Question #43523

A newly formed solid spherical planet of total mass $M$ (in $\text{kg}$) and radius $R$ (in $\text{m}$) is in hydrostatic equilibrium under its own gravity. The planet can be assumed to have a uniform density. The material of the planet has a bulk modulus $B$ (in $\text{N/m}^2$) and a coefficient of linear thermal expansion $\alpha$ (in $\text{K}^{-1}$). Due to intense internal radioactive decay, the temperature of a small central core region of the planet increases by $\Delta T$ (in $\text{K}$), while the massive outer layers remain at their initial temperature. Assuming the outer layers act as a perfectly rigid constraint that prevents any volume expansion of the core, the total compressive stress at the center of the planet reaches the material's breaking stress $S_b$ (in $\text{N/m}^2$). Which of the following expressions correctly represents $S_b$? (Here $G$ is the universal gravitational constant in $\text{N}\cdot\text{m}^2/\text{kg}^2$).