这是一份UCL伦敦大学学院PHAS00036作业代写的成功案
$$
\dot{M}{\mathrm{c}}=\frac{L}{Q} \approx 1.2 \times 10^{-11} \frac{L}{L{\odot}} \quad M_{\odot} \mathrm{yr}^{-1}
$$
where $L$ is the luminosity (averaged over a thermal pulse cycle). Substituting $L / L_{\odot}$ from equation (8.26), we obtain
$$
\frac{d M_{\mathrm{c}}}{M_{\mathrm{c}}-0.5 M_{\odot}}=7.2 \times 10^{-7} \mathrm{yr}^{-1} d t
$$
At the beginning of the asymptotic giant phase the core has some mass $M_{\mathrm{c}, 0}$ (> 0.5 $M_{\odot}$ ). Assuming that as a result of contraction the electrons become degenerate, the maximal mass the core could reach is the Chandrasekhar mass $M_{C h}$. Integrating between $M_{\mathrm{c}}=M_{\mathrm{c}, 0}$ and $M_{\mathrm{c}}=M_{\mathrm{Ch}}$, we obtain an upper limit to the duration of the asymptotic giant phase,
$$
\tau_{\mathrm{AGB}}<1.4 \times 10^{6} \ln \left(\frac{M_{\mathrm{Ch}}-0.5 M_{\odot}}{M_{\mathrm{c} .0}-0.5 M_{\odot}}\right) \quad \mathrm{yr}
$$
Evolutionary calculations show that a relation exists between the initial core mass and the initial mass of the star $M_{0}$, of the form
$$
M_{c, 0} \approx a+b M_{0}
$$
PHAS00036 COURSE NOTES ：
As we have already mentioned, the energy source of a white dwarf is the thermal energy of the ions in the isothermal core (the outer layer’s contribution being negligible):
$$
U_{\mathbf{I}}=\frac{3}{2} \frac{\mathcal{R}}{\mu_{\mathrm{I}}} M T_{\mathrm{c}}
$$
Hence the rate of energy emission $L$ must equal the rate of thermal energy depletion:
$$
L=-\frac{d U_{\mathrm{I}}}{d t}=-\frac{3}{2} \frac{\mathcal{R}}{\mu_{\mathrm{I}}} M \frac{d T_{\mathrm{c}}}{d t}=-\frac{3}{7} \frac{\mathcal{R}}{\mu_{\mathrm{I}}} M \frac{T_{\mathrm{c}}}{L} \frac{d L}{d t},
$$
where we have used the $T_{\mathrm{c}}(L)$ relation (8.39). It is easily shown that this implies
$$
-\frac{d L}{d t} \propto M T_{\mathrm{c}}^{6}
$$