In order to consider both the photoionization and radiation-pressure effects from the accreting protostar, we solve the frequency-dependent radiative transfer for stellar irradiation by the following method. We inject photons from the sink cell to the computational domain at the rate
(10)$$\begin{eqnarray*}
L_{\rm *}^{\rm tot} = L_* + L_{\rm acc},
\end{eqnarray*}$$where L* is the stellar luminosity and $L_{\rm acc} \equiv G M_* \dot{M}_* / R_*$ is the accretion luminosity. The radiation spectrum is assumed to be thermal blackbody $L_{*, \nu }^{\rm tot} \propto B_\nu (T_{\rm eff})$, where Teff is the effective temperature defined as
(11)$$\begin{eqnarray*}
T_{\rm eff} = \left(\frac{L_*^{\rm tot}}{4 \pi \sigma R_*^2} \right)^{1/4} .
\end{eqnarray*}$$Assumption of the thermal blackbody radiation could overestimate the emissivity of ionizing photons at Z ∼ Z⊙ owing to the line-blanketing effect, especially for stars with M* ∼ 10 M⊙ (e.g. Diaz-Miller, Franco & Shore 1998). This approximation should be valid in our cases since we mainly consider the feedback from more massive stars exceeding 100 M⊙. As discussed in Section 2.2.3, the effective temperature is higher for lower metallicity at a given mass: at M* = 100 M⊙, for instance, Teff ≃ 105 K for Z = 0, Teff ≃ 7 × 104 K for Z = 10−2 Z⊙, and Teff ≃ 5 × 104 K for Z = Z⊙. We use a hybrid method where the direct light emitted from the central star and diffuse light re-emitted from the accretion envelope are separately solved (e.g. Kuiper et al. 2010a, b). We solve the transfer of the stellar direct component by means of the ray-tracing method,
(12)$$\begin{eqnarray*}
F_{\nu }(r) = \frac{L_{*,\nu }^{\rm tot}}{4 \pi r^2} \mathrm{ e}^{-\tau _{\nu }},
\end{eqnarray*}$$where Fν(r) is the flux at the radial position r and τν is the optical depth
(13)$$\begin{eqnarray*}
\tau _{\nu } = \int _{r_{\rm in}}^{r} [ n_{\rm H\,{\small I}} \sigma _{\rm H\,{\small I}}(\nu) + \rho \kappa _{\rm \mathrm{ d}} (\nu)] \mathrm{ d}r,
\end{eqnarray*}$$where σH i(ν) is the H i photoionization cross-section (e.g. Osterbrock 1989), κd(ν) the dust opacity given by Laor & Draine (1993), and rin the sink radius. We assume that the dust-to-gas mass ratio decreases with decreasing metallicities, linearly scaling with Z. The dust opacity is determined by the scaled dust-to-gas mass ratio. We also assume that photons freely travel without attenuation for R* r rin. We set 200 logarithmically spaced wavelength bins between 0.03 cm and 1.5 nm, achieving the higher resolution for the shorter wavelengths. The hydrogen photoionization rate is calculated by integrating the contributions by photons below the Lyman limit, i.e. 91.2 nm (Appendix A2.3). We do not include the diffuse ionizing photons, as their contribution to the disc photoevaporation is negligible compared with the stellar direct component in the massive star formation (e.g. Hosokawa et al. 2011; Tanaka, Nakamoto & Omukai 2013). As in Hosokawa et al. (2016), we do not incorporate helium ionization for simplicity. We assume that helium is in the atomic state everywhere including H ii regions. We consider the transfer of He ionizing photons above 54.4 eV emitted from the protostar, but they are consumed to photoionize hydrogen in our simulations. Regarding the photodissociation of hydrogen molecules, we do not use Fν(r) because our wavelength grids are too sparse to resolve the Lyman–Werner bands. We instead consider another monochromatic direct component representing FUV (11.2 eV ≤ hν ≤ 13.6 eV) photons
(14)$$\begin{eqnarray*}
F_{\rm FUV} (r) = \frac{L_{\rm FUV}}{4 \pi r^2} \exp (- \tau _{\rm d, FUV}),
\end{eqnarray*}$$where LFUV is the luminosity in the FUV wavelength range, and τd,FUV is the dust opacity represented by the value at λ = 100 nm. The photodissociation rate of hydrogen molecules is estimated with the flux FFUV, including the $\rm H_2$ self-shielding. The details of the calculation are shown in Appendix A2.3
Citation: Diaz-Miller, Franco & Shore 1998
Citation function: Uses