These two planes are inclined to each other at an angle of 23.5° — Page 387, Lesson 338

Let's start with the geometry that drives everything in this chapter: the Plane of the Ecliptic and the Plane of the Equinoctial. These two planes are inclined to each other at an angle of 23.5°. That angle has a proper name — it's called the obliquity of the ecliptic. Now, what are these two planes? The Plane of the Equinoctial is simply the plane of the Earth's Equator, extended out into space. The Plane of the Ecliptic is the plane in which the Earth orbits the Sun. So the Earth's axis is tilted, and that tilt is the 23.5° obliquity. That single tilt is the root cause of the seasons. Here's the key idea: at any given time of year, the Sun sits at some angle above or below the Equator. That angle determines the season and it directly affects the length of daylight and night. That angle has a precise name — it's called DECLINATION. Think of declination as the sky's equivalent of latitude on the Earth. Just as latitude tells you how far north or south you are on the planet, declination tells you how far north or south the Sun is in the sky, measured from the celestial Equator. The Sun's declination changes over the course of a year. It swings between 23.5°N, when the Sun is overhead the Tropic of Cancer, down through 0°, when the Sun is overhead the Equator, and on to 23.5°S, when the Sun is overhead the Tropic of Capricorn. Then it comes back up through 0° to 23.5°N again. That's the full annual cycle. One term you'll meet here: when the Sun, or any planet, is directly overhead an observer — meaning at an altitude of 90° — it is said to be at its Zenith. So when the Sun's declination equals your latitude, the Sun is at your zenith. Now, there's a beautiful way to model this. The declination of the Sun may be considered to follow a sine wave. Its peak amplitude is 23.5°, and its cyclic period is one year. So the declination curve is a smooth, sinusoidal oscillation between the two tropics. You'll often see the dates of the solstices and equinoxes quoted as about the 21st or 22nd of the appropriate month. That's not sloppiness — it's because the precise date depends on the relationship between the year being considered and the leap year cycle. You are not expected to calculate the precise date of these solar events; just know they fall around those dates. Now let's connect declination to daylight. The length of daylight and night at a given latitude varies with the declination of the Sun. And here's the crucial relationship: the rate of change of the length of daylight will be greatest when the rate of change of declination is greatest. Look at the sine wave — where is the slope steepest? At the equinoxes, around March 21st and September 21st. That's when the Sun's declination is changing fastest, so that's when the length of daylight is changing fastest. But — and this is important — that relationship is not true in several special cases. Those special cases are when the latitude considered is either the Equator itself, or latitudes above 66°N or 66°S. We'll come back to those special cases later in the chapter. Let me show you the geometry. — this is the Earth's orbit, showing the ecliptic plane. adds the inclination of the Earth's axis. shows how that tilt produces the seasons. is a side view of the orbit. And is the declination of the Sun — that's the sine wave we've been talking about. So to tie it all together: the 23.5° obliquity of the ecliptic causes the Sun's declination to swing between 23.5°N and 23.5°S over the year. Declination is the sky's latitude. When the Sun is overhead, it's at your zenith. The declination follows a sine wave with amplitude 23.5° and period one year. And the rate of change of daylight is greatest at the equinoxes, when declination changes fastest — except at the Equator and above 66°N/S, which we'll examine later.