Some recently discovered planets in other solar systems are so hot that they glo

Question

Some recently discovered planets in other solar systems are so hot that they glow in the visible: they are literally "red hot" (e.g., do a Google search for "HD 149026b"). a) How many atmospheric layers would the Earth need before it glowed in the visible? (Assume S = 1, 360 W/m^2, and alpha = 0.3.) To answer this, you must first estimate what temperature the Earth has to be to begin glowing. b) Alternatively, what would the solar constant have to increase to for a one-layer planet with an albedo alpha = 0.3? c) How far would the Earth have to be from the Sun in order to have this solar constant?

Explanation / Answer

6. a. wavelength of red light is lambda = 650 nm

for a planet to glow red hot, this lambda should be the prominent one in spectrum

so from weins displacement law

lambda = b/T

here b = 2.897*10^-3

so, T = b/lambda = 4456.92 K

for earth to glow this hot, lent number of layers be n

then T = sqroot(SQROOT((n + 1)S(1 - alpha)/4*sigma))

S = solar constant = 1360 W/m^2

alpha = albedo = 0.3

sigma = stefans constant = 5.67*10^-8

so 4456.92^4 = (n + 1)*1360*(1 - 0.3)/4*5.67*10^-8

hence n = 65801

so earth will need 65801 layers of atmoshpere

b. if S changes, and n = 1

4456.92^4 = 2*S*(1-0.3)/4*5.67*10^-8

S = 63922590.9 W/m^2

c. distance for this solar constant = d

then 1360*4*pi*(1 AU)^2 = 63922590.9*4*pi*d^2

d = 0.0046125 AU

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