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It's Getting Hot In Here

In a recent post, we used trigonometry to derive the length of a day on the Earth as a function of the observer's latitude and the time of year. As promised, I want to continue modelling the Earth's orbit to see what it can tell us about temperature. The simplest explanation for the seasonal variation of temperature comes from the concept of solar flux. To see what this means, think about taking a ray of sunlight shining on the Earth and decomposing it into two components - one parallel to the Earth's surface and one perpendicular.
Decomposing a ray of sunlight into two components to find the flux.
Most of the sunlight going into the ground means this location will be hot, whereas most of the sunlight going along the ground means this location will be cold. The temperature due to direct sunlight is therefore proportional to the cosine of the angle between the ray and the outward normal to the Earth. To actually solve for a temperature, we would have to multiply the intensity of the light by $ \cos \theta $ and integrate this over a region of interest. Our main concern will be solving for $ \cos \theta $, allowing us to express the temperature at one time relative to the temperature at another time without knowing the absolute intensity.

The Sunrise Equation

In my quest to watch a sunrise recently, I had to search the web to find out the time before which I needed to get up. Predicting the sunrise is something that I had pondered before. I'm sure numerical simulations are more accurate, but I began deriving a simple formula. Everyone knows the gist of why the Sun rises and sets and why this experience depends on your location on the globe. The answer is that it all depends on the tilt of the Earth's axis.

The relationship between the tropics and the solstices.

The Earth's axis of rotation makes a certain angle with the normal to the plane in which it orbits. This angle of inclination is $ 23.5^{\circ} $. It is not hard to picture how this affects the seasons and why the tropics are offset from the equator by $ 23.5^{\circ} $ as well. I found it much harder to visualize the effect that the inclination has on the length of a day and most people I've talked to simply take it on faith that the Arctic and Antarctic circles are at $ 66.5^{\circ} = 90^{\circ} - 23.5^{\circ} $ latitude.

So now is our chance to overcome this hurdle. Together, you and I will figure out how to calculate the length of a day as a function of time for all latitudes. Hint: it is not a smooth function!

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