⁻² M_☉ yr⁻¹, the evolution becomes similar to that of the fiducial case for 0.1 M_☉ yr⁻¹ (MD1e1); the stellar radius monotonically increases with mass. The protostars undergo adiabatic accretion in the early stage. As Equation (6) shows, the stellar radius is larger for higher accretion rates at a given stellar mass, say, at M_* = 10 M_☉.

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Even after t_KH becomes shorter than t_acc, the stellar radius continues to increase for accretion rates ≳ 10⁻² M_☉ yr⁻¹. The variations of radii among cases with different accretion rates gradually disappear. The stellar radii finally converge to a unique mass–radius relation with R_*∝M^1/2_* in all the cases. We can derive the approximate mass–radius relation from the following simple argument. First, the stellar luminosity is generally written as

$$\begin{equation} L_* = 4 \pi R_*^2 \sigma T_{\rm eff}^4, \end{equation} \tag{ 10 }$$

where σ is the Stefan–Boltzmann constant. As discussed in Section 3.1 for $\dot{M}_*= 0.1 \,M_\odot \,{\rm yr}^{-1}$ (case MD1e1), the stellar luminosity generally approaches the Eddington value L_Edd(M_*) for M_* ≳ 100 M_☉. Figure 6(b) and Figure 7 show the evolution of the stellar effective temperature as a function of the stellar mass and on the H-R diagram, respectively. We see that the effective temperature stays constant at T_eff ≃ 5000 K for the high accretion rates considered. Substituting these relations into Equation (10), we obtain

$$\begin{equation} R_* \simeq 2.6 \times 10^3 \,R_\odot \left(\frac{M_*}{100 \,M_\odot } \right)^{1/2}. \end{equation} \tag{ 11 }$$

Figure 5 shows that our numerical results approximately follow this relation. The stellar luminosity is actually a bit lower than the Eddington value, which is consistent with the fact that the numerical results show a slightly smaller stellar radius than the analytic relation (11).

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Begelman (2010) also considered stellar evolution with very rapid mass accretion using simple analytic arguments. His model predicts that the stellar radius is proportional to the mass accretion rate (his Equation (24)), which does not agree with our numerical results. Begelman (2010), however, does not take into account the detailed structure of the outermost layer of the star, where H⁻ opacity is important. The fact that the strong T dependence of H⁻ opacity keeps the stellar effective temperature almost constant is essential for our results.

Cases MD3e2 and MD6e3 for intermediate accretion rates (3 × 10⁻² and 6 × 10⁻³ M_☉ yr⁻¹, respectively) exhibit a different behavior than the higher accretion-rate cases described above (see Figure 5). Figure 6(a) shows the evolution of radial positions of the stellar surface and the photosphere. In this case, the photospheric radius is located outside the stellar surface, i.e., the accreting envelope remains optically thick after the onset of KH contraction. For an accretion rate of 3 × 10⁻² M_☉ yr⁻¹ (MD3e2), the protostar initially contracts after the adiabatic accretion stage.

At M_* ≃ 70 M_☉, however, the stellar radius sharply increases and eventually converges to the mass–radius relation given by Equation (11). The case with 6 × 10⁻³ M_☉ yr⁻¹ exhibits an oscillatory behavior of the stellar radius for M_* ≳ 70 M_☉, but its photospheric radius still follows the mass–radius relation R_ph∝M^1/2_*. Figures 6(b) and 7 show that the effective temperature also assumes the constant value T_eff ≃ 6000 K. These evolutionary features for $\dot{M}_*\lesssim 10^{-2} \,M_\odot \,{\rm yr}^{-1}$ have also been found in previous studies (e.g., Omukai & Palla 2001, 2003).

We have seen that the stellar radius at M_* ≃ 10³ M_☉ is almost independent of accretion rate as long as $\dot{M}_*\gtrsim 3 \times 10^{-2} \,M_\odot \,{\rm yr}^{-1}$. However, the stellar interior structure at this moment is not identical among these cases. This is seen in Figure 8, which displays the radial distributions of physical quantities within the 10³ M_☉ models for the different accretion rates. Although each of these stars has a radiative core and a convective envelope, the mass is more centrally concentrated for the lower accretion rates; the less massive envelopes have a higher entropy and inflate more to achieve the same stellar radius.

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This can be understood by the following consideration. Recall that after the KH timescale t_KH falls below the accretion timescale t_acc, the star becomes more and more centrally condensed with increasing stellar mass as explained in Section 3.1. Since this timescale inversion occurs earlier, i.e., at a lower stellar mass for the lower accretion rate (also see the Appendix), the prolonged t_KH < t_acc phase until the stellar mass reaches 10³ M_☉ causes a more centrally condensed structure.

Figure 9(a) shows the evolution of the maximum temperature in the stellar interior, which is helpful for understanding the variation of the stellar interior structure with accretion rates. As Equation (A1) shows, the central part of the star begins to contract and release gravitational energy at a lower stellar mass for the lower accretion rates. The central temperature quickly increases with stellar mass once the KH time becomes shorter than the accretion time. The maximum temperature T_max reaches 10⁸ K at a lower stellar mass for the lower accretion rate. After that, T_max assumes an almost constant value due to the strong T dependence of the energy production rate of hydrogen burning. For the case with 3 × 10⁻² M_☉ yr⁻¹ (MD3e2), hydrogen burning begins at M_* ≃ 200 M_☉; the resulting central convective core is seen in the profiles in Figure 8. This feature is not seen for the case with 0.3 M_☉ yr⁻¹, because hydrogen has not yet ignited by the time M_* = 10³ M_☉ for $\dot{M}_*\gtrsim 0.3 \,M_\odot \,{\rm yr}^{-1}$.

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Figure 5 shows that for $\dot{M}_*\gtrsim 6 \times 10^{-3} \,M_\odot \,{\rm yr}^{-1}$, the protostar does not reach the ZAMS stage by KH contraction. Omukai & Palla (2003) explained this by the fact that the total luminosity L_tot ≡ L_* + L_acc becomes close to the Eddington limit during the contraction to the ZAMS. Figure 9(b) shows the evolution of the Eddington ratio L_tot/L_Edd for several cases. We see that for the cases with 6 × 10⁻³ and 3 × 10⁻² M_☉ yr⁻¹ (cases MD6e3 and MD3e2), for example, the abrupt expansion terminates the KH contraction when the total luminosity approaches the Eddington limit. Omukai & Palla (2003) analytically derived the maximum accretion rate ≃ 4 × 10⁻³ M_☉ yr⁻¹ with which the protostar can reach the ZAMS following KH contraction. Figure 5 indicates that there is another critical accretion rate ≃ 6 × 10⁻² M_☉ yr⁻¹, above which the stellar evolution changes qualitatively; the KH contraction stage disappears entirely at higher rates. This critical rate can also be derived from a similar argument as the one above.

Note that the increase of stellar mass during the KH contraction stage is smaller for the case with 3 × 10⁻² M_☉ yr⁻¹ than with 6 × 10⁻³ M_☉ yr⁻¹. Extending this fact for our critical case, the total luminosity would nearly reach the Eddington limit just at the end of the adiabatic accretion stage, i.e., when t_KH ≃ t_acc. Since the opacity in the surface layer is higher than from Thomson scattering during this epoch, having the total luminosity only slightly lower than the Eddington value causes the star to expand. Thus, the condition for the critical case is

$$\begin{equation} 2 L_{\rm max} \simeq C_{\rm Edd} L_{\rm Edd}, \end{equation} \tag{ 12 }$$

where C_Edd is a factor less than the unity and we have used the fact that the total luminosity is written as L_tot ≃ 2L_max when t_KH ≃ t_acc. Using C_Edd = 0.25 as a fiducial value (Figure 9(b)) and Equation (4) for L_max, the stellar mass which satisfies the condition (12) is

$$\begin{equation} M_{\rm *,Edd,teq} \simeq 21.7 \,M_\odot \left(\frac{C_{\rm Edd}}{0.25} \right)^{0.5}. \end{equation} \tag{ 13 }$$

On the other hand, Equation (A1) also gives the stellar mass when t_KH ≃ t_acc for a given accretion rate. Equating M_{*, teq} and M_{*.Edd, teq} with Equations (A1) and (13), we obtain the critical mass accretion rate

$$\begin{equation} \dot{M}_{\rm cr} \simeq 4.7 \times 10^{-2} \,M_\odot \,{\rm yr}^{-1}\left(\frac{C_{\rm Edd}}{0.25} \right)^{1.9}, \end{equation} \tag{ 14 }$$

which agrees with our numerical results.

We have used the shock outer boundary condition for the stellar models presented and discussed above. As discussed in Section 2.1, the shock boundary condition, which implies that the accreting gas joins the star with relatively high entropy, would be valid for cases with the very rapid mass accretion considered in this paper. If the accreted gas had lower entropy, however, then the stellar radius would be reduced because of the resulting lower entropy throughout the stellar interior. Here, we examine potential effects of the colder mass accretion by adopting the photospheric boundary conditions (e.g., Hosokawa et al. 2010, 2011a).

Figure 10(a) shows the evolution of the stellar radius for three cases with 0.3 M_☉ yr⁻¹, whereby the shock boundary condition is used throughout in one case (MD3e1), whereas the outer boundary condition is changed to the photospheric one for >10 M_☉ (MD3e1-HCm10) and >50 M_☉ (MD3e1-HCm50), respectively. The stars are still in the adiabatic accretion stage when the boundary condition is changed in both cases. The different outer boundary conditions do affect the stellar evolution. For the case where the photospheric boundary condition is adopted at M_* = 10 M_☉ (MD3e1-HCm10), for example, the star initially contracts after the boundary condition is switched at M_* = 10 M_☉, and then abruptly inflates at M_* ≃ 45 M_☉. The stellar radius exceeds 10³ R_☉ and gradually increases with the stellar mass thereafter. In spite of the different behaviors in the early stages, the subsequent evolution for M_* ≳ 100 M_☉ is quite similar to that in the case with the shock boundary condition throughout (MD3e1). The evolution when the boundary condition switching occurs at M_* = 50 M_☉ (MD3e1-HCm50) is much closer to that in the shock-boundary case.

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The uniqueness of the mass–radius relation for M_* ≳ 100 M_☉ can be explained by the fact that the argument leading to the analytic expression, Equation (11), does not assume a specific boundary condition. When the boundary condition is switched at M_* = 10 M_☉ (MD3e1-HCm10), the stellar interior temperature is higher and thus the opacity in the stellar interior (∝T^−3.5 according to Kramers' law) is lower than for the shock-boundary case (MD3e1) at the same stellar mass. As a result, the star begins to release its internal energy earlier than for the shock-boundary case. Indeed, the timescale equality between t_KH and t_acc occurs at M_* ≃ 40 M_☉, earlier than for the shock-boundary case, which occurs at the time of abrupt expansion of the stellar radius.