If a star's mass is $M$ solar masses, how is its main sequence lifetime $T$ approximately proportional?
$1/M^{2}$ or $1/M^{2.5}$
The relationship linking a star's initial mass ($M$) to its duration on the main sequence ($T$) is governed by a steep power law rather than a simple linear or direct relationship. Because higher mass translates into substantially greater core pressure and temperature, the rate at which hydrogen fuel is consumed accelerates much faster than the mass increases. Specifically, the lifetime $T$ scales inversely with a high power of the mass $M$. The text indicates this relationship is approximated by $T vert vert 1/M^2$ or $T vert vert 1/M^{2.5}$. This mathematical dependence signifies that small increases in initial mass budget result in profoundly shorter stellar lifetimes, emphasizing the non-linear nature of stellar aging.
