Each form of energy generation in stars depends on temperature. a. The rate of hydrogen fusion (proton-proton chain) near \(10^{7} \mathrm{K}\) increases with temperature as \(T^{4} .\) If the temperature of the hydrogen- burning core is raised by 10 percent, how much does the hydrogen fusion energy increase? b. Helium fusion (the triple-alpha process) at \(10^{8} \mathrm{K}\) increases with an increase in temperature at a rate of \(T^{40}\). If the temperature of the helium-burning core is raised by 10 percent, how much does the helium fusion energy increase?

Energy generation within stars heavily relies on a process known as hydrogen fusion. This process, primarily happening through the proton-proton chain reaction, is most active in stars like our Sun. Here, hydrogen atoms fuse together to form helium, releasing an immense amount of energy. The rate at which this fusion occurs is incredibly sensitive to the star's core temperature. For hydrogen fusion, the rate of energy production increases proportionally to the temperature raised to the fourth power, represented mathematically as \[ R \propto T^4 \]. For instance, if the core temperature increases by 10%, the new temperature becomes 1.1 times the original temperature. Calculation shows that the fusion energy increases by \[ 1.1^4 = 1.4641 \]. Hence, a small rise in temperature significantly boosts the hydrogen fusion energy.

Temperature plays a crucial role in the rate of fusion. In the hydrogen fusion process within stars, even a slight increase in temperature leads to a considerable rise in the fusion rate. This is because the particles, hydrogen atoms in particular, move faster at higher temperatures, and collisions that lead to fusion happen more frequently. Specifically, for a 10% increase in temperature, the reaction rate increases by a factor of 1.4641 for hydrogen fusion. Likewise, for helium fusion, a similar temperature rise causes a dramatic increase in energy output, illustrating the massive impact temperature has on stellar fusion processes.

After significant hydrogen has fused into helium in the star's core, temperatures can soar further, enabling helium fusion. This process, also known as the triple-alpha process, requires much higher temperatures compared to hydrogen fusion, typically around \[ 10^8 \ \text{K}\]. During helium fusion, three helium nuclei combine to form carbon, releasing even more energy. The rate of helium fusion is exponentially dependent on temperature, described by \[ R \propto T^{40} \]. Thus, a 10% increase in temperature leads to an energy production increase by \[ 1.1^{40} \approx 45.2593 \]. This immense sensitivity to temperature shows how helium fusion can greatly influence a star's energy output during the later stages of its life cycle.