# Write out the energy balance equations for an atmosphere with an emissivity

## Write out the energy balance equations for an atmosphere with an emissivity1 ,

### science

##### Description

Directions: Please submit a PDF write-up with your answers to questions and any required figures. For data analysis, also submit the file containing your analysis. For example, submit the code / output of your coding language of choice (e.g., a Stata “log file”, or R-script, or Excel sheet). Using Excel is okay, but not preferred – it is often hard to replicate work done in Excel, and it’s easy to make mistakes and not be able to track them down. The next assignment will need more work in R, so it’s a good idea to attempt these simple exercises for this assignment. There’s a sample R-script for question 4 on canvas. See note about R at the end of this problem set. This problem set looks longer than it is!

Greenhouse effect with a transparent atmosphere

1. In class, we derived the Earth’s surface temperature using a simple energy balance model with a one-layer atmosphere. Now, you’ll derive the energy balance with a partly transparent atmosphere.

(a) Draw a diagram like the one on slide 52/130 in “L1” lecture slides

(b) Write out the energy balance equations for an atmosphere with an emissivity1 ,

(c) Solve the equations for TS, the temperature of the Earth’s surface

(d) Assume the following values and calculate TS: S = 1370 Wm−2 ; = 0.78; A = 0.3

(e) One of our neighbours, Mars, has an average solar constant of SM = 589 Wm−2 , an albedo of AM 0.24, and a surface temperature of -46◦C. What would its atmospheric emissivity, M, need to be to achieve that surface temperature?