What causes the seasons?

In this brief paper, we look at the what causes the seasons. Specifically, why is it hot in the summer and cold in the winter?

The primary source of heat for the surface of earth is the Sun. The Sun is a giant thermonuclear fusion reactor, releasing large amounts of energy as a byproduct of its fusion reactions. Let us assume that the Sun is perfectly spherical. Let us also assume that its surface temperature is uniform (ignoring sun spots and solar flares). Then the energy radiation will be isotropic and traveling away from the sun. Because this emitted energy is conserved, all the energy on the surface of some sphere of radius r_1 will be the same on some other sphere of radius r_2 . Therefore, we conclude that the energy per unit area is proportional to the square of the distance from the Sun.

To relate this to temperature, we posit that the temperature of an object increases with the more energy per unit area it receives in a given area. For a flat surface, this is proportional to the flux of the energy through the surface.

 

T∝E∙s

Where T  is the temperature at some point, E is a vector describing the direction of energy travel and the amount of energy and s  is the surface normal to the point of energy capture.

  

The mean Earth-Sun distance is D_S=1.5E8 km whereas the radius of the earth is only R_E=6.37E3 km. We see that D_S/R_E>>1 and can approximate the energy propagation as constant in direction in the region of the earth. Further, earth has aphelion D_A=1.52E8 km  and perihelion D_P=1.52E8 km. This time, we see that D_A/D_P=1.03 . There’s about a 3% difference, but is that enough to account for the drastic temperature change throughout the seasons? 

Another important factor to consider is the tilt of the Earth. The rotation of the Earth actually happens at tile of about 23.4° compared to the plane of its orbit. This tilt always points in the same direction relative to the stars. Therefore, a location on Earth can be directly above the Sun at one moment in time, but only get 66.6°  of its impact in another season. This change is about cos 0°/cos 23.4°. This is a least-case scenario. If we went 60° N , then we can get a change of about cos 60°/cos 83.4°=4.35. This is a much more dramatic change compared to Earth’s proximity to the Sun.