Luminosity (L)
Astrophysics & Cosmology - OCR A-Level Physics
Key Definition
Luminosity (L)
The total power radiated by a star across all wavelengths. Units: watts (W).
The total power radiated by a star across all wavelengths. Units: watts (W).
$$L = 4\pi r^2 \sigma T^4$$
- Stefan's lawThe total power (luminosity) radiated by a black body: L = σAT⁴, where σ is the Stefan–Boltzmann constant. (Stefan-Boltzmann law) applies to a perfect black bodyAn idealised object that absorbs all incident radiation and emits a characteristic spectrum depending only on its temperature. radiator. Stars are good approximations to black bodies.
- LuminosityThe total powerThe rate of energy transfer. Measured in watts (W). radiated by a star across all wavelengths. Measured in watts (W). depends on \(T^{4}\), so a small increase in temperature produces a large increase in luminosityThe total powerThe rate of energy transfer. Measured in watts (W). radiated by a star across all wavelengths. Measured in watts (W)..
- A star with twice the temperature but the same radius radiates \(2^{4}\) = 16 times the luminosityThe total powerThe rate of energy transfer. Measured in watts (W). radiated by a star across all wavelengths. Measured in watts (W)..
- A star with twice the radius but the same temperature radiates \(2^{2}\) = 4 times the luminosity.
- The Sun's luminosity is $L_{\odot} \approx$ $3.85 \times 10^{26}$ W.
- The factor $4\pi$\(r^{2}\) is the surface area of a sphere.
Common Mistake
MEDIUM
Wrong: Using diameter instead of radius in Stefan's law, or forgetting to raise temperature to the fourth power.
Right: L = 4\pi \(r^{2}\) \sigma \(T^{4}\) uses RADIUS (not diameter) and T to the FOURTH power. If given diameter, divide by 2 first.
Right: L = 4\pi \(r^{2}\) \sigma \(T^{4}\) uses RADIUS (not diameter) and T to the FOURTH power. If given diameter, divide by 2 first.
Examiner Tips and Tricks
- When comparing stars, use ratios to avoid large numbers.
- L_1/L_2 = (r_1/r_2)^$2 \times$(T_1/T_2)^4.
- This is a very common exam technique.