Betelgeuse, a bright red star in the constellation Orion, has a luminosity that is 100,000 times that of the Sun. However, the surface temperature of Betelgeuse is only 3000 K. Assuming that the emissivity of stars is 1, What is the radius of Betelgeuse?

Betelgeuse, a bright red star in the constellation Orion, has a luminosity that is 100,000 times that of the Sun. However, the surface temperature of Betelgeuse is only 3000 K. Assuming that the emissivity of stars is 1, What is the radius of Betelgeuse?

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Expert Solution

Step 1

$L u m i n o s i t y o f t h e s u n i s 3.916 \times 10^{26} W B e t e l g u e s e h a s a l u m i n o s i t y, 100000 \times 3.916 \times 10^{26} = 3.916 \times 10^{31} W$

$T h e f l u x e m i t t e d b y a s t a r i s d e f i n e d a s t h e e n e r g y e m i t t e d p e r u n i t a r e a a n d i s g i v e n i n t e r m s o f t e m p e r a t u r e a s F = σ T^{4} σ i s S t e f a n - B o l t z m a n n c o n s \tan t = 5.67 \times 10^{- 8} W / m^{2} K^{4} T i s t h e t e m p e r a t u r e S o, F = 5.67 \times 10^{- 8} \times {(3000)}^{4} F = 5.67 \times 10^{- 8} \times 8.1 \times 10^{13} F = 4.5927 \times 10^{6} W / m^{2}$