Question

An asymptotic giant branch star has a surface temperature of 3500 K and a luminosity of 4200 L ⊙. What is this star's radius in unit of solar radius?

Answer 1

The luminosity of star is given by relation,

L= 4pR2sT^4

Where L is luminosity of star, T is temperature of star in kelvin, R is radius of star in meters . s is Stefan- Boltzmann constant

The calculation is actually somewhat easier if we try to find the ratio of another star's radius to that of our Sun. Let Ls be the luminosity of the Sun, L be the luminosity of another star, Ts be the temperature of the Sun, T be the temperature of the other star, Rs be the radius of the Sun, and R be the radius of the other star.

We can then write the ratio of their luminosities as

L/Ls = (4pR2sT4)/(4pRs2sTs4)
= (R/Rs)2(T/Ts)4

Solving for the ratio R/Rs yields

R/Rs = (Ts/T)² √(L/Ls)
Here,
Ts= 5778K
T= 3500K
L= 4200 Ls........ L/Ls = 4200
We have to find star radius in unit of solar radius
R/Rs = ( 5778/3500)² * √(4200
= 2.723 * 64.81
= 176.62
Radius of star, R= 176.62 Rs

i.e 176.62 times that of sun