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DE SPHAERA

Understanding the celestial sphere

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WHAT IS THE CELESTIAL SPHERE?

The abstract concept of a sphere, of which the earth is at the center, on the surface which the sun, moon, stars and planets all appear to lie

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WHY TEACH THIS CLASS?

The concept of the celestial sphere is fundamental to astronomy to this day

Perhaps the easiest answer is that it is so fundamental. The title of this class comes from a book published in 1230 by Johannes de Sacrobosco (also known as John of Holywood) and, while it goes beyond the simple description of the sphere, it was so popular it is perhaps the most common astronomical text to have extant copies of today even though it wasn’t first printed until 1472. That’s over 250 years after it was written before it received a printing, meaning people were copying this by hand. And after it was published, it continued to be regularly printed for several hundred more years, having over 80 editions in the next 200 years. In medieval universities, it was required reading as the core of an astronomical curriculum. It’s so fundamental, that if you wanted to become a navigator on a ship, understanding the first two chapters of Sacrobosco’s text was one of the first things you’d be assigned. It’s so fundamental that if you open Kepler’s Astronomia Nova to Chapter 1, page 1, you’ll see the diagram here. In fact, if you open nearly any astronomy 101 textbook even today, the celestial sphere is almost certain to be discussed in chapter 1.

But this explanation begs the question: Why is it so fundamental?

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WHY TEACH THIS CLASS?

The concept of the celestial sphere is fundamental to astronomy to this day
The math we learned in high school generally does not work when applied to curved surfaces
We are never taught how to think about problems on such surfaces

Simply put, it’s because this is how astronomers actually conceptualize the cosmos. Projecting objects onto this hypothetical sphere allows us to work in an unmoving coordinate system for the fixed stars, trace the paths of objects that move with respect to those stars, and ultimately build mathematical models that can be used to predict the positions of objects far into the future. This is the core objective of Ptolemy's Almagest, as well as the works of Copernicus, Kepler, and so many other astronomers. So to follow and understand their work, you must understand how they think about the cosmos.

Which leads me to my second point on why I’m teaching this class. This is that much of the math used in works like the Almagest would be entirely unfamiliar to most people for a few reasons.

The first is that many of the mathematical tools we have at our disposal and were taught in high school, had not yet been invented in Ptolemy's time. For example, much of Ptolemy's work would be made simpler through the use of trigonometric functions (sin, cos, etc...), but those weren't developed until several centuries after Ptolemy died. As such, Ptolemy had to rely on methods that were ultimately more complicated while using simpler tools. But because those tools are simpler, if they are taught at all in modern times, we tend to forget them since we have better ones available.

The second is that what tools we are familiar with, are meant for cartesian coordinates, which is to say flat surfaces. When trying to perform math on the surface of a sphere, even these modern tools break down as the axioms we learned in school no longer apply. For example, the sum of the angles in a spherical triangle no longer add up to 180º as they do in a flat triangle. This is an entirely different field of mathematics, known as Spherical Geometry, which even math majors don't often learn because the applications for it are quite limited.

As such, before trying to teach classes on Ptolemy's work, I will need to teach these esoteric math tools. And while they aren't especially complicated, trying to do math on the surface of a sphere requires some practice wrapping your head around since it forces us to think in 3D. And that's the purpose of this class - The goal is to start developing a mathematical language and thinking about the topic although we're not going to start doing any actual math in this class.

So with that bit of context out of the way, why a sphere?

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WHY A SPHERE?

The sky appears to be a dome above us
Ancient astronomers knew that the earth is a sphere
A dome would require “one whole area of the earth [to have] a kindling nature, and another an extinguishing one, or rather that the same part [of the earth] kinds for one set of observers and extinguishes for another set; and that the same stars are already kindled or extinguished for some observers while they are not yet for others.” �~Ptolemy, Almagest 1.3

Because a dome doesn’t make sense

The short answer is that nothing else makes sense.

When we look at the night sky, it appears to be a dome above us - half a sphere - meeting the horizon. But it was well known to ancient astronomers that the earth is not flat and that the horizon is simply an illusion. Since they knew the earth was a sphere, and there was no such report of anyone ever seeing the "edge" of a celestial dome, it stood to reason that it must be more than a dome it extended the whole way around the world thus being a sphere.

Ptolemy went into even more detail on this in the first book of the Almagest stating that, if the sky were a dome, it would require one portion of the sky to produce the stars and other celestial objects, and another portion of the sky to extinguish them, always recreating and extinguishing them in precisely the same arrangement the next night. This was a ridiculous concept and that these objects simply rotated on the surface of a sphere was a more parsimonious solution.

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WHY A SPHERE?

The stars move in set paths along circles, rotating around a pole
“[T]he revolution of the ever-visible stars … was observed to be circular, and always taking place about one center, the same [for all]. … [T]hose stars which were closer to it revolved on smaller circles, those that were further away described circles ever greater in proportion to their distance.”�~Ptolemy, Almagest 1.3

Because we see the sky move like one

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THIS IS STILL A SIMPLIFICATION

Ancient astronomers fully understood that the objects that moved with respect to the stars were at different distance
As such, when talking about the position of an object on the celestial sphere, we are talking about its projection onto the sphere

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AN OUTSIDE VIEW

Trying to understand the position of objects on the sky is hard when the sky keeps moving

Picturing the celestial sphere as unmoving makes it much easier to understand

In that previous class, we began exploring the concept of coordinates on the celestial sphere as viewed from the inside. But there, it was difficult to grasp because the sky is in constant motion from the point of view of an observer on the earth. Thus, when discussing the celestial sphere, we instead will extricate ourselves from this view and consider the sphere from the outside where we can maintain a fixed vantage point if we so choose, or move around the sphere to get whatever view suits us best. Either way, objects that do not move relative to one another, will have fixed coordinates. This produces the advantage that we can more easily conceptualize the motion of the heavens in their own context ignoring the rotation of the earth, but means we lose the view from the perspective of the observer on earth as their view will constantly be whizzing around. We'll occasionally make reference to the observer's view to not lose sight of it entirely and circle back to reconcile these two views at the end of the class.

With all that in mind, we can begin building the celestial sphere from scratch, beginning with a blank sphere.

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THE SPHERE

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THE NORTH CELESTIAL POLE

The point around which all stars seem to rotate
Currently near Polaris

Although you’re probably familiar with the term pole, what you may not realize is that the term has a much broader meaning. Specifically, it is related to the concept of a great circle. Taking a moment to define that, a great circle is any circle, drawn on the surface of the sphere whose center is also the center of the sphere. Another way of putting that is that , if you imagine that circle cutting the sphere, it cuts the sphere in half.

This is related to the pole because every great circle will have a pair of poles. Conversely, if you have two points diametrically opposite, there exists a circle around the sphere that is a great circle for those poles. This means that we could draw in an infinite number of great circles and poles, but not all of them are useful. However, the great circle associated with this pole is important and it’s known as the celestial equator.

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CELESTIAL EQUATOR

A great circle
90º from the celestial poles

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THE FIXED STARS

Stars have no perceptible motion (to the naked eye) on the celestial sphere - even over hundreds of years

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ECLIPTIC

The path of the sun.
Tilted by ~23.5º with respect to the celestial equator

Autumnal Equinox

Vernal Equinox

Winter Solstice

Summer Solstice

Here, I’ve added the sun’s path in red. This path is known as the ecliptic and, since it is a great circle, also has a pair of poles. For now, I’ve only drawn in the northern ecliptic pole. Now that we have all this in place, there’s a few things we can stop to discuss.

The first is that the sun’s path cuts through several constellations: Just under Leo on the left, then Virgo, Libra, Scorpio, and Sagittarius. These constellations likely sound familiar because they are the zodiac constellations. If we rotated the sphere, we would see it head into Capricorn, Aquarius, Pisces, Aries, Taurus, Gemini, and Cancer before heading back to Leo. The sun moves along the ecliptic throughout the year going counterclockwise if we were to be looking down on this sphere. So, in this view, it would move from left to right.

And there’s a few important things to note about that path and the sun’s position. First, let’s consider the two points where the ecliptic intersects the celestial equator. Whenever there are two great circles on a sphere, they will always intersect at two points known as nodes. There’s one on the near side in Virgo and another on the far side which, since I didn’t draw in the far constellations, we can’t see what it’s in, but is in Pisces. These two points are known as the equinoxes and on the dates when the sun is at those points, the day and night are at equal length. The one where the sun is going from above the celestial equator to below is the autumnal equinox and the one where the sun rises back above the celestial equator is the vernal equinox.

In addition, we can also ask where the sun is most distant from the celestial equator. We can see the low extreme on this side somewhere between Scorpio and Sagittarius. The high extreme is on the far side and is between Taurus and Gemini. These are the solstices. The low one is the winter solstice and the high one is the summer solstice.

Now, I should pause and note that this is the present-day configuration. The earth’s axis slowly wobbles, and since the north celestial pole is the extension of the terrestrial north pole, it means this point, and therefore the celestial equator wobble as well. So I’ll illustrate that as well

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PRECESSION OF THE EQUINOXES

The North Celestial Pole rotates around the equatorial pole over a period of ~26,000 years

This was known to ancient astronomers although they were off in the period

Has the effect of changing the constellations the sun is in on a given date

~30º

One difference we can immediately see is that the north celestial pole is no longer anywhere near Polaris. Rather, it is sort of in between two constellations where there’s no bright stars. So, during Ptolemy’s time, there was no “north star”.

Next, what we see is that the point over the equinoxes and solstices have changed. The Autumnal equinox is now on the right side of Virgo instead of the left. If we measured along the ecliptic, the difference between the old autumnal equinox and the new one is about 30º.

This has astrological implications because the sun signs with which everyone is familiar, is based on this system in which the sun was at the beginning of Virgo instead of towards the end. As such, your actual sign is about one month off from what you’ve been told. But there’s another more important consideration. And to understand that, we’ll need to start understanding the celestial coordinate systems for which I’ll need to flip over to the other side of the sphere.

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CELESTIAL COORDINATES

Two systems:

Equatorial coordinates – Based on the celestial equator
Ecliptic coordinates – Based on the ecliptic

Both measure along their respective great circle, from the vernal equinox, counter-clockwise, and then above or below the line along a great circle through their respective pole

As we said at the beginning of this class, the concept of the sphere allows us to use a spherical coordinate system. In these systems, we’ll have an up-down and a left-right, but no in-out since we said we are ignoring distance for the purposes of this talk. As such, we’ll need to choose a line to measure our left-right along and up-down from, as well as a point on that line to start from.

Astronomers today tend to favor the celestial equator as the primary line, although astronomers in period preferred the ecliptic. They had that preference because the moon and planets all congregate around the ecliptic, thus making it the logical choice if those are your intended objects of study.

In both cases, we use the vernal equinox as our zero point and measure relative to that, going counter-clockwise until you intersect with a great circle that goes through the pole and star. Let’s take an example:

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CELESTIAL COORDINATES

Two systems:

Equatorial coordinates – Based on the celestial equator
Ecliptic coordinates – Based on the ecliptic

Both measure along their respective great circle, from the vernal equinox, counter-clockwise, and then above or below the line along a great circle through their respective pole

Here, I’ve drawn in the bright star Almach in Andromeda.

To determine its celestial coordinates in the equatorial system. We start at the vernal equinox, which is currently towards the beginning of Pisces. We measure along the celestial equator and then up to the star. Here, we’ve gone about 30º around and 42º up. However, for historic reasons, instead of expressing the left-right in degrees, it’s expressed in hours, minutes and seconds where 24 hours = 360º. So each hour would be 15º, meaning the coordinates for this star is 2h, +42º, the plus denoting it’s north of the celestial equator. These are called the right ascension and declination respectively.

We can also do the same for the ecliptic coordinates. Here, we see we had to move much further along the ecliptic than we did the celestial equator. It’s about 44º. Then, from the ecliptic, we went up about 28º. The ecliptic coordinates don’t have any funny conventions about their units, so this is just 44º, 28º N, which are the ecliptic longitude and latitude, respectively.

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CELESTIAL COORDINATES

Two systems:

Equatorial coordinates – Based on the celestial equator
Ecliptic coordinates – Based on the ecliptic

Both measure along their respective great circle, from the vernal equinox, counter-clockwise, and then above or below the line along a great circle through their respective pole

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HISTORIC CELESTIAL COORDINATES

Due to precession of the equinoxes coordinate systems have shifted although the stars remain fixed.

While we can still measure stars the same way, our zero point – the vernal equinox – has shifted. This means that Ptolemy’s coordinate system was shifted in relation to ours despite the stars themselves not moving in any appreciable way.

In addition, Ptolemy’s method of describing coordinates also bears some explanation because he rarely described things in terms of the ecliptic longitude. Rather, like all astronomers of the time, divided the ecliptic into the 12 zodiac signs, each taking up 30º spans of the ecliptic (despite being very different actual sizes). So, if Ptolemy wanted to describe the position of an object, he would say what sign it was in, as well as how far into that sign. (CLICK) This importance of the zodiac signs is why we often see the ecliptic drawn as a thicker band in images: it allows for the zodiac signs to be denoted.

As we can see here, during Ptolemy’s time, the vernal equinox was near the beginning of Aries, so Aries was the first constellation through which astronomers would measure (again, noting we measure counter-clockwise). Because of this, astronomers to this day sometimes refer to the position of the vernal equinox as the “first point in Aries” even though it’s not longer actually in Aries.

At this point, we’ve developed a pretty good understanding of the celestial sphere in its own right. However, the ultimate goal of this series of classes will be to build a model of the solar system that is based on observations. And since is observing is done here on earth, we need to understand one more great circle: The horizon – which is certainly the hardest to understand because it is constantly moving. In my opinion, it is easiest to first imagine the earth at the center of the sphere.

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HORIZON

Based on observer’s position

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HORIZON

Based on observer’s position
Start with zenith

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HORIZON

Based on observer’s position
Start with zenith
Add in horizon as great circle of pole

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HORIZON

Based on observer’s position
Start with zenith
Add in horizon as great circle of pole
Cardinal directions can be found

North and south always lie along the meridian (a great circle that goes through the north celestial pole and zenith)
East and west are always on the celestial equator

Next, we can label the cardinal directions on the horizon, starting with north and south. These are always easy to find because if you imagine a great circle through the north celestial pole and the zenith, north and south will fall on that line. East and west will always fall at the intersection of the horizon and celestial equator.

In this instance, east is on the far side, so we can’t really see what constellations are just rising, but because west is towards us, we can see what constellations are setting. For example, just to the north of west, we see that Aquila is half-way below the horizon.

Lastly, once the horizon is set up, we can also place the sun on the ecliptic, since we know where it is on a given date. For example, consider the sun in late winter when it is before the vernal equinox.

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HORIZON

Based on observer’s position
Start with zenith
Add in horizon as great circle of pole
Cardinal directions can be found

North and south always lie along the meridian (a great circle that goes through the north celestial pole and zenith)
East and west are always on the celestial equator

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IMAGES

The Great Brass Globe – Tycho Brahe, Astronomiae Instauratae Mechanica, 1602
Handheld Celestial Sphere – Johannes de Sacrobosco, De Sphaera, 1488
Celestial Sphere – Johannes Kepler, Astronomia Nova, 1609
Flammarion Engraving – Camille Flammarion, L’atmosphere: meteorology populare, 1888
Star Trails over Windmill – sagesolar
Aristotelian Cosmos – Peter Apian, Cosmographia, 1524
Sacrobosco Holding Armillary Sphere – Wellcome collection, 1584