A star is five times as luminous as the Sun and has a surface temperature of 98,000 K. What is its radius, compared to that of the Sun?

The relationship between a star's color and temperature is as follows: Yellow stars, like our sun, have temperatures between 5000 and 6000 degrees Celsius, while orange stars, like the Supergiant Betelgeuse, have lower temperatures than main sequence stars, between 3000 and 4000 degrees Celsius. The hottest stars are blue stars, with temperatures exceeding 10,000 degrees Celsius. However, if a star has a temperature of 98,000 degrees Celsius, it must be extremely luminous in comparison to our sun.

The star R136a1, a Blue Hypergiant in the constellation Dorado, is approximately 9 million times brighter than our Sun and has a temperature of roughly 53000 degrees Celsius.

The luminosity (L) of a star is related to its radius (R) and surface temperature (T) through the Stefan-Boltzmann law, (L = 4\pi R^2 \sigma T^4), where (\sigma) is the Stefan-Boltzmann constant. Since the star is five times as luminous as the Sun, we can write (5L_{\odot} = 4\pi R^2 \sigma (T_{\text{star}})^4), where (L_{\odot}) is the luminosity of the Sun and (T_{\text{star}}) is the surface temperature of the star. Rearranging the equation to solve for the radius (R), we have (R^2 = \frac{5L_{\odot}}{4\pi \sigma (T_{\text{star}})^4}). Since the luminosity (L_{\odot}) and surface temperature (T_{\text{star}}) are known for the Sun, we can substitute these values into the equation along with the given values for the star's luminosity and surface temperature to find the radius (R) of the star compared to that of the Sun.