The goal in the next chapter in the Almagest, Ptolemy’s goal is to is to find the angle between the celestial equator and ecliptic. These are both features on the celestial sphere which, while fundamental to astronomy, are not terms we’ve yet explored (aside from a brief mention in the first chapter of Astronomia Nova). So before continuing, we’ll explore the celestial sphere a bit. In addition, if we’re to start measuring angles on that sphere, we will need to understand the coordinate systems by which we do so.
The first thing to understand is that the whole system is set up around spherical coordinates. In school, we deal so much in Cartesian coordinates (x, y, z) that thinking spherically ($\phi$, $\theta$) is challenging at first. But notice something about the variables I’ve just listed. In Cartesian, I’ve listed 3, but for spherical coordinates I’ve only listed 2.
In the Cartesian coordinates, the x, y, and z represent the left-right, back-forth, and up-down from a point of origin. In spherical, $\phi$ is the angle around, and $\theta$ is the up-down. To be complete, we could also include r, but since we cannot easily perceive distance to celestial objects, this is typically omitted for astronomical purposes.
However, as with the Cartesian system, the spherical coordinates need a starting point, an origin. To begin, we take one locally. We use our horizon and the point directly north on that horizon. So if a star was located exactly on the horizon at north, it would be located at (0º,0º). The first of these numbers is the altitude (angle above the horizon). So if a star was located due north but was half way to the point straight up (the zenith), it would be (45º, 0º). While the coordinate system itself is set on the north point, we only allow that the altitude go up to 90º which indicates that we always take it from the closest point on the horizon instead of truly north.
The second coordinate is the azimuth. This is measured around the horizon starting at due north and going through east-south-west, in that order. So if a star was half way up the sky and due east, it would be at (45º, 90º).
The problem with this system is that it is tied to our local point of view. To most astronomers in period, the Earth was fixed and the sky turned around it. In modern understanding it is the Earth that turns. But either way you look at it, the position of every object in the sky is changing from second to second.
Worse, the Earth is spherical which means everyone will have a zenith pointing somewhere else. So you can’t even compare notes with friends. So for a truly useful system, we’ll need one that’s fixed to the sky. But as with all coordinate systems, we’ll need a point from which we can measure anything else. So before developing the new coordinate system, we’ll need to develop the celestial sphere itself.
First, we must understand that the dome of the sky we see above us is only half the picture. The other half of us is simply hidden beneath the ground. But two halves put together make it appear as if we’re at the center of a sphere. I could draw this, but being a two dimensional representation, it would simply look like a circle at this point. So to add a bit of reference, we’ll extend Earth’s equator out to that sphere. Not so cleverly, this is the celestial equator.
This is a good start because, much like our horizon, it provides a plane from which we could measure an object’s position above or below. However, we still need a point on the celestial equator to serve as our origin for the around.
For this, astronomers used the position of the sun. If the path of the sun is marked out on the celestial sphere throughout the year, it traces a path known as the ecliptic.
Note that the ecliptic crosses the celestial equator at two points. These are the equinoxes (vernal and autumnal). When the sun is at its highest point on the ecliptic, that’s the summer solstice and conversely, the low point is the winter solstice. For this coordinate system, the vernal equinox was the point chosen for the origin. Since (at the time1) the sun was then in the constellation of Aries the sign for the vernal equinox is often the symbol for Aries.
This system is known as the RA-Dec system which is short for Right Ascension and declination. RA takes the place of our azimuth. It’s measured around the celestial equator starting from the point of the vernal equinox, towards the summer solstice, autumnal equinox, and winter solstice in that order. However, instead of measuring this in terms of degrees, astronomers to this day measure it in hours, minutes, and seconds – a throwback to the sexagesimal system.
Meanwhile, the declination is measured just like the altitude: up from the ecliptic in degrees with 90º being the maximum.
Let’s take a quick example:
Here I’ve sketched in a star. Starting from the vernal equinox, it’s closest point on the celestial equator is a little past the summer solstice but not yet to the autumnal equinox. So since the equinoxes and solstices are every $\frac{1}{4}$ of the 24 hours, the summer solstice would be at 6 hours. Thus, the star is somewhere around 7-8 hours RA. Let’s call it 7.5 hours which is 7 hours, 30 minutes, abbreviated 7h30m.
From there, it’s $\frac{2}{3}$ way between the celestial equator and the 90º point I’ve sketched in. Let’s call it 65º. But it’s also above the equator. That gets noted by the fact that the declination is positive. Objects below the ecliptic have a negative declination.
So it’s RA-dec position would be 7h30m, 65º.
Another thing to note in the diagram above is that I’ve drawn it with north being at the top and south at the bottom. This places the vernal equinox on the far side in this sketch because, when viewed top down, the Sun seems to move around the celestial sphere counter clockwise. We’ll be seeing this behavior a lot more as we start getting into the various models of Ptolemy, Copernicus, and Brahe, so I wanted to get everyone used to it early.
While it doesn’t directly pertain to the celestial sphere, this is also a good time to look at some topics related to the spherical geometry since we’re very shortly going to need to be going through some math in this coordinate system.
The first is the idea of great circles and small circles. A great circle is a circle whose center goes through the center of the sphere. So the ecliptic and celestial equator or both great circles. But consider the apparent path of the star:
Its nightly path does not pass through the center of the sphere. Thus, it is known as a small circle.
I’ll also remind the reader that the sun’s apparent path from day to day is also not a great circle; the ecliptic is the path the sun takes with respect to the background stars over the course of the entire year. On the day-to-day, the Sun does not etch a great circle either. This is an important distinction.
So why are great circles important? Although I didn’t explicitly call it out previously, we’ve already been using them in a sense. When we connected the star to its closest point on the celestial equator, that path was actually part of a great circle. It turns out that the shortest distance between any two points on a sphere is to take the great circle that intersects them both2. Thus, when I looked at the angular separation between between the Sun and Mars in Astronomia Nova, Ch 1, that distance was in part defined by the great circle distance between the two.
Another reason to care about great circles is that the intersection of three great circles defines a spherical triangle, shown below in red.
Something to note about spherical triangles is that they don’t particularly behave like planar triangles. Their angles don’t add up to 180º. Trig functions don’t work on them directly. So when dealing with this topic in modern astronomy, a large chunk of learning to work in spherical coordinates also involves relearning how geometry and trigonometry work on spheres which includes building new tools. That would then allow for things like the conversion between the alt-az system and the RA-dec system.
However, as we noted previously, Ptolemy didn’t have trigonometric functions available to him. So we’ll have to see exactly how he approaches this issue…
- The vernal equinox not occurs in Pisces due to the Earth wobbling. As such all the astrological “zodiac signs” are off by nearly a month. Just another reason astrology is nonsense.
- This is why if you view an airplane’s path over long distances on a normal, flattened map (known as a Mercator map), they often take what appear to be long arcs instead of straight lines – it is a straight line, but when distorted to be on a flat surface, it appears to be an arc.