(Part E) A red dwarf star named Gliese 581 has radius of 201,700 km and acts lik

Question

(Part E) A red dwarf star named Gliese 581 has radius of 201,700 km and acts like a blackbody of temperature 3,500 K. Just for fun, let's imagine that a planet like Earth orbits this star (Earth radius = 6,370 km, and distance 1.5 x 108 km). If it takes this planet 2.5 Earth-years to orbit Gliese 581 once, then how much energy is incident on the planet at all wavelengths during one orbital period, assuming the orbit is circular? Note: Your answer is assumed to be reduced to the highest power possible. Your Answer: x10 Answer units

Explanation / Answer

given radius of star, R = 201,700,000 m
temperature of this star, which acts as a blackbody, T = 3500 K
so power radiated by thios star, P = sigma*A*T^4
P = 5.67*10^-8*4*pi*R^2*T^4 = 5.67*10^-8*4*pi*(201700000)^2*3500^4
P = 4.349*10^24 W

Now distance of planet from the star, d = 1.5*10^11 m
radius of planet = r = 6370,000 m

so, power recieved by earth = P*pi*r^2/4*pi*d^2 = Pr^2/4d^2 = 4.349*10^24*(6370,000)^2/4*(1.5*10^11)^2 = 1.9611*10^15 W
Energy recieved in 2.5 years = 1.9611*10^15*2.5*365*24*60*60 J = 1.546*10^23 J

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