The main sequence lifetime of a star is determined by its mass and luminosity: more massive stars burn hydrogen much faster due to higher core temperatures, resulting in lifetimes proportional to M^(-2.5). The empirical mass-luminosity relation, L ∝ M^3.5 for main sequence stars, combined with the finite hydrogen fuel supply, determines how long each star spends on the main sequence.
Compare lifetimes of known stars (Sun, Sirius, Betelgeuse) using their masses and luminosities; calculate age estimates for star clusters by finding the main sequence turnoff point.
More massive stars are NOT longer-lived; they burn fuel faster and die younger despite having more fuel. The relationship is counterintuitive: doubling stellar mass reduces lifetime by a factor of ~6.
You already know that stars on the main sequence are fusing hydrogen into helium in their cores and that a star's position on the Hertzsprung-Russell diagram is determined by its surface temperature and luminosity. The mass-luminosity relation connects these observable properties to the star's mass through a remarkably simple power law: for main sequence stars, luminosity scales as approximately L ∝ M^3.5. A star twice the Sun's mass is not twice as luminous — it is roughly 11 times more luminous. A star ten times the Sun's mass is about 3,000 times brighter. This steep relationship arises because higher mass means higher core pressure and temperature, which dramatically accelerates the rate of nuclear fusion.
The main sequence lifetime follows directly from two facts: how much fuel a star has and how fast it burns it. The total hydrogen fuel available is proportional to the star's mass M (more massive stars have proportionally more fuel). The rate of fuel consumption is the luminosity L, which scales as M^3.5. The lifetime is therefore proportional to fuel divided by burn rate: t ∝ M/L ∝ M/M^3.5 = M^(-2.5). This inverse power law means that more massive stars live dramatically shorter lives. The Sun, with a main sequence lifetime of about 10 billion years, is a middle-aged star. A star of 10 solar masses burns through its hydrogen in roughly 30 million years — over 300 times faster. A star of 0.5 solar masses, by contrast, will remain on the main sequence for roughly 50 billion years, far longer than the current age of the universe.
This relationship has a powerful observational application: determining the ages of star clusters. Stars in a cluster form at roughly the same time from the same gas cloud, so they all begin on the main sequence together. As time passes, the most massive (and most luminous) stars exhaust their hydrogen first and evolve off the main sequence, becoming red giants. The point on the HR diagram where the main sequence "turns off" — the main sequence turnoff point — tells you the mass of stars currently leaving the main sequence, and from the mass-luminosity-lifetime relation, you can calculate the cluster's age. A cluster whose turnoff is at high-luminosity, blue stars is young; one whose turnoff has retreated to Sun-like stars is billions of years old.
The mass-luminosity relation also explains why the night sky looks the way it does. Although low-mass red dwarfs are by far the most common stars in the galaxy (comprising roughly 75% of all stars), they are so faint that none are visible to the naked eye. The bright stars you see — Sirius, Rigel, Betelgeuse — are massive, luminous stars that are cosmically rare but spectacularly visible. They are also cosmically short-lived: Rigel, at roughly 20 solar masses, has a main sequence lifetime of only a few million years and is younger than many dinosaur fossils. The mass-luminosity relation thus governs not only individual stellar lifetimes but the observable character of the stellar population as a whole.