Problem $6$: $(17\%$ of Assignment Value$)$

The core of the Sun has a temperature of $1\mathrm{.}5{10}^{7}K,$ while the surface of the Sun has a temperature of $4190$ K $($which varies over the surface, with the sunspots being cooler$).$ Treat the core of the Sun and the surface of the Sun as two large reservoirs connected by the solar interior. Nuclear fusion processes in the core produce $3\mathrm{.}8{10}^{26}J$ every second. Assume that $100\%$ of this energy is transferred from the core to the surface.Part $($a$)$

Calculate the change in the entropy $S,$ in joules per kelvin, of the Sun every second.

Part $($b$)$

Rigel is a blue giant star with a core temperature of $5\mathrm{.}0{10}^{7}K$ and a surface temperature of $10600$ K $.$ If the core of Rigel produces $60,000$ times as much energy per second as the core of the Sun does, calculate the change in the entropy $S_R,$ in joules per kelvin, of Rigel every second.

Part $($c$)$

Barnard's Star is a red dwarf star with a core temperature of $7\mathrm{.}0{10}^{6}K$ and $20$ adiffegce.temperature of $380\mathrm{.}0$ K $.$ If the core of Barnard's Star produces $5\%$ as much energy per second as the core of the Sun does, calculate the change in the entropy $S_B,$ in joules per kelvin, of Barnard's Star every second.