Stellar Luminosity

A star's luminosity is set by exactly two things: how much surface it has, and how hot that surface glows. The Stefan–Boltzmann law says every square metre of a blackbody at temperature T radiates σT⁴ watts, and a sphere of radius R has 4πR² square metres — multiply them and the star's entire power output falls out. The Sun makes the arithmetic vivid: R = 6.957 × 10⁸ m and T = 5772 K give L = 4π × (6.957 × 10⁸)² × 5.670 × 10⁻⁸ × 5772⁴ ≈ 3.828 × 10²⁶ W, which is precisely why the IAU could fix its nominal solar radius, effective temperature and luminosity as a mutually consistent set.

The fourth power of T against the square of R is the tension that organises the whole Hertzsprung–Russell diagram. A red giant at 3500 K radiates each square metre feebly, yet at 100 solar radii it pours out over a thousand Suns; a white dwarf at 25 000 K is four times hotter than the Sun but, squeezed to the size of the Earth, shines at barely 4 % of solar output. Same law, opposite corners of the diagram.

Astronomers run it backwards constantly. A spectrum gives T, a distance plus apparent brightness gives L, and R = √(L/4πσT⁴) — the radius of a star nobody can resolve — falls out. That is how we know Betelgeuse would swallow the orbit of Mars, and how the radii of thousands of exoplanet host stars were pinned down well enough for transit depths to become planet sizes.