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3. When a star is on the Horizontal brunch it burns Helium via triple alpha reactions. During a triple alpha reaction we know that 3 Helium are transformed into Carbon as in $3 \times{ }^4 \mathrm{He} \rightarrow{ }^{12} \mathrm{C}$. Estimate the duration of the horizontal branch phase in a $2 M_{\odot}$ star through the following steps:
(a) Compute the energy released in a net triple alpha reaction.
(b) Assume that at the beginning of the horizontal branch phase, $15 \%$ of the original mass of the star is in the form of ${ }^4 \mathrm{He}$ in the stellar core. Estimate the total energy released by fusing this amount of helium into carbon via the triple-alpha process.
(c) Assume that during the horizontal branch phase $L=175 L_{\odot}$. If all this luminosity is provided by the fusion of helium to carbon in the core, how long (in years) will the horizontal branch last?
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