An Introduction to Modern Astrophysics · Problem 10.11

The Sun — Carroll & Ostlie

Physics — Astrophysics · triple-alpha-temperature-dependence

Problem

Beginning with Eq. (10.62) and writing the energy generation rate in the form ϵ(T)=ϵ′′T8α,\epsilon(T) = \epsilon'' T_8^\alpha, show that the temperature dependence for the triple alpha process, given by Eq. (10.63), is correct. ϵ′′\epsilon'' is a function that is independent of temperature. Hint: First take the natural logarithm of both sides of Eq. (10.62) and then differentiate with respect to ln⁡T8\ln T_8. Follow the same procedure with your power law form of the equation and compare the results. You may want to make use of the relation dln⁡ϵdln⁡T8=dln⁡ϵ1T8dT8=T8dln⁡ϵdT8.\frac{d \ln \epsilon}{d \ln T_8} = \frac{d \ln \epsilon}{\frac{1}{T_8} dT_8} = T_8 \frac{d \ln \epsilon}{dT_8}.

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