Is the longest day the warmest day?

I woke up to a snowy day on the 30th of December, here in New Jersey and immediately realized two things! It was colder and darker than at the same time on the shortest day of the year, the 21st of December. I suppose you could blame the colder weather on the polar vortex swinging its awful oscillation down towards us yokels in the New York metro area, rather than simply harass a bunch of Brahminical New Englanders. But was it really darker? Was I missing something? Is the shortest day for us folks living at $41^{\circ}$ North ${\bf different}$ from the $21^{st}$ of December? After all, the winter solstice is defined as the day when the sun is over its extreme southward excursion, when it is over the Tropic of Capricorn.

And if you average out the vagaries of weather patterns, is the shortest day really the coldest day? Conversely, is the longest day (June $21^{st}$) actually the warmest day?

Let’s look at why we have solstices and equinoxes, first. If you look at the Earth’s orbit around the sun, it looks like this

This is a view looking down over the North pole (a colorful view is here). The earth is then known to be spinning counter-clockwise, and also revolving around the sun in the counter-clockwise direction. The Angular Momentum associated with the spinning of the earth around its own axis is conserved. This phenomenon, which we know from tops and gyroscopes, keeps the axis pointed in the same absolute direction (towards Polaris, the Pole Star). While this is not exactly true due to something called precessional motion that causes the axis to slowly point along different directions, which I will discuss later – it is rather slow so we can ignore it for now.

In position marked ${\bf A}$, the Northern Hemisphere experiences summer, as the Sun is directly overhead the Tropic of Cancer (an imaginary line that is drawn at roughly $22.5^{\circ}$) – this is on the $21^{st}$ of June. At position marked ${\bf B}$, which follows after the Northern Summer, the Sun is directly above the Equator; this is called the autumnal (or fall) equinox ($21^{st}$ September}. Next, the position marked ${\bf C}$ is the Northern winter ($21^{st} December$) which also corresponds to the Southern summer, while position ${\bf D}$ corresponds to the Spring Equinox (the $21^{st} of March$).

Why is the summer solstice the longest day? And why is it called a “sol-stice” – which is Latin for “the sun is still”. Look at the picture below for the earth on that day.

The yellow lines on the right depict the Sun’s rays. The vertical (blue) line is called the Terminator, not after the eponymous 90’s movie, but a line that separates those folk that can see the Sun from those that cannot. It is clear that at the maximum angle to the sun’s rays, as depicted, the folks in the Northern hemisphere experience the longest daytime, from leaving the Terminator behind them on one side to entering it on the other.

Why does the sun seem stationary? Well, it certainly still rises in the east and sets in the west due to the Earth’s rotation, so it does not seem stationary in that respect. However, it is stationary insofar as its North-South oscillation is concerned. Notice that on the $21^{st}$ of June, the Sun has proceeded (due to the Earth’s orbital motion of course) as far North as it can possibly get. That is a maximum of the sun’s latitudinal displacement and as you might remember from high school math, if something is at a maximum, the slope (the rate of change) is zero. ${\it Ergo}$, the sun appears to not move along the North-South direction.

Is the Northern hemisphere the warmest on this day. Let’s ignore the fact that water and soil take time to heat up; this is called the Lag of the Seasons, but that is only part of the answer. For it to feel warm, the sun has to be right atop your head, at the zenith of the sky. It is very clear that when the Tropic of Cancer has the sun at its zenith, other parts of the Earth, to the south, do ${\bf not!}$. So, there are going to be two hottest days of the year for parts of the earth between the Tropic of Cancer and the Equator (and similarly between the Tropic of Capricorn and the Equator). They will be the days that the Sun is overhead, but on its journey to the higher latitudes, then on its journey away from them! So May and July might actually be warmer than June!

There is one small problem – the Earth is also not always the same distance from the Sun. It is closest to the Sun in January and furthest in July. That also causes the temperature to vary a lot. But one can learn a lot by looking at annual charts of Sun Hours and Average Temperature for two cities, one at the Tropic of Cancer and the other between that and the Equator. Indore (India) is at ${23 \frac{1}{2}}^{\circ}$ while Port Blair (also an Indian territory, but close to Indonesia) is at ${11 \frac{1}{2}}^{\circ}$, halfway between. Look at the dip in June between May and July for Port Blair vs. Indore (which has no such dip).

By the way, several websites as well as people discuss this topic, as does the physicist Hitoshi Murayama in a neat essay. Other websites discuss the  solstice , and you can find a list of cities by latitude here. You will find temperature data here.