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A star's luminosity, surface temperature, and radius are linked by the Stefan-Boltzmann law: luminosity equals surface area times the fourth power of temperature. Apparent magnitude measures a star's brightness as seen from Earth; absolute magnitude measures its intrinsic brightness at a standard distance of 10 parsecs. Wien's displacement law connects peak emission wavelength to surface temperature: hotter stars appear blue-white, cooler stars red. These properties are derived from spectroscopy and photometry — not direct measurement — making distance estimates essential for most stellar parameters.
Apply Wien's law to calculate temperatures from peak emission wavelengths, and use the distance modulus (m − M = 5 log d − 5) to convert apparent to absolute magnitudes. Compare the properties of well-known stars (Sirius, Betelgeuse, Proxima Centauri) to build intuition about the range of stellar parameters.
Most of what we know about stars comes entirely from the light they emit. You cannot touch a star, fly past it, or measure its diameter with a ruler. Instead, astronomers extract an astonishing range of physical properties — temperature, luminosity, radius, composition — from the spectrum and brightness of the light that reaches Earth's detectors. The tools that make this possible are Wien's displacement law, the Stefan-Boltzmann law, and the magnitude system.
Wien's displacement law connects a star's color to its surface temperature: the wavelength at which a star emits most strongly is inversely proportional to its temperature (λ_max = b/T, where b ≈ 2.9 × 10⁻³ m·K). A star at 6,000 K (roughly solar temperature) peaks in visible yellow-green light. A star at 30,000 K peaks in the ultraviolet, appearing blue-white in the optical. A cool 3,000 K star peaks in the infrared and appears red. This is exactly opposite to the intuition you may have from faucets or warning lights: in the star world, blue means hot and red means cool.
The Stefan-Boltzmann law links luminosity, temperature, and radius: L = 4πR²σT⁴. This says that total energy output per second grows with the fourth power of temperature and with the square of the radius. Two stars at the same temperature but different sizes will differ enormously in luminosity — a red giant at 4,000 K with 50 times the Sun's radius can outshine the Sun despite being cooler, because the surface area effect dominates. This law lets astronomers calculate stellar radii once they know L and T from observations.
Luminosity and apparent brightness are related by distance. The magnitude system (inherited from ancient astronomy) measures brightness logarithmically: a difference of 5 magnitudes corresponds to a factor of 100 in brightness. Apparent magnitude (m) measures what you see from Earth; absolute magnitude (M) is defined as the apparent magnitude a star would have at 10 parsecs. The distance modulus — m − M = 5 log(d/10 pc) — converts between them once distance is known (e.g., from parallax). Together, these tools form the pipeline: measure apparent brightness and spectrum → derive temperature and absolute luminosity → infer radius and place the star in its correct position on the HR diagram.
The most important conceptual point is that nearly all stellar parameters are derived, not measured directly. A star's apparent brightness is the one raw observable; everything else — temperature, luminosity, radius, mass (from binary orbits), distance (from parallax) — is inferred through physical models. This chain of inference is robust but introduces uncertainties at every step, which is why stellar astrophysics requires careful error propagation and independent cross-checks.