### Activity

#### Note:

For this assignment, you will need to record your work on a word processing document. Your work must be submitted in Word (.doc or .docx) or PDF (.pdf) format so the instructor can open it.

The documents associated with this problem set, including a formatted answer sheet, can be found on CANVAS.

- Be sure also to download the answer sheet from CANVAS (Files > Problem Sets > PS#4).

Each problem (#2 to #6) will be graded on a quality scale from 1 to 10 using the general rubric as a guideline. For this problem set, each problem is equally weighted. Thus, a score as high as 50 is possible, and that score will be recorded in the grade book.

The objective of this problem set is for you to work with some of the concepts and mathematics around one-layer energy-balance models (1-D EBMs) covered in Lesson 5. You may find Excel useful in this problem set, but you may use any software you wish, keeping in mind that the instructor only can provide help with Excel.

- Re-read the derivation of the equations for planetary surface temperature and planetary atmospheric temperature under the framework of a one-layer energy-balance model, as laid out in Lesson 5. After re-reading the derivation,
**write the two aforementioned equations, and, for each equation,****explain what each variable in the equation is. Summarize also the assumptions behind the derivation; in other words, explain what a one-layer EBM is and how it differs from a 0-D EBM**. This is as straightforward as it seems, but this problem is here to ensure that you understand the underlying concepts because they are referenced often in climate science.**Report your discussion on the answer sheet.** **Calculate values of global surface temperature and planetary atmospheric temperature for varying values of the solar constant, ranging from 0 to 2,000 W m**Assume that the value of planetary albedo is 0.32, emissivity is 0.77, and Stefan-Boltzmann constant is 5.67 x 10^{-2}, incremented by 100 W m^{-2}.^{-8 }W m^{-2}K^{-4}.**Construct a table to show your results. Using the values that you calculated, plot both global surface temperature and global tropospheric temperature as a function of solar constant; use the same grid for both sets of temperature values**. That is, the horizontal axis of the plot should give values of solar constant, and the vertical axis should give values of temperature.**Report all results on the answer sheet.****Calculate values of global surface temperature and global tropospheric temperature for varying values of planetary albedo, ranging from 0 to 1, incremented by 0.05**. Assume that the value of the solar constant is 1,360 W m^{-2}, emissivity is 0.77, and Stefan-Boltzmann constant is 5.67 x 10^{-8}W m^{-2}K^{-4}.**Construct a table to show your results. Using the values that you calculated, plot both global surface temperature and global tropospheric temperature as a function of planetary albedo; use the same grid for both sets of temperature values.**That is, the horizontal axis of the plot should give values of planetary albedo, and the vertical axis should give values of temperature.**Report all results on the answer sheet.****Calculate values of global surface temperature and global tropospheric temperature for varying values of planetary emissivity, ranging from 0 to 1, incremented by 0.05.**Assume that the value of the solar constant is 1,360 W m^{-2}, planetary albedo is 0.32, and Stefan-Boltzmann constant is 5.67 x 10^{-8}W m^{-2}K^{-4}.**Construct a table to show your results. Using the values that you calculated, plot both global surface temperature and global tropospheric temperature as a function of planetary emissivity; use the same grid for both sets of temperature values.**That is, the horizontal axis of the plot should give values of planetary albedo, and the vertical axis should give values of temperature.**Report all results on the answer sheet.****Explain the relationship between each of the independent variables (solar insolation, planetary albedo, planetary emissivity) and planetary temperature. As well, discuss the various energy-balance models covered in Lessons 4 and 5, comparing and contrasting their assumptions. Write your discussion on the answer sheet**, limiting it to one to two paragraphs.