2. What processes determine the magnitude of temperature shifts in climate associated with alterations of the Earth's orbit. Does the change in solar insolation at the top of atmosphere necessarily correspond strongly to the change in Earth's temperature? Why or why not?
While slight changes in top of atmosphere solar radiation are important causes of the ice ages, their magnitude does not correspond well to the eventual change of global temperatures. Feedbacks within the atmosphere tend to amplify the impact of the lower frequency components of the Milankovitch cycles while having little to no effect on the higher frequency components. One example of a feedback that evolves slowly and may amplify the lower frequency components of the Milankovitch cycle is the ice-albedo feedback on Earth.

3. Suppose the extent of the ice caps increases by a facor of 2 during the Ice Ages which in turn results in a globally averaged cooling of 10 W m^-2. How rapidly will the atmosphere and surface oceans cool if half of the cooling acts on the atmosphere and half of the cooling acts on the surface oceans? The surface area of the globe is 1.27 x 10¹⁴ m². 
The cooling acting on the atmosphere and surface oceans amounts to 6.35 x 10¹⁴ W^-2.
The heat capacity of the atmosphere is 5.0 J gram^-1 J K^-1 x 10⁶ x 10¹⁵ gram.
The reservoir for the surface ocean is 15.2 J gram^-1 K^-1 x 10⁷ x 10¹⁵ g
The rate of atmospheric cooling is therefore 1.27x10^-7 K-1 s^-1 or 4^oK year^-1
The rate of cooling of the surface oceans is 4.18x10^-9 K-1 s^-1 or 0.13^oK year^-1

4. Derive the relationship between surface temperature and average atmospheric temperature for a single layer atmosphere of emissivity, e. Use the fact that the longwave radiation emitted by the atmosphere is equal to the longwave radiation it absorbs from surface emissions.

We know that the energy absorbed by the atmosphere is equal to the energy emitted by the atmosphere in the simple one layer model and can be expressed by :