A red giant star has a temperature of 3,000 K and a luminosity #10^5# solar luminosities. How do you calculate the radius of the red giant in terms of the radius of the sun?

Question

A red giant star has a temperature of 3,000 K and a luminosity #10^5# solar luminosities.  How do you calculate the radius of the red giant in terms of the radius of the sun?

Asher Cohen · Accepted Answer

#=>R approx 33.33 " R"_ "sun"#

The relationship between luminosity, radius, and temperature can be reduced to the following when working with solar units:

#=>L = R^2T^4#

Solving for #R#:

#=>R = sqrtL/T^2 #

We are given #T = 3000 "K"# and #L = 10^5 "L"_ "sun"#. #R# in units of #"R"_ "sun"# is then:

#=>R = sqrt((10^5)^2)/3000 "R"_"sun"#

#=>R = 10^5/(3 xx 10^3) "R"_ "sun"#

#=>R = (1/3 xx 10^2)"R"_ "sun"#

#=>R = (100/3)"R"_ "sun"#

#=>color(green)( R approx 33.33 " R"_ "sun" )#

Leah Abbate · Answer

undefined The red giant star's radius can be found using the following formula: $$ L = 4 \pi \sigma R^2 T^4 $$ where: - $ L $ is the luminosity; - $ \sigma $ is the Stefan-Boltzmann constant ($5.67 \times 10^{-8} \, \text{W m}^{-2} \, \text{K}^{-4}$); - $ R $ is the star's radius; - $ T $ is the temperature in Kelvin. Rearrange the formula to solve for $ R $: $$ R = \sqrt{\frac{L}{4 \pi \sigma T^4}} $$ Enter the given values to find the red giant star's radius in terms of the sun's radius.