Understanding the Relationship
The luminosity of a star is directly related to its temperature. The relationship is described by the Stefan-Boltzmann Law:
* L = σAT⁴
Where:
* L is luminosity (energy emitted per second)
* σ is the Stefan-Boltzmann constant (a constant value)
* A is the surface area of the star (we'll assume it's roughly the same for both stars)
* T is the surface temperature in Kelvin
Calculating the Difference
1. Set up a Ratio: Since we're interested in the *difference* in luminosity, we can set up a ratio of the luminosities of the two stars:
L(hot) / L(cool) = (σAT⁴(hot)) / (σAT⁴(cool))
2. Simplify: The surface area (A) and the Stefan-Boltzmann constant (σ) cancel out:
L(hot) / L(cool) = T⁴(hot) / T⁴(cool)
3. Plug in Temperatures:
L(hot) / L(cool) = (30000 K)⁴ / (3000 K)⁴ = 10⁴
4. Result: This means the hotter star is 10,000 times more luminous than the cooler star.
Important Note: This calculation is a simplification. In reality, stars have different sizes, and their luminosity depends on both temperature and size. However, for a first-order estimate, the Stefan-Boltzmann Law provides a good starting point.
