(Solved): 5. Estimate the energy available and the lifetime for the helium-burning phase in a \( 1.5 \mathrm ...

5. Estimate the energy available and the lifetime for the helium-burning phase in a \( 1.5 \mathrm{M} \odot \) star: (a) Calculate the energy release per net reaction \( 3{ }^{4} \mathrm{He} \rightarrow{ }^{12} \mathrm{C} \) in the triple-alpha process. (Note: The weight of \( { }^{4} \mathrm{He} \) is \( 4.0026 \), and the weight of \( { }^{12} \mathrm{C} \) is \( 12.0000 \).) (b) What fraction of the available mass of 3 helium nuclei is liberated in the form of energy in the triplealpha reaction? Compare this to the fraction of available mass liberated in the proton-proton reaction. (c) Assume that approximately \( 10 \% \) of the original mass of the star is in the form of \( { }^{4} \mathrm{He} \) in the stellar core during the helium-burning phase. Estimate the total energy available from the triple-alpha process. (d) During the helium-core-burning phase, some hydrogen burning is also occurring in a shell. Thus the star's luminosity is not due only to helium burning. Assuming that the part of the luminosity which comes from helium burning is \( 100 \mathrm{~L} \) ? (one hundred times the current luminosity of the Sun), estimate the lifetime of the helium-core-burning phase.