9.2.6 Stefan's Law

Stefan's Law

The total power P radiated by a perfect black body depends on two factors:
- It's absolute temperature
- It's surface area

The relationship between these is known as Stefan's Law or the Stefan-Boltzmann Law, which states:

The total energy emitted by a black body per unit area per second is proportional to the fourth power of the absolute temperature of the body

$P = σ A T^{4}$

Where:
- P = total power emitted across all wavelengths (W)
- σ = the Stefan-Boltzmann constant
- A = surface area of the body (m)
- T = absolute temperature of the body (K)

The Stefan-Boltzmann law is often used to calculate the luminosity of celestial objects, such as stars
The surface area of a star (or other spherical object) is equal to A = 4πr²
- Where r = radius of the star

$L = 4 π r^{2} σ T^{4}$

Where:
- L = luminosity of the star (W)
- r = radius of the star (m)
- σ = the Stefan-Boltzmann constant
- T = surface temperature of the star (K)

The surface temperature of Proxima Centauri, the nearest star to Earth, is 3000 K and its luminosity is 6.506 × 10²³ W.

Calculate the radius of Proxima Centauri in solar radii and show your working clearly.

Solar radius R_☉ = 6.96 × 10⁸ m

Answer:

Step 1: List the known quantities:

Step 2: Write down the Stefan-Boltzmann equation and rearrange for radius r

$L = 4 π R^{2} σ T^{4}$

$R = \sqrt{\frac{L}{4 π σ T^{4}}}$

Step 3: Substitute the values into the equation

$R = \sqrt{\frac{6.506 \times 10^{23}}{4 π \times (5.67 \times 10^{- 8}) \times 3000^{4}}}$

Radius of Proxima Centauri: R = 1.061 × 10⁸ m

Step 4: Find the ratio of the radii of Proxima Centauri and the Sun

$\frac{R}{R_{☉}} = \frac{1.061 \times 10^{8}}{6.96 \times 10^{8}} = 0.152 R_{☉}$