Revise Stellar radii for Cambridge 9702 (25.2)

First, let's distinguish between how much light a star truly emits and how much we actually receive on Earth. A star's luminosity (L) is its total power output, like the wattage of a light bulb. Radiant flux (F) is the power of that radiation arriving per unit area at a specific location, for instance, on your telescope lens.

$F = \frac{L}{4\pi d^2}$

This formula is the inverse square law for flux. It tells us that flux decreases with the square of the distance (d) from the star. The $4\pi d^2$ term represents the surface area of a sphere, assuming the star's energy spreads out uniformly in all directions. If we can determine the distance (d) by other means (like parallax) and measure the flux (F), we can find the star's luminosity. Conversely, this is how we use 'standard candles' – objects of known luminosity, such as Cepheid variable stars or Type Ia supernovae. By finding these objects in distant galaxies and measuring their flux, we can calculate the distance to those galaxies.

Stars are often modelled as blackbodies, which are ideal objects that perfectly absorb and emit all electromagnetic radiation. The spectrum of a blackbody is a continuous curve of radiation intensity versus wavelength, known as the blackbody radiation curve. The shape of this curve and its peak wavelength (λ_max) depend only on the object's temperature. Hotter objects not only have their peak shift to shorter wavelengths (Wien's Law) but also emit more energy at all wavelengths, resulting in a higher and broader curve. This model allows us to link a star's surface temperature to the spectrum of light it emits. Specifically, hotter stars emit more blue light (shorter wavelengths), while cooler stars appear redder (longer wavelengths).

$\lambda_{max} T = \text{constant}$

This is Wien's Displacement Law. Here, $λ_{max}$ is the peak wavelength at which the star emits most of its radiation, and T is its absolute surface temperature in Kelvin. The constant value is approximately $2.9 \times 10^{-3} \text{ m K}$. By observing a star's peak emission wavelength, we can accurately determine its surface temperature.

Once we know a star's total luminosity (L) and its surface temperature (T), we can finally determine its radius (r). This is where the Stefan-Boltzmann Law comes into play. It quantifies how a star's power output depends on its surface area and temperature.

$L = 4\pi r^2 \sigma T^4$

Here, $L$ is luminosity, $r$ is the star's radius, $T$ is its absolute surface temperature, and $\sigma$ is the Stefan-Boltzmann constant ($5.67 \times 10^{-8} \text{ W m}^{-2} \text{ K}^{-4}$). Notice that luminosity is proportional to the square of the radius and the fourth power of the temperature. This means even a small change in temperature dramatically affects a star's brightness!

The properties we've discussed – luminosity, temperature, and radius – are not independent. They are famously related on the Hertzsprung-Russell (H-R) diagram, which plots luminosity versus temperature for stars. Most stars, including our Sun, lie on the 'main sequence'. Stars in the upper right (cool but very luminous) are giants or supergiants, like Betelgeuse. They must be enormous to be so bright despite being cool. Stars in the lower left (hot but dim) are white dwarfs; they must be very small. The Stefan-Boltzmann law is the key to understanding these relationships: for a given temperature, a higher luminosity implies a much larger radius.