Question

Consider a one layer atmosphere for the planet Venus. 19% of incoming solar radiation is absorbed by the atmosphere. 99% of the outgoing radiation is absorbed by the atmosphere. The incoming radiation, E, as measured by satellite is 615 Wm-2.

1) Derive two equations, one for the top of the atmosphere and the other for the Venusian surface in terms of x, y and E. Solve for the surface and atmospheric temperatures.

Answer 1

That value of E is wrong. It is lower than the value at the earth's orbit. It should be higher.

See
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Answer 2

Assuming your value of E (unlikely):

At equilibrium, the same amount is radiated back as hits the ground or the ground would keep heating forever.
(1-.19) * 615 w/m^2 hits ground
Therefore at what Kelvin temperature is the radiation 498 w/m^2 ? That is the ground temperature.

The atmosphere is abosrbing .19(615) = 117 w/m^2 inbound + .99 (498)= 493 outbound so it radiates back a total of 610 w/m^2. Again find the temp in deg. Kelvin.