gen-define-luminosity-and-the-stefan-boltzmann-law-and-give-one-073d

Define luminosity and the Stefan-Boltzmann law and give one short worked example.

# Luminosity **Luminosity** is the total amount of energy radiated by an object (such as a star) per unit time across all wavelengths. It is measured in watts (W) or in solar luminosities (L☉), where 1 L☉ = 3.828 × 10²⁶ W. # Stefan-Boltzmann Law The **Stefan-Boltzmann law** relates the luminosity of a blackbody to its surface temperature and surface area: $$L = \sigma A T^4 = 4\pi R^2 \sigma T^4$$ Where: - **L** = luminosity (W) - **σ** = Stefan-Boltzmann constant = 5.67 × 10⁻⁸ W m⁻² K⁻⁴ - **A** = surface area (m²) - **R** = radius of the spherical object (m) - **T** = surface temperature (K) # Worked Example **Problem:** Find the luminosity of a star with radius R = 2R☉ and surface temperature T = 8000 K. (Given: R☉ = 6.96 × 10⁸ m) **Solution:** $$L = 4\pi R^2 \sigma T^4$$ $$L = 4\pi (2 × 6.96 × 10^8)^2 × (5.67 × 10^{-8}) × (8000)^4$$ $$L = 4\pi × 1.936 × 10^{18} × 5.67 × 10^{-8} × 4.096 × 10^{15}$$ $$L = 5.66 × 10^{27} \text{ W} ≈ 15 L☉$$ The star is approximately **15 times more luminous** than the Sun.