Binary stars (see masses from visual binaries) let astronomers measure masses, and for main-sequence stars a striking pattern appears: luminosity rises very steeply with mass. That explains the main-sequence lifetimes you met before, and it follows from energy transport.
Why it works
Energy diffuses out by radiation. The luminosity is roughly the radiant energy stored, which grows as $T^4$, times the volume, divided by the time it takes to random-walk out, $R^2/(\ell c)$. With the mean free path $\ell = 1/(\kappa\rho)$ (opacity $\kappa$), $\rho \sim M/R^3$ and $T \sim GM\mu m_p/(kR)$ from the central-conditions estimate: $$L \sim \frac{aT^4R^3 \cdot \ell c}{R^2} \sim \frac{ac}{\kappa}\,\frac{T^4R^4}{M} \propto \frac{(M/R)^4R^4}{M} = M^3.$$ The radius cancels. Opacity that changes with temperature and density steepens this a little, so on the main sequence roughly $L \propto M^{3.5}$.
Key idea
For main-sequence stars, roughly $L/L_\odot \approx (M/M_\odot)^{3.5}$. A star's fuel is proportional to $M$, so its lifetime goes as $M/L \propto M^{-2.5}$: about $10\text{ Gyr} \times (M/M_\odot)^{-2.5}$.
Worked examples
Worked example: A two-solar-mass star
Find the luminosity and main-sequence lifetime of a $2\,M_\odot$ star.
- $L = 2^{3.5} = 11.3\,L_\odot$.
- $t = 10\text{ Gyr} \times 2^{-2.5} = 10/5.66$ Gyr.
Answer: About $11\,L_\odot$ and 1.8 Gyr: twice the fuel, burned eleven times as fast.
Worked example: A massive star
Repeat for $10\,M_\odot$.
- $L = 10^{3.5} \approx 3200\,L_\odot$.
- $t = 10 \times 10^{-2.5}$ Gyr $= 0.032$ Gyr.
Answer: About 3,200 $L_\odot$ and 32 million years. (Very massive stars fall below the $M^{3.5}$ line, but the trend holds.)
Worked example: Dating a cluster
A cluster's main-sequence turnoff is at $2.0\,M_\odot$. About how old is it?
- The turnoff stars are just ending their lives: $t = 10 \times 2.0^{-2.5}$ Gyr.
Answer: About 1.8 billion years.
The relation is a rough guide. Below about half a solar mass it is shallower (stars there are partly or fully convective), and above about 50 solar masses luminosity rises only about in proportion to mass, because those stars approach the limit in the next topic. The smallest stars, near 0.08 solar masses, shine at about a thousandth of the Sun's luminosity or less and will last for trillions of years.
Common mistake
Thinking a star twice as massive is twice as luminous. Mass sets the central temperature and how easily energy escapes, and luminosity responds steeply: twice the mass gives about $2^{3.5} \approx 11$ times the luminosity. So massive stars, despite having more fuel, die far sooner. Check: the most massive stars live only a few million years, while the smallest red dwarfs will outlast the present age of the universe.