A.) If the solar constant I_e= 3 kW/m^2 is the incident energy flux from the sun

Question

A.) If the solar constant I_e= 3 kW/m^2 is the incident energy flux from the sun at the Earth's surface at a distance r_e=2*10^11m. What is the total power released by then sun?

B.) Building off of part a, Use the Stefan-Boltzmann law (I=4*sigma*T^4, where sigma=5.7*10^-8W/m^2K^4) to estimate the mean temperature of the earth's surface (T_e), assuming radiation heat fluxes are balanced on the earth's surface.

c.) Water-based life is constrained by the planetary habitable zone where water freezes at 273K and boils at 373K. In our solar system, what are the corresponding orbital distances from the sun (in AU). Compare to the orbits of of three other planets with r=0.4AU, r=0.8AU r=1.3AU respectively.

Explanation / Answer

a) P = I A = I 4 pi r^2 = 3.0E3*4*pi*2.0E11^2= 1.51E27 W

b) so we want I in = I out

I = I in

4*5.7E-8*T^4 = 3.0E3

T=338.7 K

c)

I = P/(4 pi r^2)

so I = Iearth 1^2/r^2 with r in AU

3.0E3*1^2/r^2 = 4*5.7E-8*273^4

r=1.54 AU

now for max

3.0E3*1^2/r^2 = 4*5.7E-8*373^4

r= 0.824 AU

if r = 0.4

3.0E3*1^2/0.4^2 = 4*5.7E-8*T^4

T=535.5 K

r = 0.8 AU
T = 378.7 K

r=1.3 AU
T= 297.05

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