Venus actual surface temperature is much hotter than its eff

Venus actual surface temperature is much hotter than its effective temperature due to the greenhouse effect. Use the relationship between optical depth and surface temperature to estimate the optical depth (at IR wavelengths) of Venus atmosphere. List two important assumptions this method depends on -- and why they are probably not valid.

Solution

Te = T_eff is the EFFECTIVE temperature - that\'s the \"temperature\" that is derived by measuring all the (blackbody - or thermal radiation) IR flux from a planet

The OPAQUE SLAB MODEL - sometimes called \"grey model\" - does not depend on wavelength.

Assumptions:

Starting at the top of the atmosphere and working down layer by layer, equating the flux into each layer with the flux out

Layer 0 - T₁ = T_eff

Layer 1 - sT₂⁴ = sT₁⁴ + sT₁⁴ ....so, T₂⁴ = 2T₁⁴

Layer 2 - sT₃⁴ + sT₁⁴ = sT₂⁴ + sT₂⁴ ....so, T₃⁴ = 2T₂⁴ - T₁⁴ = 4T₁⁴ - T₁⁴= 3T₁⁴

Layer N - T_N⁴ = NT₁⁴

So, if you imagine you keep going down the layers until you reach the surface of the planet - where the temperature is Tg - then you can envisage an expression

Tg⁴ = (1+t) T_eff⁴

where we call t = OPTICAL THICKNESS or OPTICAL DEPTH

Venus: t = (Tg/Teff)⁴ -1 = (750K/238K)⁴ -1 = 98 - Venus atmosphere is very thick

Thus, the Flux that is absorbed is e s Tg⁴ Watts m^-2 and the amount that is radiated to space is then (1-e) s Tg⁴ = s Teff ⁴ Watts m^-2

This then means that we can define

OPACITY e = = 1 - (Teff/Tg)⁴

which is not to be confused with OPTICAL DEPTH t = (Tg/Te)⁴ -1

from the above equation we calculate the optical depth for venus if we know the effective temperature (243 C) and the ground temperature (700 C) and if we assume that the atmosphere is in radiative equilibrium,

so we get, optical thickness = (700/243)^4 - 1 = 68