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.

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

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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.

To derive the equations for the top of the atmosphere and the Venusian surface, we need to use the energy balance equation. Let's define the following terms:

E = incoming solar radiation at the top of the atmosphere (given as 615 W/m^2)
x = fraction of incoming solar radiation absorbed by the atmosphere (given as 19%)
y = fraction of outgoing radiation absorbed by the atmosphere (given as 99%)
T_t = temperature at the top of the atmosphere
T_s = temperature at the Venusian surface

The energy balance equation at the top of the atmosphere can be written as:

E = (1 - x)E - yσT_t^4 [Equation 1]

where σ is the Stefan-Boltzmann constant.

The energy balance equation at the Venusian surface can be written as:

E(1 - x) - σT_s^4 = yσT_t^4 [Equation 2]

Now let's solve these equations for T_t and T_s.

From Equation 1:
E = (1 - x)E - yσT_t^4

Rearranging, we get:

yσT_t^4 = xE [Equation 3]

Substituting Equation 3 into Equation 2, we can eliminate T_t:

E(1 - x) - σT_s^4 = yσT_t^4

E(1 - x) - σT_s^4 = xyE

Simplifying, we have:

T_s^4 = (E(1 - x)) / (σ(1 - y)) [Equation 4]

Now we have an equation for the surface temperature (T_s) in terms of x, y, and E.

Similarly, substituting Equation 3 into Equation 1, we can solve for T_t:

yσT_t^4 = xE

T_t^4 = (xE) / (yσ) [Equation 5]

Now we have an equation for the top of the atmosphere temperature (T_t) in terms of x, y, and E.

To find the surface and atmospheric temperatures, plug in the values of x, y, and E into Equations 4 and 5 and solve for T_s and T_t respectively.