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?

Hydrogen fusion is the primary process by which stars generate energy. It occurs through the proton-proton chain reaction, primarily happening in the cores of stars like our Sun, at around \(10^7 \) K. In this reaction, four hydrogen nuclei (protons) are fused to form a single helium nucleus, releasing a significant amount of energy in the form of radiation and subatomic particles. The rate of this fusion reaction is highly temperature dependent, following T^4, which denotes that even small increases in temperature can hugely elevate the fusion rate. For instance, if the temperature is increased by 10%, the fusion rate boosts by a factor of (1.1)^4, which is approximately 1.4641. This translates to a 46.41% increase in energy output, showcasing the sensitive nature of fusion reactions to temperature changes.

Once a star has exhausted its hydrogen fuel, it begins to burn helium in a process called the triple-alpha reaction. This process occurs at much higher temperatures, around \(10^8 \) K. In helium fusion, three helium nuclei combine to form carbon, releasing energy in the process. The rate of helium fusion follows a T^{40} dependence, which is extremely sensitive to temperature changes. To illustrate, if the temperature of a helium-burning core is increased by just 10%, the rate of helium fusion energy increases dramatically. Specifically, the reaction rate increases by a factor of (1.1)^{40}, which is approximately 45.259. Therefore, the fusion energy output soars by 4,425.9%, indicating how a small temperature increase can lead to a massive surge in energy production during helium burning.

Understanding the temperature dependence of fusion reactions is crucial in astrophysics. Both hydrogen and helium fusion rates are exponents of temperature but with different growth rates. The proton-proton chain (hydrogen fusion) follows a T^4 dependence, meaning that a 10% increase in temperature causes a roughly 46.41% rise in energy output. On the other hand, the triple-alpha process (helium fusion) with its T^{40} dependence shows that a similar 10% temperature rise can cause a staggering 4,425.9% increase in energy output.
This significant temperature dependence explains why stars undergo different stages of burning: starting with hydrogen fusion at lower temperatures and shifting to helium fusion at higher core temperatures as they evolve. It also emphasizes the delicate balance maintained in stellar cores to prevent runaway reactions, which could lead to explosive scenarios known as novae or supernovae.