Betelgeuse Written Homework The spherical star Betelgeuse has the following prop

Question

Betelgeuse Written Homework

The spherical star Betelgeuse has the following properties:

peak wavelength = 807 nm
radius = 6.3E11 m
mass = 2.31E31 kg
distance to Earth = 6.85E18 m
emissivity = 1

Determine the following based on the above information and relevant equations:

a) Temperature of the light-emitting surface of Betelgeuse, assumed constant (as an aside, note that there isn't really a single temperature of a star).
b) Power of Betelgeuse.
c) Intensity of light from Betelgeuse reaching Earth (ignore the effect of absorption by the atmosphere)

The light from Betelgeuse hits a lens of a telescope on Earth with a diameter of 30 cm for one minute.
d) Power of this light.
e) Energy of this light.
f) Energy of a single photon of the peak wavelength.
g) Approximate number of photons absorbed by the telescope (assume all are of peak wavelength).

Explanation / Answer

(A) Wein's displacement law,

lambda_peak T = 2.898 x 10^-3 m K

(807 x 10^-9) T = 2.898 x 10^-3

T = 3591 K  

(B) P = e sigma A T^4

P = (1)(5.67 x 10^-8) (pi (6.3 x 10^11)^2) (3591)^4

P = 1.176 x 10^31 W

(C) I = P / (4 pi r^2)

I = (1.176 x 10^31) / (4 pi (6.85 x 10^18))

I = 2 x 10^8 W/m^2

(D) P = I A

P = (2 x 10^8) (pi (0.15^2))

P = 1.41 x 10^-9 W

(E) E = P t = 8.46 x 10^-8 J  

(F) E_photon = h c / lambda

= (6.626 x 10^-34)(3 x 10^8)/ (807 x 10^-9)

= 2.463 x 10^-19 J  

(g) n = E / E_photon

E = 3.43 x 10^11 photons ...Ans

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