Question

Consider a red giant star with a luminosity of 200 Lᴀᴏs and a radius of 50 Rᴏᴏʀ. Using the luminosity-temperature-radius relationship (L proportional to R^2T^4), calculate how hot this star's surface temperature will be compared to the Sun, whose temperature is 5,800 K.

a) 7,200 K
b) 2,900 K
c) 1,450 K
d) 725 K

Answer 1

To determine the surface temperature of the red giant star, we use the luminosity-temperature-radius relationship provided in the question and find that it has a temperature of 2,900 K compared to the Sun's temperature of 5,800 K.

Step-by-step explanation:

The question asks us to calculate the surface temperature of a red giant star with a known luminosity and radius, using the luminosity-temperature-radius relationship (L ≈ R²T⁴), and compare it to the Sun's temperature. To find the temperature of the red giant star, we rearrange the equation to solve for temperature (T), taking into account the known values for the Sun's luminosity (Lsun), the Sun's radius (Rsun), and the Sun's surface temperature, which is 5,800 K.

Following the formula L = 4πR²σT⁴, where σ is the Stefan-Boltzmann constant, we can compare the luminosities and radii of the red giant star and the Sun. Since the red giant has a luminosity of 200 Lsun and a radius of 50 Rsun, we can set up the following proportion:

200 Lsun = (50 Rsun)² T⁴ / (1 Rsun)² (5,800 K)⁴

Through this equation, we will solve for T to find the surface temperature of the red giant star in comparison to the Sun. The correct answer would be (b) 2,900 K, which means the surface temperature of the red giant star is 2,900 K when compared to the Sun's temperature of 5,800 K.