- An objects luminosity depends on two factors:
- Its surface temperature
- Its 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
- So Stefan's Law shows that the luminosity of a star is directly proportional:
- To its radius L ∝ r2
- To its surface area L ∝ 4πr2
- To its surface absolute temperature L ∝ T4
- Stefan's Law equation is given by:
L = 4πr2σT4
- L = luminosity of the star (W)
- r = radius of the star
- σ = the Stefan-Boltzmann constant
- T = surface temperature of the star (K)
- The surface area of a star (or other spherical object) can be calculated using: A = 4πr2
- Where r = radius of the star
The surface temperature of Proxima Centuri, the nearest star to Earth, is 3000 K and its luminosity is 6.506 × 1023 W.
Calculate the radius of Proxima Centuri in kilometres and show your working clearly.
Step 1: List the known quantities:
- Surface temperature, T = 3000 K
- Luminosity, L = 6.506 × 1023 W
- Stefan's constant, σ = 5.67 × 10−8 W m−2 K−4
Step 2: Write down Stefan's Law
L = 4πr2σT4
Step 3: Rearrange the equation for r
Step 4: Substitute into the equation
= 106 173 971 m
Step 5: Write the final answer to the correct amount of significant figures
- The radius of Proximal Centuri is 106 200 km (4 s.f.)
Remember to convert temperatures into Kelvin.
Check the values you obtain in your calculations. Do they make sense? Do they fit in with the magnitudes of other stars/objects that you know about?