Effect of rotation and metallicity on the explodability of massive stars

During the late stages of massive stellar evolution, failed supernovae (FSN) may form through core-collapse processes. The traditional evaluation criterion ξ_2.5 $=$ 0.45, where ξ_2.5 is the compactness parameter, defined as the mass enclosed within 2.5 M_⊙ divided by the radius at that mass in units of 1000 km, is primarily established using nonrotating progenitor models. It is significantly inaccurate when it is applied to rotating pre-supernova systems. The effects of metallicity and rotation on the explodability landscapes of massive stars lack a robust quantification. We investigate the effect of rotation and metallicity on the explodability of massive stars. Using the the code called modules for experiments in stellar astrophysics (MESA), we simulated stars with initial rotational velocities of V_ ini $, and $600 at three metallicities (Z_⊙,1/10,Z_⊙$, and $1/50,Z_⊙$) and tracked their evolution from the zero-age main sequence (ZAMS) until iron core collapse at $1000 $. For each MESA model at the onset of core collapse, we extracted the key parameters (the enclosed mass, temperature, density, radial velocity, electron fraction, and angular velocity) and input them in the 1D supernova collapse simulation code GR1D to simulate the core-collapse supernova (CCSN) phase. Through an iterative procedure, we determined the critical heating parameter f_̊m heat within 1% of the explosion threshold. We then defined the corresponding time-averaged heating efficiency barη_ heat crit at this f_̊m heat to estimate the progenitor explodability. By correlating the explosion outcomes with ξ_2.5, we derived an explodability criterion based on ξ_2.5 and also investigated the correlation between explodability and the ZAMS mass and CO-core mass for the rotational velocities and metallicities. We obtain new critical values of ξ_2.5 for pre-supernova star explodability under different rotation rates and metallicities: 0.45 for models with V_ ini =0; 0.48 for the V_ ini =300 km s^-1 group; 0.47 for V_ ini =600 km s^-1 at Z=Z_⊙, and 0.59 for low metallicity (Z=1/10 and 1/50 Z_⊙). These criteria enable the rapid assessment of the progenitor explodability for equation of state configurations resembling LS220, which is an equation of state with a nuclear incompressibility of 220 MeV. The upper limit of the pre-supernova star compactness for producing CCSNe is significantly higher in chemically homogeneous evolution (CHE) cases than in non-CHE scenarios. This discrepancy primarily arises because the centrifugal force generated by rotational motion in a pre-supernova star more effectively facilitates explosions than in nonrotating scenarios. According to the explodability criterion of the compactness ξ_2.5, we give the ZAMS mass ranges for FSN in different models. We also determined the position of the CO-core mass corresponding to the compactness peak. Our results show that CHE undergone by rapidly rotating low-metallicity massive stars leads to a significant decrease in the ZAMS and CO-core mass range for FSN. Rotation substantially affects the explodability of low-metallicity massive stars. This underscores that it is important to incorporate rotational effects in models of CCSN progenitors.