Stellar luminosity

Looking up at the night sky, we can see stars of varying brightness. However, our eyes can only distinguish about six different levels of brightness, which is insufficient for scientific purposes. Astronomers use a more precise measure called luminosity to classify the actual brightness of stars.

Luminosity is defined as the rate at which energy of all types is radiated by an object in all directions. It depends on two key factors:

1. The object's size
2. The object's temperature (more significant factor)

Black body radiation and temperature

Objects emit electromagnetic radiation across a wide range of wavelengths. A perfect black body radiator emits energy across the entire electromagnetic spectrum, following a specific distribution based on its temperature.

Key points about black body radiation:

• Hotter objects emit more energy overall
• The peak of the emission curve shifts to shorter wavelengths as temperature increases
• The shape of the curve is determined by the object's temperature

The Stefan-Boltzmann Law

The Stefan-Boltzmann Law describes the relationship between an object's temperature and the total amount of energy it radiates. It states that the power output (luminosity) from a black body is proportional to its surface area and the fourth power of its temperature in Kelvin:

$L=A\sigma T^4$

Where:

• L = Luminosity (W)
• A = Surface area (m²)
• σ = Stefan-Boltzmann constant (5.67 × 10^-8 W m^-2 K^-4)
• T = Temperature (K)

For a spherical object like a star, we can express the surface area in terms of its radius:

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

Where:

• r = Radius of the sphere (m)

Worked example - Calculating star luminosity

Calculate the luminosity of a star with a radius of 7 × 10⁸ m and a surface temperature of 5,800 K.

Step 1: Formula

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

Step 2: Substitution and correct evaluation

$L=4\pi {(7\times {10}^8)}^2\times (5.67\times {10}^{-8})\times {(\mathrm{5,800})}^4{\text{ = 3.95 x 10}}^{26}\text{ W}$