Math 5421 Spring 2025
Introduction to Climate Models

Assignment 7
due March 4, 2025

Earth's Eccentricity

For these exercises, assume that the Earth orbits the Sun on an elliptical orbit with semimajor axis \(a = 1.5\times 10^{11}\) meters and eccentricity \(e\). Assume that Earth's obliquity is a constant \(23.5^\circ\).

Recall from the slides of February 20 that the annual average solar flux along the orbit is \[ \frac{\hat K a^{-2}}{\sqrt{1 - e^2}}\text{Wm}^{-2}, \] where \(\hat K\) is a constant depending on the solar intensity. Since we are assuming that Earth's semimajor axis remains constant, for now we can assume that the average solar flux is \( \dfrac{P_0}{\sqrt{1 - e^2}}{\text{Wm}^{-2}} \) and that the average insolation is \( \dfrac{P_0}{4\sqrt{1 - e^2}}{\text{Wm}^{-2}} , \) where \(P_0 = 1368\text{ Wm}^{-2}\) is the solar flux at the distance \(a\) from the Sun. For Exercise 4, use Chylek and Coakley's quadratic approximation for the distribution of insolation across latitudes: \( s(y) = 1 = 0.241(3y^2 - 1). \)

Exercise 1

Compute the maximum and minimu distances from the Sun to the Earth for eccentricities of \(e = 0\), \(e = 0.03\), and \(e = 0.06\). Compute the solar flux at these distances.

Exercise 2

Assume that the northern hemisphere summer solstice occurs at perihelion (the point on the orbit of minimum distance to the Sun). For each of the eccentricities in Exercise 1, compute the insolation at noon on the summer solstice for latitudes of 0° (the Equator), 23.5°N (the Tropic of Cancer), 45°N (the latitude of the Twin Cities, 66.5°N (the Arctic Circle), and 90°N (the North Pole).

Exercise 3

Repeat Exercise 2 with the assumption that the northern hemisphere summer solstice occurs at aphelion (the point on the orbit of maximum distance to the Sun).

Exercise 4

For each of the eccentricities in Exercise 1 and each of the latitudes in Exercise 2, compute the annual average insolation.

Exercise 5

Compare and discuss your answers to Exercises 2, 3, and 4.