Effective $\beta$-decay rates of $r$-process waiting points in realistic stellar environments

Reliable nuclear weak rates are key inputs for understanding the origin of heavy elements and constraining the environments of the corresponding stellar nucleosynthesis. We present the effective stellar $\beta^-$-decay rates of the $N=50, 82, 126$ $r$-process waiting-point nuclei in realistic stellar environments with high temperature, high density and strong magnetic field. Both allowed and first-forbidden transitions are considered, and transitions from the low-lying states of parent nuclei due to the thermal population are taken into account properly. The stellar $\beta^-$-decay rates of the $N=50, 82$ waiting points are not sensitive to stellar temperature, while those of the $N=126$ waiting points increase rapidly with stellar temperature. With the increase of stellar density, the electron chemical potential increases accordingly, which leads to reduction of the stellar $\beta$-decay rates. Besides, the stellar $\beta$-decay rates are found to increase rapidly with the magnetic field $B$ when $B \gtrsim 10^{14}$ G. Depending on the stellar temperature, density and magnetic field, the rates may vary by several orders of magnitude, which indicates that dynamic $\beta$-decay rates for corresponding stellar conditions may be indispensable inputs for understanding the $r$-process nucleosynthesis.