Understanding the Relationships
* Luminosity: The total amount of energy a star emits per second.
* Temperature: The surface temperature of the star.
* Radius: The physical size of the star.
Key Equations
* Stefan-Boltzmann Law: L = 4πR²σT⁴
* L = Luminosity
* R = Radius
* σ = Stefan-Boltzmann constant (a constant value)
* T = Temperature
Calculations
1. Let's use subscripts:
* Sun: L_sun, T_sun, R_sun
* Star: L_star, T_star, R_star
2. We know:
* T_star = 0.5 * T_sun (half as hot)
* L_star = 100 * L_sun (100 times more luminous)
3. Apply the Stefan-Boltzmann Law to both the Sun and the star:
* L_sun = 4πR_sun²σT_sun⁴
* L_star = 4πR_star²σT_star⁴
4. Divide the star's equation by the Sun's equation:
* (L_star / L_sun) = (4πR_star²σT_star⁴) / (4πR_sun²σT_sun⁴)
* Simplify: L_star / L_sun = (R_star²/R_sun²) * (T_star⁴/T_sun⁴)
5. Substitute the known values:
* 100 = (R_star²/R_sun²) * (0.5T_sun⁴ / T_sun⁴)
* 100 = (R_star²/R_sun²) * (0.5⁴)
6. Solve for R_star / R_sun:
* 100 = (R_star²/R_sun²) * (0.0625)
* (R_star²/R_sun²) = 100 / 0.0625 = 1600
* R_star / R_sun = √1600 = 40
Answer:
The star's radius is 40 times larger than the Sun's radius.
