An Explicit M1 Radiation-hydrodynamics Scheme for Three-dimensional Protostellar Evolution

TL;DR

The paper tackles the challenge of simulating protostellar evolution within fully resolved 3D radiation-hydrodynamics by extending an explicit M1 closure with a reduced speed of light (RSLA). It introduces a hybrid RT approach combining RSLA with a non-RSLA component, incorporates neighboring-cell optical-depth information to handle diffusion and leakage across steep gradients, and evolves photon-number densities to reconstruct the radiation spectrum on the fly. Implemented in SFUMATO, the method is validated through tests on energy conservation, diffusion, advection, radiation pressure, and spectral reconstruction, and is then applied to a high-redshift direct-collapse scenario revealing a continuous swollen protostar connected to its disk, a feature not captured by 1D models. The framework enables realistic protostellar feedback studies in 3D cosmological environments and provides a path toward RMHD extensions and broader applications in star formation and related radiative processes.

Abstract

We present a radiation-hydrodynamics (RHD) scheme that enables three-dimensional simulations resolving both protostellar interiors and their surrounding accretion flows within a single framework, to clarify how a protostar evolves while interacting with the accretion flow. The method builds on an explicit M1 closure scheme with a reduced speed of light approximation (RSLA) for massively parallel computation. Our scheme introduces a complementary non-RSLA radiation component that dominates in optically thick regions. This hybrid treatment restores physical energy conservation inside protostars, which would otherwise be violated under the RSLA, while retaining the advantage of large time steps. To overcome the limitation of the conventional M1 closure in solving radiative transfer in extremely optically thick regions inside protostars and across steep optical-depth gradients near their surfaces, we incorporate the optical-depth information of neighboring cells into the radiative transfer calculation. We further evolve photon number densities in addition to radiation energy densities to reconstruct an effective local spectrum on the fly without resorting to costly multi-frequency transport. We implement this scheme in the adaptive mesh refinement code SFUMATO and verify its validity through a series of test calculations. As an application, we follow the early evolution of a massive protostar formed at high redshift, within a full cosmological context. The results reveal a continuous structure connecting the swollen protostar and its surrounding disk, which cannot be captured in conventional one-dimensional models. This explicit RHD scheme opens a path to studies of protostellar evolution and its interaction with the accretion flow in realistic three-dimensional environments.

Figures (8)

• Figure 1: Energy conservation test during gas-radiation interaction. The left and right panels show the results without and with the inclusion of the non-RSLA radiation component $E^\mathrm{N}_\mathrm{rad}$, respectively. The upper panels display the time evolution of the gas temperature, and the lower panels show the evolution of the energy density of each component. In the upper panels, the gray dashed line indicates the analytic equilibrium temperature when the system evolves while conserving its total energy (see text). In the lower panels, the blue solid line represents the total energy (gas plus radiation), the orange and green dashed lines denote the gas and radiation energy densities, respectively, and the red dotted line indicates the radiation energy density corresponding to the Planck distribution, $aT^4$.
• Figure 2: Radiative transfer test with the diffusion approximation. The blue solid line represents the spherically averaged radial profile in the steady-state from our simulation. The orange dashed line denotes the analytic solution. The gray dotted line indicates the computational cell size, which corresponds to the finite volume over which radiation is injected.
• Figure 3: Advecting radiation pulse test. The top and middle panels show the distributions of density and temperature. The blue solid and orange dashed lines represent the static and moving cases, respectively, while the gray dotted lines denote the initial conditions. The bottom panel presents the relative errors between the static and moving cases. The green and red lines indicate the relative errors in density and temperature, respectively. All quantities are evaluated at $t = 4.8 \times 10^{-4}$ s.
• Figure 4: Radiation pressure tube test. In the top panel, the blue and orange lines show the density $\rho$ and temperature $T$ profiles along the $x$-axis after ten sound-crossing times, while the dashed and dotted lines represent the analytic hydrostatic equilibrium solutions for density and temperature, respectively. In the bottom panel, we show the relative errors of the numerical results presented in the top panel with respect to the analytic solutions. The blue and orange lines represent the relative errors in density and temperature, respectively.
• Figure 5: Radiation transfer test with steep optical depth gradient. We show spherically averaged radial profiles of outward luminosity in the steady state. The blue line represents the result obtained by accounting for the cell optical depth $\tau_\mathrm{cell}$ of adjacent cells, while the orange line shows the result obtained using only the local value of $\tau_\mathrm{cell}$. The horizontal dashed line marks the expected luminosity, $4\pi R_\mathrm{c}^2 \sigma_\mathrm{SB} T_\mathrm{c}^4$. The vertical dotted line indicates the clump radius, $R_\mathrm{c}$.