# Assignment 01: planetary energy balance#

For these exercises you’ll need some pen and paper and your notes from the class. I also recommend reading one of (or both) of these book chapters (they explain exactly the same thing but in different styles, you can choose yours):

John Marshall & R. Alan Plumb: Atmosphere, Ocean, and Climate Dynamics (Chapter 2)

Andrew Dessler: Introduction to Modern Climate Change (Chapters 3 & 4)

In the exercises below I use mostly the notations from Marshall & Plumb.

```
# These are the modules you will need
import numpy as np
import matplotlib.pyplot as plt
```

**Constants**

```
sigma = 5.67e-8 # W / (m2 K4)
Q_Sun = 3.87e26 # W
dist_sun_earth = 150e9 # m
albedo_earth = 0.3
```

**Ex 0**: Compute the solar constant at the location of Earth, given \(Q = 3.87 \times 10^{26} W\) and \(r = 150 \times 10^{9} m\)

```
# your answer here, or on pen & paper
```

**Ex 1**: A planet in another solar system has a solar constant S = 2,000 W/m 2 , and the distance between the planet and the star is 100 million km.

a) What is the total power output of the star? (Give your answer in watts.)

b) What is the solar constant of a planet located 75 million km from the same star? (Give your answer in watts per square meter.)

```
# your answer here, or on pen & paper
```

**Ex 2**: Compute \(T_e\) and \(T_s\) of the Earth using the constants defined and computed above, assuming a one-layer planet opaque to longwave radiation but transparent to shortwave radiation.

```
# your answer here, or on pen & paper
```

**Ex 3**: Using the “leaky” atmosphere model, determine \(\epsilon\) so that \(T_S\) is equal to the observed surface temperature on Earth, about 15°C.

```
# your answer here, or on pen & paper
```

**Ex 4**: As we will discover later, one way to address global warming is to increase the reflectivity of the planet. To reduce the Earth’s temperature by 1 K, how much would we have to change the Earth albedo? (assume a one-layer planet with an initial albedo of 0.3 and solar constant of 1367 W/m 2 ).

```
# your answer here, or on pen & paper
```

**Ex 5**: Either on your own of with the help of one of the books (Exercise 5 in Marshall & Plumb, or Textbook in Dessler), determine that when a very opaque atmosphere as N layers opaque to longwave radiation, the equilibrium surface temperature is:

Now plot the surface temperature of earth as a function of the number of opaque layers in the atmosphere, with N in [1, 100].

```
# your answer here
```