$10\mathrm{.}21$ Estimate the hydrogen$-$burning lifetimes of stars near the lower and upper ends of the main

sequence. The lower end of the main sequence ${?}^{32}$ occurs near $0\mathrm{.}072M_{o.},$ with $lo{g}_{10}{T}_e=3\mathrm{.}23$

and $lo{g}_{10}(\frac{L}{L_{o.}})=-4\mathrm{.}3.$ On the other hand, an $85M_{o.}$ star ${?}^{33}$ near the upper end of the main

sequence has an effective temperature and luminosity of $lo{g}_{10}{T}_e=4\mathrm{.}705$ and $lo{g}_{10}(\frac{L}{L_{o.}})=$

$6\mathrm{.}006,$ respectively. Assume that the $0\mathrm{.}072M_{o.}$ star is entirely convective so that, through

convective mixing, all of its hydrogen, rather than just the inner $10\%,$ becomes available for

burning.

In the case of the $85M_{o.}$ star, assume that $10\%$ of the star's mass is available for hydrogen burning.

Using the information given, calculate the radii of a $0\mathrm{.}072M_{o.}$ star and a $85M_{o.}$ star.

What is the ratio of their radii?

Recall the expression that relates luminosity $L,$ effective temperature $T_e$ and stellar radius $R.$