Question

A blue supergiant star has surface temperature 27 kK and has total power output 100000 times that of the Sun.

a. Where is the Wien peak of this star's blackbody curve?​ λmax =

b. Given the Sun's 6.96×108m radius, what's the radius of the supergiant? r =

Answer 1

Answer:

Wien peak ( λmax ) is 107.40 nm

radius of super giant is 1.086 ×$10^{10}$ m

Explanation:

given data

temperature 27 kK

power = 100000 times of Sun

Sun radius =  6.96 × 10^8 m

to find out

Wien peak ( λmax ) and radius of supergiant (r)

solution

we will apply here first wien law to find Wien peak  that is

λmax = b / t

λmax = 2.9 × $10^{-3}$ / 27000 = 1.0740 × $10^{-7}$

so Wien peak ( λmax ) is 107.40 nm

and

now we apply steafay law that is

P = σ × A × $T^{4}$   .........................1

and we know total power output 100000 time of Sun

so we say  

4πr²s$T^{4}$  = 100000 × 4πR²s$Ts^{4}$

r² = 100000  × R²$Ts^{4}$ / $T^{4}$

put here value

r² = 100000  × (6.96×$6000^{8}$ )² × $6000^{4}$ / $27000^{4}$

r² = 1.18132 ×$10^{20}$

r = 1.086 ×$10^{10}$ m

so radius of super giant is 1.086 ×$10^{10}$ m